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Infrared absorption in zinc oxide crystals
J. Phys. Chem. Solids
INFRARED
Pergamon
Press 1959. Vol. 10. pp. 47-51.
ABSORPTION
IN
Printed in Great Britain.
ZINC
OXIDE
CRYSTALS
D. G. THOMAS Bell Telephone
Laboratories,
Inc., Murray
Hill, New Jersey
(Received 6 November 1958)
Abstract-Optical absorption in the 1-12 /I region has been observed in single crystals of ZnO at 300 and 78°K. Lattice bands are observed at 10.1 and 11.5 p at 300°K. It is likely that impurities also cause some absorption. Strong absorption can arise from the presence of free electrons, and by using crystals doped with lithium or indium this absorption has been measured as a function of electron concentration and wavelength. At constant wavelength the absorption is proportional to the electron concentration, and for A < 6 p at constant electron concentration, the absorption varies as As.
1. IIWRODUCTION
zinc oxide is transparent in the visible region. Intrinsic ionization of the lattice occurs in the near ultraviolet region. This paper reports absorption of light in single crystals of ZnO in the 1-12 TV region of the spectrum, which will be shown to be caused by free carriers, lattice vibration bands, and probably by photo-ionization of impurities. TOLKSDORF@) has observed absorption in the 2-22 p region, using powdered ZnO. Her results were interpreted on the basis of an analysis of the possible vibration modes of the lattice. However it is now felt that other absorption mechanisms should have been considered. PURE
2. EXPERIMENTAL
Optical transmission measurements were made on single crystals of ZnO. One crystal was obtained from commercial sources and was optically polished on two sides, but most ‘measurements were made with crystals in the form of thin ribbons, grown in these laboratories. In these the use of a polarizing microscope shows that the sixfold C axis lies in the plane of the ribbon and at an angle of about 5” to the length of theribbon. Unpolarized light was used, since a few experiments with polarized light showed that any anisotropy in the absorption coefficient amounted to less than 5 per cent. The ribbons were about 3x 10-s cm across, 1 cm long, and 5~ 10-s cm thick. The thickness was measured directly with a microscope and was uniform over the crystal to within about 47
20 per cent, the average values being taken for the crystals used. The ribbons were not polished, as their surfaces were almost optically perfect when grown. Transmission at normal incidence was measured by the crystal-in-crystal-out technique in conjunction with a Perkin-Elmer double monochromator equipped with a NaCl prism and a thermocouple as a detector. Slits of O-5 mm were generally used, giving a resolution better than O-13 p. An image of the slit was focussed onto the crystal outside the monochromator. The radiation transmitted by the crystal was then focussed onto the thermocouple. The crystal was mounted on a holder illustrated in Fig. 1. There are two slots which open and close together as an adjusting screw is turned. A mask covers the two slots and over one slot on this mask the crystal is mounted. This holder is mounted so that it can be slid between two stops which are adjusted so that either the crystal slot or the blank slot is in the beam. The slots were of almost identical size, and slight differences were allowed for by measuring the beam intensity through the two slots when neither was covered with a crystal. The adjusting screw allowed the maximum area of the crystal to be exposed. The different masks allowed different crystal lengths to be exposed; either 6- or 8-mm lengths were employed. For measurements near 78°K a metal Dewar with NaCl windows was used. The crystal was stuck to a metal mask which was in contact with copper at 78°K. A radiation shield
D.
48
G.
THOMAS
CRYSTAL SUPPORT 0.002” THICK I
FIG. 1. Crystalholder used at 300°K. with two holes for the light beam surrounded the crystal. A crystal was first observed at room temperature, then put in the Dewar and the transmission remeasured at the same temperature. In this way the system was calibrated and absolute transmission measurements made. Because this method is somewhat indirect, the results are not as precise as those taken at room temperature. The absorption coefficient, u, was determined from the transmission data as follows. FAN@) gives: +
(l-R)2 eax_R2e-a~
(1)
for cc < 103 cm-l. Here T is the transmission, Ia the intensity of the incident beam, I that of the transmitted beam, R the reflectivity from one surface, and x the crystal thickness. This equation allows for reflection from the front and back crystal surfaces as well as for absorption. The reflectivity is given by: R = (n-l)2+@ (a+ l)a+ks’
(2)
where n is the refractive index and k the extinction coefficient. For a < 103 cm-l, k2 < 1 and so, in the wavelength region considered, is negligible in equation (2). MOLLWO~) has determined the refractive index of ZnO as a function of wavelength by using a crystal as a prism and determining the angle of minimum deviation. In the region of interest it is 2.0, and is not altered by the absorption for a < 103 cm-l.
