Infrared reflectivity of zinc oxide

J. Plzys. CJzem. Solids

Pergamon

Press 1959. Vol. 11. pp. 190-194.

INFRARED

REFLECTIVITY R. J. COLLINS Bell Telephone

Printed in Great Britain.

OF ZINC

OXIDE

and D. A. KLEINMAN

Laboratories,

Murray Hill, New Jersey

(Received 4 March 1959) Abstract-The reflection spectra of single crystals of ZnO were studied between 1 and 45~. In samples with free-carrier concentration below lOis cm-s, the usual reflection spectrum of an ionic crystal was observed. An excellent fit to the data is achieved by using a value of the highfrequency dielectric constant co = 2.0, a static dielectric constant Ed= 8.15, and a transverse optical mode frequency tt = 414 cm-l. The contribution to the optical constants by the free carriers becomes significant in the reflection spectra for carrier concentrations larger than -1 >: 1017 cm-3. Using a method similar to that of SPITZER and FAN an effective mass of m* = 0.06 has been measured. The reflection of ZnO containing carriers can be fitted remarkably \I-ell over the range 1-45~ by the classical theory for free carriers and a lattice oscillator. The effective charge per ion site, as defined by SZEGETTI,is found to be q* = 1.06e. 1. INTRODUCTION

IN recent years optical measurements have been used in the study of semiconductors to gain knowledge concerning: width of energy gap, effective mass of carriers, shapes of energy bands, and details of shallow impurity levels. The results of such studies in germanium, silicon, and A”’ BV compounds have been discussed in review articles most recently by FAN and HROSTOWSKI(~). The oxides, and in particular zinc oxide, have not received as much attention as the valence semiconductors, largely due to the lack of single crystals of controlled impurity content. Work at this laboratory on the preparation of zinc oxide by D. G. THOMAS and J. J. LANDER has now reached the point where optical experiments are feasible. The infrared reflection spectrum of ZnO was observed by STRONGc2), using natural zincite, and PARODIc3), using evaporated films. In both cases residual-ray monochromators were used which gave widely spaced points and only qualitative values of the reflectivity. In addition to the reflectivity, TOLKSDORFF(4) (also using evaporated films of ZnO) reported the transmission spectra between 1 and 20~. Her data showed a large number of bands that are not observed in the single-crystal data in this paper. Based on the work of STRONG, PARODI and TOLKSDORFF, an estimate of the transverse optical mode frequency of 343 cm-r was made by KRUEGERand MEYERc5). 190

In the present paper the absolute reflectivity of single crystals with different concentrations of free charge carriers in the range l--40~ are given. From the data values for the optical mode frequencies, the effective charge per ion site and the carrier mass will be found. Also a slight modification of the SPITZER and FAX@) susceptibility method for determining the effective mass will be discussed. 2. EXPERIMENTAL The reflection spectra were taken on single crystals of zinc oxide from two sources: (a) crystals grown in our own laboratory in the form of ribbons (-1.0~ 0.1 x0.01 cm) and used both in an “as-grown” condition and after being etched in HI;, and (b) samples obtained from the cinder piles of the New Jersey Zinc Company and optically polished by conventional methods. Most of the crystals obtained from either source contained a sufficiently high electron concentration to affect the reflectivity. Free-carrier effects, however, could be reduced to a negligible value by the diffusion of lithium, which acts as an acceptor. The introduction of lithium was accomplished by allowing a solution of lithium hydroxide to dry on the crystal surface, followed by baking in air at 700°C for 1 hr. For the samples used in the free-carrier mass determinations, where some control over the carrier concentration

INFRARED

REFLECTIVITY

was desired, indium was used as the impurity.(7) In this case a similar diffusion technique was used which differed only in that the baking temperature allowed the amount of indium in the sample to be controlled. Following the diffusion process, the thin ribbons were etched to remove surface contamination and the larger crystals were ground and polished. In the case of the indium-doped samples, several microns had to be removed before steady-state surface conditions were reached. The actual carrier concentration was found from Hall-effect measurements. Current contacts to the ends of the sample were made with pools of gallium, and the Hall probes were pressure contacts of phosphor-bronze wire. The reflectivity measurements were made with a double-pass Perkin-Elmer monochromator equipped with prisms of NaCl, KBr and CsI. A toroidal mirror was used to form a 1 : 1 image of the exit slit in the plane of the sample with the center ray 8” from the normal. The reflected beam from the sample or the reference, a freshly evaporated aluminum film, was then imaged onto a widearea thermocouple. Previous measurements had shown that the absolute reflectance of a fresh aluminum film is >96 per cent in the wavelength region of interest. As could be expected from the small size of the samples, energy limitations in the far infrared made the measurements somewhat uncertain. It was not possible to employ microscope optics to alleviate this situation, since reflectance measurements should be made at a well-defined angle of incidence. 3. LATTICE

vo2-v2

N0e2 &2&f

(“02-v)2+y2,2

(1)

and c=--where

No is the

ZINC

191

OXIDE

reduced mass of the oscillator, y the width of the resonance at half maximum, and vg is the resonance frequency. These quantities and the highfrequency dielectric constant, ~0, completely determine the optical constants n and K, and the reflectivity R through the expressions :

d--k2

= q)+47ix

nk =-

o/v

and R = (n-l)2+k2 (3) (n+l)+@ The observed reflectivity at 300°K of zinc oxide for unpolarized light is shown by the points in SINGLE

