Optical properties of cubic and hexagonal CdSe

Optical properties of cubic and hexagonal CdSeSusumu Ninomiya and Sadao Adachi Citation: J. Appl. Phys. 78, 4681 (1995); doi: 10.1063/1.359815 View online: http://dx.doi.org/10.1063/1.359815 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v78/i7 Published by the American Institute of Physics. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Optical properties of cubic and hexagonal CdSe Susumu Ninomiya and Sadao Adachi Department qf Electronic Engineering, Faculty of Engineering, Gunma University, Kityu-shi, Gunma 376, Japan

(Received 1 May 1995; accepted for publication 19 June 1995)

The complex dielectric. function, e(E) = e1 (E) + I’ez(E), of hexagonal CdSe has been measured by spectroscopic ellipsometry in the photon-energy range between 1.2 and 5.3 eV at room temperature. The measured spectroscopic-ellipsometry data are analyzed on the basis of a simplified model of the interband transitions. The model is based on the Kramers-Kronig transformation and includes the E, (Eon ; ff=A,B,C), E, @,a; . a=A,B, Cj, and EA gaps as the main dispersion mechanisms. The recent SE data of cubic, zinc-blende-type CdSe have also been analyzed with the same model by considering the critical points for the cubic phase (i.e., Ea, EafAa, El, E,+A,, and EJ. Results are in satisfactory agreement with the experimental data over the entire range of photon energies. To facilitate design of various optoelectronic devices, dielectric-function-related optical constants, such. as the complex refractive index, absorption coefficient, and normal-incidence reflectivity, of these crystals are also presented. 0 199.5 American institute of Physics.

I. INTRODUCTION

Cadmium selenide (CdSe) is a member of the II-VI semiconducting compounds. At normal temperature and pressure it crystallizes in the hexagonal, wurtzite lattice [space group=P6smc(C&)]. CdSe can also crystallize in the cubic, zinc-blende modification [space group =F&m(Z’$], although it is metastable under normal condi- tions. By using the technique of molecular-beam epitaxy (MBE) and suitable substrates it is now possible to grow cubic CdSe relatively easily. ’

Spectroscopic ellipsometry (SE) is an advantageous technique to measure the optical constants of solids. This technique is unquestionably more powerful for a number of reasons.” For example, both the real and imaginary parts of the complex dielectric function, E(E) = e1 (E) + iq( E), of a material can be obtained directly on a wavelength-by- wavelength basis without having to resort to multiple measurements or to Kramers-Kronig (KK) analysis. On-line digitization of the data also permits fast and efficient analysis of the structure observed in the E(E) spectra in terms of theoretical line shapes for interband critical points (CPs). Numerical differentiation of the data facilitates this analysis. While most of the SE works have been confined to optically isotropic materials, the usefulness of this technique to study anisotropic systems have also been demonstrated.3

Logothetidis et al.” reported the dielectric behaviors of hexagonal CdSe for both Elc and E]]c measured by SE in the 1.3-5.6 eV spectral region for temperatures between 15 and 550 K. The structures observed in the E(E) spectra were attributed to edge excitons and interband CPs. These structures were analyzed in terms of a standard Lorentzian line shape (edge excitons) or an analytical line shape [CPs) of

E(E)=C-A ln(E-E,+iI’)e’~, (1) where A is the CP amplitude, E, is the CP energy, F is a damping energy, and 4 is an excitonic phase angle. Unfortu- nately, however, they reported neither graphical nor tabular information on the optical constants of hexagonal CdSe at room temperature. This is because their work was mainly

focused on the analysis of the CP structures and their temperature dependences. Janowitz et al.* have also measured E(E) spectra for hexagonal CdSe in the 4-23 eV range using SE at low temperature (90 K), but not at room temperature. Needless to say, knowledge on the room-temperature optical constants is required for the application of hexagonal CdSe in various optoelectronic devices, such as photoconductive devices and photovoltaic solar cells.

