Physica status solidi / A.: Volume 11, Number 2 June 16 [Reprint 2021 ed.] 9783112472767, 9783112472750

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plrysica status solidi ca)

V O L U M E 11 . N U M B E R 2 • 1 9 7 2

Classification Scheme 1. Structure of Crystalline Solids 1.1 Perfectly Periodic Structure 1.2 Solid-State Phase Transformations 1.3 Alloys. Metallurgy 1.4 Microstructure (Magnetic Domains See 18; Ferroelectric Domains See 14.4.1) 1.5 Films 1.6 Surfaces 2. Non-Crystalline State 3. Crystal Growth 4. Bonding Properties 5. Mossbauer Spectroscopy 6. Lattice Dynamics. Phonons 7. Acoustic Properties 8. Thermal Properties 9. Diffusion 10. Defect Properties (Irradiation Defects See 11) 10.1 Metals 10.2 Non-Metals 11. Irradiation Effects (X-Ray Diffraotion Investigations See 1 and 10) 12. Mechanical Properties (Plastic Deformations See 10) 12.1 Metals 12.2 Non-Metals 13. Electron States 13.1 Band Structure 13.2 Fermi Surfaces 13.3 Surface and Interface States 13.4 Impurity and Defect States 13.6 Elementary Excitations (Phonons See 6) 13.5.1 Excitons 13.5.2 Plasmons 13.5.3 Polarons 13.5.4 Magnons 14. Electrical Properties. Transport Phenomena 14.1 Metals. Semi-Metals 14.2 Superconductivity. Superconducting Materials and Devices 14.3 Semiconductors 14.3.1 Films 14.3.2 Surfaces and Interfaces 14.3.3 Devices. Junctions (Contact Problems See 14.3.4) 14.3.4 High-Field Phenomena, Space-Charge Effects, Inhomogeneities, Injected Carriers (Electroluminescence See 20.3; Junctions See 14.3.3) 14.4 Dielectrics 14.4.1 Ferroelectrics 15. Thermoelectric and Thermomagnetic Properties 16. Photoconductivity. Photovoltaic Effects 17. Emission of Electrons and Ions 17.1 Field Emission Microscope Investigations 18. Magnetic Properties 18.1 Paramagnetic Properties 18.2 Ferromagnetic Properties 18.2.1 Ferromagnetic Films 18.3 Ferrimagnetic Properties 18.4 Antiferromagnetic Properties (Continued

on cover three)

physica status solidi (a) applied research

B o a r d of E d i t o r s S. A M E L I N C K X , Mol-Donk, J. A U T H , Berlin, H. B E T H G E , Halle, K. W. B Ö E R , Newark, P. G Ö R L I C H , Jena, G. M. H A T O Y A M A , Tokyo, C. H I L S U M , Malvern, B. T. K O L O M I E T S , Leningrad, W. J. M E R Z , Zürich, E. W. M Ü L L E R , University Park, Pennsylvania, D. N. N A S L E D O V , Leningrad, A. S E E G E R , Stuttgart, G. S Z I G E T I , Budapest, K. M. VAN V L I E T , Montreal Editor-in-Chief P. G Ö R L I C H Advisory Board L. N. A L E K S A N D R O V , Novosibirsk, W. A N D R A , Jena, E. B A U E R , Clausthal-Zellerfeld, G. C H I A R O T T I , Rom, H. C U R I E N , Paris, R. G R I G O R O V I C I , Bucharest, F. B. H U M P H R E Y , Pasadena, E. K L I E R , Praha, Z. M À L E K , Praha, G. O. M Ü L L E R , Berlin, Y. N A K A M U R A , Kyoto, T. N. R H O D I N , Ithaca, New York, R. S I Z M A N N , München, J. S T U K E , Marburg, J. T. W A L L M A R K , Göteborg, H. W E I S S , München, E. P. W O H L F A R T H , London

Volume 11 • Number 2 • Pages 375 to 790, K97 to K162, and A9 to A16 June 16, 1972

A K A D E M I E - V E R LAG

BERLIN

Subscriptions and orders for single copies should be addressed to AKADEMIE-VERLAG GmbH, 108 Berlin, Leipziger Straße 3 - 4 or to Buchhandlung K U N S T U N D WISSEN, Erich Bieber, 7 Stuttgart l,Wilhelmstr. 4 - 6 or to Deutsche Buch-Export und -Import GmbH, 701 Leipzig, Postschließfach 160

Editorial Note: "physica status solidi (a)" undertakes that an original paper accepted for publication before the 23r and holes, E* Under steady-state conditions the concentration of condensed electrons and holes, n0, corresponds to the minimum free energy of the condensed phase. For the highly degenerate electron-hole gas (F e , Fh kT) the free energy coincides

Ya. Pokbovskei

388

approximately with the interior energy per electron-hole pair and may be written as [16] o (Ve I J/7 \ I jpee i rihh . pee . phh "g" (• e ~ r ^h)

I •O'exch

I -O'exch +

Here .Fe, are the Fermi energies of electrons and holes, exchange energies of electrons and holes, respectively. Fe and Fb are given by the expression F

"

F v. h —

2 m „ m .h\32n2)

(1)

&cor T •"cor •

-Belch. -®exch

the

(2)

r\ v2/3 '

where m e and m h are the effective density-of-states masses, v is the number of equivalent valleys, 4/3 n rjj = w" 1 . I n calculating the exchange energies it is necessary to take into account t h e many-valley band structure of Ge and Si. I t may approximately be taken t h a t the dominant term in the exchange energy is the result of t h e electron interaction in one valley only. Therefore Eiee -C*exch

0,158 e 2 1/3

s r0 v

phh •C'exch

=

0.458 e 2

(3)

e rn

where e is the elementary charge and e the dielectric constant. The correlation energy was calculated in a large number of papers. I t may be carried out exactly only when the charge carrier concentration is small (a 1) or large (a wj' 3 1) (ae, ah are the Bohr radii of electrons and holes). When a n ] j 3 as 1, different interpolating expressions are useful, which give similar results. I n the calculation of the correlation energy we use Wigner's expressions [16] Eiee -tt cor —