CRYSTAL

WAVELENGTH

IN

in0

M,CRONS

FIG. 1. Reflectivity of single crystal ZnO. Fig. 1. The parameters

solid

curve

is calculated

from

the

REFLECTION

The dominant feature in the reflection spectrum of an ionic crystal is the residual ray band of the fundamental lattice absorption. According to the classical dispersion theory(*) of Lorentz, the susceptibility x and the conductivity G of the lattice are given by:

x =

OF

iVoe2

2Tryv2

4+M (v$-v2)2+y%2 ion pair

concentration,

(2)

M the

y = 3.1 X lo11 see-Iand $0 = 414 cm-1(X0 24.2 p). For the trial-and-error method of selecting the best values for the three parameters, the equations were programmed for an IBM 650 calculator. It is clear from the agreement in Fig. 1 between the data and the calculated curve that the estimate of KR&ER and MEYER@) for the ZnO residual-ray frequency of 343 cm-l was too low. The value of the static dielectric constant corresponding to the theoretical fit is cg = 8.15, which compares with the value of 8.5 measured at 1 cm by HuTSON(

192

R.

J.

and

COLLINS

Since zinc oxide has the wurtzite structure, it would be expected to have two distinct lattice frequencies. Using light polarized along (extraordinary ray) and perpendicular (ordinary ray) to the c-axis, these two frequencies are separately excited. These experiments showed only very slight differences, the shift being less than 0.1~~ and hence the frequencies are nearly equal. According to the LYDDANE-SACIIS-TELLER(~@ relation, the longitudinal optical mode frequency is

The effective charge per ion site can be deduced from the parameters of the resonance by means of the SZIGETTI(~~)relation :

which gives q* = 1*06e. In the lattice absorption of an ionic crystal the first-order and strongest absorption will take place at the transverse optical frequency and secondorder or two-phonon processes will take place with combinations between branches of the zn

5

o

LATTICE

ABSORPTION

15 20 25 WAVELENGTH IN

30

35

40

MICRONS

FIG. 2. Lattice absorption spectra of ZnO.

45

D.

A.

KLEINMAN

optical modes or between one optical and one acoustical mode. Fig. 2 shows the lattice absorption spectrum of ZnO. The points from 6 to 14i~ represent data obtained from transmission measurements on thin samples, while the points at longer wavelengths are calculated from the residual-ray band reflection. Aside from the main absorption at 24.2~, only two strong absorption lines are present in the spectrum of zinc oxide, one at 11.4~ and the other at 10.2~. The assignment of twice the transverse optical mode for the 11.4~ band and the combination of the transverse optical and longitudinal optical for the 10.2~ band is suggested. Higher-order combinations are probably present, but will bc of a much lower magnitude. In the early work of TOLKSDORFF(4) on evaporated films of zinc oxide, many bands were present which were not observed in the single-crystal specimens, and speculation on their origin would not be fruitful. The reflection data STRONG(~)obtained from natural zincite, although not in complete agreement with our own work, probably differ only because of the effect of carriers in his samples. 4. FREE-CARRIER EFFECTS The presence of free charge carriers affects both the real and imaginary parts of the index of refraction. In valence semiconductors, i.e. germanium and silicon, the free-carrier absorption has been studied in detail both theoretically and experimentally, with the result that no present theory seems adequate to allow measurement of the effective mass from absorption data. SPITZER and FAN@) showed, however, that under the condition w~>l a value of effective mass could be deduced in a straightforward manner from the free-carrier susceptibility without knowledge or assumptions concerning the relaxation time. Using the values of susceptibility determined from absolute reflectivities, they obtained effective masses for carriers in germanium, silicon, Ins, and InAs which are in good agreement with the cyclotron-resonance values. In the case of zinc oxide the free-carrier absorption has been reportedus) with marked deviations from theory and therefore cannot be used to determine the carrier effective mass. The susceptibility method can be applied, however, if enough carriers are present to produce a

INFRARED

REFLECTIVITY

well-defined minimum in the reflectivity. If the minimum reflectivity is very low, the extinction co-efficient K is very small, and the reflectivity minimum therefore corresponds to the condition n(h) = 1. The contribution of the carriers to the susceptibility x may be found from equation (1) by setting vs = 0 and interpreting NO as the carrier density, M as the carrier effective mass, and (%ry)-1 as the carrier relaxation time 7. Therefore if NO is known from independent Halleffect experiments, the effective mass can be calculated from the position of the reflectivity minimum by the relation: esN&&n

m* =

7r(<- l)c2 It is not essential absolute reflectivity, low reflectivities.

