JOURNAL OF APPLIED PHYSICS

VOLUME 86, NUMBER 8

15 OCTOBER 1999

Temperature dependence of exciton peak energies in ZnS, ZnSe, and ZnTe epitaxial films R. Pa¨sslera) Technische Universita¨t Chemnitz, Institut fu¨r Physik, D-09107 Chemnitz, Germany

E. Griebl, H. Riepl, G. Lautner, S. Bauer, H. Preis, and W. Gebhardt Universita¨t Regensburg, Institut fu¨r Festko¨rperphysik, D-93040 Regensburg, Germany

B. Buda, D. J. As, D. Schikora, and K. Lischka Universita¨t Paderborn, FB 6-Physik, D-33095 Germany

K. Papagelis and S. Ves Aristotle University of Thessaloniki, Solid State Physics Section, Thessaloniki, Greece

共Received 11 May 1999; accepted for publication 13 July 1999兲 High-quality ZnS, ZnSe, and ZnTe epitaxial films were grown on 共001兲-GaAs-substrates by molecular beam epitaxy. The 1s-exciton peak energy positions have been determined by absorption measurements from 2 K up to about room temperature. For ZnS and ZnSe additional high-temperature 1s-exciton energy data were obtained by reflectance measurements performed from 300 up to about 550 K. These complete E 1s (T) data sets are fitted using a recently developed analytical model. The high-temperature slopes of the individual E 1s (T) curves and the effective phonon temperatures of ZnS, ZnSe, and ZnTe are found to scale almost linearly with the corresponding zero-temperature energy gaps and the Debye temperatures, respectively. Various ad hoc formulas of Varshni type, which have been invoked in recent articles for numerical simulations of restricted E 1s (T) data sets for cubic ZnS, are discussed. © 1999 American Institute of Physics. 关S0021-8979共99兲06420-8兴

ample, for cubic ZnS the uncertainties in published E 1s 共300 K兲 and E g 共300 K兲 values are on the order of 170 meV.1–5,19–22 More recent data for E g 共300 K兲 ranged between 3.723 eV9 and 3.74 eV,8 which corresponds to an uncertainty of 20 meV. In addition, different shapes of E 1s (T) and E g (T) curves have been measured for cubic ZnS: the data in Refs. 2, 3, 7, and 9 suggest a nearly linear dependence between 120 and 300 K, whereas, the data in Ref. 8 show a rather strong nonlinearity in the same temperature region. Thus, one of the goals of this work is to reliably obtain the detailed shape of the E 1s (T) curve from very low up to room temperature. The other goal of this study is to extend experimental measurements beyond room temperature, to obtain data to test various analytical models which were proposed previously 共Refs. 23–31兲. In particular we discuss the results of numerical fittings to E 1s (T) data based on an analytical fourparameter representation developed recently by one of the authors 共Ref. 27兲.

I. INTRODUCTION

The wide-gap binary II–VI materials ZnS and ZnSe have many potential applications in short-wavelength lightemitting devices. This is also true for the ternary systems, as well as the superlattice and multiple-quantum-well structures, due to the great variety of alloys and layered structures composed of various zinc and cadmium chalcogenides. For this reason, there has been considerable activity in improving epitaxial growth procedures and in detailed studies of excitonic and other near-band-edge emission and absorption features of such systems. To be able to control the emission-line energy positions in different II–VI material structures under various operating conditions it is necessary to know the corresponding energy shifts over a wide temperature range. Near room temperature the magnitudes of these thermal shifts are known to be larger than 0.4 meV/K in ZnS, ZnSe, and ZnTe.1–4 These shifts in zinc chalcogenides are markedly stronger than, e.g., in cadmium chalcogenides, group-IV materials, and many III–V compounds. The available experimental data on the temperature dependence of the lowest (1s) exciton line, E 1s (T), or the associated fundamental band gap, E g (T), in ZnS,5–9 ZnSe,5,10–15 and ZnTe16–18 bulk crystals or layers is in the range of cryogenic to room temperatures. Most of these E(T) data sets5–18 are not adequate, however, for detailed analytical and numerical descriptions or reliable extrapolations above room temperature. For ex-

II. SAMPLE PREPARATION, EXPERIMENTAL DETAILS, AND RESULTS

The ZnSe and ZnTe samples were prepared at the University of Regensburg and the ZnS samples at the University of Paderborn. All samples were grown on 共001兲 GaAs substrates by molecular beam epitaxy 共MBE兲 and have zinc-

a兲

Electronic mail: [email protected]

0021-8979/99/86(8)/4403/9/$15.00

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© 1999 American Institute of Physics

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FIG. 2. The exciton peak positions derived from absorption measurements ˜ i et al. 共see Ref. 33兲 共solid by means of a fitting scheme according to Gon lines: experimental data; dashed lines: fit兲. In general, the experimental curve is reproduced well, except in the region of low absorption due to interference effects.

FIG. 1. Absorption spectra of a free-standing 900-nm-ZnS layer at different temperatures. At low temperatures the discrete 1s exciton lines of the E 1s and E 1s(SO) gap are visible. The E 2s -exciton absorption is partially superposed by the E 1s(SO) transition.

blende structure with good structural quality, as monitored by in situ reflection high energy electron diffraction 共RHEED兲 and ex situ high resolution x-ray diffraction. 关The latter yielded full widths at half maximum of about 800, 250, and 600 arcsec for the 共004兲-reflection signal of the ZnS, ZnSe, and ZnTe samples, respectively.兴 The thickness d of the films ranged between 200 and 1000 nm. Samples of freestanding films were prepared from about 900-nm-thick films. The substrates were selectively etched off by a mixture of 82% 1N–NaOH and 18% H2O2 共30% solution in water兲. The films were kept on fine copper nets in order to avoid strain effects. Absorption measurements were carried out in a He cryostat at temperatures between 2 K and about 390, 280, and 290 K for ZnS, ZnSe, and ZnTe, respectively. The transmitted light was monochromized by a 1-m-double monochromator and detected by a multichannel diode array with a resolution of 0.1 Å, which corresponds to about 0.05 meV. The detector covers an energy region of more than 50 meV. We were thus able to carry out measurements at up to 60 K without moving the grating of the monochromator, avoiding deviations caused by the tuning mechanism. The absorption coefficient ␣ ( ) was determined from the Lambert–Beer rule33

冉 冊

1 I共 兲 ␣ 共 兲 ⫽⫺ ln ⫺␣c , d I 0共 兲

共1兲

where I( ) and I 0 ( ) are the intensity of light transmitted through the sample and the reference intensity measured without the sample, respectively; ␣ c denotes losses by reflection and scattering. Equation 共1兲 is a good approximation in regions of high absorption. Since the temperature broadening is relatively weak, especially for T⬍80 K, the peak positions could be determined

to within 0.5 meV. Figure 1 shows typical absorption spectra of ZnS recorded between 1.7 and 354 K. The 1s exciton peak is clearly visible up to room temperature. Unfortunately, the 2s-exciton absorption peak in ZnS could not be well resolved and the spin-orbit 1s 共SO兲 exciton absorption peak was observable only up to 200 K 共Fig. 1兲. The energy of the 1s exciton line E 1s (T) was obtained from the absorption spectra by using a fitting procedure de˜ i et al.33 共Fig. 2兲. This descripvised by Toyozawa34 and Gon tion uses the exciton model developed by Elliott35 and includes Lorentz broadening. The absorption coefficient is given by

␣共 兲⫽

再兺 ⬁

C 0 R 1/2 ប ⫹

冋

1 2 2 ⬁

⫺

n⫽1

⌫n

2R

n 3 共 ប ⫺E n 兲 2 ⫹⌫ 2n

⫹arctan

冉

ប ⫺E g ⌫c

冊册

⌫c

R

兺 3 2 2 n⫽1 n 共 ប ⫺E n 兲 ⫹⌫ c

⫹

⫹

sinh共 2u 兲

2 cosh共 2u ⫹ 兲 ⫺cos共 2u ⫺ 兲

冎

共2兲

,

with u ⫾⫽

冑 冉冑 R

共 ប ⫺E g 兲 2 ⫹⌫ 2c ⫾ 共 ប ⫺E g 兲

2

共 ប ⫺E g 兲 2 ⫹⌫ 2c

冊

,

共3兲

and C 0⫽

4 共 2 兲 3/2e 2 兩 M VB,CB兩 2 ncប 2 m 20

⬀E 3/2 g .

共4兲

兩 M VB,CB兩 denotes the matrix element for optical transitions between the valence and conduction bands, ⌫ n the Lorentz broadening 共line width兲 of the discrete exciton state n, ⌫ c the broadening of the excitonic continuum, R the effective Rydberg energy, E g the gap energy, and E n ⫽E g ⫺R/n 2 are the

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TABLE I. Exciton energies and fundamental energy gaps of the ZnS, ZnSe, and ZnTe epitaxial films at T⫽0 and 300 K. The exciton binding energies R are obtained from Eq. 共6兲. The absolute-zero energy gap values E g (0) are given by E 1s (0)⫹R. In the last column we have listed the first derivatives dE g (T)/dT at 300 K as obtained by fitting Eq. 共9兲 to the experimental data 关Figs. 4共a兲–4共c兲兴.

ZnS ZnSe ZnTe

R (this work) 共meV兲

R 共Ref.兲 共meV兲

E 1s (0) 共meV兲

E 1s 共300 K兲 共meV兲

E g (0) 共meV兲

E g 共300 K兲 共meV兲

34⫾2 20.0⫾0.3 12.8⫾0.2

34.2a 19.9b 12.7c

3806.9⫾0.5 2805.2⫾0.1 2381.6⫾0.2

3706.3⫾1.5 2702.2⫾1.0 2276.7⫾1.0

3841.1⫾1.0 2825.2⫾0.4 2394.4⫾0.4

3741⫾4 2722⫾3 2290⫾2

dE(T)/dT 兩 T⫽300 共meV/K兲

K

⫺0.505⫾0.005 ⫺0.476⫾0.003 ⫺0.453⫾0.003

a

Reference 41. Reference 42. c Reference 43. b

energies of transitions due to the individual exciton states. The associated line widths have been approximated by the empirical relation36 ⌫ n ⫽⌫ c ⫺

⌫ c ⫺⌫ 1 n2

.

