Temperature dependence of the dielectric function and of the parameters of critical point transitions of CdTe

Optical Materials 12 (1999) 143±156

Temperature dependence of the dielectric function and of the parameters of critical point transitions of CdTe J.T. Benhlal

a,* ,

K. Strauch a, R. Granger a, R. Triboulet

b

a b

Laboratoire de Physique des Solides, INSA, F. 35043 RENNES Cedex, France Laboratoire de Physique des Solides, CNRS, F. 92195 MEUDON Cedex, France

Received 28 October 1997; received in revised form 6 February 1998; accepted 6 March 1998

Abstract The dielectric function e of CdTe is deduced from spectroscopic ellipsometry measurements performed from 0.7 to 5.5 eV. For the ®rst time e is given for temperatures increasing from 20 to 370 K. e variations with photon energy are analysed within the standard critical point model (SCP). The variations, with temperature, of the CP parameters are given, discussed and compared with known data. For T < 80 K the dielectric response is completely governed by the exciton near the fundamental gap E0 . The contribution of this bound state decreases with T but remains high at room temperature. The transition at E1 is likely to correspond to a quasi bound state interacting with the continuum in the entire temperature range studied. Only one critical point is seen in the vicinity of E2 which behaves like a two dimensional transition at a saddle point. The weak variations of the spin±orbit splittings D0 and D1 with temperature are explained as resulting from the choice of lineshapes used to ®t experimental data. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: 7820D; 7125T; 060F

1. Introduction CdTe is used as a substrate for the epitaxial growth of HgCdTe compounds [1]. It is one of the end binaries of the Hg1ÿx Cdx Te and Cd1ÿx Znx Te systems, the latter ternary is replacing CdTe for the epitaxial growth of HgCdTe [2]. CdTe is used in optoelectronic devices such as detectors of high energy photons [3] and in heterostructures [4]. Re¯ectivity measurements at room and low temperatures on cleaved samples led to the determination of the energy of the critical transitions [5,6] and e has been deduced from the Kramers±Kronig *

Corresponding author.

inversion. The di€erent results are gathered together in ref. [7]. The variations of the fundamental gap with temperature T were determined by Camassel et al. [8]. Those of the transition energies E1 and E1 + D1 were deduced from optical re¯ectivity and absorbtion experiments on thin ®lms at some temperatures [9,10]. More recently, the dielectric function has been deduced from ellipsometric measurements at room temperature below 5.6 eV [11±14]. The variations of the dielectric function with photon energy E were calculated with di€erent models of dielectric function (MDF) [12,15]. Kim and Sivananthan tentatively calculated e(E) at 77 K with their model which appears the most accurate. They have not taken into account the excitonic contributions near the

0925-3467/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 3 4 6 7 ( 9 8 ) 0 0 0 1 9 - 6

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critical transitions, however, no comparison was possible as no experimental data were available. The experimental dependence of e(E) on temperature is known only for some elemental [16] and important III±V semiconductors [17]. This paper gives, for the ®rst time, e(E) as measured with spectroscopic ellipsometry (SE) from 0.7 to 5.5 eV and in the temperature range 20±370 K. e(E) is analysed within the standard critical point model (SCP) around the main transitions. The next section is devoted to the necessary experimental details, the third section describes the ®tting procedures. Section four gives the variations of the parameters associated with each critical point and their discussion. The paper ends with a conclusion. 2. Sample preparation and low temperature experiments A CdTe sample of (1 0 0) orientation has been cut from an ingot grown in a three-zone furnace by the vertical Bridgman technique under controlled Cd vapour pressure [18]. The sample was given a stoichiometric anneal at 700°C, with Cd at 680°C, for 100 h. It is n type with a concentration of 1 ´ 1015 cmÿ3 , its mobility increases from 730 cm2 Vÿ1 Sÿ1 at 300 K to 13 250 cm2 Vÿ1 Sÿ1 at 35 K where it is maximum. This sample was then mechanically polished with alumina of a grain size 0.3 lm and then with alumina of grain size 0.04 lm. Finally the surface was chemomechanically polished with Br2 in methanol at a concentration of 0.1% [13] and carefully rinsed with water-free methanol. The last rinsing is performed in situ on the ellipsometer to check that the sample has a re¯ectivity corresponding to the best known results. Other stripping processes have been used but the one just described above gives our best results as compared to the published data. Ellipsometric measurements are performed with a phase modulated spectroscopic ellipsometer UVISEL (Instrument S.A.). The energy mesh is usually 10 meV and the spectral width 5 meV, except for the region of the edge exciton around E0 where the mesh and the spectral width are decreased to 1 meV at low temperature. The ellip-

