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The optical absorption edge in layer structures
J. Phys.
Chem. Solids
Pergamon
Press 1964. Vol. 25, pp. 1427-1433.
Printed in Great Britain.
THE OPTICAL ABSORPTION
EDGE
IN LAYER STRUCTURES J. L. BRBBNER Cyanamid European
Research
Institute,
Cologny-Geneva
(Received 15 May 1964)
Abstract-The layer structures GaSe and GaS are composed of isomorphic, 4-fold atomic layers, and differ only in the manner in which these layers are stacked. The Se and S atoms occupy equivalent sites and it is possible to obtain crystals with a continuous range of composition between pure GaSe and pure Gas. The optical absorption of a series of mixed crystals of the system GaSezS1-Z has been measured at 12” K near the absorption edge. The line structure associated with the edge, generally held to be excitonic in nature, is present for all compositions but shows a rapid fall in intensity with increasing sulphur content. Absorption and reflectivity measurements on the pure crystals indicate that the edge arises from a direct transition in GaSe and from an indirect transition in Gas. The small thickness of the individual layers significantly affects the behaviour of excitons in such structures and the strong carrier-phonon interaction gives rise to polarization of the lattice by optically excited carriers. It is suggested that the observed lines in GaSe, GaS and the absence of hydrogenlike behaviour may then result. Absorption and photoconductivity show a marked dependence on the orientation of the polarization of the incident illumination with respect to the c-axis of the crystals. A model is proposed to account for this dependence.
1. INTRODUCTION
of GaSe and GaS are composed of isomorphous four-fold atomic layers and differ only in the manner in which these layers are stacked.(l) The Se and S atoms occupy equivalent sites and it is possible to obtain crystals with a continuous range of composition between pure GaSe and pure Gas. The bonding within a multiple layer is predominantly covalent with a small ionic contribution. Because the bonding between layers is of the van der Waals type the optical properties@-@ are largely determined by the structure of the individual four-fold layers. The optical absorption in the neighbourhood of the absorption edge has been determined for the system GaSe&_, where 0 < x < 1. Photoconductivity measurements on a Zn-doped sample of GaSe between 300” and 90°K have been made. A discussion of the nature of the absorption and photoconductive maxima associated with the absorption edge is given, and the influence of the characteristic layer structure on the optical absorpTHE
CRYSTAL structures
tion considered for different directions of polarization of the incident radiation. 2. EXPBRIMBN’TAL PROCBDURB The crystals were grown by transport reaction(s) and the composition of the mixed crystals was taken to be that of the starting material. Samples of the appropriate thickness, between 10 and 65 CL, were obtained by cleaving thicker flakes along the plane of the layers, i.e. normally to the c-axis. They were mounted in a cryostat equipped with suitable windows and clamped between two copper plates provided with apertures to allow for the passage of the radiation. The sample temperature was taken to be that of the plates. For absorption and photoconductivity measurements with unpolarized radiation the samples were illuminated at normal incidence. For measurements with polarized radiation the samples were illuminated at an angle of incidence of 45”. The radiation was polarized either perpendicular to or parallel to the plane of incidence. The reflectivity of freshly cleaved GaSe and
1427
J. L. BREBNER
1428
FIG. 1. The absorption coefficient of a series of mixed crystals Ga at 12’K, where x has the following values (1) 0.95; (2) 0.90; (3) 0.80; (4) 0.60; (5) 0.40; (6) 0.20; (7) 0.00.
SeZS1-z
GaS was measured for unpolarized radiation at an angle of incidence of 7”. Photoconductivity measurements were made on a sample of GaSe doped with 0.5 per cent Zn to increase the conductivity. Nevertheless as the resistances at low temperatures were of the order of 10X it was necessary to use a specially designed 4-probe high resistance bridge.(lO) The photocurrent and the potential drop across the sample as a function of the wavelength of the radiation was registered directly on a recorder. In all the measurements the illumination was provided by a Perkin Elmer 112 U spectrometer equipped with a silica prism, and Polaroid sheet was employed as polarizer. The mean resolution was 0.005 eV for absorption and O-050 eV for reflectivity and photoconductivity measurements.
