H. K. WLLER: Electrical and Optical Properties of In,O, Films (11) 733

phys. stat. sol. 27, 733 (1968)

Subject classification: 20.1; 14.3.1; 22.6

Institut fiir angewandte Physik, Universitat Karlsruhe

Electrical and Optical Properties of Sputtered h , O , Films

11. Optical Properties in the Nea.r Infrared

BY H. K. MUUER

The influence of conduction electrons on the optical properties in the near infrared has been investigated for sputtered In,O, films. The results are interpreted using the present theories of absorption and dispersion by free carriers. The optical properties in the investi- gated frequency range are considerably influenced by the anisotropy of the predominant scattering mechanism (scattering at charged defect centres) and by the low mean free time of the electrons (t w 1.5 x s) .

In dunnen, durch reaktive Kathodenzerstaubung hergestellten hO,-Schichten wurde der EinfluB der Leitungs-Elektronen auf die optischen Konstanten im nahen Ultrarot untersucht. Die Ergebnisse werden auf der Grundlage der bestehenden Theorien der Absorption und Dispersion durch freie Elektronen diskutiert. Dabei zeigt sich, daB die Anisotropie des vorherrschenden Elektronen-Streuprozesses (Streuung an ionisierten Storstellen) und die kleine mittlere StoBzeit der Elektronen (t w 1,5 x s) die optischen Eigenschaften im untersuchten Wellenliingen-Gebiet wesentlich beeinflussen.

1. Introduction

Electrical properties and the intrinsic absorption of sputtered In,O, films were subjects of part I of this paper [l]. The films are excess electron conductors; as a consequence of deviations from stoichiometric composition, excess indium atoms or oxygen vacancies serve as donors, the concentration of which can be increased by a reducing heat treatment in vacuum.

The properties of the films are decisively influenced by an additional disorder which obviously is due to the particular structure of In,O, and the relatively low temperature during the preparation of the films. As a consequence, there is a strong scattering effect of the conduction electrons a t the charged defect centers. Thereby the mobility of the charge carriers is lowered considerably (Hall mobility a t room temperature: ,uH < 10 cm2/Vs).

This paper deals with the optical properties in the near infrared. In this fre- quency range, the optical constants (refractive index a;, absorption coefficient k or absorption constant K = 4 n k/A) are predominantly conditioned by optical intraband transitions. They also depend on the properties of the electron gas such as carrier density n,, mean free time z and effective mass m* of the conduc- tion electrons, i.e. they should be influenced by the low electron mobility ,u and the low mean free time t, respectively. In particular it will be discussed in how far the experimental results can be interpreted on the ground of the present theories of absorption and dispersion by free carriers.

734 H. K. MULLER

2. Experimental Procedure and Results

For a number of polished films (as sputtered or after different heat treatments in vacuum, i.e. for different electron concentrations), the optical transmissivity and reflectivity were measured a t room temperature for nearly normal light incidence. The measuring set-up differed from that for investigating the intrin- sic absorption [l]: as light source an incandescent lamp ( I 5 2 pm) or a globar ( I 2 2 pm) were used; the radiation was modulated (12.5 Hz) before entering the double monochromator. A spectral dispersion was achieved using a set of flint prisms for I 5 2 pm and a set of fluorite prisms for I 2 2 pm. The receiver consisted of a thermal radiation detector and a narrow-band display amplifier. The optical constants were obtained from the measured data using Murmann's method [2] as bef0re.l)

The result of IR measurements of a sample after different heat treatments is shown in Fig. 1 and 2. The real part of the complex dielectric constant (Fig. 1) varies as Iz in the short-wave region (A2 < I : = 8 pm2) and is independent of I in the long-wave region (Iz > I : = 8 pm2) except for curve VI in Fig. 1. The relatively sharp transition between these two regions occurs a t I = 2.8 pm; there is no dependence on the electron concentration.

The product n, K , too, shows a different behaviour in these two regions: for I < 2.8 pm the absorption curve ns K varies as A3, for I > 2.8 pm there is a weaker dependence on wavelength.

This behaviour of the optical constants will be compared in section 4 with the results of the present theories for the IR dispersion. But first, the theoretical results will be summarized and discussed in section 3.