Only very recently some SE data have become available for cubic CdSe.5.6 Kim et al.’ have reported the room- temperature SE data of cubic Zn,Cd,-,Se films (OeG1.0) grown on (100)GaAs by MBE in the 1.5-6.0 eV spectral region. They identified the E,, Eo+A,, El, E,+A,, and E, CPs in the E(E) spectra of cubic CdSe (x=0). They ,also reported that the E, and E, +A, CP amplitudes cannot be understood by the one-electron approximation, confirming the existence of strong excitonic effects. Janowitz et aL5 have also reported SE E(E) data of cubic CdSe grown on (100)GaAs by MBE measured at room temperature and at 90 K. These authors identified several interband CPs (at E < 10 eV) and assigned them to the results of band-structure calculations. These SE data on cubic CdSe, however, seem to have one disadvantage with respect to their usefulness: they are not expressed in closed analytical functions of CP energies or even of photon energy.

The purpose of this paper is twofold:

(9 to report the room-temperature SE data of hexagonal CdSe, and

(ii) to analyze these data and published cubic CdSe data using a simplified model of the interband transitions, namely, the model dielectric function (MDFj17

It is of both technical and scientific interest to obtain the analytic expression for the optical constants of semiconductors. In Sec. II we briefly describe our model, which includes the E~[E~WO+A~)I, EIIEIWI+AIjI, and E&Q gaps as the main dispersion mechanisms. In Sec. III we present experimental details and an SE data analysis method for the optically anisotropic (uniaxial) crystal. In Sec. IV the experi-

J. Appl. Phys. 78 (7), 1 October 1995 0021-8979/95/78(7)/468119/$6.00 Q 1995 American Institute of Physics 4681


Cubic Hexagonal

rs

r6

r7

r7

rQ(A) r7Rv

r7(c) Elc E II c

FIG. 1. Energy gaps at the I? point of the zinc-blende- (cubic) and wurtzite- type (hexagonal) crystals. Note that the EOA transitions are forbidden for Elk.

mental results are presented and analyzed. Dielectric-related optical constants, such as the complex refractive index n*(E)=n(E)+ik(E), absorption coefficient a(E), and normal-incidence retlectivity R(E), of hexagonal and cubic CdSe are also presented in Sec. IV. Finally, in Sec. V the conclusions obtained from this study are briefly summarized.

II. THEORETICAL MODEL

A. E,,[ /&I( EO + A,)] transitions

The fundamental absorption edge of hexagonal CdSe (-1.7 eV) corresponds to direct transitions from the highest valence band to the lowest conduction band at the F point (k =O).’ In Fig. 1 we schematically show the conduction- and valence-band structures at the l? point of the zinc-blende (cubic) and wurtzite (hexagonal) crystals. The valence band maxima are derived from p-like orbitals, pX, p?, , and pz. When electron spin is included there are altogether six states. In the zinc-blende lattice, these states are split by spin-orbit interaction into a four hold degenerate I‘s state and a twofold degenerate I’s state (splitting energy Aa)?

The essential difference between the potential that an electron experiences in an ideal hexagonal lattice and that in a cubic, zinc-blende lattice is the relatively small difference in “crystal field” due to sites beyond the next nearest neighbors. lo The detailed differences in the energy bands then arise from the difference in the crystal field and from the difference in the Brillouin zone (BZ). In the actual valence band structure of a hexagonal crystal, the three valence bands, F9, F7, and F7, illustrated in Fig. 1 can be thought of as arising from the simplest tetrahedral lattice by the combined effect of both “spin-orbit” and “crystal-field” pertur- bations, while the wave function for each band may be written as a linear combination of px, py , and pr and spin functions, provided the interaction of the F7 levels with the conduction band is neglected. This result is known as the “quasi-cubic” model.”

The EO gaps in the hexagonal crystals are of the three- dimensional (3D) MO CP. Assuming the bands are parabolic and using the KK relations, we obtain the contribution of these transitions to c(E):~

4-U = ,=FB c Ao,‘%?f(xoa)r G9 , , with

15 2 Aoru=%botuj . Ponr (3)

xoa= (Ef ir)lEo, , t5> where pou,, is the combined density-of-states mass, P& is the squared momentum-matrix element, F is the damping energy of the E,, gap, and (Y stands for the r,(A)-+r7, .iY,(B)-+r,, and r,(C) tlY7 transitions, respectively.