Hence by using equation (l), a may be determined for any T value. It was found useful to calculate a graph of T versus ccx. If a = 0 there are only reflectivity losses and T = 040. The observed T values in the visible spectral region fell within about 5 per cent of this value, indicating quite high perfection of the surfaces. The crystals as grown have, at room temperature, electron concentrations in the range lOIs1017/cc. This may be reduced several orders of magnitude by coating the crystal with a dilute solution of LiOH and heating in air for 1 hr near 800°C. The lithium diffuses into the crystal and in substitutional sites becomes an electron acceptor. The electron concentration is increased by diffusing in indium by the technique already described.@) Equilibrium was achieved at any particular doping temperature, so that the electron concentration was the same throughout the crystal. The optical quality of the crystal surfaces remained almost unchanged before and after doping providing that sufficiently dilute solutions of the doping agents were used. The conductivity and Hall effect of the crystals were measured at room temperature in a holder in which two wires pressed against the opposite edges of the crystal and formed Hall contacts while current passed along the crystal. A calibrated permanent magnet and a Kintel microvoltmeter were used for observation of the Hall effect. The electron concentration n was derived from the relation :
&=3?rL 8 ne
INFRARED
ABSORPTION
IN
ZINC
OXIDE
CRYSTALS
49
where RJJis the Hall coefficient and e the electron charge. However, it is by no means certain that the factor 3+ is applicable over the whole range studied, as the scattering mechanism may not remain constant. 3. RESULTS (a> Room temperature Most results were obtained at room temperature with the crystals surrounded by air. A lithiumdoped crystal virtually free of conduction electrons showed no absorption between the intrinsic absorption near O-4 TVand lattice absorption bands in the infrared with peaks at 10.1 and 11.5 p as shown by crystal 3 in Fig. 2. These lattice bands FIG. 3. Free carrier absorption in ZnO for crystals with different electron concentrations. The wavelengthdependence of the absorption is indicated for the linear sections. Notice that crystal 19 was observed at 78°K as well as at 300°K.
40 0 0
10 12 14 A P, .,c,,“,, FIG. 2. Absorption in ZnO crystals at 3OO’K. r
4
were observed in all crystals in which they were not masked by free-carrier absorption. At longer wavelengths much stronger lattice absorption occurs.@) Fig. 2 also shows the results for crystal 24, which was undoped. Absorption is appreciable and increasing beyond 2 p, but there are small maxima and minima before the lattice bands are reached. Fig. 3 shows the results obtained with four crystals doped to different electron concentrations with indium. The logarithm of the absorption coefficient is plotted against the logarithm of the wavelength, and it can be seen that over an appreciable range’s varies approximately as hs, the maxima and minima no longer being present. The slopes of the straight lines are indicated in the figure. The irregularities at the small a values arise D
from experimental inaccuracies. Lattice absorption can be detected in crystal 10 near 9 CL,but it could not cause the deviation from a straight line which begins near 6 CL.Absorption in crystal 7 containing more electrons could only be followed to 9 p, and it too showed a deviation from linear behavior near 6 I_L.Crystals 15 and 19 could only be observed in the linear region.
FIG. 4. Absorption in ZnO at three different wavelengths as a function of electron concentration. The lines drawn have slopes of 45”. Notice the deviation from the lines at low electron concentration. All crystals were doped to equilibrium values with indium except crystals 24 and R4, which were used as-grown. Crystals 20 and 28 contain precipitated indium (see reference 4).