3. EXPERIMENTAL RESULTS The logarithm of the inverse transmission of the series of mixed crystals at 12°K is shown as a function of energy in Fig. 1. A reduction in intensity of the main absorption line(7-s) with increasing sulphur content is observed except in Ga Se0.s S0.s whose anomalous behaviour may be attributed to a poor quality sample. A second broad weak line is found in pure GaSe and in the Se-rich
crystals up to GaSeo.db The absorption level at energies just tibove that of the line is the same for all samples and begins to rise more rapidly as the sulphur content increases. At energies less than that of the main line, the absorption falls rapidly to a very low level for low concentrations of sulphur, but an absorption tail appears as the concentration is increased beyond 40 per cent. Figure 2 shows the energy at which the maximum absorption in the line occurs as a function of the sulphur content. Figure 3 shows the absorption of a 65-p thick sample of GaS between 300°K and 77°K. The absorption tail extends over several tenths of an electron volt and there is a sudden increase immediately after the onset of absorption. The slope of the main portion of the tail decreases as the temperature decreases. The reflectivity of GaSe and GaS at 300°K and 77°K is shown in Fig. 4. The absolute accuracy is poor due to the difficulty of obtaining sufficiently plane samples. Below the edge the reflectivities are in reasonable agreement with those derived from the refractive indices in this region of GaSe and Gas, which are 2.7 and 3-O respectively. These values were obtained from multiple interference in samples of known thickness. The reflectivity of GaSe shows an abrupt increase at energies corresponding to the position of the main absorption
THE
OPTICAL
ABSORPTION
EDGE
line whereas that of GaS shows a smaller more gradual rise. Figure 5 shows the photoconductive behaviour of a Zn-doped sample of GaSe at 170°K. The resistance change as a function of the wavelength for normally incident unpolarized radiation was similar to that for radiation incident at 45” and
IN
LAYER
STRUCTURES
1429
polarized perpendicular to the plane of incidence. A photocurrent maximum was observed at temperatures between 250°K and 90”K, corresponding to the position of the main absorption line. Due to the very high sample resistance and the resultant extremely long response time it was not possible to obtain reproducible results below 90°K. For radiation incident at 45” and polarized parallel to the plane of incidence the photocurrent maximum occuring at the same energies as before was strongly enhanced, as compared with the photocurrent at higher energies. Under similar conditions of illumination the room temperature absorption of GaSe in the line and at higher energies increased by an order of magnitude for polarization parallel to the incident plane over that for polarization normal to the plane. The increase became greater at higher angles of incidence. The same effect was found in Gas. 4. DISCUSSION
20
40
60
GaSe
80
Gas
At%
-
FIG. 2. Position of the energy of the maximum in the main absorption line of the crystals GaSezS1_z as a function of the sulphur content.
The layer structure of compounds such as GaSe, Gas, Moss, MoTez, WSes, HgIs and PbIs gives rise to marked anisotropy in many of their properties. Free charge carriers occuring within one layer may be considered as being confined to that layer. In what follows the potential in which the carriers move is taken to be the sum of two parts, one dependent on z, the coordinate normal to the
FIG. 3. The square root of the absorption coefficient of pure GaS between 300°K and 78°K as a function of energy.
1430
J.
L.
BREBNER
layers, the other dependent on x and y. The electron wave functions are written as \rp&,
Y, z> = SP+,
Y)%&+
(1)
The S-functions are solutions of a Schrijdinger equation containing that part of the potential periodic in x and y. The 2, are wave functions of an electron in a one-dimensional potential well. The transition probability for an electron being excited from a state with wave function $t to a state with wave function #f is governed by the square of the matrix element S?VrHradYg dxdYdz
Ga S
where H rad
=
-
ieh -A ??u
*grad
A being the vector potential characterizing the radiation. Experiment shows that the absorption increases as the component of A along the c-axis increases and it is assumed to be a maximum when A is parallel to this axis. The transition probability is then proportional to FIG. 4. The reflectivity of pure GaSe and pure GaS at 300” K and 77” K as a function of energy.