0.44 e 2 e (r0 + 7.8

jTihh

ae)

•"cor = —

0.44 e 2 e~(r0 + 7.8

ah)

The other expressions for the correlation energies also give a weak dependence on r0 when rja « 1. Therefore it is not important what specific expression for the correlation energy is used in the determination of the energy minimum (1). The dependence of Fe + Fb, Etlch + -®exch, 2?cor + ¿3cor, and their sum on n in Si is represented in Fig. 2. Using equations (1) to (4) and the values of the constants for Ge (me = 0.22 m, mb = 0.39 m, v = 4, s = 16) and for Si (me = 0.33 m, rah •

10" 10" nlan3!-

Fig. 2. The dependence of kinetic (1), correlation (2), and exchange (3) energy per electron-hole pair on the free carrier concentration n ior Si. (4) is the total energy

Condensation of Non-Equilibrium Charge Carriers in Semiconductors Fig. 3. a) Energy scheme after formation of the condensed phase of non-equilibrium carriers; b) the electron transitions in the condensed phase

389

"T"

4 a

= 0.55 m,v = 6, s = 12) we obtain the carrier concentration n0, corresponding to the minimum energy (1):

x 1 0 " cm- 3 for Ge , n 0 = 3 x 1018 cm- 3 for Si . n„ = 2

The energy gap variation ¿?exch + -®exch + -®cor + -Scot corresponding to these concentrations is approximately equal to —10 meV for Ge and —30 meV for Si. Therefore the mean energy per one electron-hole pair in the condensed phase is smaller than the free exciton energy at least by 2 meV for Ge and 5 meV for Si, and the condensation of the non-equilibrium carriers in these semiconductors leads to an energy lowering at any concentration of excitons. Now the spectral distribution of recombination radiation is considered as the result of electron-hole recombination in the degenerate non-equilibrium plasma. We assume that the probability of radiative transitions is practically independent of the energy of electrons and holes, E [18], and that the densities of states in the conduction and valence bands, NE and NH, are proportional to E 1! 2. We also take into account that the momentum conservation law is satisfied for the radiative recombination of electrons and holes due to the emission of phonons with momenta corresponding to the minima of the conduction band. Therefore, in the calculation of the spectral distribution of the recombination radiation we must consider all band-to-band transitions with different momenta corresponding to a given photon energy value, h v (Fig. 3). In this case the spectral density of recombination radiation in each type of phonon peak is given by the equation [10] I(hv

-hv0)

where O^hv

— hv0 where EQ = 744 meV, hco = 27.5 meV. I t is seen from Fig. 4 t h a t the decrease of the energy gap of Ge due to the formation of the condensed phase is equal to ^exch + -®exch + -®cor + EC0I as 11 meV in agreement with the value obtained from the minimum energy condition (1). The spectral distribution of the LO and TO components of the recombination radiation for pure silicon at 4.2 °K is shown in Fig. 5. The open circles correspond to the calculation according to (5) and (6) for n0 = 3 . 7 x l 0 1 8 c m - 3 [17], Here we also have a good agreement between theoretical and experimental data. I t is seen from Fig. 5 t h a t there is only a slight discrepancy at low energies where the density-of-states "tails" manifest themselves. I n this case there is also a good agreement of values n 0 determined from the spectral distribution and from the energy minimum condition. The decrease of the energy gap of Si as obtained from Fig. 5 is equal to —44 meV. This value is a little larger t h a n t h a t determined from equations (1) to (4). However, the approximations used in (1) lead to a lower value of the binding energy of the condensed phase. 3. Dependence of the Energy Spectrum of the Condensed Phase on Uniaxial Stress and Strong Magnetic Field The dependence of the recombination radiation spectra in Ge on uniaxial stress and strong magnetic field have been studied by Bagaev et al. [19, 20]. Under uniaxial stress along [111] direction the energy gap of Ge is decreased and the density of states in the conduction and the valence bands are varied. The decrease of the energy gap is proportional to the stress. Three minima of the conduction band are shifted towards higher energy, and one minimum towards lower energy. If the deformation is large enough, the splitting of the conduction band is larger t h a n the Fermi energy of the condensed electrons.

Condensation of Non-Equilibrium Charge Carriers in Semiconductors

700

0

500

391

mo PI Hp Icm2)

Fig. 6. The dependence of the radiation maximum position of the free exciton ( O ) and the condensed phase ( • ) in Ge at deformation along the direction [111] at 4.2 ° K

Fig. 7. a) Variation of the condensed phase spectra in magnetic fields at 1.5 ° K . (1) B= 0; (2) B = 38.4; (3) J S = 4 6 ; (4) £ = 6 0 ; (5) £ = 100 kG. b) Dependence of the radiation maximum position of free excitons, Hex, and condensed phase, E c , on magnetic field

m 710 706 *—hvirneV)

Therefore all electrons should be concentrated in the lowest minimum of the conduction band. In this case the energy of free electrons increases and the equilibrium between the condensed phase and the free exciton gas disappears. So the carrier concentration in the condensed phase has to decrease until it reaches a new equilibrium state. According to an estimation taken from (1) to (4) the concentration n0 is about two times lower, leading to a decrease of the binding energy of the condensed phase. The dependence of the position of the radiation peaks of the free exciton and of the condensed phase on uniaxial stress is represented in Fig. 6. One can see that the free exciton radiation peak is shifted towards lower energy proportionally to the stress, according to the variation of the energy gap of Ge. The position of the radiation peak of the condensed phase is independent of the stress up to as 350 kp/cm2, but then this peak is shifted towards lower energy similar to the exciton peak. Therefore the binding energy of the condensed phase is decreased approximately by 3 meV under stress up to 350 kp/cm 2 , but then it becomes constant. The strong decrease of the energy distance between the exciton radiation peak and that of the condensed phase cannot be explained by the excitonic molecule model. Indeed under high stress the distance between the radiation peaks becomes a few times smaller than the free exciton binding energy. This fact is in disagreement with Haynes' model. A similar influence of uniaxial stress along [100] direction was found in Si [21]-