共5兲

关Note that the total set of adjustable parameters for fitting a given ␣ ( ) dependence by Eq. 共2兲 is confined in this way to a quartet of energy parameters, E 1 , R, ⌫ 1 , and ⌫ c , in combination with the proportionality factor C 0 .] The 2s-peaks were clearly visible in ZnSe15 and ZnTe from 2 K 共Fig. 2兲 up to about 110 and 60 K, respectively. The separation between the 1s and 2s lines was found to be temperature independent. Thus we find for the effective Rydberg energy R⫽ 43 关 E 2s 共 T 兲 ⫺E 1s 共 T 兲兴

共6兲

共Table I兲. A weak temperature dependence could have been expected due to the dependence of the exciton binding energies on the reduced effective mass 37 and the static dielectric constant .22 For the zinc chalcogenides considered here the reduction of R⬀ / 2 共Refs. 37 and 38兲 between 2 K and room temperature should be smaller than 2 meV.15 Within this uncertainty the energy gap E g (T) can thus be estimated from E 1s (T) by E g (T)⫽E 1s (T)⫹R(T→0). ZnS has the highest Debye temperature, ⌰ D ⬇440 K,32 of all II–VI compounds investigated here. In order to find the E 1s (T) dependence up to about (4/3)⌰ D 共Sec. IV兲 we have measured, between 300 and 550 K, the transition energy to within ⫾6 meV by using a reflection difference spectrometer 共RDS兲 attached to a MBE chamber. The sample, which remained on the substrate, was fixed to the substrate manipulator and heated from room temperature up to about 550 K 共mostly in steps of 10 K兲. The sample temperature was measured by a thermocouple. The uncertainties of these measurements near 共and below兲 room temperature are ⫾0.3 K, but increase significantly 共nearly linearly兲 up to ⫾5 K for T ⬎500 K. This corresponds to a maximum systematic uncertainty of about ⫾3 meV for the experimental E 1s (T) data points between 450 and 550 K, which means a possible systematic error of less than ⫾3% with respect to the slope of the associated high-temperature asymptote. In order to avoid short period Fabry–Perot interferences in the transparent spectral region, the thickness of the sample

was chosen to be only about 200 nm. Typical reflectivity spectra for ZnS are shown in Fig. 3. The transition energy was obtained from these spectra by a fit function developed for thick layers. A double layer model39 yielded practically the same transition energies. The refractive index, n 0 ( ), which controls the normal reflectance, (n 0 ⫺1) 2 /(n 0 ⫹1) 2 , was derived from the dielectric function of a damped harmonic oscillator:39 共 兲 ⫽ H ⫹

A 21

21 ⫺ 2 ⫺i ⌫

共7兲

,

where H represents the background dielectric constant, A the polarizability, ⌫ the broadening and 1 ⫽E 1s /ប the circular frequency associated with the 1s transition. In addition to the latter we have also taken into consideration the contributions due to the 2s and 3s transitions. Thus we have used an expression of the form 3

共 兲 ⫽ H ⫹

A n 2

n 兺 2 2 n⫽1 ⫺ ⫺i ⌫ n

,

共8兲

n

where n ⫽(E g ⫺R/n 2 )/ប are the corresponding circular frequencies, A n ⫽A 1 /n 3 the polarizabilities,40 and ⌫ n the associated broadenings 关Eq. 共5兲兴.

FIG. 3. Reflection spectra of ZnS at high temperatures. The 200-nm-ZnS layer was left on the GaAs substrate. The curve in the inset shows a sample fit 共for 352 K兲 by Eq. 共8兲.

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Although the ZnS film is fully relaxed at growth temperature there may remain some residual biaxial strain that tends to split the excitonic transition into a heavy-hole and a light-hole one 共cf. Ref. 8, where a splitting of 5.4 meV was observed兲. A detailed quantitative description of such a splitting requires a more elaborate analytical model with two series of excitonic oscillator functions. In our case, however, the possible splitting is estimated to be very small 共less than 3 meV兲, i.e., even smaller than the inhomogeneous broadening (⌫ inh⬇8 meV兲. Thus, it was impossible, even at liquid helium temperatures, to detect some splitting of the measured absorption spectra into heavy- and light-hole exciton lines. For ZnS, both absorption and reflectance measurements were performed in the temperature region between 300 and 390 K. In general the uncertainty in both data sets is smaller than ⫾3 meV 关Fig. 4共a兲兴. Hence, it is legitimate to use simultaneously both data sets for the fitting procedure. The same is true for ZnSe 关Fig. 4共b兲兴.

E 共 T 兲 →E 共 0 兲 ⫺ ␣

III. ANALYTICAL FOUR-PARAMETER MODEL AND NUMERICAL ANALYSES

In this article we perform numerical analyses of our experimental E 1s (T) data for ZnS 关Fig. 4共a兲兴, ZnSe 关Fig. 4共b兲兴, and ZnTe 关Fig. 4共c兲兴 on the basis of an analytical model27 proposed recently by one of the authors. The corresponding semiempirical theory25,27 is based on the generally accepted interpretation that the gap shrinkage effect in semiconductors is caused by the cumulative effects of thermal lattice expansion and electron–phonon interaction, where the latter is usually the dominating one. By considering the corresponding spectral function25,27 in the form of a combination of a linear and a singular part 共as approximate representations of the contributions of long-wavelength acoustical phonons, on the one hand, and of short-wavelength acoustical plus optical phonons on the other hand兲, we have obtained an analytical expression of the form27 E 共 T 兲 ⫽E 共 0 兲 ⫺

再 冉冑 冉 冊 冉 冊 冊 冋 冉 冊 册冎

␣⌰ 2 2

⫹ 共 1⫺ 兲 coth

4

1⫹

⌰ ⫺1 2T

2 4T 6 ⌰

2

⫹

.

4T ⌰

4

⫺1

共9兲

Here the parameter 0⭐ ⭐1 determines the relative weights27 of the contributions of the linear (⬀ ) and singular 关 ⬀(1⫺ ) 兴 parts of the spectral function to the magnitude of the limiting 共high-temperature兲 slope ␣ 关see also Eq. 共11兲兴. The quantity ⌰ is defined in terms of the effective phonon temperature,25,27 ⌰, and the parameter as ⌰ ⬅

⌰ 1⫺ 12

.

共10兲

In the high temperature limit, TⰇ⌰ , Eq. 共9兲 tends to the usual linear asymptote,25,27 E 共 T 兲 →E 共 0 兲 ⫺ ␣ 共 T⫺ 21 ⌰ 兲 ⬅H 共 ⬁ 兲 ⫺ ␣ T,

that the magnitude of the effective phonon temperature ⌰ is related in a simple way to a prominent geometrical feature of the asymptote, namely: the half value of the phonon temperature, ⌰/2, is the point on the temperature scale where the high-temperature asymptote 关Eq. 共11兲兴 crosses the zerotemperature position, E(0), of the E(T) curve in consideration. These characteristic points are indicated in Figs. 4共a兲– 4共c兲 by crosses. However, the measured E(T) dependence approaches the calculated high-T asymptote 关Eq. 共11兲兴 关dotted lines in Figs. 4共a兲–4共c兲兴 only at relatively high temperatures, T⭓2⌰. At intermediate temperatures, T⬇⌰/2, there are large differences between the measured slope, S(T) ⬅⫺dE(T)/dT, and its limiting value, ␣ ⬅S(⬁). Estimates of empirical parameters such as ␣ and ⌰ are thus questionable in cases where the available experimental data are restricted to temperatures far below 2⌰. In the region of very low temperatures, TⰆ⌰, we see that Eq. 共9兲 tends to a quadratic asymptote of the form

共11兲

where H(⬁)⬅E(0)⫺ ␣ ⌰/2 represents the high-temperature limit of the associated enthalpy.23–25,29 From Eq. 共11兲 we see

共 1⫺ 21 兲 2 T 2 6⌰

,

共12兲

关using again Eq. 共10兲兴. This low-T asymptote 关Eq. 共12兲兴 clearly shows the crucial role of the weighting parameter within this four-parameter model, namely: the variation of between 0 and 1 allows for a material-specific magnitude of the quadratic term in Eq. 共12兲 from 0 up to ( 2 /12) ␣ T 2 /⌰. This ability to vary the curvature of the low-temperature asymptote, for given ␣ and ⌰ 关i.e., at fixed slope and point of crossing of the asymptote Eq. 共11兲兴, has been proven in many instances to be necessary for adequate numerical fittings of a large range of E(T) data for different materials. The experimental E 1s (T) data obtained from absorption or reflectance measurements for ZnS, ZnSe, and ZnTe epitaxial films are represented by empty or filled circles in Figs. 4共a兲–4共c兲. By fitting the three sets of E 1s (T) data using Eq. 共9兲 关solid curves in Figs. 4共a兲–4共c兲兴 we have obtained the material-specific parameter values E 1s (0), ␣ , ⌰, and listed in Table II. For ZnS 关Fig. 4共a兲兴 we note that the shape of the E 1s(SO) (T) curve due to the splitoff band is slightly different from the shape of the E 1s (T) curve associated with heavy- and light-hole excitons. The set of parameters ␣ , ⌰, and listed for ZnS in Table II refers exclusively to the E 1s (T) curve. It is obvious that the temperature range for the measured E 1s(SO) curve is not broad enough 共in analogy to Ref. 8兲 for an unambiguous determination of a band-specific parameter set. At least we can deduce from the lowtemperature positions E 1s (T→0)⫽3806.7 meV and E 1s(SO) (T→0)⫽3874.6 meV 关Fig. 4共a兲兴 a split-off parameter value of ⌬ SO⫽67.7 共⫾0.5兲 meV, in good agreement with earlier observations.21,22,32 Figure 6 shows the temperature dependence of the slopes, S 1s (T)⫽⫺dE 1s (T)/dT, of the E 1s (T) curves, which follow for ZnS, ZnSe, and ZnTe from Eq. 共9兲 关in combination with Eq. 共10兲兴, which follow from using the E 1s (0), ␣ , ⌰, and values of Table II. IV. DISCUSSION