PARAMETERS

I? = S x T =

I .29

T = 0.5,

FOR

10”

to measure which may

CALCVLATEO

CURVES:

ELECTRONS/CM: “,*=

0.06

x lo-‘4

SEC

-----

x lo-‘4

SEC

-

accurately the be difficult at

m

FIG. 3. Reflectivity of ZnO containing 5 x lo’* electron/ cm3. Fig. 3 (points) shows the reflectivity of an indium-doped sample of zinc oxide containing 5.0 x 101s electrons/ems. The reflectivity has been modified by the carriers even in the region of the residual-ray band. From the minimum at 6.0~, the point where the susceptibility of the carriers has caused the total dielectric constant to equal unity, one calculates m* = 0.06 m. For one sample it was verified that the reflectivity on the short-wavelength side of the minimum obeys the classical equations for w+l. N

OF

ZINC

OXIDE

193

Experiments in which the light was polarized parallel and perpendicular to the c-axis revealed differences in the carrier mass which were not significantly larger than the experimental error. In principle, to calculate the total reflection from an ionic lattice containing free carriers, a knowledge of the wavelength-dependence of each dispersion mechanism is needed. For zinc oxide the lattice dispersion seems to be well accounted for by the classical relations (1) and (2), but the free-carrier contribution is not so well understood. However, the simplest assumption which can be made for the carrier absorption is a constant relaxation time, which leads to equations of the form of (1) and (2) with vs = 0. In this model the only adjustable parameters for the free carriers are the effective mass and the relaxation time. Fig. 3 shows the calculated reflectivity for a sample containing 5 x 1018 carriers/cm3 with the effective mass as determined above and for two values of relaxation time. The fit obtained for the shorter relaxation time is quite good over most of the range. It is interesting to note that a long relaxation time 7 = 1.2X lo-l4 set gives a very good fit on the short-wavelength side of the minimum, but fails completely on the longwavelength side, suggesting that near the frequency of the minimum an additional relaxation process may become effective. The same type of change in relaxation time has been observed -40~ for a sample containing 3.7 X lOI electrons/ cm3. At low frequencies the coulomb field of carriers in an ionic crystal causes a polarization of the lattice to accompany the carrier; the combination of carrier and polarization is called the poZuron. The polaron effective massoa) mp, is given in terms of the bare electron mass m* by: mp =

m*( 1 +a/6)

where

is the coupling constant between the carriers and the lattice. Since the coupling of the electron to the lattice responsible for the polaron takes place through the longitudinal optical mode, the polaron mass will be observed for V ~1. Unfortunately,

194

R.

J.

COLLINS

and

zinc oxide does not offer an easy way of checking this prediction, since an a M 0.5 will produce in effective mass on onlv ~8 per cent increase passing through vl.

D.

A.

KLEINMAN

the points show the measured sample, and the solid curve reflectivity.

reflectivity of the is the calculated

Acknozuledgements-The authors wish to thank Dr. D. G. THOMAS for supplying the crystals used in this work. REFERENCES

WLWELENGTH

FIG.

4. Reflectivity

IN

MICRONS

of ZnO containing electronicm3.

3.7 x 1Ol7

Using the method described to determine the carrier effective mass from the location of the reflectivity minimum, for a sample containing 3.7 x 1017 electronsjcma, a value of m+ = 0.06 m is obtained. The value of the effective mass for V
1. FAN 1~. Y., Rep. Progr. Phys. 19, 107 (1956). HROXOWSKI H. J., Semiconductors (Ed. I3. HASNAY) Reinhold, New York. 2. STRONGJ., Phvs. Rev. 38, 1818 (1931). 3. PARODIM., C.R. Acad. Sci., Paris 204, 1111 (1937). 4. TOLKSDORPFS., Z. Phys. Chem. 132, 161 (1928). 5. KRUEGER F. A. and MEYER H. J. G., Physica’s Gmo. 20, 1149 (1954). 6. SPITZER W. G. and FAN H. Y., Phys. Reo. 106, 8X2 (1957). 7. T~IOMASD. G., J. Phys. Chem. Solids 9, 31 (1958). 8. SEITZ I:., Nlodern Theory of Solids 633, McGraw Hill, New York (1940). 9. Hurso~ A. R., Phys. Rev. 108, 227 (1957). 10. LVDDANER. H., SACHSR. G. and TELLER E., Phys. Rev. 59, 673 (1941). 11. SZIGEWI Is., Trans. Faraday Sot. 45, 155 (1949). 12. THOZIASD. G., J. Phys. Chem. Solids 10,47 (1959). ARNETH R., Natunaissenschaften 45, 282 (1958). 13. LEE, 1’. D., Low F. E. and PINES D., Phys. Rev. 90, 297 (1953).