+ I I

Fig. 1. Real part ng -ka of the complex dielectric constant as a function of the square of the wavelengtli A' for different Hall concentrations (after different heat

treatments of one and the same sample)

1) The calculations using a computer have been done by Dipl. Phys. A. Jesse and Dipl. Math. K. Rittmann. Other data have been made available by Dr. H. Finkenrath and Dipl. Phys. M. Volkmann (Darmstadt).

Electrical and Optical Properties of Sputtered InzO, Films (11)

3. Theories

The classical theory of absorption and dispersion of a free electron gas (start- ing with the well-known collision statement by Drude and Zener) leads to the following expressions for the optical constants [3] :

735

( e - electronic charge, E~ - influence constant, co - velocity of light, E~ - di- electric constant without the contribution xe of the conduction electrons to the electric susceptibility).

These equations may be simplified for the limiting cases of high and low fre- quencies. I n the first case (ma > 1/t2), the well-known proportionality of both the susceptibility x e and the product n, K t o the square of the wavelength is obtained :

e2ne 1 corn* W* ' n: - k2 = cg + xe = E g - -- (3)

whereas for low frequencies (w2< 1/?) both xe and n, K do not depend any more on A.

This theory is valid as long as the energy of the electron after absorbing a pho- ton is changed but slightly, i.e. the energy of the photon must be small as com- pared with the energy of the electron in the conduction band ( h w < kT for a non-degenerate electron gas and h o < E , for a degenerate electron gas, E , Permi energy). This prerequisite is usually not fulfilled in the case of absorption and dispersion by conduction electrons in semiconductors ; never- theless, this theory is generally applicated when interpreting measurements.

l/t2 show that the absorption follows equation (4) to a, good approximation far beyond the given limits, namely up to h o = 15 kT and 8 E,, respectively; it is assumed, however, that the electron mobility is given by an isotropic scattering mecha- nism, e.g. scattering a t acoustical phonons. Since n, K and n," - k2 are closely related by the Kramers-Kronig relations, a validity of equation (3) beyond the stated limits is to be expected. However, for an anisotropic scattering mecha- nism the deviations from equation (4) will be stronger according to Dumke.

The classical theory has been modified [6 to 71 in order to take into considera- tion the quantum nature of the absorption process to a certain extent; thereby a description of the I R absorption may be obtained which even for an aniso- tropic scattering process is valid in the region of larger photon energies ( h w 2 E , or kT, respectively). The collision time t of an electron is a function of its energy. Decisive for the probability of an indirect optical transition should not be only its collision time in the initial state but a certain averaged value of the collision times of the absorbing electron in the initial and final state (energy El and El + A o, respectively). Therefore the constant mean free time t in the equa- tions (l), (2) and (4) is replaced by a mean free time z (El, El + h 0). The terms

In fact, theoretical considerations by Dumke [4] for the region w2

736 H. K. Muum

containing t, are considered as averaged values over all electrons of different initial energies El

--) ; (5 ) e2n, / ?(L- 2 E , E , - ! - h w )

eo m n: - k2 = E~ - \ 1 + w2 ?*(El, ET+ h w )

These equations may be simplified without further assumptions :

t l ( w ) and ~ ~ ( 0 ) being (in general different) real-valued functions of the frequen- cy w , which are characteristic for a definite semiconductor.

A quantitative comparison, however, between equations (5) and (6) or (7) and (8); respectively, and experimental results indicates that the above pro- cedure is not correct. This is shown by the following considerations:

Deducing t 2 ( w ) according to equation (8) from absorption curves and the respective electron concentrations leads to complex values of t2 for CdO [6] and SnO, [7] as well as for the present results unless the effective mass is very low (m* 5 0.15 m, for In,O,); i t should be lower a t least by a factor 1/2 as compared with the effective mass which is calculated from the observed fre- quency dependence of the electrical susceptibility xe using equation (3) (m* = = 0.30 m, for In,O,; cf. section 4). Now, equation (3) is also the limiting case of equation (7) for large w , and a contradiction results between equations (7) and (8) or between these equations and the observed behaviour of the optical constants.