PI, = j” S;‘&,,dxdY
jz;+
FIG. 5. The photocurrent of a sample of GaSe doped with 0.5 per cent of Zn at 170°K as a function of the energy of the exciting radiation incident at 45’. (1) Radiation polarized normal to the incident plane; (2) Radiation polarized parallel to the incident plane.
%I,
dz.
(3)
THE
OPTICAL
ABSORPTION
EDGE
From the definition of 2 in order that P,, should not be zero Z,, # Z,,. Z,, is the ground state 2s of the well and 2, the first excited state 21. The valence band thus contains a series of S-functions associated with 2s and the conduction band the same S-functions associated with 21. The situation for both GaSe and GaS is represented schematically in Fig. 6. The transition 1 represents a transition whereby an dectron is excited from the top of the series A of overlapping S-bands associated with 2s to the bottom of the series A associated with 21. The transition 2 between series A and B of the S-bands, both associated with 20, occurs at a higher energy than transition 1. In a crystal such as Moss where the layer thickness is smaller than in either GaSe or Gas, the energy separation of 2s and 21 is
fJ &(a)
D (Ef(b)
Fro 6. Schematic representation of the density of states of the band system of GaSe and Gas. The S-bands A and B on the left hand side of the figure are associated with 20, the same bands on the right hand side with 2~. In transition 1 both the S- and Z-functions change. In transition 2 only the S-functions change.
IN
LAYER
STRUCTURES
1431
larger and hence transition 2 may occur at a smaller energy than transition 1. Since the series A is made up of several S-bands then in the transition 1 the S-functions must change as well as the Z-functions. Thus in equation (3) S,, is taken to be different from SPf. Since these functions are orthogonal the first integral in equation (3) might be expected to be zero for direct transitions but this is so only if we neglect the finite value of the radiation wavevector. Therefore the integral is small but nonzero. For radiation polarized perpendicular to the c-axis, along x say, we have
From the definitions of S and .Z it follows that the term involving the gradient is larger for Pu than for P_L. Furthermore the orthogonality of the Z-functions is likely to be better than that of the S-functions. Therefore it is seen that PII > PJ_ in accord with experiment. From the behaviour of the absorption and reflectivity the transition is seen to be direct in GaSe and, at energies below the absorption line, indirect in Gas. If the rise in absorption close to the onset in Fig. 3 is interpreted as being due to the emission of only one type of phonon the corresponding temperature is 420%. The transition at the line and at higher energies is direct in both compounds for the following reasons. If the line is regarded as arising from the formation of excitons then the transition must be direct otherwise a step-function rather than a line would be found. If on the other hand the line is regarded as being due to the excitation of electrons to impurity or self-trapped states as suggested below, then a twophonon process must be invoked if the transition is not direct. This would give rise to a much smaller value of the absorption than is found. Assuming the transition probability to be independent of the energy the absorption coefficient is proportional to the density of states of the valence and conduction bands. In our case, the density of states depends on the energy through the S-bands only. If these S-bands are assumed to be isotropic in x and y and quadratic in k, it is found that an indirect transition gives a linear relationship between the absorption and the photon energy whereas a direct transition leads to a step-function,
1132
J. L.
BREBNER
i.e. &&.ect independent &ndirect
CL(hv-
of hv, step-function
AE* &a).