It is well known that the excitonic interaction in the neighborhood of the lowest-direct band edge dramatically changes the optical spectra.” The discrete series of the exciton lines at the E,, gap can be simply written, with Lorent- zian line shape, as

E(E)= I2 a=A,B,C

where A?& is the exciton strength parameter and Gzi is the exciton Rydberg energy.

The one-electron contribution to E(E) of the Eol(Eo+Ao) transitions in cubic CdSe can be written as7

(7)

with

xo=(EfiryEo,

Pb)

wheref(xo) andf(xso) are functions defined by Eq. (4). The excitonic contribution to E(E) at the EoI(Eo+ A,)

edges of cubic CdSe can also be given by

m A;;x 4E)=n~l yci-

2 Eo- ( GiD/rz2) -E- ir

1 +

I Eo+Allo-(G;%z2)-E-ir ’ (10)

6. 15, [El/( El + A,)] transitions

The structures found in hexagonal CdSe in the higher- energy region than E. are labeled El, (a=A,B,Cj. The original assignment of these structures was made by Cardona13 by analogy to the E, /(El + A r) peaks of the zinc- blende-type crystals. As schematically shown in Fig. 2, the E,, and E,, peaks may be related to transitions along the A axis (A5--+A3) of the BZ which are split by spin-orbit interaction.‘3 These peaks are forbidden for El/c (i.e., only the Elc peak appears for the Ellc polarization). The Et, structure may originate from transitions at the (I point of the BZ (U,+U, for Elc and U3-+lJ3 for Ellc).

4682 J. Appt. Phys., Vol. 78, No. 7, 1 October 1995 S. Ninomiya and S. Adachi


Cubic Hexagonal or U

Elc E II c

FIG. 2. E, transitions in the zinc-blende- (cubic) and wurtzite-type (hexagonal) cqstals. Note that the E,, and E,, transitions are forbidden for Elk.

As in the case of the diamond- and zinc-blende-type materials, the E,, structures in hexagonal materials may be associated with the 3D M, CP.3 Since the M ,-CP longitudinal effective mass is usually much larger than its transverse counterparts, one can treat the 3D M, CPs as two- dimensional (2D) minima MO. The contribution to E(E) of this type of 2D minima can be given by7

4EkE~B c W1&,2 W-x:,)1, 5 * with

-- Xla-- El, . w

E(Ej=a=~‘cn?I (?n-lj3~E1,-[G~~(~n-1j2]-E-iT)’ 9 .

In Bq. (11) Bt, and I? are the strength parameter and damping energy, respectively.

Equation (11) is a consequence of the one-electron approximation. Excitonic states should, in principle, exist at each type of CP, since the Coulomb interaction is always present between the electrons and the holes.‘2X14-21 Optical spectra at the EII(E1 + Al j region of some III-V semiconductors, such as GaAs (Ref. 22) and InP (Ref. 23), become sharp when the temperature is lowered. Such spectral change cannot be explained within the framework of the one- electron approximation with lifetime broadening corrections. This fact clearly suggests an evidence for the contribution of the excitonic effects to the E,I(E,+Al) transitions.24 The excitonic effects are usually stronger in II-VI semiconductors than those in III-V ones. This is because of the larger binding energies (smaller values of the static dielectric constant) for the II-VI semiconductors. We can, therefore, successfully neglect the one-electron term [Eq. (ll)] in the case for the II-VI semiconductors (see, e.g., CdS3 ZnSe,= and ZnS26).

The contribution of the 2D-exciton transitions to E(E) in hexagonal CdSe can now be written, with Lorentzian line shape, as

where BTX, is the exciton strength ‘parameter and GTE is the exciton Rydberg energy.

The 2D-exciton contribution to E(E) at the EII(E, + A 1) gaps of cubic CdSe can also be given by

E(E)= 5 1 B:

nml (2r~-l)~ El-[c3fD/(2n-1)2]-E-ir

B ex -I- Id 1 E,+A.1-[~2n-l)2]-E-iI’ ’ (14)

where By(BTX,) and GiD(GT$) are the strength parameter and Rydberg energy of the E, (E, + A r > excitons, respectively.