Inspecting Table II we observe a general trend of a monotonic increase in the magnitudes of the parameters ␣

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FIG. 4. Exciton energy positions in 共a兲 ZnS, 共b兲 ZnSe, and 共c兲 ZnTe epitaxial films determined at lower temperatures by absorption measurements 共empty circles兲 and at higher temperatures by reflectance measurements 共filled circles兲. The solid curves represent numerical fits using Eq. 共9兲 共with parameters listed in Table II兲. The dotted lines correspond to the hightemperature asymptotes 关Eq. 共11兲兴. The insets compare the low-temperature data of Eq. 共9兲 with analytical models: solid curve Eq. 共9兲, dashed Eq. 共14兲, dotted-dashed Eq. 共15兲.

and ⌰ with increasing exciton energy and fundamental energy gap 共in analogy to the monotonic increase of the 1s exciton binding energies, R, listed in Table I兲. In Fig. 6 we have shown the material-specific pairs of parameters ␣ and E g (0). This comparison shows that, within the sequence of zinc chalcogenides in question, the limiting slope parameter ␣ increases weakly with increasing energy gap, E g (0). Moreover, it appears from Fig. 6 that this weak monotonic

dependence is nearly linear 共indicated by the dotted line兲. For comparison, we have included in Fig. 6 the results of analogous numerical analyses of E g (T) data available for CdS,23 CdSe,45 and CdTe46 关where a similar empirical relationship between the parameters ␣ and E g (0) could not be found兴. We now consider the monotonic change of the materialspecific values listed in Table II for the effective phonon

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FIG. 6. Correlation between the magnitudes of limiting slopes, ␣ ⬅S(⬁) 共Table II兲, of the E 1s/g (T) curves and the low-T values of the fundamental energy gaps, E g (0), in zinc chalcogenides and cadmium chalcogenides.

FIG. 5. Temperature dependence of the slopes, S(T)⫽⫺dE(T)/dT 共⫽entropies兲 of the E 1s (T) curves shown in Figs. 4共a兲–4共c兲. The dotted lines represent the limiting values, ␣ ⬅S(⬁), the magnitudes of which increase monotonically with the energy gap widths 共cf. Table II and Fig. 6兲.

temperature, ⌰.25,27 An instructive relationship is obtained by connecting the latter parameter with the asymptotic (T ⬎300 K兲 value of the Debye temperature, ⌰ D , 22,32 for the given material. We find for ZnS, ZnSe, and ZnTe that the effective phonon temperatures, ⌰, are nearly proportional to the Debye temperatures, ⌰ D , that is ⌰⬇ 32 ⌰ D .

共13兲

This finding for zinc and cadmium chalcogenides is analogous to results obtained for various III–V compounds as well as Si, Ge, and 15R-SiC.25 These strong changes of the effective phonon temperature, ⌰⬀⌰ D , from one material to the other provide, among other things, a reasonable explanation

for the inverse order of the materialspecific slopes, S(T)⬅ ⫺dE(T)/dT 共Fig. 5兲, at T⬍150 K. Actually, according to Eq. 共12兲, the low-T asymptote of S(T) is generally given by S(T)→ ␣ (2⫺ )( 2 /6)(T/⌰), where the parameters ␣ and change weakly from one material to another 共Table II and Fig. 6兲. Thus the inverse dependence, S(T)⬀T/⌰, on the effective phonon temperature ⌰ is dominant at low T 共cf. also the inset to Fig. 5兲. The approximate proportionality25 between ⌰ and ⌰ D 关Eq. 共13兲兴 allows us to estimate the temperature range necessary to examine for a trustworthy determination of the parameters ␣ and ⌰, especially for wide band-gap materials. From Fig. 5 we see that at higher temperatures the slopes, S(T), of the measured E(T) curves approach their limiting values, ␣ ⬅S(⬁) 15,23–27 共dotted lines in Fig. 5兲. Yet, from the latter we see that the difference between S(T) and its limiting value ␣ ⬅S(⬁) is reduced to a few % only at temperatures T⭓2⌰. In view of Eq. 共13兲 we can thus say that ␣ ⬅S(⬁) can be properly determined only when the temperature range inspected extends to about (4/3)⌰ D . This could be easily done for ZnTe by performing absorption measurements within the usual temperature interval 共i.e., from cryogenic up to about room temperature兲, because the Debye temperature is relatively low, ⌰ D(ZnTe) ⬇260 K 共cf. Fig. 8 in Ref. 32兲. This is in accordance with the corre-

TABLE II. Parameter sets obtained by fitting the temperature dependence of 1s exciton peak positions in ZnS, ZnSe, and ZnTe epitaxial films using the four-parameter Eq. 共9兲 关combined with Eq. 共10兲; cf. Ref. 27兴 compared with the results of the three-parameter models Eq. 共14兲 共Refs. 24–27,30,31兲 and Eq. 共15兲 共Refs. 28,29兲. In the last column we have listed the corresponding effective phonon energy values, k B ⌰.

Eq.

E 1s (0) 共meV兲

␣ 共meV/K兲

⌰ 共K兲

⌰ 共or  ) 共K兲

k B⌰ 共meV兲

ZnS

共9兲 共14兲 共15兲

3806.9 3805.3 3810.0

0.548 0.539 0.632

350 – –

0.389 共0兲 –

282 272 254

24.3 23.4 –

ZnSe

共9兲 共14兲 共15兲

2805.2 2804.3 2807.1

0.499 0.494 0.558

259 – –

0.330 共0兲 –

216 209 187

18.6 18.0 –

ZnTe

共9兲 共14兲 共15兲

2381.6 2380.9 2383.2

0.467 0.460 0.549

206 – –

0.370 共0兲 –

168 163 159

14.5 14.1 –

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spondingly low phonon temperature, ⌰ ZnTe⬇170 K 共Table II兲. In contrast, for ZnSe and ZnS we have found here that the effective phonon temperatures are ⌰ ZnSe⬇220 K and ⌰ ZnS⬇280 K 共Table II兲. This agrees with the relatively high Debye temperatures, ⌰ D(ZnSe) ⬇340 K and ⌰ D(ZnS) ⬇440 K 共Figs. 15 and 16 in Ref. 32兲. Consequently, it is necessary to perform measurements for ZnSe and ZnS up to 450 and 550 K, respectively, which is what we did. In this way we succeeded in determining ␣ and ⌰ to within a few percent. The ⌰ values in Table II are physically plausible. This follows from a comparison of the effective phonon energy values, k B ⌰, with characteristic peaks of the phonon energy spectra. The highest phonon energy peaks, ប LO , are known to be at 45.5 meV for ZnS, 27.6 meV for ZnSe, and 25.6 meV for ZnTe.22,32 Thus we can satisfy ourselves that, as usual,24–27 the effective phonon energies are located in the upper half 共a few meV above the middle兲 of the total phonon energy spectra. Of special interest is a comparison with the highest LA peak energies, ប LA , which are at 27.4 meV for ZnS, 23.6 meV for ZnSe, and 17.7 meV for ZnTe.22,32 Thus our estimated effective phonon energies are about 3–5 meV below the cutoff energies, ប LA . However, the nonvanishing values ( ⬇0.3– 0.4 for the materials in question; cf. Table II兲 indicate that phonons with energies considerably lower and higher than k B ⌰ also make significant contributions to the measured E(T) dependencies. Thus the estimated magnitudes of the effective phonon energies k B ⌰ 共listed in the last column of Table II兲 are obviously a nontrivial result of the inherent averaging processes25,27 over the contributions of all phonon branches and energies.

V. ASSESSMENT OF PUBLISHED THREE-PARAMETER MODELS

Taking the →0 limit in Eq. 共9兲, which corresponds to ⌰ ⫽⌰ 关according to Eq. 共10兲兴, we come immediately to a simple equation of the familiar form24–27 E 共 T 兲 →E 共 0 兲 ⫺ ⬅E 共 0 兲 ⫺

冋 冉 冊 册

␣⌰ ⌰ coth ⫺1 2 2T ␣⌰ . exp共 ⌰/T 兲 ⫺1

共14兲

These expressions are equivalent to the Bose–Einstein model ˜ a et al.30 and O’Donnell and functions proposed by Vin 31 Chen 兵and their equality arises from 2 关 exp(x)⫺1兴⫺1 ⫽关coth(x/2)⫺1 兴 其 . We have fitted our E 1s (T) data 关Figs. 4共a兲–4共c兲兴 using Eq. 共14兲 and obtained the ␣ and ⌰ values listed in the corresponding rows of Table II. Equation 共14兲 yields values that are 1% to 2% lower in the case of ␣ and about 3%–5% lower for ⌰ than those derived with Eq. 共9兲. Thus, despite the simplicity of Eq. 共14兲, it is obviously suitable for obtaining reasonable estimations of the basic parameters ␣ and ⌰. 共This is true at least for cases where is much less than unity; cf. Table II.兲 On the other hand, we see from the insets in Figs. 4共a兲–4共c兲 that due to the plateau-like behavior of Eq. 共14兲 at TⰆ⌰, this simple model is not capable of providing satisfactory fits in the cryogenic region.