These conclusions refer to allowed transitions. When the transitions are forbidden we have
Experiment shows that for GaS a plot of the square root of the absorption coefficient vs. energy yields the nearest approach to a straight line (Fig. 3), and therefore the transition is forbidden. The change in absorption with composition in the mixed crystal series indicates that the mechanism producing the line absorption is the same in GaSe and Gas. NIKITINE and GROS&*) have ascribed the lines in GaSe to the formation of excitons but they were unable to fit a hydrogenlike series to them. Following the proposed model it is easy to see that since the carriers are regarded as remaining within a layer the excitons will be profoundly modified for orbital diameters greater than the layer thickness, the first orbital being the one least perturbed. Electrical measurements, reported elsewhere@lf have shown that the effective mass of the holes is close to unity and that of the electrons is likely to be large. In such a case the excitons would have a reduced mass close to unity. Taking a value of the dielectric constant of 7.6, derived from the infrared refractive index of GaSe of Z-76, the first exitonic state should lie 0.24 eV from the absorption edge. In fact the energy separation of the main line from the edge, whiIe difficult to determine accurately is not greater than 0.07 eV. In this respect GaSe resembles PbIz where the first prominent line is found closer to the edge than simple exciton theory predicts. It is somewhat surprising that the energies of the remaining Iines in PbIs fit a hydrogen-like series. It has been suggested that electrons excited to the conduction band of GaSe are self-trapped.(ll) The main absorption line may then be attributed to the excitation of electrons from the valence band to these self-trapped states. As the electrons have a long lifetime in these states the resultant holes remain free giving rise to the p-type photo~ondu~tivi~ found by &3BE and Lm.f5) The difference in the line strengths in GaSe and GaS
is thought to arise from the difference in the coupling constants between the carriers and the lattice in the two compounds. Previous work(‘) on the photoconductivity of one sample of a GaSe crystal grown by pulling from the melt suggested that there is a photoconductive peak at the position of the main absorption line at 4~2°K. Photoconductive maxima have been observed corresponding to absorption lines in crystals such as CdS where the formation of excitons is generally accepted. In order to account for the maxima the excitons are supposed to move to ionization centres or to ionize through exciton-exciton collisions, since at low temperatures the probability of ionization by phonons is negligible. It has been shownQ1) that the electron mobility in GaSe is likely to be small, and is negligible in the case where the electrons are self-trapped. Since the exciton mobility cannot be much different from that of the less mobile carrier it is difficult to account for the low temperature photoconductive maximum in Case in terms of processes involving exciton migration. On the other hand if the electrons are excited to selftrapped states the maximum can be attributed to the resulting free holes. The present experiments show that between 250°K and 90°K the centre of the broad photoconductive maximum in GaSe lies at the energy of the main absorption line. Since the phonon energies available at 90°K are much less than the separation of the line from the absorption edge, it is unlikely that the mechanism giving rise to the rn~~rnurn is that of the creation of excitons subsequently ionized by lattice vibrations. A peculiar feature of the photoconductivity in GaSe is the considerable enhancement of the max~um when radiation incident at 45” is polarized in the plane of incidence, without an accompanying increase at energies greater than that of the maximum. On the other hand the absorption in the main Iine and at higher energies both increase for similar conditions of illumination and it is not clear why the photoconductivity shows different behaviour. The increase in the number of electron-hole pairs due to increased absorption at energies above the photoconductive maximum may be offset by the increase in the probability of direct recombination, found in general as the absorption increases. Such a recombination is less liiely to occur if the electrons are excited to self-trapped states.
THE
OPTICAL
ABSORPTION
This paper represents an attempt to introduce in a simple fashion the effect on the optical properties of semiconductors such as GaSe, GaS etc. of their markedly anisotropic crystal structures. Two main points are made: firstly it is shown that many properties may be explained assuming the electrons to move in a potential made up of the sum of two terms which vary independently of one another. This brings out the two-dimensional aspect of the structures. Secondly an alternative explanation to that of excitons is proposed to account for the lines associated with the absorption edge in GaSe, and GaS which may also be valid for the lines found in HgIs, PbIs, Moss, MoTes and WSes.(rsJa)
Acknowledgements-The author would like to thank Mr. H. BOELSTEFUJfor providing the crystals and Mr. R. FIVAZ and Dr. E. MOOSERfor many helpful discussions.
EDGE
IN
LAYER
STRUCTURES
1433
REFERENCES 1. BA~INSKI Z. S., DOVE D. B. and MOOSER E., Helv. Phys. Acta 34, 374 (1961). R. I., Sow. Phys. 2. RWKIN S. M. and KHANSEVAROV Tech. Phys. 1, 2688 (1956). 3. FIELDING P., F&CHER G. and MOOSER E., J. Phys. C?aem. Solids 8.434 11959). 4. GROSS E. F., N&IKO; B. V., RAZBIRIN B. S. and SUSLINA L. G., Opt. and Spec. 6, 364 (1959). 5. BUBE R. H. and LIND E. L., Phys. Rev. 115, 1159 (1959). 6. BUBE R. H. and LIND E. L., Phys. Rev. 119, 1535 (1960). 7. BREBNER J. L. and FISCHER G., Int. Co& Physics of Semiconductors, Exeter, p. 760, Institute of Physics and the Physical Society (1962). 8. NIKITINE S., NITSCHE R., SIFSKIND M. and VOCT J., J. Chim. Phys. 60, 667 (1963). 9. BOELSTERLI H. U. and MOOSER E., Helw. Phys. Acta 35, 539 (1962). 10. FIVAZ R., Helv. Phys. Acta 36. 1052 (1963). 11. FIVAZ R. and MOUSER E., to-be published in the Proc. Int. Conf.. Phvsics of Semiconductors, Paris (1964). 12. FRINDT R. F. and YOFFE A. D., Proc. Roy. Sot. 273, 69 (1963). 13. FRINDT R. F., J. Phys. Chem. Solids 24,1107 (1963).