C. Eh( E2) transitions

The peak in the E(E) spectra of hexagonal CdSe in the region higher in energy than Et is the so-called EA structure. The Eh structure (-6 eV) in hexagonal CdSe may originate from transitions at the M point of the BZ.4 SE measurements also reveal several additional peaks, such as F, , E2, Ei , C’, and D ‘, at energies higher than 7 eV5 These transitions correspond to higher CPs and core excitons.

J. Appl. Phys., Vol. 78, No. 7, 1 October 1995

(13)

I

The relatively strong structure found in the E(E) spectra of cubic CdSe at -6.1 eV can be labeled E2.5 The E2 structure contains contributions from different regions of the BZ, mostly from the 2 direction and in the T-X- U-L plane.

Here we simply assume these CPs (EA and E2) as the damped harmonic oscillator (DHO):

C E(E)= (l--J%+ixy (15)

where C and y are the strength and nondimensional broadening parameters of the DHO, respectively. The DHO is considered to be a different representation of a broadened 2D-M 1 CP.”

Ill. EXPERIMENTAL DETAlLS.AND SE DATA ANALYSIS

A. SE measurement and data analysis

The automatic ellipsometer used was of the polarizer- sample-rotating-analyzer type. The 150 W xenon lamp was

S. Ninomiya and S. Adachi 4683


used as the light source. The SE measurements were per- formed over the photon energy range of 1.2-5.3 eV in a flowing N2 atmosphere at room temperature (300 K). The angle of incidence and the polarization azimuth were set at 75” and 20”, respectively.

The dielectric functions, c=(E)=et(E) +i~z(E), for the ordinary (El c) and extraordinary (Ej]c) rays were obtained by analyzing the ellipsometric data using a uniaxial model.4 Let us assume that the light is incident at an angle 4 from the ambient (air) onto a uniaxially anisotropic crystal without any overlayer and with the optic axis c parallel and perpendicular to the interface. Fresnel’s formulae then allow us to calculate the two complex reflection coefficients, rp and r, , for light polarized parallel and perpendicular to the plane of incidence, respectively. If the c axis is parallel to both the interface and the plane of incidence (S=O”), Fresnel’s reflection coefficients can be simply written as4

!Gi cm 4-G rp= JElEll cos qs+x,‘-

(174

cos q5-x, Is=cos 4+x,’ i17b)

with

x,= , (18)

where e,, and qi represent the complex dielectric constant (E= e1 +i+) for El c and Ellc, respectively.

If the c axis is parallel to the interface and perpendicular to the plane of incidence (S=90°), Fresnel’s expressions can be given by4

El cos 4-x, Q=q cos (b+x,

cos 4-x, rs=cos 4+x,’

with ._

M-4

X,= , Go

and X, is given by Eq. (18). Ellipsometry yields the complex ratio p between

Fresnel’s reflection coefficients rp and r, :

where A and $ are the ellipsometric variables. At -each photon energy E=ho,.the four optical param-

eters (E,~,+, E,,,,E~,,) can be determined from these two pairs of SE measurements [pmea(G=O” and 90”)] by minimizing the errorfunction G(E~~,E~,E~~~,E~~~):

+ [hipy) -Im(pc,a’)12}, (22)

where p”“l is the calculated reflection coefficient ratio:

4684 J. Appl. Phys., Vol. 78, No. 7, 1 October 1995

ta) . . . . . . . . .

t CY (b) --- OC (4 -

%--T-i---- 5 Photon energy (eV)

I ’ I * I

?xagonal CdSe

FIG. 3. Pseudodielectric-function spectra for CdSe (S=O”) taken after (a) mechanical-polishing-degreasing treatment only (d&ted lines), (b) chemomechanical polishing with a solution of 0.01 ~017% Br2 in methanol, followed by rinsing with deionized water (dashed lines), and (c) chemomechanical polishing with a solution of 0.01 ~0195 Br, in methanol, followed by rinsing with deionized water and further with organic solvents (solid lill.TS).