This simple variation of the radiation spectra takes place only if the deformation is uniform. At non-uniform deformation and a maximum not coinciding with the photo-excitation area, a strong decrease of the radiation intensity of the condensed phase is observed. There is a simple explanation of this phenomenon. Really, the non-equilibrium condensed carriers should interact with the lattice vibrations very weakly because of the high degeneration of the electron-hole gas. An estimate for Ge gives the scattering time due to collisions with phonons as ss 10~7 to 10~8 s at 4.2 ° K [19]. Therefore, the drops can be 26

physica (a) 11/2

392

Y A . POKROVSKII

accelerated up to sound velocity in the non-uniform deformation field. I n this case the drops should leave the excitation area before reaching steady-state dimensions. This leads to the strong decrease of the radiation intensity of the condensed phase. The dependence of the radiation spectra of the condensed phase on the magnetic field is shown in Fig. 7 [20]. I n magnetic fields ranging from 40 to 70 kG a splitting of the radiation peak is observed. The presented data can be explained by the radiative recombination of electrons occupying only the lowest Landau subband with holes occupying the two upper Landau subbands. If the Landau splitting of the valence band becomes larger than the Fermi energy of holes in the condensed phase (it corresponds to H as 80 kG at n 0 = = 2 X 1017 cm - 3 ), the single peak of the recombination radiation of the condensed phase remains because only one upper Landau subband is occupied by holes in this case. 4. Dependence of the Intensity of the Recombination Radiation of the Condensed Phase on Excitation Level and Temperature The dependence of the intensity of the new recombination radiation on the excitation level should be quite different for the cases of excitonic molecules and of the condensed phase. Indeed the concentration of the excitonic molecules has to be proportional to the square of the free exciton concentration n at low temperatures, where the thermal dissociation of the excitonic molecules is negligible. Therefore the intensity of the recombination radiation, 7 em , produced as the result of the radiative annihilation of the excitonic molecule may be described as follows: lem ~ ' f x ,

(7)

where J e x is the intensity of the free exciton radiation. The experimental data of Haynes [8] correspond to a linear dependence of 7 ex on the excitation level g, but Iem ~ g2 in agreement with equation (7) (Fig. 8a). To check more exactly this conclusion we have presented the dependence I e m o n / e x i n F i g . 8 b, using the data of Fig. 8a. One can see in Fig. 8 b t h a t / e m ~ / | x at low excitation level (the dashed line in Fig. 8 b). The slope is reduced at higher excitation level. This fact may be explained by the heating of the Si sample at high exciting radiation intensity. Therefore Haynes' experimental data do not confirm the formation of excitonic molecules in Si.

10

100

gfrel. units)- —

Fig. 8. a) Dependence of the radiation intensity of free excitons, lex (A), and biexcitons, Iem ( O ) , on excitation level g according t o [8] at r » 3 °K. b) Dependence of Iem on lex obtained from Fig. 8 a

Condensation of Non-Equilibrium Charge Carriers in Semiconductors

393

To analyse the dependence of the intensity of recombination radiation on the excitation level we shall use the following simple model [15]: The condensed phase is assumed to be constituted of spherical drops of radius R, where R is substantially less than the mean free path of excitons. Then the current of free excitons at the surface of the drop, n R2 v n, under steady-state conditions must be equal to the sum of the recombination rate inside the drop. (4/3) n -R3(m0/t0), and of the current of carriers through the surface of the drop into the volume of the crystal due to thermal emission, 4 n R A exp (— q>jkT): 2

n

R * v n

=

3

r0

+

i

n

R

2

A e x v l - ^ ) .

\

kl /

(8)

Here v is the mean thermal velocity, r 0 and n0 are lifetime and concentration of non-equilibrium charge carriers in the condensed phase, (p is the work function, A a coefficient which is only slightly temperature dependent. Iex is proportional to n, while the radiation intensity of the condensed phase, / c , is proportional to (4/3) n R3 (w0/rr) N, N being the concentration of drops and r r the radiative electron-hole lifetime in the condensed phase. Thus at sufficiently low temperatures, when the thermal emission is negligible, from equation (8) follows /c ~ n n x . (9) If the concentration of drops is controlled by some nucleation centres, it is expected to be only slightly dependent on excitation level and temperature. In this case Ie ~ /|x . (10) Such a dependence was typical both for Si and Ge. The dependence for pure Si determined at 4.2 °K and steady-state photoexcitation is shown in Fig. 9 [10]. The discrepancy between the experimental data and the cubic law was observed only at high photoexcitation levels and was ascribed to heating of the samples. This heating was registered by the broadening of free exciton radiation peaks.

Fig. 9. The dependence of 7 C on lex for pure Si at 4.2 " K and under steady-state GaAs laser excitation 26'

K g . 10. The dependence of lex ( A ) and / 0 ( O) on excitation level g for pure Si at 4.2 ' K

394

Y A . POKBOVSKII

We obtained a similar dependence also from Haynes' experimental data shown in Fig. 8b. The dependence of 7 ex and 7C on the excitation level g is more complicated (Fig. 10), but the cubic law (10) is satisfied quite well even when the variation of the dependence of 7 ex and 7C on g becomes strong. I n the following we consider in detail the dependence of the radiation intensity of the condensed phase, 7C, on excitation level g. Under steady-state conditions g= ^ +

(11)

T o

T0

where r is the lifetime of excitons. From (8) and (11) we obtained 2 = ^4 — 4 A expv(-(p/kT) n ( 1- + N v t i R \ R . x 3 v r00 \ x J

(12)

Equation (12) gives evidence that R ^ 0 if 4 A exp (~ 0

I t follows that the condensed phase can appear only if the threshold values of excitation level, gth, and temperature, Tth, are reached. These values are related i>y gth _ ^ e x P ( (p/kTth) (13) If g is small enough and the quantum efficiency of the condensed phase radiation is also small, we can assume N v n R2 1/r. In this approximation the solution of (12) is R ~ (g — g th ), and IB ~ (9 - 0 th ) 3 • (14) Therefore the radiation intensity of the condensed phase, 7C, should be proportional to the cube of the excitation level g after reaching threshold conditions. On the other hand, g gih and 7C ~ g at sufficiently low temperature and sufficiently high excitation level, so that N v n R 2 ^ > 1/r and most of non-equilibrium carriers are concentrated in the condensed phase. So, in this case 7C depends linearly on the excitation level g. The condition N v n R2^> 1/r has a simple physical meaning. The radiative recombination process should predominate in a condensate only if the exciton lifetime in the gaseous phase, r, is considerably longer than the characteristic time for the capture of excitons by the condensate drops, l/(N vn R2). This condition sets the upper limit to the value of R. Indeed, (4/3) n Ra (w0/r0) N ^ g and R

< * -

4

x

y

g

n0

.

as)

Now we consider the dependence of the radiation intensity of the excitons, 7 ex , and the condensed phase, 7C, on temperature. Equation (8) can be written as a7

e x

- 6 7i/3 =

e x p

( - ^ .