Chem. Solids
Pergamon
Press 1964. Vol. 25, pp. 1427-1433.
Printed in Great Britain.
THE OPTICAL ABSORPTION
EDGE
IN LAYER STRUCTURES J. L. BRBBNER Cyanamid European
Research
Institute,
Cologny-Geneva
(Received 15 May 1964)
Abstract-The layer structures GaSe and GaS are composed of isomorphic, 4-fold atomic layers, and differ only in the manner in which these layers are stacked. The Se and S atoms occupy equivalent sites and it is possible to obtain crystals with a continuous range of composition between pure GaSe and pure Gas. The optical absorption of a series of mixed crystals of the system GaSezS1-Z has been measured at 12” K near the absorption edge. The line structure associated with the edge, generally held to be excitonic in nature, is present for all compositions but shows a rapid fall in intensity with increasing sulphur content. Absorption and reflectivity measurements on the pure crystals indicate that the edge arises from a direct transition in GaSe and from an indirect transition in Gas. The small thickness of the individual layers significantly affects the behaviour of excitons in such structures and the strong carrier-phonon interaction gives rise to polarization of the lattice by optically excited carriers. It is suggested that the observed lines in GaSe, GaS and the absence of hydrogenlike behaviour may then result. Absorption and photoconductivity show a marked dependence on the orientation of the polarization of the incident illumination with respect to the c-axis of the crystals. A model is proposed to account for this dependence.
1. INTRODUCTION
of GaSe and GaS are composed of isomorphous four-fold atomic layers and differ only in the manner in which these layers are stacked.(l) The Se and S atoms occupy equivalent sites and it is possible to obtain crystals with a continuous range of composition between pure GaSe and pure Gas. The bonding within a multiple layer is predominantly covalent with a small ionic contribution. Because the bonding between layers is of the van der Waals type the optical properties@-@ are largely determined by the structure of the individual four-fold layers. The optical absorption in the neighbourhood of the absorption edge has been determined for the system GaSe&_, where 0 < x < 1. Photoconductivity measurements on a Zn-doped sample of GaSe between 300” and 90°K have been made. A discussion of the nature of the absorption and photoconductive maxima associated with the absorption edge is given, and the influence of the characteristic layer structure on the optical absorpTHE
CRYSTAL structures
tion considered for different directions of polarization of the incident radiation. 2. EXPBRIMBN’TAL PROCBDURB The crystals were grown by transport reaction(s) and the composition of the mixed crystals was taken to be that of the starting material. Samples of the appropriate thickness, between 10 and 65 CL, were obtained by cleaving thicker flakes along the plane of the layers, i.e. normally to the c-axis. They were mounted in a cryostat equipped with suitable windows and clamped between two copper plates provided with apertures to allow for the passage of the radiation. The sample temperature was taken to be that of the plates. For absorption and photoconductivity measurements with unpolarized radiation the samples were illuminated at normal incidence. For measurements with polarized radiation the samples were illuminated at an angle of incidence of 45”. The radiation was polarized either perpendicular to or parallel to the plane of incidence. The reflectivity of freshly cleaved GaSe and
1427
J. L. BREBNER
1428
FIG. 1. The absorption coefficient of a series of mixed crystals Ga at 12’K, where x has the following values (1) 0.95; (2) 0.90; (3) 0.80; (4) 0.60; (5) 0.40; (6) 0.20; (7) 0.00.