B. Samples and chemical treatment effect

The hexagonal CdSe single crystals used were grown by a vapor-phase method, not intentionally doped, and had the surface orientation parallel to the c axis. The crystals were mechanically polished with diamond paste (0.25 pm) and degreased in organic solvents (trichlene, acetone, and methanol). They were then studied with various methods of surface chemical treatment using SE.

Ellipsometry is a very surface-sensitive technique. It is known that the dielectric function eZ of semiconductors at the energy of the E2 peak provides a sensitive indication of the dielectric discontinuity between the sample and ambient. The sample with the cleanest surface, as well as the dielectric function spectrum most accurately representing that of the pure bulk material, can be identified simply from e2(E) values by using the “biggest is best” criterion.28 We have studied’various methods of surface chemical treatment using SE, and found that this simple criterion (es value at the Et) is also applicable to hexagonal crystals, such as CdSe and CdS iRef. 3).

Figure 3 shows, as examples, the pseudodielectric- function spectra for CdSe (S=O”) taken after

ia) mechanical-polishing-degreasing treatment only (dot- ted lines),

(b) chemomechanical polishing with a solution of 0.01 ~01% Br, in methanol, followed by rinsing with deion ized water (dashed lines), and

(c) chemomechanical polishing with a solution of 0.01 ~01% Brz in methanol, followed by rinsing with deionized water and further with organic solvents (solid lines).

We see in Fig. 3 that the best spectra can be obtained by the chemical treatment of (c). Because neither the thickness nor

S. Ninomiya and S. Adachi


Photon erkrgy (eV)

FIG. 4. Pseudodielectric-function spectra of hexagonal CdSe for both (a) Elc and (b) Elk measured by SE at 300 IS (solid linesj. For comparison, the SE data taken at 210 K by Logothetidis et al. (Ref. 4) are also shown by the dashed lines.

the dielectric propeitiks of the residual surface layer(s) are known, we did not attempt any corrections for them. The SE data were also ‘taken without correction for any s’urface roughness.

IV. RESULTS AND DISCUSSION

4. E(E) spectra

In Fig. 4 we show the pseudodielectric-function spectra of hexagonal CdSe for both El c and El/c measured by SE at room temperature (solid lines). For comparison, the SE data taken at 210 K by Logothetidis et al. (Ref. 4) are also shown by the dashed lines.-As seen in the figure, the experimental f! E) data for El c reveal thr’ee clear structures at the 4 -5 eV region. These structures are originating from transitions at the I?, edges (E,,,,E,,, and E,,). The E(E) data for Ellc, on the other hand, show only a broad peak (E, & in this spectral region. The &-edge region also reveals several CPs (EOA, E,,, and E,, for Elc; E,, and E,, for Ellc).

We see in Fig. 4 that our measured data are similar to those reported by Logothetidis et aZ. (Ref. 4). We can also understand that the CP, features in the E(E) spectra become sharper as the temperature is lowered. If we label the electron (exciton)-LO-phonon coupling as the main broadening mechanism, its parameter value can be expressed by a sum of two different contributions: l?(T) = lY,+ r=(T), where r0 is independent of the tkmperature T, arising from such as crystalline imperfections, Auger process, and surface scatter- ing, and rg(T) is a contribution through emission and absorption of LO phonons of averagtfrequency E, propor- tional to (2nB+l) where ng=(e*‘T-l)-l. The above expression ensures a relatively constant I? value from 10~ T up to, in many cases, T-50-200 K, at which point a com-

5

0

5 w

0

5

0

1 2 3 4 5 6 Photon energy (eV)

FIG. 5. q(E) spectra for hexagonal and cubic CdSe measured by SE (solid circles). The cubic CdSe data are taken from the literature.6 The best-fitted results of MDF to the experimental q(E) are shown in the figure-s by the solid lines. The fit-determined CP parameters are listed in Tables I (hexagonal) and II (cubic). Individual contributions to q(E) of the various energy gaps are also shown by the dashed lines. -

ponent r=(T)- becomes,discernible, and then increases nearly proportionally to T for high T.