someter gives the ellipsometric angles w and D which are de®ned by the ratio rp /rs ˆ tan w eiD where rp and rs are the complex re¯ectivities for the polarizations of the light parallel and perpendicular to the plane of incidence [19]. The value of e is easily deduced if no overlayer is considered on the surface (two phase model). The CdTe sample is transferred from the last rinsing bath to the ellipsometer in a windowless cell under ¯owing Ar gas. The sample is rinsed again with pure methanol. This process gives the best value of ei at the energy E2 corresponding to the peak of optical absorbtion of highest energy [13,14,17]. Our value of ei (E2 ) ˆ 12.15 is as high as the best values already reported (ei (E2 ) ˆ 12.1 for CdTe (1 0 0)), see Refs. [11,12,14,20]. The sample is placed on the cold ®nger of a ultrahigh vacuum cryostat with strain free silica windows [21]. The cryostat is evacuated with turbomolecular pumps and baked up to 160°C to reach a pressure lower than 10ÿ7 Pa at room temperature. The pressure of residual gases must be low to avoid gas condensation on the sample when it is cooled [6,21]. The surface oxidizes during the sample transfer in the cryostat, which is done partly under the ambient atmosphere and lasts about 15 min [13,16,17]. The sample temperature can be adjusted between 20 and 400 K with an uncertainty of 0.2 K below 150 K and of 0.5 K up to 370 K. 3. Ellipsometric measurements and dielectric function analysis The ellipsometric angles w and D are monitored at an angle of incidence of 70°. After each change of the sample temperature the optical alignment is checked and a new polarization calibration is performed to take into account the possible polarization contribution of the windows as the optical beams may be not accurately perpendicular to them. In a ®rst step, the thickness of the overlayer, grown during the transfer in the cryostat, and its dielectric function are deduced from the comparison of the ellipsometric angles measured, on the one hand, on CdTe just after the last rinsing with

J.T. Benhlal et al. / Optical Materials 12 (1999) 143±156

methanol performed in situ on the ellipsometer, and, on the other hand, after it has been mounted in the cryostat. Both experiments are done at the same temperature, Fig. 1 shows the two corresponding pseudodielectric function spectra. The composition of the overlayer which has not been characterized is not known, so we prefer to describe its dielectric function with the phenomenological MDF of Zollner [22] whose expression is: e…E† ˆ ai ‡ bi E ÿ1

‡ Bi exp…ihi †…E ÿ Ei ‡ iCi † ;

…1†

where ai , bi are complex, hi is a phase angle, Ei and Ci are the energy and the broadening parameter associated with a transition, Bi is an oscillator strength. The best ®t is obtained if two layers are considered, labelled 2 for the uppermost layer and 1 for the underlying one. Table 1 gives the thicknesses and the values of the parameters entering Eq. (1). Fig. 2 shows the variations of er and ei for both layers. The spectrum of e for the upper layer looks like that of the anodic oxide grown on CdTe [20] which has an optical absorbtion which increases steeply above 4.5 eV. The underlying layer is rather thin and its dielectric function appears far from that of amorphous Te although the uncertainty in e remains high. Angle resolved X-ray photoelectron spectroscopy (XPS) characterizations show that the surface of the samples is cov-

Fig. 1. Real (r ) and imaginary (i ) parts of the pseudodielectric function of CdTe: (±±±±±±) after the last rinsing, (±__±__) after the transfer to the cryostat.

145

Table 1 Values of the parameters entering relation (1) and describing the dielectric function of the two layers present at the CdTe surface (cf. Fig. 2) di (nm) Bi (eVÿ1 ) Ei (eV) Ci (eV) hi (deg) Re(ai ) Im(aj ) Re(bi ) Im(bi )

Layer 1

Layer 2

0.34 1.677 4.338 10.2 49.3 17.6 0 ÿ1.23 0

0.94 5.419 5.419 0.001 65.8 ÿ0.49 0.09 0.24 0

Fig. 2. Dielectric function of the two layers present on the CdTe surface after the sample transfer to the cryostat: (±_±_) er1 and (- - - - - - - -) ei1 of the layer lying above CdTe, (±±±±±±) er2 and (±__±__) ei2 of the uppermost layer.

ered with a layer containing mainly carbon after a chemomechanical polish with Br2 and rinsing [13,23,24]. However it appears surprising that this carbon rich layer stays at the CdTe interface during the oxidation. We assume that the dielectric functions of these overlayers and their thicknesses remain independent of temperature [16,17]. For each temperature e is corrected for the contribution of the overlayer to obtain the dielectric function of CdTe. Figs. 3 and 4 show respectively er and ei spectra of CdTe for some selected temperatures. The critical transitions are located by arrows in Fig. 4, they are named E2 , E1 + D1 , E1 , E0 + D0 , which is clearly seen at low temperatures on er and E0 . The feature near E0 which can be better seen in the

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Fig. 3. Real part of the CdTe dielectric function at: (±±±±±±) (T ˆ 20, (±±±_±±±) 208 and (- - - - - -) 294 K.

inset of Fig. 4 reveals the strong excitonic contribution due to the Coulomb correlation between excited carriers [25]. Fig. 5 reproduces the electronic structure of CdTe at 0 K calculated with the empirical pseudopotential method [26] in which the form factors are adjusted to ®t the energies at critical points. Table 2 recalls the critical transitions already cited with their energies and symmetries. Previous calculations by the same group