4409

For a variety of group-IV and III–V semiconductor materials,28,29,44 as well as for ZnS,47,48 ZnSe,12–14,47,48 and ZnTe,16–18,47,48 Varshni’s formula:28 E Var.共 T 兲 ⫽E 共 0 兲 ⫺

␣T2 ,  ⫹T

共15兲

has typically been used for data analysis. Here the parameter ␣ is again the T→⬁ limit of the gap entropy29 and  was expected28,29 to be comparable with the Debye temperature ⌰ D in the given material. The parameter values ␣ and  obtained by fittings to our E 1s (T) data with Eq. 共15兲 are listed in the corresponding rows of Table II. For ZnTe our Varshni parameters are relatively close to ␣ Var.⫽0.52 meV/K and ⫽165 K given by Langen et al.16 At the same time we see from Table II that ␣ Var. is generally larger than the ␣ values of Eq. 共9兲 by 12%–18%. This means that, in general, Eq. 共15兲 is not suited15 for extrapolating E 1s (T) curves to significantly higher temperatures. Moreover we see from the insets to Figs. 4共a兲–4共c兲 that, as usual,23–27 Varshni’s model 关Eq. 共15兲兴 is not capable of providing satisfactory fits of the experimentally observed E 1s (T) curves in the cryogenic region. The cause of this discrepancy can be understood by considering in more detail the temperature dependence of the slopes, S(T). For zinc chalcogenides these S(T) curves 共Fig. 5兲 show an unfamiliar feature 关as does the S(T) curve for GaAs in Ref. 26兴, namely: these curves are concave at T ⬍30 K and convex at T⬎80 K. The inflection points of the S(T) curves for ZnS, ZnSe, and ZnTe are at about 70, 53, and 41 K, respectively. In our analytical model it is the parameter which, by virtue of its variability between 0 and 1, controls the material-specific shape of these curves. 关The concave part of the low-T section of an S(T) curve disappears in the →1 limit and becomes extremely pronounced in the →0 limit.兴 Consider now Varshni’s ansatz 关Eq. 共15兲兴, from which the entropy follows the form S Var.(T)⫽ ␣ T(T ⫹2  )/(T⫹  ) 2 . 24,29 The second derivative of S Var.(T) is d 2 S Var.(T)/dT 2 ⫽⫺6 ␣ 2 /(T⫹  ) 4, i.e., its sign remains unchanged along the whole T axis 共for all values of ␣ and 兲. This is in stark contrast to the observed behavior of S(T) in Fig. 5, proving the limitations of Varshni’s model. This deficiency in Varshni’s model is responsible for the poor fits in the cryogenic region, the large uncertainties reported for ␣ and ,15,25 and the physically unreasonable values of 1.1 meV/K ⬍ ␣ Var.⬍1.7 meV/K and 500 K⬍  ⬍1050 K quoted for zinc chalcogenides in Refs. 47 and 48. The limitations of Varshni’s model have been repeatedly observed, especially for ZnS.7–9 Various attempts have been made to improve the fittings 共numerical simulations兲 by including modifications to the formula. Abounadi et al.7 and subsequently Tran et al.9 used the ansatz E Ab.共 T 兲 ⫽E 共 0 兲 ⫺

␣T4  ⫹T 3

.

共16兲

By fitting experimental E 1s (T⭐250K兲7 or E 1s (T⭐320K兲9 data sets for cubic ZnS, these authors arrive at ␣⫽0.401 meV/K and  ⫽2⫻106 K3 and ␣⫽0.4201 meV/K and  ⫽2.172⫻106 K3 , respectively. It is obvious that neither

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nor, e.g., its third root, (  ) 1/3⬇125 or 130 K, can be related in a reasonable way to an empirical temperature parameter such as ⌰ or ⌰ D . Moreover, even the parameter ␣ following from fittings using Eq. 共16兲 obviously does not give the actual high-T limit of the slope of the E 1s (T) curve in ZnS. Table II shows that the ␣ values given in Refs. 7 and 9 are as much as 27% and 23% lower than the actual magnitude of the high-T slope in cubic ZnS 关as is clearly seen from Figs. 4共a兲 and 5兴. This large discrepancy is not surprising, however, in view of the ad hoc nature of ansatz 关Eq. 共16兲兴. By calculating the first and second derivatives of Eq. 共16兲 we find that E Ab.(T) has a point of inflection at T i ⬅(2  ) 1/3 ⬇159 or 163 K 共cf. Fig. 2 in Ref. 7 and Fig. 3 in Ref. 9兲. This means that the slope, S Ab.(T)⫽ ␣ (4  T 3 ⫹T 6 )/(  ⫹T 3 ) 2 , of the E Ab.(T) curve is maximum at this inflection point and decreases towards higher temperatures. Thus, Eq. 共16兲 directly contradicts many experiments on cubic ZnS. The maximum slope at the inflection point is S Ab.(T i ) ⫽(4/3) ␣ , that is, it is about 33% higher than the associated high-T limit, S Ab.(T→⬁)⫽ ␣ . This large difference explains the significant underestimation of the limiting slope by Eq. 共16兲. A more convenient modification 共generalization兲 of Varshni’s ansatz 关Eq. 共15兲兴 has been given by Ferna´ndez et al. in Ref. 8: E Fer.共 T 兲 ⫽E 共 0 兲 ⫺

␣T4 共  ⫹T 兲 3

.

共17兲

Equation 共17兲 provided good fits to the experimental E 1s(LH) (T), E 1s(HH) (T), and E 1s(SO) (T) curves from cryogenic to about room temperature (T⭐290 K兲 for a strained ZnS layer. This is primarily due to the fact that the shape of the entropy function, i.e., S Fer.(T)⬅⫺dE Fer.(T)/dT⫽ ␣ (T 4 ⫹4  T 3 )/(  ⫹T) 4 , changes from concave at very low T, to convex at higher T. The second derivative of this entropy function readily shows that the inflection point is at T i ⫽2  /3. Thus it follows from the fitted  value of about 75 K that the inflection point of S Fer.(T) is at about 50 K. This value is comparable to our result 共of about 70 K兲. However, there is no physically plausible connection between the parameter  in an ad hoc model like Eq. 共17兲 关or Eqs. 共15兲 or 共16兲兴 and a feature of a phonon energy spectrum 共like cutoff energy positions or moments25,27 of the electron-phonon spectral function兲.

VI. CONCLUSIONS

We have determined the temperature dependence of the 1s exciton peak positions in high-quality ZnS, ZnSe, and ZnTe epitaxial films grown on 共001兲-GaAs substrates by MBE. These data are analyzed via a semiempirical expression that provides, as a rule, a good fit to the observed temperature dependence. We succeeded in limiting the uncertainty for basic parameters to within a few percent by extending the experiments from the cryogenic temperature range up to about 4/3 of the corresponding Debye temperature ⌰ D , i.e., far beyond room temperature. This has been achieved by combining the results of absorption measure-

ments with those of reflectance measurements performed below and above room temperature, respectively. We suggest that an analogous procedure should be undertaken for other wide band-gap materials, including diamond, SiC, GaN, Alx Ga1⫺x N, etc., whose Debye temperature is much higher than 300 K.

ACKNOWLEDGMENTS

The authors acknowledge the financial support of the groups at Paderborn and Regensburg by the Deutsche Forschungsgemeinschaft. The authors would also like to thank D. Decker and P. Simon of the Technical University Chemnitz, for preparing Figs. 4–6 of this article. 1