In Figs. 5 and 6 we show the pseudodielectsic-function spectra, q(E) (Fig. 5) and e2(E) (Fig. 6), for hexagonal and cubic CdSe (solid circles) measured by.SE. The cubic CdSe data are taken from the literature.” The best-fitted results of MDF to the experimental e(E) spectra ,ae shown in the fig-

IO ’ I I I . I t*

hexagonal Cd% ----7 EM

1 2 3 4 5 6 Photon energy (eV)

FIG. 6. As Fig. 5, but for c2(E).

J. Appl. Phys., Vol. 78, No. 7, 1 October 1995 S. Ninomiya and S. Ad&hi 4685


TABLE I. CP parameters used in the calculation of optical constants for hexagonal CdSe.

Parameter Elc Elk

EoA WI 1.732 4~ W) 1.761 1.761 EOC WI 2.161 2.161 G% W 0.015 0.015 A, (eV’.5) 7.0

:.

Aos (eV’.‘) 2.7 8.6 Aoc (eV’,5) 4.3 5.4 AE (4 0.016 A6 W) 0.006 0.020 4% WI 0.010 0.012 WOA) (4 0.03 REoB) W) 0.03 0.03 VE0d (4 0.06 0.05 &A cev) 4.10 4, WI 4.55 EL, @VI 4.85 4.78 BEI W) 0.36 B;XB WI 3.6 BE @VI 0.90 6.5 r&A) (ev) 0.18 uw WI 0.80 w,,) iev) ~ 0.40 0.70 E,i WI 5.6 6.0 c 0.38 0.10 Y 0.12 0.12 4m 2.20 2.20

ures by the solid linesThey are obtained from the sum of Eqs. (2), (6), (13), and (15) for the case of hexagonal CdSe and from the sum of Eqs. (7), (lo), (14), and (15) for cubic CdSe. The fit-determined CP parameter values are listed in Tables I and II. The experimental Ed spectra are found to be usually somewhat larger than those of the MDF calculation. In order to improve the fit, therefore, we included in the calculation an additional term, elm, in et(E) (see Tables I and II). This term is assumed to be nondispersive and may arise from other higher-lying interband transitions and core-level transitions.

TABLE II. CP parameters used in the calculation of optical constants for cubic CdSe.

Parameter Numerical value

El, W Eo+& W) GiD (eV) A0 (eV’.‘) A;;X (eV) ~~EoI(Eo+Ao)l (eV) El (eV) El+4 ieV) B;” (eV) BE W) WJ W) I’(El+Al) (eV) & W) c Y 4m

1.74 2.15 0.015 4.7 0.01 0.03 4.28 4.48 3.0 1.5 0.50 0.90 6.08 1.4 0.21 1.40

The lowest-direct band gap E. of wurtzite can be obtained from that of zinc blende by the action of a small hexagonal crystal field.” Under the spin-orbit interaction, represented by the matrix element $, , and that of the crystal field, represented by AC, the energy differences of the split- off valence bands are given by*l

EBA=&B--&IA=

2 1 0.5

- ,&A 7 (234

As,+& ~ - EcA=&c-&A= 2 + A,,+& ’

Ii 1 2

2 1 0.5

- +A- . (23b)

The corresponding wave functions of the three valence bands axe

IA): S+T, i244

1B): %S-t+%%J~, W-4

IO a&-? + +%i, (24~)

where 7 and J. represent spin up and spin down, respectively, and S+ , S- , and So are functions defined by the p-like basis functions. The admixture coefficients CQ and ac can now be given by

0,=[1+$(2-~E,,i’]-‘.‘, (25a)

~Y~=[l+f(2--&E~~)~]-~~~, @b)

where r~$ + CY$= 1. Using the EBA and E, values reported in Ref. 29, we obtain A,,=4128 meV, A,=45.2 meV, ~~~=0.620, and ac=O.785.

The corresponding squared p-matrix elements 1 (c ]p ] u) ] ’ between the p-like valence and the s-like conduction band are

for El c, and

PiA,, = 0, Pi,,, = &P2, P&,,= &;P2 i2W

for Ellc. We thus obtain the strength .ratio

P~A,:P~B,:P~,:P~A~~:P~B~~:P~~~

=1.0:0.38:0.62:0.0:1.23:0.77. (27)

Both the Ao, [Eq. (2)] and AZ [Eq. (6)] are constrained to satisfy the strength ratio of Eq. (27) in the present analysis (see Table I). We can understand that these values give rea- sonably good agreement with the experimental data (Figs. 5 and 6).