[5] or more recent ones [27] give slightly di€erent values of the transition energies but with the same assignments of the critical points. Four structures have been found in the calculated density of states between 4.5 and 5.5 eV [5] but only two were apparent in the low temperature re¯ectivity spectrum at 5.18 and 5.53 eV. The derivatives of the room temperature dielectric function of CdTe reveal two transitions near 5 and 5.27 eV [11]. However more recent ellipsometric data show only one broad transition around 5.13 eV at room temperature [12,13]. The spectral analysis of e(E) follows the usual SCP model. Around a critical point of energy E` , e can be described by the general expression [28]: n

ih` L` …E† ˆ C` Cÿn ` e …E ÿ E` ‡ iC` † ‡ F …E†;

…2†

C` Cÿn `

is the where C` is the broadening parameter, oscillator strength and n is the order of the transition (n ˆ 12 for a 3D transition, n ˆ 0 (logarithmic) for a 2D transition, n ˆ ÿ 12 for a 1D transition and n ˆ ÿ1 for a discrete exciton). F(E) is a slowly varying function of E resulting from remote transitions; its derivatives are often assumed to be negligible [28]. h` is a phase angle

Fig. 4. Imaginary part of the CdTe dielectric function at: (±±±±±±) T ˆ 20, (±±±_±±±) 208 and (- - - - - -) 294 K as obtained with a spectral width of 10 meV. The inset gives a magni®ed view of ei around the fundamental gap as obtained with a spectral width of 1 meV.

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147

Fig. 5. Electronic band structure of CdTe along the directions of high symmetry in the Brillouin zone, from [26].

whose value is a multiple of 12 p for transitions between uncorrelated electron states. Other values are attributed to a mixture of contiguous CPs due to correlation e€ects [16,17,25]. For each critical transition, the parameters de®ned in Eq. (2) are adjusted by ®tting simultaneously the second derivatives of experimental er and ei to those of the real and imaginary parts of L` …E†. A slight smoothing with an exponential regression is applied after the ®rst and the second di€erentiations of the experimental data in the analysis of the weak transition E0 + D0 for temperatures above 150 K. In order to avoid possible distortions in the numerical calculations, the lineTable 2 Transitions in CdTe, their location in the Brillouin zone, their symmetry and their calculated energy from [26] Critical point

Location

Symmetry

Energy (eV)

E0 E0 + D0 E1 E1 + D1 E2 (X) 0 E0 E2 (R) 0 E2 (X)

Cv8 ® Cc6 Cv7 ® Cc6 Lv4:5 ® Lc6 Lv6 ® Lc6 Xv7 ® Xc6 Cv8 ® Cc7

M0 M0 M1 M1 M1 M0 M2 M1

1.65 2.48 3.49 4.00 5.16 5.36

Xv6 ® Xc6

5.46

shapes given by Eq. (2) are also numerically calculated and, if necessary, smoothed with the same procedures as those applied to calculate the second derivatives [29]. The ®t is performed with the Levenberg±Marquardt method [30]. The transition E2 , which is broad, is ®rst ®tted. Then E1 and E1 + D1 are ®tted simultaneously [13]. For this second ®t the weak contribution of the transition E2 to the second derivative of L` …E† is therefore taken into account. For E0 + D0 and E0 the contributions of the three upper transitions to the second derivatives of L` …E† are also introduced. An example of the second derivatives of er and ei experimental data is given in Fig. 6 with ®tted curves deduced from Eq. (2). The ®gure shows the good agreement between the experimental data and the lineshapes obtained after the ®ts. 4. Results and their analyses The values of the parameters of a transition depend on the type of lineshape chosen for the ®t. The type of lineshape used is indicated as a sux placed in parentheses at the end of the letters designating the parameter. `Ex' labels the excitonic lineshape.

148

J.T. Benhlal et al. / Optical Materials 12 (1999) 143±156

Fig. 6. Example of second derivatives at 20 K of (s) er and of (h) ei with their ®t with an exciton lineshape for E0 and E0 + D0 and a 2D lineshape for the other transitions: (±±±±±±) er , (- - - - - - - -) ei .

The energies of the critical transitions as deduced from ®ts like those given in Fig. 6, are plotted as a function of temperature T in Fig. 7(a) and (b). The shift of electronic-state energies with temperature comes from two contributions [31]: the ®rst is from the thermal expansion of the lattice and the second is due to the thermal vibrations of the atoms which spreads their potential. The E` …T † can be well ®tted with phenomenological Varshni relations [32]. However, we prefer to describe them with expressions based on more physical grounds [17,33] as they are related to the excitation of phonons leading to the E` …T † variation.   2 ; …3† E` …T † ˆ E`0 ÿ a` 1 ‡ h =T e ` ÿ1 where E`0 ; a` and h` are adjustable parameters, h` is an e€ective temperature which describes the excitation of phonons interacting with the electronic states considered. The values of the parameters de®ned in Eq. (3) are gathered in Table 3. The variations of the E` …T † are almost linear above 100 K. The temperatures coecients dE` =dT jm are calculated from the experimental results between 100 and 370 K. Their values are reported in Table 4 together with those already published. After the ®t of the E` …T † data with expression (3), it is possible to deduce the high temperature limit of dE` =dT from dE` =dT jh ˆ ÿ2a` =h` . This high temperature coecient is also given in Table 4.