(16)

The constants a and b can be determined in the following way: Equation (8) can be satisfied only for R ^ 0, 7C 0, and T •

01 02 03 at OS

06

t,a>-

07

Fig. 3. a) Typical relaxation curve of 90°-sappbire specimen; b) effect of temperature and deformation rate on flow stress in 60°-specimen during extension. Vt = 10~6 s - 1 , V2 - 4.35 x 10~ 6 s - 1

In the relaxation stress experiments the specimens were heated to a present temperature at which they were maintained for 15 to 20 min, and then deformed by extending at a rate of F 0 = 8.7 X 10" 5 s _ 1 to e0 = 1 to 1.5%, thereafter the specimens were allowed to make a transient relaxation (about 3 min). Then the specimens were reloaded at a rate F„ > F 0 to create a large stress, and then allowed to relax. Thus, the spectra of relaxation curves at different stresses were recorded for the same specific temperature. Then the temperature was lowered and the spectra of relaxation curves '«rere once again recorded for other values of the stress. The relaxation curves of a-Al 2 0 3 crystals were recorded in the range s = 1 to 2 % for the 0°- and 90°-specimens, and in the range e = 1 to 1 6 % for the 60°specimens in which the area of slipping with a deformation hardening coefficient close to zero is greater than 2 0 % of e [10], The rate of drop in the stress was determined from the slope 55 °K for the glass, T > 30 °K for the crystal) can be connected with the increase of the probability of non-radiative transitions via luminescence levels. Nevertheless the nature of the activated quenching is unclear at present. As to the changes of the spectral characteristics at high temperatures as well as at low temperatures, they can be explained by the existence of a group of luminescence levels. Obviously, both for the glass and the crystal the values of the activation energy for thermal quenching corresponding to each level of the group are close but different in both temperature ranges (below ¡^115 °K and above fa 115 °K). This fact can lead to the shift of the maxima of the radiation bands to higher energies for the glass and the crystal as well as to the narrowing of the luminescence band with increasing temperature for the crystal. The weak broadening of the luminescence band of the glass at high temperature is apparently connected with the stronger interaction of electrons with "lattice" vibrations. Thus the temperature dependence of steady-state luminescence of vitreous and single crystalline As2Se3 indicate the analogy as well as the distinguishing features of the electronic spectra of disordered and ordered states of this material. References [1] B. T. KOLOMIETS, T. N. (1968).

MAMONTOVA,

[ 2 ] B . T . KOLOMIETS, T . N . MAMONTOVA,

and V. V. and

NEGRESKUL,

A . A . BABAEV, J .

phys. stat. sol. 27, K15

non-cristall. Solids

4,

289

(1970). [ 3 ] B . T . KOLOMIETS, T . N . MAMONTOVA, sol. (a) 7, K 2 9 [4]

R.

[5]

B.

I.

A . DOMORYAD,

and

A . A . BABAEV,

phys. stat.

(1971).

FISCHER, V . H E I N , F . S T E R N , and K . W E I S E R , Phys. Rev. Letters 2 6 , 1 1 8 2 ( 1 9 7 1 ) . T. KOLOMIETS, T. N. MAMONTOVA, and A. A. B A B A E V , Proc. IV. Internat. Conf. Phys. Amorphous and Liquid Semicond., Ann Arbor, Michigan; J. non-cristall. Solids, in the press. [6] E. I. ADIROVICH, Nekotorye voprosy teorii luminestsentsii kristallov, Gostekhizdat, Moskva 1956. [ 7 ] T. F. MASETS, Avtoref. cand. dissert., Leningrad 1964. [8] B. T. KOLOMIETS, V. M. L Y U B I N , and V. L . AVERYANOV, Mater. Res. Bull. 5, 655 (1970). [9] B. T. KOLOMIETS and G. I. STEPANOV, Fiz. tverd. Tela 7, 2698 (1965).

(Received

March 16,

1972)

M. ZOUAGHI et al. : Near Infrared Optical and Photoelectric Properties (III)

• 449

phys. stat. sol. (a) 11, 449 (1972) Subject classification: 16 and 20.1; 10.2; 13.4; 22.6 Laboratoire de Spectroscopie et d'Optique du Corps Solide, Groupe de Recherche du C.N.R.8., Strasbourg

Near Infrared Optical and Photoelectric Properties of Cu 2 0 III. Interpretation of Experimental Results1) By M. ZOUAGHI, B . PREVOT, C. CARABATOS, and M. SIESKIND Experimental results given in Part I and II are interpreted with the help of a band scheme including five energy levels: (3 (1.91 eV above the VB, due to exciton-neutral acceptor complexes creation with 0.1 eV binding energy), A (0.55 eV above the VB, copper vacancies Vcu), B (0.76 eV above A, [VCu-VQ"] vacancy associations), C (0.97 eV above A, [Vcu _ Vcu] vacancy associations) and D (0.38 eV below the CB, due to oxygen vacancies V + ) . The strongest absorption bands, as well as the minima in the photoconductivity spectrum are explained by optical transitions from A to B and C levels. The defect density responsible for the level C is about 1017 cm" 3 . This scheme is consistent with results obtained by other authors on the electrical properties and the luminescence of Cu20. On interprète les résultats décrits dans les parties I et II de cette étude à l'aide d'un schéma de bandes comprenant cinq niveaux d'énergie: (3 (1,91 eV au-dessus de la BV, dû à la formation de complexes exciton-accepteur neutre avec une énergie de liaison de 0,1 eV), A (0,55 eV au-dessus de la BV, vacances de cuivre Vcu)> B (0,76 eV au-dessus de A, associations de vacances [ V c u - V j ] ) , C (0,97 eV au-dessus de A, associations de vacances [Vçu-VcJ) et D (0,38 eV au-dessous de la BC, vacances d'oxygène VQ). Les bandes d'absorption les plus intenses ainsi que les minimums de photoconductivité correspondants sont attribués à des transitions du niveau A vers les niveaux B et C. La densité des niveaux C est évaluée à 1017 cm -3 . Ce schéma de bandes ne contredit pas les résultats obtenus par d'autres auteurs sur les propriétés électriques et la luminescence de Cu20.