SeZS1-z
GaS was measured for unpolarized radiation at an angle of incidence of 7”. Photoconductivity measurements were made on a sample of GaSe doped with 0.5 per cent Zn to increase the conductivity. Nevertheless as the resistances at low temperatures were of the order of 10X it was necessary to use a specially designed 4-probe high resistance bridge.(lO) The photocurrent and the potential drop across the sample as a function of the wavelength of the radiation was registered directly on a recorder. In all the measurements the illumination was provided by a Perkin Elmer 112 U spectrometer equipped with a silica prism, and Polaroid sheet was employed as polarizer. The mean resolution was 0.005 eV for absorption and O-050 eV for reflectivity and photoconductivity measurements.
3. EXPERIMENTAL RESULTS The logarithm of the inverse transmission of the series of mixed crystals at 12°K is shown as a function of energy in Fig. 1. A reduction in intensity of the main absorption line(7-s) with increasing sulphur content is observed except in Ga Se0.s S0.s whose anomalous behaviour may be attributed to a poor quality sample. A second broad weak line is found in pure GaSe and in the Se-rich
crystals up to GaSeo.db The absorption level at energies just tibove that of the line is the same for all samples and begins to rise more rapidly as the sulphur content increases. At energies less than that of the main line, the absorption falls rapidly to a very low level for low concentrations of sulphur, but an absorption tail appears as the concentration is increased beyond 40 per cent. Figure 2 shows the energy at which the maximum absorption in the line occurs as a function of the sulphur content. Figure 3 shows the absorption of a 65-p thick sample of GaS between 300°K and 77°K. The absorption tail extends over several tenths of an electron volt and there is a sudden increase immediately after the onset of absorption. The slope of the main portion of the tail decreases as the temperature decreases. The reflectivity of GaSe and GaS at 300°K and 77°K is shown in Fig. 4. The absolute accuracy is poor due to the difficulty of obtaining sufficiently plane samples. Below the edge the reflectivities are in reasonable agreement with those derived from the refractive indices in this region of GaSe and Gas, which are 2.7 and 3-O respectively. These values were obtained from multiple interference in samples of known thickness. The reflectivity of GaSe shows an abrupt increase at energies corresponding to the position of the main absorption
THE
OPTICAL
ABSORPTION
EDGE
line whereas that of GaS shows a smaller more gradual rise. Figure 5 shows the photoconductive behaviour of a Zn-doped sample of GaSe at 170°K. The resistance change as a function of the wavelength for normally incident unpolarized radiation was similar to that for radiation incident at 45” and
IN
LAYER
STRUCTURES
1429
polarized perpendicular to the plane of incidence. A photocurrent maximum was observed at temperatures between 250°K and 90”K, corresponding to the position of the main absorption line. Due to the very high sample resistance and the resultant extremely long response time it was not possible to obtain reproducible results below 90°K. For radiation incident at 45” and polarized parallel to the plane of incidence the photocurrent maximum occuring at the same energies as before was strongly enhanced, as compared with the photocurrent at higher energies. Under similar conditions of illumination the room temperature absorption of GaSe in the line and at higher energies increased by an order of magnitude for polarization parallel to the incident plane over that for polarization normal to the plane. The increase became greater at higher angles of incidence. The same effect was found in Gas. 4. DISCUSSION
20
40
60
GaSe
80
Gas
At%
-
FIG. 2. Position of the energy of the maximum in the main absorption line of the crystals GaSezS1_z as a function of the sulphur content.
The layer structure of compounds such as GaSe, Gas, Moss, MoTez, WSes, HgIs and PbIs gives rise to marked anisotropy in many of their properties. Free charge carriers occuring within one layer may be considered as being confined to that layer. In what follows the potential in which the carriers move is taken to be the sum of two parts, one dependent on z, the coordinate normal to the
FIG. 3. The square root of the absorption coefficient of pure GaS between 300°K and 78°K as a function of energy.
1430
J.
L.