D. G. THOMAS
50
In Fig. 4 the logarithm of a at various wavelengths is plotted against the logarithm of the electron concentration, for several crystals. The electron mobility of the crystals is also tabulated. The lines drawn all have a slope of 45”, which is expected if a is proportional to 12. Most of the points fall near these lines except at low electron concentrations, where the absorption is higher than expected. It is, however, in this region where irregularities show in the curves and the Asrelation is not obeyed. (b) Low temperature Crystal 19 was studied near 78°K with results shown in Fig. 3. The behavior is almost identical
0
FIG. 5.
CRYSTAL 33 THICKNESS=6xlO-3CM 3000K 6.3qOHM
CM)-’
Percentage transmission versus wavelength at 300 and 78°K. Crystal No. 33, undoped.
to that at room temperature. At this high electron concentration the conductivity is practically independent of temperature (oaiJo’K = 185(&J cm)-1; o?s’K = 175(&Z? cm)-1) because the shallow donor states merge into the conduction band, and the mobility does not change as the temperature is lowered. Fig. 5 illustrates the behavior of an undoped crystal at room temperature and at 78°K. Because the absolute accuracy at 78°K is not very high, the percentage transmission is plotted against wavelength over part of the observed wavelength interval. At 300°K the conductivity is 3 -2 (a cm)-’ and at 78°K 0.06 (Q cm)-l, SO that cooling lowered
the electron concentration by nearly a factor of 100. However the absorption is stronger at low than at high temperature. Clearly, therefore, there is another mechanism of absorption besides that provided by free carriers. Probably this consists of excitation of electrons from states within a few tenths of an electron volt from the conduction band. There may be a variety of such states arising from various imperfections in the crystal. This explanation is in accord with the presence of irregularities in the absorption, which become somewhat more marked at low temperatures. However, further work is necessary to confirm this view. The irregularities cannot be explained in terms of interference effects. It may be pointed out that absorption due to excitation of bound electrons which is stronger than that due to free electrons has been observed before in n-type silicon.(s) 4. DISCUSSION In the wavelength range examined, two lattice bands were observed as noted above. These are probably subsidiary bands of the main restrahlen peak near 24 p.(b) The irregularities in the absorption of crystals containing 101s-1Or7 electrons/ems are not due to lattice bands, but are probably due to excitation of electrons obt of bound states. If the crystal is doped with lithium, no absorption occurs, as shown in Fig. 2. Lithium introduces a level probably below the center of the gap0 and so removes electrons not only from the conduction band but also from states near it. Excitation from lithium states to the conduction band has not yet been observed. It may be concluded that the absorption observed in crystals with an electron concentration greater than about 5 x 1017/emsis to be ascribed to free electrons. At lower electron concentrations lattice bands or bound states play a prominent role in the absorption. The characteristics of the freecarrier absorption in the range studied may be summarized as follows: (a) The absorption coefficient, a, varies approximately as X3 for wavelengths shorter than about 5 CL; at longer wavelengths the dependence becomes less steep. (b) At a fixed wavelength a varies directly with the electron concentration. (c) Although the Hall mobility values show considerable variation among the do, ::i _sta s
INFRARED
ABSORPTION
there is no apparent correlation of this variation with the a values. ARNETH,@) in a recent paper concerned with infrared absorption in ZnO, has reported that at 4 p the absorption varies linearly with conductivity, in approximate agreement with conclusion (b) above. Lattice absorption peaks at 10.1 and 11.5 p were also observed. The electric field of electromagnetic radiation causes free electrons to accelerate. Absorption occurs when energy is lost by collisions between the electrons and irregularities in the host lattice. If the frequency of the oscillating electric vector, w/27r, is greater than l/r, the reciprocal of the relaxation time for electron collisions, then the loss of energy becomes less efficient with increasing frequency, and so a will decrease with increasing frequency. FAN@) has given an account of free carrier absorption. If (~7)s > 1 and if the electrons are scattered by ionized impurity centers, a is expected to vary as P, whereas if lattice vibrations predominate in the scattering, a should vary as hl.5. In ZnO for h < 6 p a A3 relation is obeyed, which is suggestive of impurity scattering, and at the impurity concentrations used previous work@) has shown that impurities do affect the Hall mobility. However, the impuri~-sca~ering theory is clearly not adequate, for if it were, a at a fixed wavelength should depend on ns, where n is the electron concentration, since for every electron there is an ionized impurity center. Further theoretical work is necessary to account for the results, and this may take into account the screening of the scattering centers by the free electrons.