1. Introduction A s seen in Part I, Prévôt et al. have shown that the absorption spectra in the near infrared of Cu 2 0 samples belonging to class B , present at low temperatures new bands; the strongest ones are at 1.91, 1.63, 1.41, 1.31, 0.97, and 0.76 eV. I t has been observed that the higher is the annealing oxygen pressure, the stronger are the absorption bands. Analogous effects have been obtained with H e + irradiated Cu 2 0 samples. I n Part I I , Zouaghi has studied the photoconductivity of all three classes of samples (A, B, C) and shown that at low temperatures, photocurrent minima appear at 1.90, 0.97, and 0.76 eV. I n the region of the remaining bands reported in Part I, he observed structures corresponding either to photocurrent thresholds or to wide steps. I n addition, in the range of 0.5 to 0.6 eV, a generally diffuse photocurrent threshold appears. The purpose of the present Part I I I is to discuss the experimental results and to suggest a model for the energy levels allowing an interpretation of the experimental results. I n Section 2, the synthesis of the results given in Parts I !) Part I: B. PREVOT, C. CARABATOS, andM. SIESKIND, phys. stat. sol. (a) 10,455 (1972). Part II: M. ZOUAGHI, phys. stat. sol. (a) 11, 219 (1972).

450

M . ZOTTAGHI, B . P B E V O T , C . CARABATOS, a n d M . S I E S K I N D

and I I is compared with photoluminescence data obtained by other authors [1, 2], I n Section 3, we recall briefly the general shape of the absorption and photoconductivity spectra predicted by the calculations of Kohn [3], Lucovsky [4] and Bebb et al. [5]. I n Section 4, we sketch an energy level scheme which will allow the description of both absorption and photoconductivity spectra as well as luminescence. Section 5 is devoted to the discussion of the nature and position of the localized energy levels, correlating to Bloem's work [1] on stoichiometry defects and to Munschy's paper [6] on four-particle complexes. I n the last section, the effective defect concentration responsible for the observed bands is evaluated. 2. Synthesis of Experimental Data A preliminary remark is necessary in order to define the three classes of samples, with the help of two parameters introduced by Zielinger et al. [7]: the partial oxygen pressure during the annealing and the cooling procedure. Following their terminology, class A samples are obtained after an annealing at 850 °C under 10~4 to 10~5 Torr followed by a slow cooling; they are highly compensated. If ^ N ^ ^ N f j denotes the total density of donor (acceptor) centres, the compensation ratio y = £ Nfy2 i

j

is yk fa 1 for class A samples.

But the conductivity of Cu 2 0 is always of p-type, therefore yA < 1. Such samples have a low photosensitivity whatever the temperature of the experiment and no photocurrent has been detected in the low energy range (hoj < < 0.8 eV). Class B samples are obtained after annealing at 1050 to 1100 °C under oxygen pressure varying between 5 and 300 Torr, followed by quenching. The corresponding compensation ratio yB is weaker t h a n yA and decreases with increasing annealing pressure; the photosensitivity of B samples is higher t h a n for samples of class A. The observed absorption and photoconductivity spectra in the near infrared are complex; not less t h a n eleven absorption bands are found at low temperatures, corresponding to either minima or thresholds, or steps in the photocurrent spectra. Class C samples are obtained after a very slow cooling (48 h) in poor vacuum (10 _1 to 10 - 2 Torr). At temperatures high enough they pass the stability limits of Cu 2 0 inducing stoichiometry modifications; the resulting compensation ratio yc is very weak (yc 1). Samples belonging to this class are most photosensitive, but the photoconductivity spectra are simpler t h a n for class B. Thus, class B samples show a more complex photoconductivity spectra compared to those of classes A and C; much more structure is observed t h a n in photoluminescence spectra [1, 2], in which only two bands at 1.35 and 1.24 eV are seen at 20 K on analogous class B samples. I n the emission spectra of samples equivalent to those of class A, two more bands appear at 1.51 and at 1.71 eV. Table 1 summarizes the near infrared optical transitions observed in Cu 2 0; the first two columns give the structures at 10 K in the photoconductivity spectra of class B and C samples. I n column 3, the energies corresponding to the absorption bands at 77 K in class B samples are reported. The positions of the luminescence bands observed at 20 K by Bloem [1] and Gorban et al. [2] are gathered in column 4. One can immediately notice a close correlation between the positions of the absorption bands and the features in the photocurrent spectra; this correlation seems natural since the photoconductivity is

Near Infrared Optical and Photoelectric Properties of Cu 2 0 (III)

451

Table 1 Recapitulatory positions in eV of the features observed in the photoresponse, absorption, and emission spectra of Cu 2 0 photoresponse (T

class B = 10 K)

1.9 Ft Th 1.8 1.77, • St 1.24 1.19 Th 1.11 Th 1.05 Th 0.97 Min 0.9 Th 0.76 Min 0.7 Max 0.61 Th 0.5 J

emission [1, 2]

absorption

class C

(T = 10 K)

class B

(T = 77 K)

1.9

Min

1.91

1.65

Th

1.63 1.41 1.3

1.03 0.97

Th Min

0.6*1 : \ 0.5 J

Th

T = 20 K

1.71 1.51 1.35 1.24

1.12 0.97 0.89 0.76

0.366 0.362 0.353 Th threshold, Ft feature, St step, Min minimum, and Max maximum. Strongest features are denoted by bold face letters.

proportional to the concentration of photocarriers, created by optical transitions which in turn determine the absorption coefficient. However, in this case, to each absorption band a photoconductivity band should correspond. Such a correspondence is not observed experimentally. Consequently a detailed examination of the phenomena will be necessary in order to explain the complexity of the absorption and photoconductivity spectra as well as the general shape of the spectral responses. 3. Comparison Between General Theoretical Shape and a Typical Photoconductivity Spectrum

After Moss [8] the spectral distribution of the absorption or photoconductivity in semiconductors, due to transitions from a localized energy level to the allowed bands, can be described by the following empirical function : /(*») =

1 + exp

k

i~T E: — h(0

.