For In,O,, this contradiction may also be shown using the Kramers-Kronig relations [8]

m

The observed frequency dependence of the product n, K (Fig. 2) follows eyua- tion (8) quite well in the investigated frequency region if putting t 2 ( w ) = t w where t is a constant the values of which are not very different from each other for curves 111-VI (Fig. 2). With this assumption equation (8) is of the form

Again, with this expression, the elect,rical susceptibility from equation (9) becomes for w2 > l / ~ i

1 A c , X e = - 2 7 '

Equation (9a) gives the correct electrical susceptibility for short wavelengths (A <A,); the values are also quantitatively correct when using the A values as

Electrical and Optical Properties of Sputtered I%O, Films (11) 73 7

obtained by fitting equation (8a) to the curves in Fig. 2. However, the quotient of the susceptibility xe and the product n, K with equations (8a) and (9a), xe/ns K = - (1/2) c, t,, is only half the value as obtained with equations (7) and (8) for the limiting case w2 > l/t;, l/ti. Whereas the observed behaviour of xe(w) and n,(w) K ( w ) is consistent with equation (9)2), there is a contradiction if equations (7) and (8) are assumed to hold simultaneously.

These considerations show that for anisotropic scattering for high photon energies ( A w 2 EF) not only the system of the classical equations (1) and (2) does not hold any more (in contrast to the case of isotropic scattering), but also the modified system of “semi-classical” equations ( 5 ) and (6), (7) and (€9, respec- tively, does not show the true behaviour of the optical constants. It seems that the absorption by conduction electrons is stronger than given by the classical Drude-Zener description.

On the other hand, the values of the effective mass as calculated with equs- tion (3) for w 2 > l/t2 from the electrical susceptibility of the free cariers have turned out correct, even for anisotropic scattering if Aw 2 EF. From this it may be inferred that equation (3) also holds for A w 2 EF if w2 > l/z2 in con- trast to the respective absorption formula (4).

This certainly is the case if n, K obeys equation (4) in a frequency range large enough, i.e. if in this region both conditions for the validity of equation (4), EF > A w and A w >A/T , are fulfilled (a necessary condition is E, > A / t ) . The strong absorption for low frequencies ( A w < EF) then determnes the electrical susceptibility by equation (9) also in the shortwave region, where equation (4) for n, K does not hold any more since ?z w 2 EF.’)

A general decision whether equation (3) also holds for A w > A/z > EF (i.e. if there is no frequency range for which n, K is given by equation (4)) can only be expected for a quantum theoretical treatment of the optical properties for free electrons. Such a treatment of the electrical susceptibility does not yet exist.

Quantum theoretical treatments of the absorption by free carriers have been performed by several authors [lo to 131 both for the case of electron scattering a t acoustical phonons and at ionized impuritie~.~) The results by Schmidt for scattering at acoustical phonons [12] and Meyer for scattering at ionized impu- rities [lo], which presumably are the best approximations, can be put into the following form, taking the limiting cases of high ( A w >2 kT) and low (Awg2kT) photon energies assuming w > l/t : scattering at acoustical phonons

hw>2kT’: n, K oc A3I2 , (11) Aw<2kT: n, K oc A2; (12)

scattering at ionized impurities A w > 2 kT: A w g 2 k T :

n, K oc As , n, K cc A2 .

2) This is the case at least in the shortwave region; nothing can be mid about the region A > A, since there could still be a considerable influence of ns K on xe in the non-inveati- gated region A > 4 pm.

8 ) This cam has been. investigated by Geserioh [9] with T40, films. 4) A comparison of the slightly deviating results has been given by Planker and Kauer [14].

48 phystca 2112

738 H. K. MULLER

As can be seen, the A2 dependence of Drude's equation (4) holds independent of the scattering mechanism, in the range of sufficiently long waves (but yet w must be larger than the reciprocal collision time of the electrons). This is consistent with Dumke's investigations [4].

The theories by Schmidt and Meyer have actually been deduced for a non- degenerate electron gas. However, a t least in the case of scattering at charged defect centres it is to be assumed that the stated wavelength dependence does not change considerably6) when the electron gas becomes degenerate ; the tran- sition from the A3 dependence to the classical La dependence occurs in the vicinity of A w = E , rather than a t A w = 2 kT. Recent investigations by Geserich [9] with T1,0, seem to confirm these conclusions.