IN ZINC
OXIDE
CRYSTALS
51
If (CUT)~is not greater than unity, different relations are expected. T may be estimated from the mobility by using the expression:(s)
p=L
cmz/V set
m”300 where m* is the effective electron mass. Using the approximate values p = 100 ems/V set and m* m O.O7m,(s) we find T RC4x 10-1s sec. The applicability of this relaxation time obtained from d.c. measurements of mobility may be questioned when frequencies greater than the testnzhlen frequency are employed; nevertheless it is interesting to note that, if this figure is accepted, at 6 p we m 1.2. It is perhaps not surprising, therefore, that in this region the ~velength-de~ndence of a changes. are due to E. A, SADOWSKI for doping the crystals and measuring the Hall effect. The author benefited from Conversations with R. J.
Acknowledgements-Thanks
COLLINS.
REFBRENCES 1. Tommow S., Z. #hp. Ch. 132, 161 (1928). 2. FANH. Y., Rep. P~ogr. Phys. 19,107 (1956).
3. MOLLWO E., Reichsber. Phys. 1,1 (1944). 4. THOMASD. G., J. Pkys. Chem. Solids 9, 31 (1959). 5. COLLINSR. J. To be published. 6. SPITZER W. G. and FAN H. Y., Phys. Rew. 108,268 (1957). 7. LANDER J. J. To be published. 8. ARNZIB R., ~o~s~c~fr~ 45, 282 (1958); see also BOGNER G. and MOLLWO E., _T. Phys. Chem. So&is 6, 136 (1958). 9. SHOCKLEYW., Electrons and Holes in Semiconductors. D. Van Nostrand, New York (1950).
INFRARED
Pergamon
Press 1959. Vol. 10. pp. 47-51.
ABSORPTION
IN
Printed in Great Britain.
ZINC
OXIDE
CRYSTALS
D. G. THOMAS Bell Telephone
Laboratories,
Inc., Murray
Hill, New Jersey
(Received 6 November 1958)
Abstract-Optical absorption in the 1-12 /I region has been observed in single crystals of ZnO at 300 and 78°K. Lattice bands are observed at 10.1 and 11.5 p at 300°K. It is likely that impurities also cause some absorption. Strong absorption can arise from the presence of free electrons, and by using crystals doped with lithium or indium this absorption has been measured as a function of electron concentration and wavelength. At constant wavelength the absorption is proportional to the electron concentration, and for A < 6 p at constant electron concentration, the absorption varies as As.