BREBNER
layers, the other dependent on x and y. The electron wave functions are written as \rp&,
Y, z> = SP+,
Y)%&+
(1)
The S-functions are solutions of a Schrijdinger equation containing that part of the potential periodic in x and y. The 2, are wave functions of an electron in a one-dimensional potential well. The transition probability for an electron being excited from a state with wave function $t to a state with wave function #f is governed by the square of the matrix element S?VrHradYg dxdYdz
Ga S
where H rad
=
-
ieh -A ??u
*grad
A being the vector potential characterizing the radiation. Experiment shows that the absorption increases as the component of A along the c-axis increases and it is assumed to be a maximum when A is parallel to this axis. The transition probability is then proportional to FIG. 4. The reflectivity of pure GaSe and pure GaS at 300” K and 77” K as a function of energy.
PI, = j” S;‘&,,dxdY
jz;+
FIG. 5. The photocurrent of a sample of GaSe doped with 0.5 per cent of Zn at 170°K as a function of the energy of the exciting radiation incident at 45’. (1) Radiation polarized normal to the incident plane; (2) Radiation polarized parallel to the incident plane.
%I,
dz.
(3)
THE
OPTICAL
ABSORPTION
EDGE
From the definition of 2 in order that P,, should not be zero Z,, # Z,,. Z,, is the ground state 2s of the well and 2, the first excited state 21. The valence band thus contains a series of S-functions associated with 2s and the conduction band the same S-functions associated with 21. The situation for both GaSe and GaS is represented schematically in Fig. 6. The transition 1 represents a transition whereby an dectron is excited from the top of the series A of overlapping S-bands associated with 2s to the bottom of the series A associated with 21. The transition 2 between series A and B of the S-bands, both associated with 20, occurs at a higher energy than transition 1. In a crystal such as Moss where the layer thickness is smaller than in either GaSe or Gas, the energy separation of 2s and 21 is
fJ &(a)
D (Ef(b)
Fro 6. Schematic representation of the density of states of the band system of GaSe and Gas. The S-bands A and B on the left hand side of the figure are associated with 20, the same bands on the right hand side with 2~. In transition 1 both the S- and Z-functions change. In transition 2 only the S-functions change.
IN
LAYER
STRUCTURES
1431
larger and hence transition 2 may occur at a smaller energy than transition 1. Since the series A is made up of several S-bands then in the transition 1 the S-functions must change as well as the Z-functions. Thus in equation (3) S,, is taken to be different from SPf. Since these functions are orthogonal the first integral in equation (3) might be expected to be zero for direct transitions but this is so only if we neglect the finite value of the radiation wavevector. Therefore the integral is small but nonzero. For radiation polarized perpendicular to the c-axis, along x say, we have
From the definitions of S and .Z it follows that the term involving the gradient is larger for Pu than for P_L. Furthermore the orthogonality of the Z-functions is likely to be better than that of the S-functions. Therefore it is seen that PII > PJ_ in accord with experiment. From the behaviour of the absorption and reflectivity the transition is seen to be direct in GaSe and, at energies below the absorption line, indirect in Gas. If the rise in absorption close to the onset in Fig. 3 is interpreted as being due to the emission of only one type of phonon the corresponding temperature is 420%. The transition at the line and at higher energies is direct in both compounds for the following reasons. If the line is regarded as arising from the formation of excitons then the transition must be direct otherwise a step-function rather than a line would be found. If on the other hand the line is regarded as being due to the excitation of electrons to impurity or self-trapped states as suggested below, then a twophonon process must be invoked if the transition is not direct. This would give rise to a much smaller value of the absorption than is found. Assuming the transition probability to be independent of the energy the absorption coefficient is proportional to the density of states of the valence and conduction bands. In our case, the density of states depends on the energy through the S-bands only. If these S-bands are assumed to be isotropic in x and y and quadratic in k, it is found that an indirect transition gives a linear relationship between the absorption and the photon energy whereas a direct transition leads to a step-function,
1132
J. L.
BREBNER
i.e. &&.ect independent &ndirect
CL(hv-
of hv, step-function
AE* &a).