Fig. 7. (a) Temperature variations of (s) E0 and (h) E0 + D0 , both deduced with an exciton lineshape. Full lines correspond to ®ts with relation (3). The dotted line gives the fundamental gap variations after an upward shift of the exciton binding energy of 9 meV. 20 K values correspond to ®ts of results given in Fig. 6. (b) Temperature variations of (´) E1 , (D) E1 + D1 , (») E2 all ®tted with a 2D lineshape. Full lines correspond to ®ts with relation (3). 20 K values correspond to ®ts of results given in Fig. 6.

The variations of the Lorentzian broadening parameters C` with T are given in Fig. 8(a) and (b). They are also described with expressions similar to Eq. (3), [17]:  C` ˆ C`1 ‡ C`0 1 ‡

2 0

eh `=T ÿ 1

 ;

…4†

h0` is di€erent from h` de®ned in Eq. (3) as thermal vibrations contribute only to C` . The values of the parameters entering Eq. (4) for each critical transition are reported in Table 5. The results are now discussed for each transition.

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149

Table 3 Values of the parameters entering relation (3) giving the energy of each critical transition versus temperature (the last sux in column 1 indicates the lineshape used for the ®t of experimental data) Transition ` and ®tting lineshape

E`0 (eV)

a` (meV)

h` (K)

Error bar on E` (meV)

(E0 )…Ex† (E0 )…3D† (E0 + D0 )…Ex† (E0 + D0 )…3D† (E1 )…Ex† (E1 )…2D† (E1 + D1 )…Ex† (E + D1 )…2D† (E2 )…2D†

1.617 1.615 2.563 2.560 3.568 3.580 4.126 4.171 5.152

21.5 19.3 25 22 43 65 28 97 27

115 106 117 107 115 169 83 244 144

2 2 3 2 3 5 4 7 6

Table 4 Temperature coecients of the critical transitions (in 10ÿ4 eV Kÿ1 ), the subscript meanings are: h high temperature limit, m as deduced between 100 and 370 K both from this work; c calculated, d other experimental determinations (numbers inside brackets give the reference) Transition ` and ®tting lineshape

ÿdE` /dT|h

ÿdE` /dT|m

ÿdE` /dT|c

ÿdE` /dT|d

E0…3D† E0…Ex† E1…2D†

3.6 ‹ 0.1 3.7 ‹ 0.1 7.7 ‹ 0.3

3.5 ‹ 0.1 3.6 ‹ 0.1 7.3 ‹ 0.3

6.2 [40]

3 [10]

E1…Ex† E2 (X)…2D† E2…2D† E2 (R)…2D†

7.4 ‹ 0.2

7.2 ‹ 0.2

5.5 [40]

5.5 [6] 2.3 [10] 5 [9]

3.7 ‹ 0.3

3.6 ‹ 0.3

4.1. E0 transition This transition, which corresponds to the fundamental gap, involves the Cv8 valence band and the Cc6 conduction band states in the double group notation. It is dominated by the edge exciton at low temperatures [8], as can be seen on ei (cf. inset of Fig. 4). The fundamental state of the exciton appears not completely separated from its excited states and the continuum, as can be seen in the ei spectrum at 20 K. Although the binding energy of the free exciton in CdTe is relatively high (Ex ˆ 9.05 meV [34,35]) no optical data have supported this separation. This feature is also found in ZnTe [36] whose free exciton binding energy is near that of CdTe. The 20 K er spectrum reported in Fig. 4, shows only a small excitonic contribution at E0 when a spectral width of 10 meV is used for the measurements. The fundamental state of the exciton dominates er when the spectral width is decreased to 1 meV as seen in the inset of Fig. 4. Thus, the incomplete separation of the funda-

5.2 [40]

4.1 [6]

mental state from the excited ones is partly due to its broadening but also to the limited resolution of the monochromator (1 meV). e second derivatives around E0 are ®tted with an excitonic lineshape (n ˆ ÿ1) but also with a 3D lineshape …n ˆ 12† letting all the other parameters de®ned in Eq. (2) vary freely. The values of E0 found with both ®ts remain the same, within ‹1 meV, in the entire temperature range studied. The phase angle h(E0 )…Ex† , which is plotted in Fig. 9, takes a value very near p at 20 K corresponding to a pure exciton as C` is always taken positive in Eq. (2). h(E0 )…Ex† decreases rapidly with temperature above 80 K describing a Fano pro®le [37] resulting from the interaction of the exciton with a continuum of states [17,38]. At room temperature, the phase angle deduced from the ®t with a 3D lineshape (cf. Fig. 9) has not reached the value h…E0 †…3D† ˆ 12 p, corresponding to transitions of M0 type between uncorrelated states. This result is in accordance with the criterion of negligible excitonic coupling which holds when C(E0 )  70Ex where Ex is the

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J.T. Benhlal et al. / Optical Materials 12 (1999) 143±156

Fig. 9. Temperature variations of the phase angle h, for the meaning of the symbols see Fig. 8(a) and (b). 20 K values correspond to ®ts of results given in Fig. 6.