1. IIWRODUCTION
zinc oxide is transparent in the visible region. Intrinsic ionization of the lattice occurs in the near ultraviolet region. This paper reports absorption of light in single crystals of ZnO in the 1-12 TV region of the spectrum, which will be shown to be caused by free carriers, lattice vibration bands, and probably by photo-ionization of impurities. TOLKSDORF@) has observed absorption in the 2-22 p region, using powdered ZnO. Her results were interpreted on the basis of an analysis of the possible vibration modes of the lattice. However it is now felt that other absorption mechanisms should have been considered. PURE
2. EXPERIMENTAL
Optical transmission measurements were made on single crystals of ZnO. One crystal was obtained from commercial sources and was optically polished on two sides, but most ‘measurements were made with crystals in the form of thin ribbons, grown in these laboratories. In these the use of a polarizing microscope shows that the sixfold C axis lies in the plane of the ribbon and at an angle of about 5” to the length of theribbon. Unpolarized light was used, since a few experiments with polarized light showed that any anisotropy in the absorption coefficient amounted to less than 5 per cent. The ribbons were about 3x 10-s cm across, 1 cm long, and 5~ 10-s cm thick. The thickness was measured directly with a microscope and was uniform over the crystal to within about 47
20 per cent, the average values being taken for the crystals used. The ribbons were not polished, as their surfaces were almost optically perfect when grown. Transmission at normal incidence was measured by the crystal-in-crystal-out technique in conjunction with a Perkin-Elmer double monochromator equipped with a NaCl prism and a thermocouple as a detector. Slits of O-5 mm were generally used, giving a resolution better than O-13 p. An image of the slit was focussed onto the crystal outside the monochromator. The radiation transmitted by the crystal was then focussed onto the thermocouple. The crystal was mounted on a holder illustrated in Fig. 1. There are two slots which open and close together as an adjusting screw is turned. A mask covers the two slots and over one slot on this mask the crystal is mounted. This holder is mounted so that it can be slid between two stops which are adjusted so that either the crystal slot or the blank slot is in the beam. The slots were of almost identical size, and slight differences were allowed for by measuring the beam intensity through the two slots when neither was covered with a crystal. The adjusting screw allowed the maximum area of the crystal to be exposed. The different masks allowed different crystal lengths to be exposed; either 6- or 8-mm lengths were employed. For measurements near 78°K a metal Dewar with NaCl windows was used. The crystal was stuck to a metal mask which was in contact with copper at 78°K. A radiation shield
D.
48
G.
THOMAS
CRYSTAL SUPPORT 0.002” THICK I
FIG. 1. Crystalholder used at 300°K. with two holes for the light beam surrounded the crystal. A crystal was first observed at room temperature, then put in the Dewar and the transmission remeasured at the same temperature. In this way the system was calibrated and absolute transmission measurements made. Because this method is somewhat indirect, the results are not as precise as those taken at room temperature. The absorption coefficient, u, was determined from the transmission data as follows. FAN@) gives: +
(l-R)2 eax_R2e-a~
(1)
for cc < 103 cm-l. Here T is the transmission, Ia the intensity of the incident beam, I that of the transmitted beam, R the reflectivity from one surface, and x the crystal thickness. This equation allows for reflection from the front and back crystal surfaces as well as for absorption. The reflectivity is given by: R = (n-l)2+@ (a+ l)a+ks’
(2)
where n is the refractive index and k the extinction coefficient. For a < 103 cm-l, k2 < 1 and so, in the wavelength region considered, is negligible in equation (2). MOLLWO~) has determined the refractive index of ZnO as a function of wavelength by using a crystal as a prism and determining the angle of minimum deviation. In the region of interest it is 2.0, and is not altered by the absorption for a < 103 cm-l.
Hence by using equation (l), a may be determined for any T value. It was found useful to calculate a graph of T versus ccx. If a = 0 there are only reflectivity losses and T = 040. The observed T values in the visible spectral region fell within about 5 per cent of this value, indicating quite high perfection of the surfaces. The crystals as grown have, at room temperature, electron concentrations in the range lOIs1017/cc. This may be reduced several orders of magnitude by coating the crystal with a dilute solution of LiOH and heating in air for 1 hr near 800°C. The lithium diffuses into the crystal and in substitutional sites becomes an electron acceptor. The electron concentration is increased by diffusing in indium by the technique already described.@) Equilibrium was achieved at any particular doping temperature, so that the electron concentration was the same throughout the crystal. The optical quality of the crystal surfaces remained almost unchanged before and after doping providing that sufficiently dilute solutions of the doping agents were used. The conductivity and Hall effect of the crystals were measured at room temperature in a holder in which two wires pressed against the opposite edges of the crystal and formed Hall contacts while current passed along the crystal. A calibrated permanent magnet and a Kintel microvoltmeter were used for observation of the Hall effect. The electron concentration n was derived from the relation :
&=3?rL 8 ne
INFRARED
ABSORPTION
IN
ZINC
OXIDE
CRYSTALS
49
where RJJis the Hall coefficient and e the electron charge. However, it is by no means certain that the factor 3+ is applicable over the whole range studied, as the scattering mechanism may not remain constant. 3. RESULTS (a> Room temperature Most results were obtained at room temperature with the crystals surrounded by air. A lithiumdoped crystal virtually free of conduction electrons showed no absorption between the intrinsic absorption near O-4 TVand lattice absorption bands in the infrared with peaks at 10.1 and 11.5 p as shown by crystal 3 in Fig. 2. These lattice bands FIG. 3. Free carrier absorption in ZnO for crystals with different electron concentrations. The wavelengthdependence of the absorption is indicated for the linear sections. Notice that crystal 19 was observed at 78°K as well as at 300°K.