Obviously the relations (12) and (14) cannot hold for very low photon energies since for long wavelengths (A + co) the product ns K must tend to a finite value which, according to the classical theory after equation (2), is n, K = cro/o/Eo c,, (cro d.c. conductivity). It may be supposed that the quantum theoretical treatment does not hold any more if the photon energy Aw decreases to the energy uncertainty h / t of the electron states ( A w w A / t ) .

4. Discussion Scattering in the In203 films occurs predominantly a t ionized centres [l].

In spite of the anisotropy of the scattering process, the wavelength dependence of the real part of the dielectric constant may be described by the limiting cases of the classical equation (l), both for the shortwave side (mi - k2 cc At) and the longwave side (n: - k2 independent of A). The transition between these two regions occurs a t A, = 2.8 pm or w, = 0.67 x 10l6 s-l.

According to the classical theory, this frequency should be equal to the reci- procal mean free time l/t of the electrons.

Also the proportionality between the electrical susceptibility and the electron concentration, as expressed by equation (3), holds quite well in the short wave- length region (Fig. 3). An estimate of the effective mass of the conduction elec- trons, using the rise of the straight line in Fig. 3, leads to m* = 0.30 mo.6) In

Fig. 3. Derivative dXe/d(l') of the electrical susceptibil- ity xeof the conduction electrons by the square of the wavelength 1' as a function of the Hall concentration f l ~ for 1 < 1, = 2.8 pm. The rise of the straight line

leads to the effective mass of the electrons m* = 0.30 mp

s, A I? dependence of the product n8 K has been found for different substances with degenerate electron gas [14, 151.

6 , Weiher [9] obtained m*=0.55 mo from measurement of theHallcoefficient and thermo- electric power. Details about the thermoelectric power unfortunately have not been published.

Electrical and Optical Properties of Sputtered b O s Films (11) 739

the present case, the validity of equation (3) is not substantiated theoretically because of the low mean free time (EB < @); nevertheless the value m* = = 0.30 m, is in good agreement with the value m* = 0.33 m, that can be evalu- ated from the position of the plasma edge for sprayed In,O, films [16], (see foot- note 8); for this case E , > hit, and equation (3) should be a good approximation in the region ma > 1/r2.

s-l, the magnitude of the mobility ,u = (elm*) z is ,u = 8.9 cm2/Vs. This value is in good agreement with the Hall mobility for high electron concentrations, which is the mobility within the crystallites El]. This indicates that the light frequency cc),, a t which the real part of the dielectric constant changes from the 1, behaviour to a value independent of A, is indeed approximately equal to the reciprocal mean free time of the electrons as claimed by the classical theory. However, the transition between the two regions is steeper than given by equation (1).

The curves in Fig. 1 seem to indicate that in these thin films no plasma edge') develops as Groth [16] observed with sprayed In,O, films; his data include n, = 4.9 x,lOao om-, and ,u = 51.5 cma/Vs for A, = 1.7 pm.*) (A, wavelength position of the plasma edge.) In fact, the sputtered films did not show a plasma edge when the reflectivity (light incidence a t the free surface) was measured up to 15 pm; the electron concentration n, was as high as 1020 om-,.

The obtained product n, K does not show any features of the classical theory in contrast t o the real part of the complex dielectric constant ; instead, the ob- served 1, dependence in the short wavelength region (Fig. 2 ) corresponds to the quantum mechanical absorption formula (13). The frequency w, = 2 n COIL, a t which the transition to the weaker 1 dependence occurs, is approximately equal t o l/z, as shown by the above-done estimate. Thus the deviation from theA3 dependence in the long wavelength region is due to the fact that the photon energy for 1 > A, = 2.8 pm is smaller than the energy uncertainty of the elec- tron states. Therefore equation (13) does not hold any more.