Fig. 8. (a) Temperature variations of the broadening parameters (s) C(E0 ), (h) C(E0 + D0 ), deduced with an exciton lineshape; (*) C(E0 ), (Å) C(E0 + D0 ) deduced with a 3D lineshape. The lines correspond to ®ts with relation (4). 20 K values correspond to ®ts of results given in Fig. 6. (b) Temperature variations of the broadening parameters: (´) C(E1 ), (D)C(E1 + D1 ), (») C(E2 ) deduced with a 2D lineshape. (}) C(E1 ), (>) C(E1 + D1 ) deduced with an exciton lineshape. 20 K values correspond to ®ts of results given in Fig. 6.

binding energy of the exciton [39] and is far from being ful®lled as C(E0 )  30 meV. The choice of the excitonic lineshape at low temperature is con®rmed by the behaviour of the corresponding osÿn cillator strength A…E0 † ˆ C…E0 †C…E0 † . Fig. 10(a) shows that A(E0 ) remains almost independent of T up to 180 K when it is calculated with an excitonic lineshape (n ˆ ÿ1). By contrast A(E0 ) calculated with a 3D lineshape varies strongly in this temperature range as can be seen in Fig. 10(b). The temperature variation of E0 appears linear for T > 100 K. The high temperature coecient of E0 as deduced from the ®t of the experimental results is dE0 =dT jh ˆ ÿ…3:7  0:1†  10ÿ4 eV Kÿ1 . It is found to be slightly lower if the temperature range considered is limited to that of the experimental results …dE0 =dT jm ˆ ÿ…3:6  0:1†  10ÿ4

Table 5 Values of the parameters entering relation (4) giving the broadening parameter for each critical transition versus temperature (the last sux in column 1 indicates the lineshape used for the ®t of experimental data) 0

Transition ` and ®tting lineshape

C`1 (meV)

C`0 (meV)

h` (K)

Error bar on C` (meV)

E0…Ex† E0…3D† (E0 + (E0 + E1…Ex† E1…2D† (E1 + (E1 + E2…2D†

ÿ107 ÿ269 1.8 ÿ32 21 ÿ8.7 39 9 ÿ212

110 269 14.5 33.6 29 36 18 23 293

674 1245 123 458 238 421 104 178 621

‹2 <1 ‹4 <1 ‹2 ‹1 ‹3 ‹3 ‹10

D0 )…Ex† D0 )…3D† D1 )…Ex† D1 )…2D†

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151

lower than that calculated in the empirical pseudopotential framework (dE0 =dT ˆ ÿ6:7  10ÿ4 eV Kÿ1 ) and older experimental data [40]. Our ®t gives E0 (0 K) ˆ 1.596 eV which is the value of the energy of the free exciton commonly given within ‹1 meV [8,34,35,41]. The energy of the fundamental gap EG of CdTe is also drawn in Fig. 7 after an upward shift of 9 meV corresponding to the binding energy of the exciton [34,35,41]. The broadening parameter C…E0 †…3D† is found extremely to be small at low temperatures. …C…E0 †…3D† ˆ 8  10ÿ5 eV at 20 K†. This low value indicates that e is governed by a local bound state at E0 . C…E0 †…Ex† ˆ 2 meV at 20 K is in fact limited by the monochromator resolution which was unfortunately limited to 1 meV near 1.6 eV. This value is twice that found for GaAs at the same temperature [16] and may appear somewhat low as the exciton fundamental state is not completely separated from the band continuum in the ei spectrum. The room temperature value of C…E0 †…3D† (8 meV) lies between those of [15] (5 meV) and [12] (10 meV) which also uses a ®t with a 3D lineshape. 4.2. E0 + D0 transition

Fig. 10. (a) Oscillator strengths versus temperature after exciton lineshape ®ts, (b) 3D lineshape ®ts and (c) 2D lineshape ®ts, for the meaning of the symbols see Fig. 8(a) and (b). 20 K values correspond to ®ts of results given in Fig. 6.

eV Kÿ1 for 370 K
This weak transition between Cv7 and Cc6 states is clearly seen in the er spectrum at low temperatures (cf. Fig. 3). h…E0 ‡ D0 †…3D† reaches the value of 180° at 20 K corresponding to a critical point of M1 type [25,28]. This type of transition does not correspond to the transition studied which is of M0 type (cf. Fig. 5). h…E0 ‡ D0 †…Ex† decreases smoothly from 150° at 20 K to 100° at 400 K (cf. Fig. 9) and indicates a bound state interacting with a continuum. Nevertheless the experimental data have been ®tted with a 3D and an exciton lineshape, the corresponding results are given in Figs. 8±10 and Tables 3 and 5. The di€erence between the E0 + D0 values deduced from both ®ts remains small (‹3 meV) when the uncertainty on the values themselves increases from 2 meV at 20 K to 7 meV at 350 K. The values found at 294 K ……E0 ‡ D0 †…Ex† ˆ 2:431 eV and …E0 ‡ D0 †…3D† ˆ 2:428 eV† are very near to those deduced from the analysis of ellipsometric data with a more complex model, but which is of the 3D type around E0 + D0 [15] (cf. Table 6).