(1)

kBT

where El is the ionization energy of the impurity centres or defects, hco the photon energy, 1cB the Boltzmann constant, and T the temperature. At temperatures low enough {kBT E { ) , f(ha>) trends to zero if hco < E i ; for hco = E f(hco) = 1/2; for hco > E u f{hco) is almost constant.

452

M . ZOUAGHI, B . P K E V O T , C . C A E A B A T O S , a n d M . S I E S K I N D

The use of such a function gave a good description of several experimental results, namely for Ge [9] and diamond [10]. However, the comparison between f(ha>) and the Fermi-Dirac function shows that Moss has taken into account only the density of energy states, but not the photoionization cross section function S(hco) [3 to 5, 11 to 13]. The different determinations of S(ha>) show the difficulty to find (in a genera] way) the shape and position of the extrinsic photoconductivity threshold and absorption edge in crystals. This is the reason for which Moss' empirical function (1) is often a sufficient first approximation for the discussion of such spectra. Its advantage is to describe in a simple way a great deal of experimental results. When a unique energy level E & is present, for example due to acceptor centres of density N a , the spectral distribution of the photoconductivity Acr(Aoj) induced by the photoionization of the acceptor centres looks like a step function. If two energy levels (2?a and E d) of relative densities iVa and N d are present, one has to consider compensation effects. If N a > N' 1, possible optical transitions occur between the valence band and the two levels E & and ( E g — E ' 1 ) , respectively, with E d the photoionization energy of the donor centres N d . If JVa = N d (full compensation) transitions from the valence band to the donor level are possible, giving rise to a threshold at the energy ( E g — E d ) . Finally if iVa the transitions occur from the valence band to E d and from E d and E & to the conduction band; three photocurrent thresholds should be observed at ( E g — E d ) , E d , and (Eg — E & ) , respectively. According to equation (1), we have plotted on Fig. 1 two thresholds defined with the help of two parameters E* = 0.61 eV and ( E g — E d) = 1.84 eV for kBT = 36 meV and assuming 2Va > N d (class B sample); the first threshold is shown entirely (dashed line), but only the rising part of the second is illustrated at the right-hand side of the plot. The comparison between the previous theoretical spectrum Aa(hco) and the experimental one at 110 K with a typical class B sample shows that important structures are observed which are not predicted — X ( f i m ) by the functions of Moss, Eagles, and Lucovs3.0 2j0 1.5 0.8 0.7 0.6 ky. In other words, even if one takes into account the cross section function S ( h c o ) , it is not possible to explain photocurrent minima, namely at 0.97 eV and at 0.76 eV which then should correspond to optical transitions without creation of free carriers accounting for bound states. Such bound states may be due to three- or four-par ticle complexes (excitons-impurities [6, 14 to 16]) or to transitions to excited states of an impurity centre [3], or to transitions between centres of different nature [17]. All

OA

OS

1.6 2.0 M e V ) -

F i g . 1. Comparison between experimental ( O ) and theoretical ( ) photoconductivity spectra for a class B sample. Two Gaussian bands are substracted from the steps ( ). The fundamental edge ( F . E . ) , the excitonic complex (0), and the four bound states (A, B , C, and D) are indicated by arrows a t their respective theoretical (experimental) energies

Near Infrared Optical and Photoelectric Properties of Cu 2 0 (III)

453

the calculations concerning such bound states show t h a t absorption lines or bands have to be observed : without considering the detail calculations, in first approximation we can assume t h a t the absorption bands are Gaussian. I n absence of any photothermal transitions [18], the photoconductivity spectrum should present dips of Gaussian shape. The two experimental dips on Fig. 1 have been fitted by superposition of two Gaussian bands centered at 0.73 eV and 0.98 eV. We notice t h a t this superposition allows for a satisfactory overall description of the experimental spectrum. 4. Origin of the Infrared Absorption and Photoconductivity The study of the absorption and photoconductivity spectra of Cu 2 0 showed t h a t the band at 1.91 eV shifts to lower energies when temperature increases whereas the other bands are insensitive. This behaviour leads to a distinction between the band at 1.91 eV and all the others at lower energies. 4.1 Band at 1.91 eV

The existence of this band does not seem to depend on annealing and cooling conditions. I t is always observed whatever the class of sample may be. The vicinity of the first red absorption edge makes difficult the calculation of the band intensity. I n the photoconductivity spectra, it appears at high temperature as a shoulder for all three classes of samples. At low and very low temperatures, it appears as a strong minimum in class C samples but still as a shoulder in class A and B samples. As mentioned, it is difficult to separate the shoulder from the fundamental edge ; it can be a photocurrent maximum as well as a dip. Photocurrent measurements made with increasing temperature showed t h a t this shoulder shifts to lower energies. The rate of shift is of the order of 5 X X 10~4 eV/K, very close to the one found by Grun et al. [19] for the yellow exciton series in Cu 2 0. These two characteristics show t h a t the band at 1.91 eV is an intrinsic property of the crystal perturbed by the presence of impurity centres or stoichiometry defects. I t is then very probable t h a t the band is due to the formation of exciton-defect complexes. The effective mass approximation shows t h a t the stability of the three-particle complexes depends upon the effective mass ratio of the electron and the hole. The complexes exist for ratios significantly smaller t h a n one. I n Cu 2 0, the ratio is very close to unity; hence such a complex should not be stable. We conclude t h a t the complexes responsible for the 1.91 eV band are rather four-particles complexes. As Cu 2 0 behaves as a p-type semiconductor, these complexes are associations of excitons with neutral acceptors. Such associations are analogous to ^-centres in the alkali halides. However, in the case of deep centres, the effective mass approximation generally does not apply; it follows t h a t the existence of three-particle complexes in Cu 2 0 suggested in [20] cannot be totally excluded. 4.2 Bands of lower

energy

The position of all these bands is temperature independent. Under the effect of annealing or He + particle bombardment, the absorption bands of lower energy appear simultaneously. Their magnitudes increase in the same ratio:

454

M . ZOTJAGHI, B . PBEVOT, C. CARABATOS, a n d M . SIESKIND

it should then be natural to attribute them to a single type of defects, that is to a single energy level. However, such an assumption is contradictory to three facts: First, conductivity and Hall effect studies [21] showed that two levels are at least necessary to explain the experimental results. For certain samples, such as class A, the presence of three levels has been suggested [7]. Second, the positions of the absorption bands do not seem to follow a simple law. They do not correspond to a hydrogen-like series, such as those predicted by Kohn's calculations [3], corrected more recently by Bebb and Chapman [5], In opposition to diamond [22] or silicon [23] they do not seem to correspond to transitions from a single type centre to its excited states with phonon absorption or emission. Third, the correspondence between the strongest absorption bands (0.97 and 0.76 eV), and the deepest photoconductivity minima exclude any optical transitions between one of the allowed energy bands and energy levels associated to a single type of defects. Consequently, it is necessary to assume the existence of several levels due to different types of defects which have to explain not only absorption and photoconductivity spectra, but also the experimental results on conductivity [7], photoluminescence [1, 2], and photomemory effects [24]. 4.3 Proposed energy scheme

On account of the results, we show on Fig. 2 a level scheme for Cu20 including the n = 1 exciton level of the yellow series, the ^-complex level as well as four additional levels, A, B, C and D. The level A corresponds to acceptors; the photoconductivity measurements show that it is located at 0.55 to 0.61 eV from the valence band. The levels B and C are situated at 0.76 and 0.97 eV from the level A, respectively. Finally, we place the donor level D at 0.38 eV below the conduction band. On Fig. 2, the transition 1 of electrons from the valence band to empty centres of the level A corresponds to a photoconductivity threshold observed at 0.55 eV; the transitions 2 and 3 of electrons from filled centres of the level A to empty centres of the levels B and C correspond to the strongest absorption bands observed at 0.76 and 0.97 eV, respectively; the transition 4 related to the dissociation of the ^-complexes corresponds to the luminescence band observed at 1.35 eY [1, 2]; finally the transitions 5 to 7 of electrons from high filled levels to empty low levels seem to correspond to the luminescence bands observed at 1.24, 1.51, and 1.71 eV by authors quoted in [1, 2]. All these transitions depend essentially upon the position of the Fermi level, that is upon conduction band —

I

n-1

1.36eV B-

1.91eV r

^

w 2\

i

g M

0.55eV J L valence

band

f

2.17ev

Fig. 2. Proposed energy level scheme for CuaO. n = 1 represents the first exciton line of the yellow series; 3, the excitonic complex due to the n = 1 exciton combined with the level A. The levels A, B, C, and D correspond to v C u ' [ v C u - v O ] + > [ v C u ~ v C u ] a n d y 0 vacancies, respectively. Upward arrows indicate infrared absorption or photocurrent features; downward arrows indicate luminescence results [1, 2]

Near Infrared Optical and Photoelectric Properties of Cu 2 0 (III)

455

the compensation ratio, determined by the sample preparation conditions. For this reason, it is necessary to discuss the nature of the levels and the influence of the annealing and cooling on their concentrations and their ionization degrees. In the following discussion, on account of the slight difference between the optical and static dielectric constants in Cu 2 0 [33], no distinction will be made between the energy levels of neutral and ionized centres. 5. Nature and Position of the Energy Levels. Discussion of the Observed Transitions Using Wagner's model [25] Bloem [1] has performed a physicochemical study of the possible different defects in the Cu 2 0 lattice. The presence of these defects is due only to the exchange between solid and vapour phases at very high temperatures. Taking into account the interactions between different defects, Bloem assumed the presence of four types of defects: neutral copper vacancies Vc u or ionized vacancies Vc u , neutral or ionized oxygen vacancies V 0 , VQ , or V o c o p p e r vacancy associations [VCu-Vcu] o r [Vcu-Vcu] and finally associations of dipolar character [Vc u _ Vo ] or multipolar character [Vcu-Vo"] + . The concentrations as well as the ionization states of these defects depend upon annealing and cooling. Vcu and YQ (or Vcu and V 0 ) vacancies are most frequently invoked to explain experimental results of electric [7], photoelectric [20] or luminescence [1, 2] studies. However, the dispersion of the conductivity results (of several orders of magnitude) and the big variety of activation energies [1, 7, 21, 30] show that these two types of defects are not sufficient for the description of the extrinsic properties of Cu 2 0. For this reason, Pasternak and Kunzel [26] assumed the presence of [Vcu-Vcu] associations to explain the photomemory effect; those associations should be formed at T = 150 to 200 °C and destroyed at T = 20 °C under illumination. This explanation is not admitted either by Fortin et al. [27] or by Zouaghi et al. [24], Schwab et al. [28] and Zielinger et al. [7] proposed the presence of [VQ-VCU] associations to explain absorption and conductivity results, respectively. Nevertheless no authors gave details for the occurrence of those associations in the extrinsic properties of Cu 2 0. On the basis of optical absorption and photoconductivity in the near infrared, we propose (Fig. 2) a level scheme for Cu 2 0 including four individual levels, A, B, C, and D. An attempt will be made to attribute these levels to the four types of lattice defects which are suggested by Bloem [1], In class C samples, for which the oxygen excess is largest relative to stoichiometry, the density of Vc u vacancies is much greater than that of VQ vacancies [24], But the conductivity of Cu 2 0 is p-type so that we can attribute the level A to Vcu (or V Cu ). The more the oxygen annealing pressure decreases, the more the density of Vo increases, inducing an increase of the VQ concentration when the samples transform from class C to class B and furthermore to class A. As these vacancies have donor character, they will be assigned to the level D. The presence of high Vc u and VQ densities favours the formation of [Vc u -Vc u ] and [Vcu-Vj] vacancy associations. As in class B and C samples, the density of Vcu IS higher than that of VQ, one can conclude that the density of [V C U -V C U ] is higher than that of [Vcu-Vo]- On account of the fact that the absorption band at 0.97 eV is stronger than the band at 0.76 eV, and that its magnitude 30