Using m* = 0.30 m, and T = l/w, = 110.67 x

6. Summary and Conclusions

The results of the optical investigations in the near infrared may be summariz- ed as follows:

The real part of the complex dielectric constant shows a few features that have been predicted by the classical theory of absorption and dispersion by free electrons. The electrical susceptibility X, of the conduction electrons is proportional to 1, in the short wavelength region and is independent of 1 in the long wavelength region if the electron concentration is not too high; the tran- sition between these two regions occurs a t or, which is approximately equal t o the reciprocal mean free time of the electrons. This transition is relatively sharp, therefore there is a considerable discrepancy in this region between the observed wavelength dependence of xe and the predicted behaviour (equation (1)). However, there are indications that a t least the agreement of the behaviour of X, in the short wavelength region (w > w,= l/t) and equation (3), also for the

') The plasma edge is the steep rise of the reflectivity to nearly lOOyo which occurs for weak absorption (k2 < l), i.e. for large r and therefore high mobility, when ni - k2 vanishes.

8 , From these values and equation (3), the effective mass m* of the electrons may be estimated: m* = 0.33 m,, with eg = 3.58.

4s.

740 H. K. MULLER: Electrical and Optical Properties of In,O, Films (11)

present case h w 2 E,, is not accidental. But the classical theory fails completely if applied to the absorption (Fig. 2).

The theory still does not f i t the experimental data if the absorption is described by introducing a frequency dependent ‘G in equations (1) and (2). This also is the case with CdO [6] and SnO, [7]. The absorption by the conduction electrons is obviously stronger than given by Drude-Zener’s theory.

For In,O, films, the observed P dependence of the product n, K in the short wavelength region does agree with the results of quantum theoretical treatments [lo]. These quantum theoretical considerations obviously only hold for high photon energies ( A w > A / t ) , in contrast t o the classical theory where the validity is restricted to low photon energies ( A w < EF or kT, resp. for anisotropic scat- tering). Because of the low electron mobility (+ > EF) and the anisotropy of the scattering process, there exists for the investigated In,03 films a frequency region (A/z > ii DJ > E F ) where both the classical and the quantum mechanical absorption theory fail.

Acknowledgements

The author wishes to express his appreciation to Prof. Dr. F. Stockmann for suggesting and supporting these investigations, t o Dr. U. Todheide-Haupt, Dr. W. Stossel and Dr. H. Finkenrath (Darmstadt) for fruitful discussions; he especially wants to thank Dr. H. P. Geserich for many discussions and good cooperation.

References H. K. MULLER, phys. stat. sol. 27, 723 (1968). H. MAYER, Physik dunner Schichten, Bd. I, Wiss. Verlagsgesellschaft, Stuttgart 1950 (p. 207). See, e.g. J. M. ZIMAN, Principles of the Theory of Solids, University Press, Cambridge 1964. W. P. DUMKE, Phys. Rev. 124, 1813 (1964). H. FINKENRATH, Z. Phys. 168, 511 (1960). H. FINKENRATH and M. VOLKMANN, phys. stat. sol. 2, 850 (1962). H. KOCH, phys. stat. sol. 3, 1619 (1963). L. D. LANDAU and E. M. LIFSHITZ, Electrodynamics of Continuous Media, Pergamon Press, 1960 (p. 256). H. P. GESERICH, phys. stat. sol. 26, 741 (1968). H. J. G. MEYER, J. Phys. Chem. Solids 8, 264 (1959); Phys. Rev. 112, 298 (1958); Advances in Semiconductor Science, Pergamon Press, New York and London 1959 (p. 264). H. Y. FAN, W. SPITZER, and R. J. COLLINS, Phys. Rev. 101, 566 (1955). H. SCHMIDT, Z. Phys. 189, 433 (1954). R. ROSENBERO and M. LAX, Phys. Rev. 112,843 (1958). K. J. PLANKER and E. KATJER, Z. angew. Phys. 9,425 (1960). See, e.g. J. R. DIXON, Proc. Internat. Conf. Semiconductor Phys. Prague 1960, Czech. Acad. of Science, Prague 1961 (p. 366); R. L. WEIHER, Phys. Rev. 162, 736 (1966). R. GROTFI, phys. stat. sol. 14, 69 (1966). R. L. WEIHER, J. appl. Phys. 33, 2834 (1962).

(Received April 5, 1968)