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Table 6 Room temperature values of the energies (El ) and broadening parameters C(l) of the critical transitions compared with those from the literature [6] E0 E0 + D0 E1 E1 + D1

2.4 2.32 3.88

E2

C(5.16)

C(E0 ) C(E0 + D0 ) C(E1 ) C(E1 + D1 ) C(E2 )

[20]

3.363 (3D) 3.968 (3D) (X)4.99 (3D) (R)5.28 (3D) 0.055 (3D) 0.077 (3D)

[14]

[12]

[15]

This work (294 K)

1.50 (3D)

1.500 (3D) 2.422 (3D) 3.372 (2D) 3.989 (2D) (X) 5.070 (2D)

1.504 2.432 3.347 3.924

3.287 (2D) 3.848 (2D)

3.32 (2D) 3.87 (2D)

5.068 (1D)

5.07 (1D) 0.01 (3D)

0.025 (2D) 0.100 (2D)

0.07 (2D) 0.12 (2D)

0.240 (1D)

0.33 (1D)

(R) 5.376 (2D) 0.005 (3D) 0.032 (3D) 0.078 (2D) 0.091 (2D) (X) 0.156 (2D) (R) 0.189 (2D)

(3D) (3D) (2D) (2D)

5.042 (2D) 0.008 0.020 0.051 0.088

(3D) (3D) (2D) (2D)

0.167 (2D)

The assignment of transition E2 in the Brillouin zone is given, in parentheses, before its value. The type of lineshape used to deduce El and Cl is given in parentheses, when it is known, after each value.

The values of D0 , as deduced from the di€erences between E0 + D0 and E0 , show a low and monotonic decrease of 14 meV between 20 and 320 K. This decrease is the same whatever the type of ®t used (3D or Ex). The variation of D0 appears supprising as this energy originates from an intraatomic spin±orbit interaction [14,17]. The decrease of D0 is a consequence of ®tting er and ei derivatives with lineshapes corresponding to pure excitonic on the one hand or 3D uncorrelated transitions on the other hand. The value of E0 + D0 depends on that of the phase angle which introduces a distortion of the lineshape when it is varied. If the ®t at 320 K is done with h…E0 ‡ D0 †…Ex† ®xed at its 20 K value, a new value for …E0 ‡ D0 †…Ex† is found which is 25 meV higher than the original one. The same di€erence is found when the ®t is done with a 3D lineshape. This di€erence is about twice the decrease of D0 between 20 and 320 K. Therefore the small variation found on (E0 + D0 )(T) originates from a systematic error introduced by the choice of the lineschapes which is kept constant for all the ®ts in the temperature range studied. The value of D0 which must stay constant of T [17], is deduced from the values of E0 and E0 + D0 at low temperatures where the uncertainties are lower. Taking into account the binding energy of the exciton at E0 of 9 meV [34,35] we deduce D0 ˆ (932 ‹ 8) meV for

CdTe. This value agrees with those already reported in the literature [15,42] and that calculated when the d bands are introduced in the evaluation of the electronic structure of CdTe [43]. The broadening parameter C(E0 + D0 ) deduced from the ®t with an exciton lineshape is probably more accurate at low temperatures (14 meV at 20 K) than the one deduced with a 3D lineshape (1.7 meV at 20 K). C(E0 + D0 )…3D† seems also too small at room temperature (C(E0 + D0 )…3D† ˆ 17 meV), however, its variations with temperature are also given in Fig. 8 with those of C(E0 + D0 )…Ex† . The values of the parameters deduced from the ®ts of C(T) variations with relation (4), are also reported in Table 5. 4.3. E1 transition This strong transition links L
v4:5 and Lc6 states or ÿ also states along K Kv4:5 ! Kc6 (cf. Fig. 5). The asymmetry in the CdTe re¯ectivity at 10 K, around E1 has been explained by the Coulomb interaction [44]. An analysis of the experimental results led to a ratio of longitudinal to transverse mass, at this saddle point, of 28 [44]. The analysis of the thermore¯ectance of CdTe at 203 K led to the same conclusion of a quasi bound exciton at E1 [45]. More detailed analyses of the dielectric response in the vicinity of a saddle point have been