Y. F. Tsay, S. S. Mitra, and J. F. Vetelino, J. Phys. Chem. Solids 34, 2167 共1973兲. 2 J. Camassel, D. Auvergne, and H. Mathieu, J. Phys. 共Paris兲 35, 67 共1974兲. 3 J. Camassel and D. Auvergne, Phys. Rev. B 12, 3258 共1975兲. 4 S. Biernacki, U. Scherz, and B. K. Meyer, Phys. Rev. B 49, 4501 共1994兲. 5 D. Theis, J. Phys. Chem. Solids 38, 1125 共1977兲. 6 L. Soonckindt, D. Etienne, J. P. Marchand, and L. Lassabatere, Surf. Sci. 86, 378 共1979兲. 7 A. Abounadi, M. Di Blasio, D. Bouchara, J. Calas, M. A. Averous, O. Briot, N. Briot, T. Cloitre, R. L. Aulombard, and B. Gil, Phys. Rev. B 50, 11677 共1994兲. 8 M. Ferna´ndez, P. Prete, N. Lovergine, A. M. Mancini, R. Cingolani, L. Vasanelli, and M. R. Perrone, Phys. Rev. B 55, 7660 共1997兲. 9 T. K. Tran, W. Park, W. Tong, M. M. Kyi, B. K. Wagner, and C. J. Summers, J. Appl. Phys. 81, 2803 共1997兲. 10 W. Maier and C. Klingshorn, Solid State Commun. 28, 13 共1978兲. 11 J. Baillou, J. Daunay, P. Bugnet, J. Daunay, C. Auzary, and P. Poindessault, J. Phys. Chem. Solids 41, 295 共1980兲. 12 Y. Shirakawa and H. Kukimoto, J. Appl. Phys. 51, 2014 共1980兲. 13 W. Krystek, L. Malikova, F. H. Pollak, M. C. Tamargo, N. Dai, L. Zeng, and A. Cavus, Acta Phys. Pol. 88, 1013 共1995兲. 14 L. Malikova, W. Krystek, F. H. Pollak, N. Dai, A. Cavus, and M. C. Tamargo, Phys. Rev. B 54, 1819 共1996兲. 15 R. Pa¨ssler, F. Blaschta, E. Griebl, K. Papagelis, B. Haserer, T. Reisinger, S. Ves, and W. Gebhardt, Phys. Status Solidi B 204, 685 共1997兲. 16 B. Langen, H. Leiderer, W. Limmer, W. Gebhardt, M. Ruff, and U. Ro¨ssler, J. Cryst. Growth 101, 718 共1990兲. 17 A. Naumov, H. Stanzl, K. Wolf, S. Lankes, and W. Gebhardt, J. Appl. Phys. 74, 6178 共1993兲. 18 H.-C. Mertin, H.-E. Gumlich, and Ch. Jung, Semicond. Sci. Technol. 8, 1634 共1993兲. 19 R. A. Soref and H. W. Moos, J. Appl. Phys. 35, 2152 共1964兲. 20 M. Cardona and G. Harbeke, Phys. Rev. A 137, 1467 共1965兲. 21 V. V. Sobolev, V. I. Donetskikh, and E. F. Zagainov, Fiz. Tekh. Poloprovodn. 共S.-Peterburg兲 12, 1089 共1978兲; Sov. Phys. Semicond. 12, 646 共1978兲. 22 O. Madelung, Landolt-Bo¨rnstein: Numerical Data and Functional Relationships in Science and Technology, New Series, Group III, Vol. 17b 共Springer, Berlin, 1982兲. 23 R. Pa¨ssler, Phys. Status Solidi B 193, 135 共1996兲. 24 R. Pa¨ssler, Solid-State Electron. 39, 1311 共1996兲. 25 R. Pa¨ssler, Phys. Status Solidi B 200, 155 共1997兲. 26 R. Pa¨ssler and G. Oelgart, J. Appl. Phys. 82, 2611 共1997兲. 27 R. Pa¨ssler, J. Appl. Phys. 83, 3356 共1998兲. 28 Y. P. Varshni, Physica 共Utrecht兲 34, 149 共1967兲. 29 C. D. Thurmond, J. Electrochem. Soc. 122, 1133 共1975兲. 30 ˜ a, S. Logothetidis, and M. Cardona, Phys. Rev. B 30, 1979 共1984兲. L. Vin 31 K. P. O’Donnell and X. Chen, Appl. Phys. Lett. 58, 2924 共1991兲. 32 O. Madelung, Landolt-Bo¨rnstein: Numerical Data and Functional Relationships in Science and Technology, New Series, Group III, Vol. 22 共Springer, Berlin, 1989兲. 33 ˜ i, A. Cantarero, K. Syassen, and M. Cardona, Phys. Rev. B 41, A. R. Gon 10111 共1990兲. 34 Y. Toyozawa, Prog. Theor. Phys. 20, 53 共1958兲. 35 R. J. Elliott, Phys. Rev. 108, 1384 共1957兲.

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VOLUME 86, NUMBER 8

15 OCTOBER 1999

Temperature dependence of exciton peak energies in ZnS, ZnSe, and ZnTe epitaxial films R. Pa¨sslera) Technische Universita¨t Chemnitz, Institut fu¨r Physik, D-09107 Chemnitz, Germany

E. Griebl, H. Riepl, G. Lautner, S. Bauer, H. Preis, and W. Gebhardt Universita¨t Regensburg, Institut fu¨r Festko¨rperphysik, D-93040 Regensburg, Germany

B. Buda, D. J. As, D. Schikora, and K. Lischka Universita¨t Paderborn, FB 6-Physik, D-33095 Germany

K. Papagelis and S. Ves Aristotle University of Thessaloniki, Solid State Physics Section, Thessaloniki, Greece

共Received 11 May 1999; accepted for publication 13 July 1999兲 High-quality ZnS, ZnSe, and ZnTe epitaxial films were grown on 共001兲-GaAs-substrates by molecular beam epitaxy. The 1s-exciton peak energy positions have been determined by absorption measurements from 2 K up to about room temperature. For ZnS and ZnSe additional high-temperature 1s-exciton energy data were obtained by reflectance measurements performed from 300 up to about 550 K. These complete E 1s (T) data sets are fitted using a recently developed analytical model. The high-temperature slopes of the individual E 1s (T) curves and the effective phonon temperatures of ZnS, ZnSe, and ZnTe are found to scale almost linearly with the corresponding zero-temperature energy gaps and the Debye temperatures, respectively. Various ad hoc formulas of Varshni type, which have been invoked in recent articles for numerical simulations of restricted E 1s (T) data sets for cubic ZnS, are discussed. © 1999 American Institute of Physics. 关S0021-8979共99兲06420-8兴

ample, for cubic ZnS the uncertainties in published E 1s 共300 K兲 and E g 共300 K兲 values are on the order of 170 meV.1–5,19–22 More recent data for E g 共300 K兲 ranged between 3.723 eV9 and 3.74 eV,8 which corresponds to an uncertainty of 20 meV. In addition, different shapes of E 1s (T) and E g (T) curves have been measured for cubic ZnS: the data in Refs. 2, 3, 7, and 9 suggest a nearly linear dependence between 120 and 300 K, whereas, the data in Ref. 8 show a rather strong nonlinearity in the same temperature region. Thus, one of the goals of this work is to reliably obtain the detailed shape of the E 1s (T) curve from very low up to room temperature. The other goal of this study is to extend experimental measurements beyond room temperature, to obtain data to test various analytical models which were proposed previously 共Refs. 23–31兲. In particular we discuss the results of numerical fittings to E 1s (T) data based on an analytical fourparameter representation developed recently by one of the authors 共Ref. 27兲.

I. INTRODUCTION

The wide-gap binary II–VI materials ZnS and ZnSe have many potential applications in short-wavelength lightemitting devices. This is also true for the ternary systems, as well as the superlattice and multiple-quantum-well structures, due to the great variety of alloys and layered structures composed of various zinc and cadmium chalcogenides. For this reason, there has been considerable activity in improving epitaxial growth procedures and in detailed studies of excitonic and other near-band-edge emission and absorption features of such systems. To be able to control the emission-line energy positions in different II–VI material structures under various operating conditions it is necessary to know the corresponding energy shifts over a wide temperature range. Near room temperature the magnitudes of these thermal shifts are known to be larger than 0.4 meV/K in ZnS, ZnSe, and ZnTe.1–4 These shifts in zinc chalcogenides are markedly stronger than, e.g., in cadmium chalcogenides, group-IV materials, and many III–V compounds. The available experimental data on the temperature dependence of the lowest (1s) exciton line, E 1s (T), or the associated fundamental band gap, E g (T), in ZnS,5–9 ZnSe,5,10–15 and ZnTe16–18 bulk crystals or layers is in the range of cryogenic to room temperatures. Most of these E(T) data sets5–18 are not adequate, however, for detailed analytical and numerical descriptions or reliable extrapolations above room temperature. For ex-

II. SAMPLE PREPARATION, EXPERIMENTAL DETAILS, AND RESULTS

The ZnSe and ZnTe samples were prepared at the University of Regensburg and the ZnS samples at the University of Paderborn. All samples were grown on 共001兲 GaAs substrates by molecular beam epitaxy 共MBE兲 and have zinc-

a兲

Electronic mail: [email protected]

0021-8979/99/86(8)/4403/9/$15.00

4403

© 1999 American Institute of Physics

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FIG. 2. The exciton peak positions derived from absorption measurements ˜ i et al. 共see Ref. 33兲 共solid by means of a fitting scheme according to Gon lines: experimental data; dashed lines: fit兲. In general, the experimental curve is reproduced well, except in the region of low absorption due to interference effects.

FIG. 1. Absorption spectra of a free-standing 900-nm-ZnS layer at different temperatures. At low temperatures the discrete 1s exciton lines of the E 1s and E 1s(SO) gap are visible. The E 2s -exciton absorption is partially superposed by the E 1s(SO) transition.

blende structure with good structural quality, as monitored by in situ reflection high energy electron diffraction 共RHEED兲 and ex situ high resolution x-ray diffraction. 关The latter yielded full widths at half maximum of about 800, 250, and 600 arcsec for the 共004兲-reflection signal of the ZnS, ZnSe, and ZnTe samples, respectively.兴 The thickness d of the films ranged between 200 and 1000 nm. Samples of freestanding films were prepared from about 900-nm-thick films. The substrates were selectively etched off by a mixture of 82% 1N–NaOH and 18% H2O2 共30% solution in water兲. The films were kept on fine copper nets in order to avoid strain effects. Absorption measurements were carried out in a He cryostat at temperatures between 2 K and about 390, 280, and 290 K for ZnS, ZnSe, and ZnTe, respectively. The transmitted light was monochromized by a 1-m-double monochromator and detected by a multichannel diode array with a resolution of 0.1 Å, which corresponds to about 0.05 meV. The detector covers an energy region of more than 50 meV. We were thus able to carry out measurements at up to 60 K without moving the grating of the monochromator, avoiding deviations caused by the tuning mechanism. The absorption coefficient ␣ ( ) was determined from the Lambert–Beer rule33

冉 冊

1 I共 兲 ␣ 共 兲 ⫽⫺ ln ⫺␣c , d I 0共 兲

共1兲

where I( ) and I 0 ( ) are the intensity of light transmitted through the sample and the reference intensity measured without the sample, respectively; ␣ c denotes losses by reflection and scattering. Equation 共1兲 is a good approximation in regions of high absorption. Since the temperature broadening is relatively weak, especially for T⬍80 K, the peak positions could be determined

to within 0.5 meV. Figure 1 shows typical absorption spectra of ZnS recorded between 1.7 and 354 K. The 1s exciton peak is clearly visible up to room temperature. Unfortunately, the 2s-exciton absorption peak in ZnS could not be well resolved and the spin-orbit 1s 共SO兲 exciton absorption peak was observable only up to 200 K 共Fig. 1兲. The energy of the 1s exciton line E 1s (T) was obtained from the absorption spectra by using a fitting procedure de˜ i et al.33 共Fig. 2兲. This descripvised by Toyozawa34 and Gon tion uses the exciton model developed by Elliott35 and includes Lorentz broadening. The absorption coefficient is given by

␣共 兲⫽

再兺 ⬁

C 0 R 1/2 ប ⫹

冋

1 2 2 ⬁

⫺

n⫽1

⌫n

2R

n 3 共 ប ⫺E n 兲 2 ⫹⌫ 2n

⫹arctan

冉

ប ⫺E g ⌫c

冊册

⌫c

R

兺 3 2 2 n⫽1 n 共 ប ⫺E n 兲 ⫹⌫ c

⫹

⫹

sinh共 2u 兲

2 cosh共 2u ⫹ 兲 ⫺cos共 2u ⫺ 兲

冎

共2兲

,

with u ⫾⫽

冑 冉冑 R

共 ប ⫺E g 兲 2 ⫹⌫ 2c ⫾ 共 ប ⫺E g 兲

2

共 ប ⫺E g 兲 2 ⫹⌫ 2c

冊

,

共3兲

and C 0⫽

4 共 2 兲 3/2e 2 兩 M VB,CB兩 2 ncប 2 m 20

⬀E 3/2 g .