The strengths of the El transitions of hexagonal materials can be estimated with the simple expressions4

4686 J. Appl. Phys., Vol. 78, No. 7, 1 October 1995 S. Ninomiya and S. Adachi


3 44 B,A=-

8 (%d"3E~~ =0.66,

3 44 B1B=8(2v~)1/3~1Bi-n.60)

5 44 Blc=-

8 G%4Y34c -0.93,

for Elc, and

44 Blc=(2~M)!f3~IC=1*51

(284

(28b)

(28~)

0%

for Ellc, where V, is the volume per molecule in A3 and E,$ in eV. We then obtain the strength ratio

B~A(~):B~B(~):B~c(~):B~c(II)= 1.0:0.9: 1.4:z.j . (30)

Pseudopotential-band calculations of Bergstresser and Cohen3’ suggest the presence of the 3D MI CPs at I’ [I’,+F,(4.3 eV; Elc)] .and at the M-U-L line [‘U,--+ U3 (5.0 eV; Elc) and Us-+ Us (5.0 eV; Ellc)] of hexagonal CdSe. The strengths of these transitions in relative units are B(l?,-+IT,)= 1.4, B( U43 .?I,) =0.7, and B( U,-+ U,) =0.9.

On the other hand, the experimental strength ratio can be written from Table I as

Thus, at present there is not enough knowledge or a rigid model which enables to fully explain the experimental El strength in hexagonal CdSe.

The strengths of the El transitions in a cubic material can also be estimated from the expressions”

(324

B,d=44 E,+(2Ar/3) uo(E,+Ad2 ’ Wb)

where a0 is the lattice constant in w and Et, A, in eV. The above expression predicts B, =1.73 and B,,=l.67 (B, + B,,=3.40) for cubic CdSe. The- present analysis, on the other hand, provides BfX=3.0 and Bfl2=1.5 (ByCBTfj=4.5; see Table II). Although one may attribute this discrepancy to the extremely crude model used, it can be attributed to excitonic enhancement of the oscillator strength (see Ref. 4 and references cited therein).

Individual contributions to q(E) and eZ(E) of the various energy gaps for hexagonal and cubic CdSe are shown in Figs. 5 and 6 (dashed lines), respectively. They are obtained from Eq. (2) [Eq. (7)] for the 3D-M, CP contribution (Eo), from Eq. (6) [Eq. (lo)] for the 3D-exciton contribution, from Eq. (13) [Eq. (14)] for the 2D-exciton contribution (E,), and from Eq. (15) for the E&gap (E,-gap) contribution.

The E, gaps are of the 3D-M, type. Hence, the line shape of the corresponding one-electron e2(E) spectrum can be characterized by a continuous absorption obeying the l/2- power law [i.e., m(E - E,)“‘]. The 3D exciton absorption spectrum consists of a series of sharp discrete lines with a

rapidly decreasing oscillator strength err -3 and a continuum absorption due to the continuum (ionized) states. It is known that the continuum-exciton transitions at the 3D-Mc CP be- have like the noninteracting electron-hole pair characteris- tics. The contribution of these transitions to E(E) can, therefore, be considered with expressions similar to Eqs. (2) and (7). The deviations of the continuum spectrum from the one- electron spectrum are usually described by the so-called Sommerfeld or Coulomb enhancement factor.31

The Et gaps may be of the 2D-Me type. The corresponding one-electron e2(E) spectrum [Eq; (1 l)] should be characterized by’ a steep low-energy side and a broader high- energy side.7 As mentioned before, however, it is possible to neglect this one-electron term in the present study. Since the 2D n = 1 exciton term in Eq. (13) [Eq. (14)] contains 95% of the total oscillator strength, we neglected the excited-state terms (1232) in our calculation. We also assumed that GfE=O eV (GTD = G:F = 0 eV). This is because no accurate value has been reported on this physical quantity so far.

Although our experimental photon-energy range did not cover the EA(E2) CP region (-6 eV), we considered here its contribution to E(E). This is because the EA(E2) CP structure provides a peculiar line shape in the E(E) spectra even when the photon energies are well below the EA(E2) CP energy.