J.T. Benhlal et al. / Optical Materials 12 (1999) 143±156

given [38,46], however, they lead to complicated expressions which must be handled numerically. Such an analysis is in progress, however, we will restrict discussion to the simple use of the SCP description given by Eq. (2), especially as Kim and Sivananthan [15] claim that e can be well described with a 2D transition at E1 . Hence, the numerical derivatives of er an ei have been ®tted with a 2D lineshape (n ˆ 0 or Ln in Eq. (2)), on the one hand, and an excitonic lineshape (n ˆ ÿ1) on the other hand. This ®t is done simultaneously for E1 and E1 + D1 . At low temperatures, the exciton lineshape gives a slightly better ®t of the second derivatives of e than the 2D lineshape, especially in the wings of the spectra. Thus the phase angle …h…E1 †…Ex† ' 260 † appears far from the value corresponding to a pure bound state. h…E1 †…Ex† decreases, as shown in Fig. 9, expressing the decreasing interaction of a bound state with a continuum [25]. The phase angle h…E1 †…2D† deduced from the ®t with a 2D lineshape stays very near 0° for T < 140 K. This value corresponds to a maximum for 2D uncorrelated transitions; h…E1 †…2D† decreases then with temperature indicating a transition which cannot be a pure 2D transition. The oscillator strength ÿn C…E1 †…2D† C…E1 †…2D† appears almost constant above 140 K (see Fig. 10(c)) showing that a 2D transition can describe e properly in the vicinity of E1 . By contrast, C…E1 †…Ex† Cÿn …Ex† stays contant for T < 140 K (cf. Fig. 10(a)) and tends to invoke a bound state at low temperatures. These results are consistent with a quasi-bound state interacting with a continuum at low temperatures. Tables 3 and 5 give the parameters deduced from the ®ts of E1 (T) and C(E1 )(T) with Eqs. (3) and (4) using both lineshapes. The di€erence E1…Ex† ÿ E1…2D† is .+6 meV for T < 100 K it reverses to ÿ8 meV from 100 to 200 K, and then remains lower than 2 meV above 200 K. The value found at 294 K (E1…2D† ˆ E1…Ex† ) ˆ 3.347 eV agrees with those already reported in the literature at room temperature as can be seen in Table 6. dE1 /dT is near (ÿ7.6 ‹ 0.4) ´ 10ÿ4 eV Kÿ1 which is larger than values already reported, which are also given in Table 4. However, it seems that these last values are deduced from measurements at only two temperatures. Our value is also larger than that calculated within the

153

pseudopotential framework [40]. The value of C(E1 )…2D† ˆ 51 meV at 294 K is in the range found by other authors, as seen in Table 6, some of them being evaluated with more sophiticated models [15]. 4.4. E1 + D1 transition The comparison of the e derivatives around E1 + D1 with the 2D and excitonic lineshapes leads obviously to similar conclusions concerning the type of transition. This can be checked on the phase angle variations h(E1 + D1 )…2D† and h(E1 + D1 )…Ex† and the oscillator strengths which are plotted respectively in Fig. 9 and Fig. 10(c) and (a). C(E1 + D1 )…2D† for T ˆ 294 K is also in the domain of values already published, as can be checked in Table 6, as are also the values for E1 + D1 . The di€erence (E1 + D1 ) ÿ E1 ˆ D1 is calculated for both lineshape ®ttings (2D and Ex). D1…Ex† stays constant for T < 160 K (D1…Ex† ˆ 571 ‹ 3 meV) and then increases with T to 0,590 eV at 370 K. D1…2D† starts from 0.55 eV at 20 K, increases to 0.573 eV at 80 K to remain constant within ‹4 meV. The variations of D1 deduced with each lineshape orginate also from the choice of them in a temperature range where they do not apply strictly. We propose D1 ˆ (572 ‹ 5) meV for the spin±orbit splitting near L and keep this value constant of temperature. d(E1 + D1 )/dT is the same as dE1 /dT, which disagrees with the existing experimental temperature coecients reported in Table 4 and the calculated value [40]. 4.5. E2 transition Several transitions have been proposed in the vicinity of 5 eV. The three most important are E2 (X) ˆ 5.4 eV [9], E0 0 ˆ 5.16 eV [9], E2 (R) ˆ 5.28 eV [20] but also other transitions are given in the literature along D near 5.10 eV and near K at 4.98 eV [5]. Ellipsometric measurements, all performed at room temperature, show only one transition named E2 [12,13]. However, the numerically calculated derivatives of er of Arwin and Aspnes [20] show two features at 4.99 and 5.28 eV. These two transitions are attributed to E2 (X) and E2 (R) transitions in [15]. We have not found several