共4兲

兩 M VB,CB兩 denotes the matrix element for optical transitions between the valence and conduction bands, ⌫ n the Lorentz broadening 共line width兲 of the discrete exciton state n, ⌫ c the broadening of the excitonic continuum, R the effective Rydberg energy, E g the gap energy, and E n ⫽E g ⫺R/n 2 are the

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TABLE I. Exciton energies and fundamental energy gaps of the ZnS, ZnSe, and ZnTe epitaxial films at T⫽0 and 300 K. The exciton binding energies R are obtained from Eq. 共6兲. The absolute-zero energy gap values E g (0) are given by E 1s (0)⫹R. In the last column we have listed the first derivatives dE g (T)/dT at 300 K as obtained by fitting Eq. 共9兲 to the experimental data 关Figs. 4共a兲–4共c兲兴.

ZnS ZnSe ZnTe

R (this work) 共meV兲

R 共Ref.兲 共meV兲

E 1s (0) 共meV兲

E 1s 共300 K兲 共meV兲

E g (0) 共meV兲

E g 共300 K兲 共meV兲

34⫾2 20.0⫾0.3 12.8⫾0.2

34.2a 19.9b 12.7c

3806.9⫾0.5 2805.2⫾0.1 2381.6⫾0.2

3706.3⫾1.5 2702.2⫾1.0 2276.7⫾1.0

3841.1⫾1.0 2825.2⫾0.4 2394.4⫾0.4

3741⫾4 2722⫾3 2290⫾2

dE(T)/dT 兩 T⫽300 共meV/K兲

K

⫺0.505⫾0.005 ⫺0.476⫾0.003 ⫺0.453⫾0.003

a

Reference 41. Reference 42. c Reference 43. b

energies of transitions due to the individual exciton states. The associated line widths have been approximated by the empirical relation36 ⌫ n ⫽⌫ c ⫺

⌫ c ⫺⌫ 1 n2

.

共5兲

关Note that the total set of adjustable parameters for fitting a given ␣ ( ) dependence by Eq. 共2兲 is confined in this way to a quartet of energy parameters, E 1 , R, ⌫ 1 , and ⌫ c , in combination with the proportionality factor C 0 .] The 2s-peaks were clearly visible in ZnSe15 and ZnTe from 2 K 共Fig. 2兲 up to about 110 and 60 K, respectively. The separation between the 1s and 2s lines was found to be temperature independent. Thus we find for the effective Rydberg energy R⫽ 43 关 E 2s 共 T 兲 ⫺E 1s 共 T 兲兴

共6兲

共Table I兲. A weak temperature dependence could have been expected due to the dependence of the exciton binding energies on the reduced effective mass 37 and the static dielectric constant .22 For the zinc chalcogenides considered here the reduction of R⬀ / 2 共Refs. 37 and 38兲 between 2 K and room temperature should be smaller than 2 meV.15 Within this uncertainty the energy gap E g (T) can thus be estimated from E 1s (T) by E g (T)⫽E 1s (T)⫹R(T→0). ZnS has the highest Debye temperature, ⌰ D ⬇440 K,32 of all II–VI compounds investigated here. In order to find the E 1s (T) dependence up to about (4/3)⌰ D 共Sec. IV兲 we have measured, between 300 and 550 K, the transition energy to within ⫾6 meV by using a reflection difference spectrometer 共RDS兲 attached to a MBE chamber. The sample, which remained on the substrate, was fixed to the substrate manipulator and heated from room temperature up to about 550 K 共mostly in steps of 10 K兲. The sample temperature was measured by a thermocouple. The uncertainties of these measurements near 共and below兲 room temperature are ⫾0.3 K, but increase significantly 共nearly linearly兲 up to ⫾5 K for T ⬎500 K. This corresponds to a maximum systematic uncertainty of about ⫾3 meV for the experimental E 1s (T) data points between 450 and 550 K, which means a possible systematic error of less than ⫾3% with respect to the slope of the associated high-temperature asymptote. In order to avoid short period Fabry–Perot interferences in the transparent spectral region, the thickness of the sample

was chosen to be only about 200 nm. Typical reflectivity spectra for ZnS are shown in Fig. 3. The transition energy was obtained from these spectra by a fit function developed for thick layers. A double layer model39 yielded practically the same transition energies. The refractive index, n 0 ( ), which controls the normal reflectance, (n 0 ⫺1) 2 /(n 0 ⫹1) 2 , was derived from the dielectric function of a damped harmonic oscillator:39 共 兲 ⫽ H ⫹

A 21

21 ⫺ 2 ⫺i ⌫

共7兲

,

where H represents the background dielectric constant, A the polarizability, ⌫ the broadening and 1 ⫽E 1s /ប the circular frequency associated with the 1s transition. In addition to the latter we have also taken into consideration the contributions due to the 2s and 3s transitions. Thus we have used an expression of the form 3

共 兲 ⫽ H ⫹

A n 2

n 兺 2 2 n⫽1 ⫺ ⫺i ⌫ n

,

共8兲

n

where n ⫽(E g ⫺R/n 2 )/ប are the corresponding circular frequencies, A n ⫽A 1 /n 3 the polarizabilities,40 and ⌫ n the associated broadenings 关Eq. 共5兲兴.

FIG. 3. Reflection spectra of ZnS at high temperatures. The 200-nm-ZnS layer was left on the GaAs substrate. The curve in the inset shows a sample fit 共for 352 K兲 by Eq. 共8兲.

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Although the ZnS film is fully relaxed at growth temperature there may remain some residual biaxial strain that tends to split the excitonic transition into a heavy-hole and a light-hole one 共cf. Ref. 8, where a splitting of 5.4 meV was observed兲. A detailed quantitative description of such a splitting requires a more elaborate analytical model with two series of excitonic oscillator functions. In our case, however, the possible splitting is estimated to be very small 共less than 3 meV兲, i.e., even smaller than the inhomogeneous broadening (⌫ inh⬇8 meV兲. Thus, it was impossible, even at liquid helium temperatures, to detect some splitting of the measured absorption spectra into heavy- and light-hole exciton lines. For ZnS, both absorption and reflectance measurements were performed in the temperature region between 300 and 390 K. In general the uncertainty in both data sets is smaller than ⫾3 meV 关Fig. 4共a兲兴. Hence, it is legitimate to use simultaneously both data sets for the fitting procedure. The same is true for ZnSe 关Fig. 4共b兲兴.

E 共 T 兲 →E 共 0 兲 ⫺ ␣

III. ANALYTICAL FOUR-PARAMETER MODEL AND NUMERICAL ANALYSES

In this article we perform numerical analyses of our experimental E 1s (T) data for ZnS 关Fig. 4共a兲兴, ZnSe 关Fig. 4共b兲兴, and ZnTe 关Fig. 4共c兲兴 on the basis of an analytical model27 proposed recently by one of the authors. The corresponding semiempirical theory25,27 is based on the generally accepted interpretation that the gap shrinkage effect in semiconductors is caused by the cumulative effects of thermal lattice expansion and electron–phonon interaction, where the latter is usually the dominating one. By considering the corresponding spectral function25,27 in the form of a combination of a linear and a singular part 共as approximate representations of the contributions of long-wavelength acoustical phonons, on the one hand, and of short-wavelength acoustical plus optical phonons on the other hand兲, we have obtained an analytical expression of the form27 E 共 T 兲 ⫽E 共 0 兲 ⫺

再 冉冑 冉 冊 冉 冊 冊 冋 冉 冊 册冎

␣⌰ 2 2

⫹ 共 1⫺ 兲 coth

4

1⫹

⌰ ⫺1 2T

2 4T 6 ⌰

2

⫹

.

4T ⌰

4

⫺1

共9兲

Here the parameter 0⭐ ⭐1 determines the relative weights27 of the contributions of the linear (⬀ ) and singular 关 ⬀(1⫺ ) 兴 parts of the spectral function to the magnitude of the limiting 共high-temperature兲 slope ␣ 关see also Eq. 共11兲兴. The quantity ⌰ is defined in terms of the effective phonon temperature,25,27 ⌰, and the parameter as ⌰ ⬅

⌰ 1⫺ 12

.