40 0 0
10 12 14 A P, .,c,,“,, FIG. 2. Absorption in ZnO crystals at 3OO’K. r
4
were observed in all crystals in which they were not masked by free-carrier absorption. At longer wavelengths much stronger lattice absorption occurs.@) Fig. 2 also shows the results for crystal 24, which was undoped. Absorption is appreciable and increasing beyond 2 p, but there are small maxima and minima before the lattice bands are reached. Fig. 3 shows the results obtained with four crystals doped to different electron concentrations with indium. The logarithm of the absorption coefficient is plotted against the logarithm of the wavelength, and it can be seen that over an appreciable range’s varies approximately as hs, the maxima and minima no longer being present. The slopes of the straight lines are indicated in the figure. The irregularities at the small a values arise D
from experimental inaccuracies. Lattice absorption can be detected in crystal 10 near 9 CL,but it could not cause the deviation from a straight line which begins near 6 CL.Absorption in crystal 7 containing more electrons could only be followed to 9 p, and it too showed a deviation from linear behavior near 6 I_L.Crystals 15 and 19 could only be observed in the linear region.
FIG. 4. Absorption in ZnO at three different wavelengths as a function of electron concentration. The lines drawn have slopes of 45”. Notice the deviation from the lines at low electron concentration. All crystals were doped to equilibrium values with indium except crystals 24 and R4, which were used as-grown. Crystals 20 and 28 contain precipitated indium (see reference 4).
D. G. THOMAS
50
In Fig. 4 the logarithm of a at various wavelengths is plotted against the logarithm of the electron concentration, for several crystals. The electron mobility of the crystals is also tabulated. The lines drawn all have a slope of 45”, which is expected if a is proportional to 12. Most of the points fall near these lines except at low electron concentrations, where the absorption is higher than expected. It is, however, in this region where irregularities show in the curves and the Asrelation is not obeyed. (b) Low temperature Crystal 19 was studied near 78°K with results shown in Fig. 3. The behavior is almost identical
0
FIG. 5.
CRYSTAL 33 THICKNESS=6xlO-3CM 3000K 6.3qOHM
CM)-’
Percentage transmission versus wavelength at 300 and 78°K. Crystal No. 33, undoped.
to that at room temperature. At this high electron concentration the conductivity is practically independent of temperature (oaiJo’K = 185(&J cm)-1; o?s’K = 175(&Z? cm)-1) because the shallow donor states merge into the conduction band, and the mobility does not change as the temperature is lowered. Fig. 5 illustrates the behavior of an undoped crystal at room temperature and at 78°K. Because the absolute accuracy at 78°K is not very high, the percentage transmission is plotted against wavelength over part of the observed wavelength interval. At 300°K the conductivity is 3 -2 (a cm)-’ and at 78°K 0.06 (Q cm)-l, SO that cooling lowered
the electron concentration by nearly a factor of 100. However the absorption is stronger at low than at high temperature. Clearly, therefore, there is another mechanism of absorption besides that provided by free carriers. Probably this consists of excitation of electrons from states within a few tenths of an electron volt from the conduction band. There may be a variety of such states arising from various imperfections in the crystal. This explanation is in accord with the presence of irregularities in the absorption, which become somewhat more marked at low temperatures. However, further work is necessary to confirm this view. The irregularities cannot be explained in terms of interference effects. It may be pointed out that absorption due to excitation of bound electrons which is stronger than that due to free electrons has been observed before in n-type silicon.(s) 4. DISCUSSION In the wavelength range examined, two lattice bands were observed as noted above. These are probably subsidiary bands of the main restrahlen peak near 24 p.