These conclusions refer to allowed transitions. When the transitions are forbidden we have
Experiment shows that for GaS a plot of the square root of the absorption coefficient vs. energy yields the nearest approach to a straight line (Fig. 3), and therefore the transition is forbidden. The change in absorption with composition in the mixed crystal series indicates that the mechanism producing the line absorption is the same in GaSe and Gas. NIKITINE and GROS&*) have ascribed the lines in GaSe to the formation of excitons but they were unable to fit a hydrogenlike series to them. Following the proposed model it is easy to see that since the carriers are regarded as remaining within a layer the excitons will be profoundly modified for orbital diameters greater than the layer thickness, the first orbital being the one least perturbed. Electrical measurements, reported elsewhere@lf have shown that the effective mass of the holes is close to unity and that of the electrons is likely to be large. In such a case the excitons would have a reduced mass close to unity. Taking a value of the dielectric constant of 7.6, derived from the infrared refractive index of GaSe of Z-76, the first exitonic state should lie 0.24 eV from the absorption edge. In fact the energy separation of the main line from the edge, whiIe difficult to determine accurately is not greater than 0.07 eV. In this respect GaSe resembles PbIz where the first prominent line is found closer to the edge than simple exciton theory predicts. It is somewhat surprising that the energies of the remaining Iines in PbIs fit a hydrogen-like series. It has been suggested that electrons excited to the conduction band of GaSe are self-trapped.(ll) The main absorption line may then be attributed to the excitation of electrons from the valence band to these self-trapped states. As the electrons have a long lifetime in these states the resultant holes remain free giving rise to the p-type photo~ondu~tivi~ found by &3BE and Lm.f5) The difference in the line strengths in GaSe and GaS
is thought to arise from the difference in the coupling constants between the carriers and the lattice in the two compounds. Previous work(‘) on the photoconductivity of one sample of a GaSe crystal grown by pulling from the melt suggested that there is a photoconductive peak at the position of the main absorption line at 4~2°K. Photoconductive maxima have been observed corresponding to absorption lines in crystals such as CdS where the formation of excitons is generally accepted. In order to account for the maxima the excitons are supposed to move to ionization centres or to ionize through exciton-exciton collisions, since at low temperatures the probability of ionization by phonons is negligible. It has been shownQ1) that the electron mobility in GaSe is likely to be small, and is negligible in the case where the electrons are self-trapped. Since the exciton mobility cannot be much different from that of the less mobile carrier it is difficult to account for the low temperature photoconductive maximum in Case in terms of processes involving exciton migration. On the other hand if the electrons are excited to selftrapped states the maximum can be attributed to the resulting free holes. The present experiments show that between 250°K and 90°K the centre of the broad photoconductive maximum in GaSe lies at the energy of the main absorption line. Since the phonon energies available at 90°K are much less than the separation of the line from the absorption edge, it is unlikely that the mechanism giving rise to the rn~~rnurn is that of the creation of excitons subsequently ionized by lattice vibrations. A peculiar feature of the photoconductivity in GaSe is the considerable enhancement of the max~um when radiation incident at 45” is polarized in the plane of incidence, without an accompanying increase at energies greater than that of the maximum. On the other hand the absorption in the main Iine and at higher energies both increase for similar conditions of illumination and it is not clear why the photoconductivity shows different behaviour. The increase in the number of electron-hole pairs due to increased absorption at energies above the photoconductive maximum may be offset by the increase in the probability of direct recombination, found in general as the absorption increases. Such a recombination is less liiely to occur if the electrons are excited to self-trapped states.
THE
OPTICAL
ABSORPTION
This paper represents an attempt to introduce in a simple fashion the effect on the optical properties of semiconductors such as GaSe, GaS etc. of their markedly anisotropic crystal structures. Two main points are made: firstly it is shown that many properties may be explained assuming the electrons to move in a potential made up of the sum of two terms which vary independently of one another. This brings out the two-dimensional aspect of the structures. Secondly an alternative explanation to that of excitons is proposed to account for the lines associated with the absorption edge in GaSe, and GaS which may also be valid for the lines found in HgIs, PbIs, Moss, MoTes and WSes.(rsJa)
Acknowledgements-The author would like to thank Mr. H. BOELSTEFUJfor providing the crystals and Mr. R. FIVAZ and Dr. E. MOOSERfor many helpful discussions.
EDGE
IN
LAYER
STRUCTURES
1433
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