physica (a) 11/2

456

M . ZOUAGHI, B . PREVOT, C . CABABATOS, a n d M . S I E S K I N D

increases with the annealing oxygen pressure, we attribute the level C to the [Vcu-Vcu] associations and the level B to [Vcu~Vo]- Keeping in mind that Cu 2 0 is always p-type and following the previous discussion, the position of the Fermi level E ¥ varies between the valence band and the middle of the energy g a p , EGJ2. 5.1 The level A

The optical transitions involving this level depend upon the position of the Fermi level EV. I f E-G < EA (class C samples), the level A is empty. The most probable optical transitions occur between the valence band and this level (transition 1 on Fig. 2). The photoconductivity threshold at about 0.55 eV is attributed to this type of transition; on the other hand the neutral V C u centres of the level A should be at the origin of the four-particle complexes formation: excitons associated to V C u vacancies responsible for the band at 1.91 eV and represented schematically by

I f E-g E a (class A and B samples), the level A consists of neutral and ionized centres. In addition to transition 1 on Fig. 2, other transitions may occur either to the conduction band or to the levels B, C, and D under conditions exposed in the following paragraph. Transitions of electrons to the conduction band have to give rise to an absorption edge and a photoconductivity threshold at 1.62 eY; the photoconductivity experiments indicate such a threshold at 1.65 eV. Transitions of electrons from the level A to the levels B , C, and D are possible only when the latter are empty. The absorption bands observed at 0.76, 0.97, and at 1.3 eV should be explained by such transitions. Furthermore, the ionized centres Vc u (filled) of the level A can trap the photoholes created in the valence band and thus prevent their recombination with photoelectrons inducing a permanent increase of the conductivity. This conclusion is consistent with the interpretation of the photomemory effect given by Fortin et al. [27] and Zouaghi et al. [24]. Finally, the luminescence band observed by Bloem [1] and Gorban et al. [2] at 1.35 eV seems to correspond to transitions from the level of the (3-centre to the level A ; during this transition, the complex should dissociate, and a free exciton is created. 5.2 The level B

As mentioned above, the level B is attributed to [VC u -VQ] associations represented schematically by + I +

Their concentration N B is always lower than N A . As for the level A the optical transitions involving the level B depend upon the position of the Fermi level E$.

Near Infrared Optical and Photoelectric Properties of Cu 2 0 (III)

457

However, as Cu 2 0 is a p-type semiconductor, EV is always smaller than EB, t h a t is the level B is rather empty. The centres are of the form [Vc u -Vo ] + I n this case, besides the transitions starting from the level A and responsible for the photoconductivity minimum at 0.76 eV, other transitions from the valence band are possible. A photoconductivity threshold at 1.31 eV must correspond to the latter ones; the higher the concentration of the ionized Vc u centres is the more intense is this threshold observed experimentally at 1.3 eV but only with class A samples. On the other hand, the level B, because of its nature, can trap photoelectrons from the conduction band and contribute to the photomemory effect as well; such an interpretation completing t h a t given by Fortin et al. [27] and Zouaghi et al. [24] is consistent with the interpretation proposed by Zielinger et al. [7]. 5.3 The level C

As seen previously, we attribute the level C to [Vc u -Vc u ] associations, schematically represented by

©© This configuration involves the fact t h a t their concentration NC is lower t h a n NA and increases with the annealing oxygen pressure. For class B and C samples, NC is larger t h a n JVB. Remembering the position of the Fermi level in the forbidden gap (E¥ < EC), we conclude that the level C is empty: the centres are of the form [Vc u -Vc u ]; in this case, the most probable optical transitions occur from the level A or from the valence band. To the former case the absorption band (and the minimum of photocurrent) observed at 0.97 eV corresponds; to the latter a photoconductivity threshold located at 1.52 eV should correspond. Such a threshold has been observed experimentally between 1.35 and 1.52 eV only with class B samples, annealed under low oxygen pressure (5 to 20 Torr). For other samples, a wide step of photocurrent is measured in this spectral range. The luminescence band at 1.51 eV reported by authors quoted in [1,2] seems to be correlated with transitions between the level C and the valence b a n d ; one may admit t h a t a free electron can be trapped by the level C before it falls down radiatively into the valence band. 5.4 The level D

We attribute the level D to VQ vacancies illustrated by

0 I t s position is given by the emission observed at 1.24 eV by Bloem [1] and Gorban et al. [2], assuming a transition from the level D to the level A; its 30»

458

M . Z O U A G H I , B . P E E V O T , C . CABABATOS, a n d M . S I E S K I N D

concentration is such t h a t JVB < Njy < N A and as E v < 2?D, the level D is empty, thus the centres are in the state Vo • In the optical absorption process the most probable transitions occur either from the valence band or from the level A. These two types of transitions should correspond to the photoconductivity threshold observed at 1.8 eV with certain class A samples and to the absorption band located at 1.3 eV with class B samples, respectively. The luminescence band observed by authors quoted in [1, 2] at 1.71 eV should take place in this scheme between the level D and the valence band. This luminescence process, as well as t h a t at 1.24 eV, starting from the level D involves electrons of the conduction band trapped by this level D. 6. Density of the Centres As pointed out in Section 5, the concentrations of the different types of defects vary with the partial oxygen pressure during annealing at high temperature. Bloem's work [1] allows an evaluation of the concentration of Vc u (level A) and Vo" (level D) vacancies at temperatures high enough to ensure a permanent equilibrium between solid and vapour phases. For example, at 1000 °C and for p0l > 100 Torr, Bloem gives NA « 1019 cm - 3 , and Ny r^ 1017 cm - 3 , as concentrations forVc u and V