共10兲

In the high temperature limit, TⰇ⌰ , Eq. 共9兲 tends to the usual linear asymptote,25,27 E 共 T 兲 →E 共 0 兲 ⫺ ␣ 共 T⫺ 21 ⌰ 兲 ⬅H 共 ⬁ 兲 ⫺ ␣ T,

that the magnitude of the effective phonon temperature ⌰ is related in a simple way to a prominent geometrical feature of the asymptote, namely: the half value of the phonon temperature, ⌰/2, is the point on the temperature scale where the high-temperature asymptote 关Eq. 共11兲兴 crosses the zerotemperature position, E(0), of the E(T) curve in consideration. These characteristic points are indicated in Figs. 4共a兲– 4共c兲 by crosses. However, the measured E(T) dependence approaches the calculated high-T asymptote 关Eq. 共11兲兴 关dotted lines in Figs. 4共a兲–4共c兲兴 only at relatively high temperatures, T⭓2⌰. At intermediate temperatures, T⬇⌰/2, there are large differences between the measured slope, S(T) ⬅⫺dE(T)/dT, and its limiting value, ␣ ⬅S(⬁). Estimates of empirical parameters such as ␣ and ⌰ are thus questionable in cases where the available experimental data are restricted to temperatures far below 2⌰. In the region of very low temperatures, TⰆ⌰, we see that Eq. 共9兲 tends to a quadratic asymptote of the form

共11兲

where H(⬁)⬅E(0)⫺ ␣ ⌰/2 represents the high-temperature limit of the associated enthalpy.23–25,29 From Eq. 共11兲 we see

共 1⫺ 21 兲 2 T 2 6⌰

,

共12兲

关using again Eq. 共10兲兴. This low-T asymptote 关Eq. 共12兲兴 clearly shows the crucial role of the weighting parameter within this four-parameter model, namely: the variation of between 0 and 1 allows for a material-specific magnitude of the quadratic term in Eq. 共12兲 from 0 up to ( 2 /12) ␣ T 2 /⌰. This ability to vary the curvature of the low-temperature asymptote, for given ␣ and ⌰ 关i.e., at fixed slope and point of crossing of the asymptote Eq. 共11兲兴, has been proven in many instances to be necessary for adequate numerical fittings of a large range of E(T) data for different materials. The experimental E 1s (T) data obtained from absorption or reflectance measurements for ZnS, ZnSe, and ZnTe epitaxial films are represented by empty or filled circles in Figs. 4共a兲–4共c兲. By fitting the three sets of E 1s (T) data using Eq. 共9兲 关solid curves in Figs. 4共a兲–4共c兲兴 we have obtained the material-specific parameter values E 1s (0), ␣ , ⌰, and listed in Table II. For ZnS 关Fig. 4共a兲兴 we note that the shape of the E 1s(SO) (T) curve due to the splitoff band is slightly different from the shape of the E 1s (T) curve associated with heavy- and light-hole excitons. The set of parameters ␣ , ⌰, and listed for ZnS in Table II refers exclusively to the E 1s (T) curve. It is obvious that the temperature range for the measured E 1s(SO) curve is not broad enough 共in analogy to Ref. 8兲 for an unambiguous determination of a band-specific parameter set. At least we can deduce from the lowtemperature positions E 1s (T→0)⫽3806.7 meV and E 1s(SO) (T→0)⫽3874.6 meV 关Fig. 4共a兲兴 a split-off parameter value of ⌬ SO⫽67.7 共⫾0.5兲 meV, in good agreement with earlier observations.21,22,32 Figure 6 shows the temperature dependence of the slopes, S 1s (T)⫽⫺dE 1s (T)/dT, of the E 1s (T) curves, which follow for ZnS, ZnSe, and ZnTe from Eq. 共9兲 关in combination with Eq. 共10兲兴, which follow from using the E 1s (0), ␣ , ⌰, and values of Table II. IV. DISCUSSION

Inspecting Table II we observe a general trend of a monotonic increase in the magnitudes of the parameters ␣

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4407

FIG. 4. Exciton energy positions in 共a兲 ZnS, 共b兲 ZnSe, and 共c兲 ZnTe epitaxial films determined at lower temperatures by absorption measurements 共empty circles兲 and at higher temperatures by reflectance measurements 共filled circles兲. The solid curves represent numerical fits using Eq. 共9兲 共with parameters listed in Table II兲. The dotted lines correspond to the hightemperature asymptotes 关Eq. 共11兲兴. The insets compare the low-temperature data of Eq. 共9兲 with analytical models: solid curve Eq. 共9兲, dashed Eq. 共14兲, dotted-dashed Eq. 共15兲.

and ⌰ with increasing exciton energy and fundamental energy gap 共in analogy to the monotonic increase of the 1s exciton binding energies, R, listed in Table I兲. In Fig. 6 we have shown the material-specific pairs of parameters ␣ and E g (0). This comparison shows that, within the sequence of zinc chalcogenides in question, the limiting slope parameter ␣ increases weakly with increasing energy gap, E g (0). Moreover, it appears from Fig. 6 that this weak monotonic

dependence is nearly linear 共indicated by the dotted line兲. For comparison, we have included in Fig. 6 the results of analogous numerical analyses of E g (T) data available for CdS,23 CdSe,45 and CdTe46 关where a similar empirical relationship between the parameters ␣ and E g (0) could not be found兴. We now consider the monotonic change of the materialspecific values listed in Table II for the effective phonon

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FIG. 6. Correlation between the magnitudes of limiting slopes, ␣ ⬅S(⬁) 共Table II兲, of the E 1s/g (T) curves and the low-T values of the fundamental energy gaps, E g (0), in zinc chalcogenides and cadmium chalcogenides.

FIG. 5. Temperature dependence of the slopes, S(T)⫽⫺dE(T)/dT 共⫽entropies兲 of the E 1s (T) curves shown in Figs. 4共a兲–4共c兲. The dotted lines represent the limiting values, ␣ ⬅S(⬁), the magnitudes of which increase monotonically with the energy gap widths 共cf. Table II and Fig. 6兲.

temperature, ⌰.25,27 An instructive relationship is obtained by connecting the latter parameter with the asymptotic (T ⬎300 K兲 value of the Debye temperature, ⌰ D , 22,32 for the given material. We find for ZnS, ZnSe, and ZnTe that the effective phonon temperatures, ⌰, are nearly proportional to the Debye temperatures, ⌰ D , that is ⌰⬇ 32 ⌰ D .

共13兲

This finding for zinc and cadmium chalcogenides is analogous to results obtained for various III–V compounds as well as Si, Ge, and 15R-SiC.25 These strong changes of the effective phonon temperature, ⌰⬀⌰ D , from one material to the other provide, among other things, a reasonable explanation

for the inverse order of the materialspecific slopes, S(T)⬅ ⫺dE(T)/dT 共Fig. 5兲, at T⬍150 K. Actually, according to Eq. 共12兲, the low-T asymptote of S(T) is generally given by S(T)→ ␣ (2⫺ )( 2 /6)(T/⌰), where the parameters ␣ and change weakly from one material to another 共Table II and Fig. 6兲. Thus the inverse dependence, S(T)⬀T/⌰, on the effective phonon temperature ⌰ is dominant at low T 共cf. also the inset to Fig. 5兲. The approximate proportionality25 between ⌰ and ⌰ D 关Eq. 共13兲兴 allows us to estimate the temperature range necessary to examine for a trustworthy determination of the parameters ␣ and ⌰, especially for wide band-gap materials. From Fig. 5 we see that at higher temperatures the slopes, S(T), of the measured E(T) curves approach their limiting values, ␣ ⬅S(⬁) 15,23–27 共dotted lines in Fig. 5兲. Yet, from the latter we see that the difference between S(T) and its limiting value ␣ ⬅S(⬁) is reduced to a few % only at temperatures T⭓2⌰. In view of Eq. 共13兲 we can thus say that ␣ ⬅S(⬁) can be properly determined only when the temperature range inspected extends to about (4/3)⌰ D . This could be easily done for ZnTe by performing absorption measurements within the usual temperature interval 共i.e., from cryogenic up to about room temperature兲, because the Debye temperature is relatively low, ⌰ D(ZnTe) ⬇260 K 共cf. Fig. 8 in Ref. 32兲. This is in accordance with the corre-

TABLE II. Parameter sets obtained by fitting the temperature dependence of 1s exciton peak positions in ZnS, ZnSe, and ZnTe epitaxial films using the four-parameter Eq. 共9兲 关combined with Eq. 共10兲; cf. Ref. 27兴 compared with the results of the three-parameter models Eq. 共14兲 共Refs. 24–27,30,31兲 and Eq. 共15兲 共Refs. 28,29兲. In the last column we have listed the corresponding effective phonon energy values, k B ⌰.

Eq.

E 1s (0) 共meV兲

␣ 共meV/K兲

⌰ 共K兲

⌰ 共or  ) 共K兲

k B⌰ 共meV兲

ZnS

共9兲 共14兲 共15兲

3806.9 3805.3 3810.0

0.548 0.539 0.632

350 – –

0.389 共0兲 –

282 272 254

24.3 23.4 –

ZnSe

共9兲 共14兲 共15兲

2805.2 2804.3 2807.1

0.499 0.494 0.558

259 – –

0.330 共0兲 –

216 209 187

18.6 18.0 –

ZnTe

共9兲 共14兲 共15兲

2381.6 2380.9 2383.2

0.467 0.460 0.549

206 – –

0.370 共0兲 –

168 163 159

14.5 14.1 –

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spondingly low phonon temperature, ⌰ ZnTe⬇170 K 共Table II兲. In contrast, for ZnSe and ZnS we have found here that the effective phonon temperatures are ⌰ ZnSe⬇220 K and ⌰ ZnS⬇280 K 共Table II兲. This agrees with the relatively high Debye temperatures, ⌰ D(ZnSe) ⬇340 K and ⌰ D(ZnS) ⬇440 K 共Figs. 15 and 16 in Ref. 32兲. Consequently, it is necessary to perform measurements for ZnSe and ZnS up to 450 and 550 K, respectively, which is what we did. In this way we succeeded in determining ␣ and ⌰ to within a few percent. The ⌰ values in Table II are physically plausible. This follows from a comparison of the effective phonon energy values, k B ⌰, with characteristic peaks of the phonon energy spectra. The highest phonon energy peaks, ប LO , are known to be at 45.5 meV for ZnS, 27.6 meV for ZnSe, and 25.6 meV for ZnTe.22,32 Thus we can satisfy ourselves that, as usual,24–27 the effective phonon energies are located in the upper half 共a few meV above the middle兲 of the total phonon energy spectra. Of special interest is a comparison with the highest LA peak energies, ប LA , which are at 27.4 meV for ZnS, 23.6 meV for ZnSe, and 17.7 meV for ZnTe.22,32 Thus our estimated effective phonon energies are about 3–5 meV below the cutoff energies, ប LA . However, the nonvanishing values ( ⬇0.3– 0.4 for the materials in question; cf. Table II兲 indicate that phonons with energies considerably lower and higher than k B ⌰ also make significant contributions to the measured E(T) dependencies. Thus the estimated magnitudes of the effective phonon energies k B ⌰ 共listed in the last column of Table II兲 are obviously a nontrivial result of the inherent averaging processes25,27 over the contributions of all phonon branches and energies.