154

J.T. Benhlal et al. / Optical Materials 12 (1999) 143±156

features in the derivatives of er and ei from 20 to 370 K as can be seen in Figs. 3, 4 and 6. The second derivatives of er and ei around 5 eV are ®tted with a 2D lineshape which can better describe a group of several transitions near the same energy [25,17]. This choice appears judicious as the phase angle h(E2 )…2D† , which is plotted in Fig. 9, stays constant in the entire temperature range considered. Moreover, its value, close to 3/2p, is consistent with a transition at a saddle point This conclusion is consistent also with the weak variation of the oscillator strength deduced from the ®t with a 2D lineshape, and which is given in Fig. 10(c). The room temperature value of E2 is, indeed, in the same range of values already published (cf. Table 6), however, the existence of several transitions in this energy domain is not con®rmed. The temperature coecient dE2 /dT . ÿ (3.6 ‹ 0.3) ´ 10ÿ4 eV Kÿ1 compares reasonably with the other determinations given in Table 4. These values of dE2 /dT for CdTe appear low compared to those measured for GaAs [17] and InSb [33], which are around ÿ6 ´ 10ÿ4 eV Kÿ1 , and cast doubt on the identi®cation and the lineshape used for the ®t. There are few experimental determinations of dE2 / dT in II±VI cubic compounds. Our value is not far from that of ZnSe (dE2 /dT ˆ ÿ4.7 ´ 10ÿ4 eV Kÿ1 ) [47]. Moreover dE2 /dT of Ge and Si [16] (<ÿ3.10ÿ4 eV Kÿ1 ) are also low. C(E2 ) variations with T are plotted in Fig. 8(b) and the corresponding parameters deduced from the ®t with Eq. (4) are given in Table 5. The 294 K value of C(E2 ) is the same as that calculated by Kim and Sivananthan [15] using the ellipsometric data of Aspnes and Arwin [11] (cf. Table 6) although a second transition (E2 (R)) has been shown 0.3 eV above E2 . These two last values of C(E2 ) are half that given by Adachi et al. [12], who considered only one transition. 5. Conclusion Ellipsometric measurements performed from 20 to 370 K allow the observation of the temperature behaviour of the di€erent transitions seen between the fundamental gap and 5.6 eV for CdTe. The

temperature variations of the energies E0 , E0 + D0 , E1 , E1 + D1 and E2 are given and described by expressions whose parameters are also given. At low temperatures, our results show the fundamental state of the exciton which is not completely separated from the ®rst excited states and the continuum. This result should be attributed to the limited resolution of the monochromator. The optical properties in the vicinity of this gap are controlled by the exciton for T < 80 K, the interaction of this bound state with the continuum increases, however, the exciton contribution remains important at room temperature. The dielectric response near E1 is controlled by a quasi-bound state interacting with the continuum in all the temperature range studied. Thus the dielectric function can be described properly with a 2D transition or with an exciton lineshape. The spin±orbit splittings D0 at C and D1 at L show a small variation with temperature which is explained by the use of lineshapes for the ®ts which are strictly valid only for limiting cases. Only one transition (E2 ) is found in the vicinity of 5 eV even at low temperatures where theoretical calculations predict several transitions in the near vicinity and one ellipsometric result reveals two transitions of equal amplitude. Kim and Sivananthan [15] tentatively calculated the dielectric function of CdTe at 77 K and 600 K. They started from the parameters deduced from the ®t of ellipsometric data, at 300 K, and their ®rst and second derivatives with their theoretical expressions. In the lack of knowledge of the temperature variation for all the necessary parameters they took the same temperature coecient dE/ dT ˆ ÿ3 ´ 10ÿ4 eV Kÿ1 for all the transition energies which have been described here. Table 4 shows that this choice is sucient for dE0 /dT, d(E0 + D0 )/dT and dE2 /dT which are actually 20% higher than their assumption. This choice is worse for E1 and E1 +D1 , where the actual temperature coecient is twice the assumed value. The broadening parameters of CdTe have been deduced from those obtained for GaAs and described by the same expression as Eq. (4) [17]. For each transition, the temperature h0` in Eq. (4) is obtained by multiplying that of GaAs by the ratio of the Debye temperatures of CdTe and GaAs. Table 7 shows

J.T. Benhlal et al. / Optical Materials 12 (1999) 143±156

155

Table 7 Values of the energy of the critical transitions and their linewidth at two temperatures as deduced from our experimental data (this work) and compared to those evaluated in [15] (cal. in [15]) Critical transition

Energy (eV)

Linewidth (meV)

At 70 K E0…3D† E0…Ex† (E0 + D0 )…3D† (E0 + D0 )…Ex† E1…2D† (E1 + D1 )…2D† E2 (X)…2D† E2…2D† E2 (R)…2d†

At 600 K

At 70 K

This work

Cal. in [15]

This work

Cal. in [15]

This work

1.585 1.586 2.526 2.526 3.503 4.068

1.569

1.396 1.393 2.303 2.311 3.115 3.689

1.410

0.08 3.2 1.8 22 28 35

5.1176

2.491 3.441 4.058 5.139 5.445

4.930

the values of the broadening parameters at 70 and 600 K calculated by the extrapolation scheme described above and the values deduced from our experimental results. These last values are given for the di€erent lineshapes used for the ®ts of the e derivatives. Among our results the excitonic values of C are likely to be taken into account at low temperatures. At high temperatures it is certainly more accurate to consider those given by the 3D lineshape for E0 and E0 + D0 and those for a 2D lineshape for E1 and E1 + D1 . The modelling, in [15] omitted the exciton contribution at 70 K where it cannot be neglected, as seen in the er and ei spectra given in Figs. 3 and 4.

Acknowledgements C. Picard is acknowledged for technical assistance.

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