B. Complex refractive index, absorption coefficient, and normal-incidence reflectivity

Optical constants, such as the complex refractive index n*(E)=n(E)+ik(E), absorption coefficient a(E), and normal-incidence reflectivity R(E), can be easily obtained from the present study in the form of practical functions, since they are’directly related to the dielectric function E(E).

The real refractive index n and extinction coefficient k can be given by

[q(E)‘+ E~(E)~]“~+ E,(E) 112

n(E)= 2 , (33a)

112

k(E)= [E~(E)~~E~(E)~]~‘~-E~(E)

2 I . W-4

We show in Fig. 7 the numerically calculated spectral dependence of the complex refractive index, n*(E)=n(E)+ik(E), for both hexagonal and cubic CdSe (MDF; solid lines). They are obtained from Eq. (33a) for 12 and from Eq. (33b) for k. The solid circles represent the experimental data measured by SE. The stronger peaks in rz (-1.7 eV) are related mainly to the E. transitions. In k, the strongest peak is associated with the El transitions (hexagonal) or E, transitions (cubic).

The absorption coefficient CY and normal-incidence reflectivity R can now be written as

[n(E)- 112+k(E)’ R’E)= [n(E)+ l]‘+k(E)‘* (3%

J. Appt. Phys., Vol. 78, No. 7, 1 October 1995 S. Ninomiya and S. Adachi 4687


- hexagonal CdSe

y 2- 6 - hexagonal Cd%

;I;;* 2 3 4 5 6

Photon energy (ev)

FIG. 7. Numerically calculated complex refract& index, n*(E) =n(E) + ik(E), for both hexagonal and cubic CdSe (solid lines). The solid circles represent the experimental data.

The calculated spectral dependences of a(E) and R(E) for hexagonal and cubic CdSe are shown in Figs. 8 [a(E)] and 9~ [R(E)] by the solid lines. The solid circles represent the experimental a(E) and R(E) spectra. Like n(E) and k(E), the a(E) and R(E) spectra are found to reveal the distinct E, and E, structures.

IO4 1 2 3 4 5 6

Photon energy (eV)

FIG. 8. Numerically calculated absorption coefficient, a(E), for both hexagonal and cubic CdSe (solid lines). The solid circles represent the experimental data.

Photon energy (eV)

FIG. 9. Numerically calculated normal-incidence reflectivity, R(E), for both hexagonal and cubic CdSe (solid lines). The solid circles represent the experimental data.

V. CONCLUSIONS

We have measured the spectral dependence of the complex dielectric function, E(E) = et (I!?) + i E~( E) , of hexagonal CdSe using SE in the 1.2-5.3 eV photon-energy range at room temperature. The measurements are made on the surface paralIe1 to the optic axis c to extract E(E) spectra for both El c and Ellc polarizations. The measured E(E) spectra show distinct structures at energies of the E, and E 1 CPs. These spectra are analyzed on the basis of a simplified model of the interband transitions. The model is based on the KK transformation and includes the E. (E,,; cu=A,B,C), El (El,; a=A,B), and EL gaps as the main dispersion mechanisms. We have aIso analyzed the recent ellipsometric data of cubic, zinc-blende-type CdSe using the same model with the CPs for the cubic phase (i.e., E,, E,,+hg, E,, E,+A,, and Ed. Since the excitonic effects may profoundly modify the CP singularity structure, the experimental SE data could not be explained within the framework of the one-electron approximation. The model is thus made to account for the excitonic effects at these CPs. Results are in satisfactory agreement with the experimental data over the entire range of photon energies. Dielectric-function-related optical constants, such as the complex refractive index n* = R + ik, absorption coefficient .ZY, and normal-incidence reflectivity R, of hexagonal and cubic CdSe are also presented.

ACKNOWLEDGMENT

We would like to express our appreciation to Dr. K. Era for his kind advice concerning this work.

4688 J. Appl. Phys., Vol. 78, No. 7, 1 October 1995 S. Ninomiya and S. Adachi


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Optical properties of cubic and hexagonal CdSe