V. ASSESSMENT OF PUBLISHED THREE-PARAMETER MODELS

Taking the →0 limit in Eq. 共9兲, which corresponds to ⌰ ⫽⌰ 关according to Eq. 共10兲兴, we come immediately to a simple equation of the familiar form24–27 E 共 T 兲 →E 共 0 兲 ⫺ ⬅E 共 0 兲 ⫺

冋 冉 冊 册

␣⌰ ⌰ coth ⫺1 2 2T ␣⌰ . exp共 ⌰/T 兲 ⫺1

共14兲

These expressions are equivalent to the Bose–Einstein model ˜ a et al.30 and O’Donnell and functions proposed by Vin 31 Chen 兵and their equality arises from 2 关 exp(x)⫺1兴⫺1 ⫽关coth(x/2)⫺1 兴 其 . We have fitted our E 1s (T) data 关Figs. 4共a兲–4共c兲兴 using Eq. 共14兲 and obtained the ␣ and ⌰ values listed in the corresponding rows of Table II. Equation 共14兲 yields values that are 1% to 2% lower in the case of ␣ and about 3%–5% lower for ⌰ than those derived with Eq. 共9兲. Thus, despite the simplicity of Eq. 共14兲, it is obviously suitable for obtaining reasonable estimations of the basic parameters ␣ and ⌰. 共This is true at least for cases where is much less than unity; cf. Table II.兲 On the other hand, we see from the insets in Figs. 4共a兲–4共c兲 that due to the plateau-like behavior of Eq. 共14兲 at TⰆ⌰, this simple model is not capable of providing satisfactory fits in the cryogenic region.

4409

For a variety of group-IV and III–V semiconductor materials,28,29,44 as well as for ZnS,47,48 ZnSe,12–14,47,48 and ZnTe,16–18,47,48 Varshni’s formula:28 E Var.共 T 兲 ⫽E 共 0 兲 ⫺

␣T2 ,  ⫹T

共15兲

has typically been used for data analysis. Here the parameter ␣ is again the T→⬁ limit of the gap entropy29 and  was expected28,29 to be comparable with the Debye temperature ⌰ D in the given material. The parameter values ␣ and  obtained by fittings to our E 1s (T) data with Eq. 共15兲 are listed in the corresponding rows of Table II. For ZnTe our Varshni parameters are relatively close to ␣ Var.⫽0.52 meV/K and ⫽165 K given by Langen et al.16 At the same time we see from Table II that ␣ Var. is generally larger than the ␣ values of Eq. 共9兲 by 12%–18%. This means that, in general, Eq. 共15兲 is not suited15 for extrapolating E 1s (T) curves to significantly higher temperatures. Moreover we see from the insets to Figs. 4共a兲–4共c兲 that, as usual,23–27 Varshni’s model 关Eq. 共15兲兴 is not capable of providing satisfactory fits of the experimentally observed E 1s (T) curves in the cryogenic region. The cause of this discrepancy can be understood by considering in more detail the temperature dependence of the slopes, S(T). For zinc chalcogenides these S(T) curves 共Fig. 5兲 show an unfamiliar feature 关as does the S(T) curve for GaAs in Ref. 26兴, namely: these curves are concave at T ⬍30 K and convex at T⬎80 K. The inflection points of the S(T) curves for ZnS, ZnSe, and ZnTe are at about 70, 53, and 41 K, respectively. In our analytical model it is the parameter which, by virtue of its variability between 0 and 1, controls the material-specific shape of these curves. 关The concave part of the low-T section of an S(T) curve disappears in the →1 limit and becomes extremely pronounced in the →0 limit.兴 Consider now Varshni’s ansatz 关Eq. 共15兲兴, from which the entropy follows the form S Var.(T)⫽ ␣ T(T ⫹2  )/(T⫹  ) 2 . 24,29 The second derivative of S Var.(T) is d 2 S Var.(T)/dT 2 ⫽⫺6 ␣ 2 /(T⫹  ) 4, i.e., its sign remains unchanged along the whole T axis 共for all values of ␣ and 兲. This is in stark contrast to the observed behavior of S(T) in Fig. 5, proving the limitations of Varshni’s model. This deficiency in Varshni’s model is responsible for the poor fits in the cryogenic region, the large uncertainties reported for ␣ and ,15,25 and the physically unreasonable values of 1.1 meV/K ⬍ ␣ Var.⬍1.7 meV/K and 500 K⬍  ⬍1050 K quoted for zinc chalcogenides in Refs. 47 and 48. The limitations of Varshni’s model have been repeatedly observed, especially for ZnS.7–9 Various attempts have been made to improve the fittings 共numerical simulations兲 by including modifications to the formula. Abounadi et al.7 and subsequently Tran et al.9 used the ansatz E Ab.共 T 兲 ⫽E 共 0 兲 ⫺

␣T4  ⫹T 3

.

共16兲

By fitting experimental E 1s (T⭐250K兲7 or E 1s (T⭐320K兲9 data sets for cubic ZnS, these authors arrive at ␣⫽0.401 meV/K and  ⫽2⫻106 K3 and ␣⫽0.4201 meV/K and  ⫽2.172⫻106 K3 , respectively. It is obvious that neither 

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nor, e.g., its third root, (  ) 1/3⬇125 or 130 K, can be related in a reasonable way to an empirical temperature parameter such as ⌰ or ⌰ D . Moreover, even the parameter ␣ following from fittings using Eq. 共16兲 obviously does not give the actual high-T limit of the slope of the E 1s (T) curve in ZnS. Table II shows that the ␣ values given in Refs. 7 and 9 are as much as 27% and 23% lower than the actual magnitude of the high-T slope in cubic ZnS 关as is clearly seen from Figs. 4共a兲 and 5兴. This large discrepancy is not surprising, however, in view of the ad hoc nature of ansatz 关Eq. 共16兲兴. By calculating the first and second derivatives of Eq. 共16兲 we find that E Ab.(T) has a point of inflection at T i ⬅(2  ) 1/3 ⬇159 or 163 K 共cf. Fig. 2 in Ref. 7 and Fig. 3 in Ref. 9兲. This means that the slope, S Ab.(T)⫽ ␣ (4  T 3 ⫹T 6 )/(  ⫹T 3 ) 2 , of the E Ab.(T) curve is maximum at this inflection point and decreases towards higher temperatures. Thus, Eq. 共16兲 directly contradicts many experiments on cubic ZnS. The maximum slope at the inflection point is S Ab.(T i ) ⫽(4/3) ␣ , that is, it is about 33% higher than the associated high-T limit, S Ab.(T→⬁)⫽ ␣ . This large difference explains the significant underestimation of the limiting slope by Eq. 共16兲. A more convenient modification 共generalization兲 of Varshni’s ansatz 关Eq. 共15兲兴 has been given by Ferna´ndez et al. in Ref. 8: E Fer.共 T 兲 ⫽E 共 0 兲 ⫺

␣T4 共  ⫹T 兲 3

.

共17兲

Equation 共17兲 provided good fits to the experimental E 1s(LH) (T), E 1s(HH) (T), and E 1s(SO) (T) curves from cryogenic to about room temperature (T⭐290 K兲 for a strained ZnS layer. This is primarily due to the fact that the shape of the entropy function, i.e., S Fer.(T)⬅⫺dE Fer.(T)/dT⫽ ␣ (T 4 ⫹4  T 3 )/(  ⫹T) 4 , changes from concave at very low T, to convex at higher T. The second derivative of this entropy function readily shows that the inflection point is at T i ⫽2  /3. Thus it follows from the fitted  value of about 75 K that the inflection point of S Fer.(T) is at about 50 K. This value is comparable to our result 共of about 70 K兲. However, there is no physically plausible connection between the parameter  in an ad hoc model like Eq. 共17兲 关or Eqs. 共15兲 or 共16兲兴 and a feature of a phonon energy spectrum 共like cutoff energy positions or moments25,27 of the electron-phonon spectral function兲.

VI. CONCLUSIONS

We have determined the temperature dependence of the 1s exciton peak positions in high-quality ZnS, ZnSe, and ZnTe epitaxial films grown on 共001兲-GaAs substrates by MBE. These data are analyzed via a semiempirical expression that provides, as a rule, a good fit to the observed temperature dependence. We succeeded in limiting the uncertainty for basic parameters to within a few percent by extending the experiments from the cryogenic temperature range up to about 4/3 of the corresponding Debye temperature ⌰ D , i.e., far beyond room temperature. This has been achieved by combining the results of absorption measure-

ments with those of reflectance measurements performed below and above room temperature, respectively. We suggest that an analogous procedure should be undertaken for other wide band-gap materials, including diamond, SiC, GaN, Alx Ga1⫺x N, etc., whose Debye temperature is much higher than 300 K.

ACKNOWLEDGMENTS

The authors acknowledge the financial support of the groups at Paderborn and Regensburg by the Deutsche Forschungsgemeinschaft. The authors would also like to thank D. Decker and P. Simon of the Technical University Chemnitz, for preparing Figs. 4–6 of this article. 1

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