(b) The irregularities in the absorption of crystals containing 101s-1Or7 electrons/ems are not due to lattice bands, but are probably due to excitation of electrons obt of bound states. If the crystal is doped with lithium, no absorption occurs, as shown in Fig. 2. Lithium introduces a level probably below the center of the gap0 and so removes electrons not only from the conduction band but also from states near it. Excitation from lithium states to the conduction band has not yet been observed. It may be concluded that the absorption observed in crystals with an electron concentration greater than about 5 x 1017/emsis to be ascribed to free electrons. At lower electron concentrations lattice bands or bound states play a prominent role in the absorption. The characteristics of the freecarrier absorption in the range studied may be summarized as follows: (a) The absorption coefficient, a, varies approximately as X3 for wavelengths shorter than about 5 CL; at longer wavelengths the dependence becomes less steep. (b) At a fixed wavelength a varies directly with the electron concentration. (c) Although the Hall mobility values show considerable variation among the do, ::i _sta s
INFRARED
ABSORPTION
there is no apparent correlation of this variation with the a values. ARNETH,@) in a recent paper concerned with infrared absorption in ZnO, has reported that at 4 p the absorption varies linearly with conductivity, in approximate agreement with conclusion (b) above. Lattice absorption peaks at 10.1 and 11.5 p were also observed. The electric field of electromagnetic radiation causes free electrons to accelerate. Absorption occurs when energy is lost by collisions between the electrons and irregularities in the host lattice. If the frequency of the oscillating electric vector, w/27r, is greater than l/r, the reciprocal of the relaxation time for electron collisions, then the loss of energy becomes less efficient with increasing frequency, and so a will decrease with increasing frequency. FAN@) has given an account of free carrier absorption. If (~7)s > 1 and if the electrons are scattered by ionized impurity centers, a is expected to vary as P, whereas if lattice vibrations predominate in the scattering, a should vary as hl.5. In ZnO for h < 6 p a A3 relation is obeyed, which is suggestive of impurity scattering, and at the impurity concentrations used previous work@) has shown that impurities do affect the Hall mobility. However, the impuri~-sca~ering theory is clearly not adequate, for if it were, a at a fixed wavelength should depend on ns, where n is the electron concentration, since for every electron there is an ionized impurity center. Further theoretical work is necessary to account for the results, and this may take into account the screening of the scattering centers by the free electrons.
IN ZINC
OXIDE
CRYSTALS
51
If (CUT)~is not greater than unity, different relations are expected. T may be estimated from the mobility by using the expression:(s)
p=L
cmz/V set
m”300 where m* is the effective electron mass. Using the approximate values p = 100 ems/V set and m* m O.O7m,(s) we find T RC4x 10-1s sec. The applicability of this relaxation time obtained from d.c. measurements of mobility may be questioned when frequencies greater than the testnzhlen frequency are employed; nevertheless it is interesting to note that, if this figure is accepted, at 6 p we m 1.2. It is perhaps not surprising, therefore, that in this region the ~velength-de~ndence of a changes. are due to E. A, SADOWSKI for doping the crystals and measuring the Hall effect. The author benefited from Conversations with R. J.
Acknowledgements-Thanks
COLLINS.
REFBRENCES 1. Tommow S., Z. #hp. Ch. 132, 161 (1928). 2. FANH. Y., Rep. P~ogr. Phys. 19,107 (1956).
3. MOLLWO E., Reichsber. Phys. 1,1 (1944). 4. THOMASD. G., J. Pkys. Chem. Solids 9, 31 (1959). 5. COLLINSR. J. To be published. 6. SPITZER W. G. and FAN H. Y., Phys. Rew. 108,268 (1957). 7. LANDER J. J. To be published. 8. ARNZIB R., ~o~s~c~fr~ 45, 282 (1958); see also BOGNER G. and MOLLWO E., _T. Phys. Chem. So&is 6, 136 (1958). 9. SHOCKLEYW., Electrons and Holes in Semiconductors. D. Van Nostrand, New York (1950).