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Exponential absorption edges in GeSe
Solid State Communications,Vol. 19, pp. 759-763, 1976.
Pergamon Press.
Printed in Great Britain
EXPONENTIAL ABSORPTION EDGES IN GeSe S.V. Vlachos, A.P. Lambros and N.A. Economou Department of Physics, University of Thessaloniki, Greece
(Received 16 January 1976 by M. Balkanski) The exponential absorption edges of GeSe were analysed and the pertinent parameters calculated as a function of temperature, using polarized light with the plane of polarization along the main axis of the (001) plane. The behaviour is anisotropic, and it was found that different LO phonons participate in the exciton-phonon interaction. Along the e II a direction at a temperature ~ 200°K a switch-over mechanism from a LO phonon of 0.022 eV to a LO phonon 0.013 eV is operating. The difference in the value of the slope parameter of the exponential rise along the two directions of polarization indicates that the binding of the atoms is different in character.
GERMANIUM SELENIDE has an indirect energy gap 1 similar with the behaviour of the isomorphic compounds SnS and SnSe. 2 At higher photon energies the absorption of GeSe follows a semilogarithmic dependence on energy, indicating that the absorption of this material in the vicinity of the direct absorption edge is best described by Urbach's rule a = ao exp
o(hp -- Eo) kT
(1)
where a is the absorption coefficient, h, u, k T have their usual meaning, ao and Eo are constants and the factor o is expressed by the relation
2kT
hw
o = Oo h w t a n h 2 k T .
(2)
The h6o term represents the energy of the phonon participating in the interaction, while Oo is describing the state of the exciton. Thus o is proportional to the energy gain due to the transfer of the exciton over the undeformed lattice and is inversely proportional to the energy gain of the localized exciton due to the interaction with lattice vibrations. 3 A value for Oo smaller than unity indicates the existence of excitonic states lying lower than free excitons in the undeformed lattice, where excitons become self trapped by a local lattice deformation induced by itself. Therefore Oo expresses in a way the ionic character of the material. The experimental procedure for measuring the absorption coefficient was described previously. 4 GeSe has an orthorhombic structure with lattice parameters a = 4.40 A, b = 3.85 A and c = 10.82 A, ~ therefore in examining its properties the anisotropy should be considered. Due to difficulties that arise from the easy shear along {001} the measurements of the optical 759
properties were confined using polarized light with the plane of polarization along the a and b axis. In Fig. 1 the relationship between log a vs hu is shown for various temperatures. Figure l(a) represents the results obtained with light polarized along the b axis. The linear relationship extending over large ranges of a and T confirms that we are actually dealing with Urbach's rule. Extrapolating the straight lines, each of which corresponds to results at a different temperature, they converge to a focal point (log ao, Eo), with values log ao = 5.004 and Eo = 1.38 eV. In the inset the steepness coefficient o is plotted vs temperature, while the solid line represents the expression of equation (2), which was obtained using the values 1.009 and 0.024 eV for the parameters Oo and hco. The value 0.024 eV (or 201 + I0 cm -~) found for the phonon participating in the exciton phonon interaction should be compared with the one phonon observed along the b direction in the far infrared spectrum of GeSe 6 which has a value equal to 208 cm-l. It is known 7 that the isoabsorption curves show a linear behaviour for temperatures above a certain temperature Tt, obeying the relation Eo = Eo -- A E - - Tt 0E~ 3T where Eo is the focal point energy, z5~7the spread of energy values that result from the curvature of the isoabsorption curves for T < Tt, Eo the exciton peak energy and OEg/OT the variation of tire energy gap with temperature. In Fig. 2(a) a plot of the results according to the above scheme is presented. The pertinent values obtained from this plot are given in Table I. In Figs. l(b) and 2(b) the results obtained by
(3)
2.8
3.2
3fi
/
2~1.21
~
,Go 2~o T (OK) b
/
//~/
/// ~
1.~3
1.~
Photon
1;,7
, . , ~ , . ~ ~
o.o
3~o
/
f/ / "
/
34
energy,
1.2e hv (eV)
1.31 -
1.33
1 : 300°K
1.3~
1.37
10, 131°K
9: 151°K
8: 177°K
7, 197°K
5 232°K
2=273°K 3: 257°K
2°~130 b)
2~,
2~
3.0
3.2
•
1.34
2~ 3/ 4/
,/
0%
5
'
!.38
/
1oo
6
/
/
,,
1~,2
/
260
/
•
/"
1~,6
)10
/
36o
Photon energy,
/
T (OK)
~~ ~ ~
1.50 hv {eV)
£11a
"~
/
//
/
/
5: 222°K
1 : 300°K 2 : 273°K 3 : 25q°K
10:
~"
1.~8
113°K
9 : 136°K
7' 177 OK 8: 155°K
/
1.~,4
/
,/
/
1.62
(a02,E02)
(aovEo)
Fig. I. The linear relationship in GeSe of log a vs hu at different temperatures (a) along the b axis, (b) along the a axis. (Inset; the steepness coefficient vs temperature.)
a)
C)
t
4~3
4.4
48
lo//
a2
(a0'E0)
30
Ellb
0.6
~.o
4.O
Z 0 oo
70
< o
Cb
bl
Z rtl
0
>
Y~
Z bl Z,-]
--3 0
\
x
x
\
xx\
26o
T#
\ ~, 360
dT
>
1.55
1.60 Eo~
O_
a)
1.20
1.22
1.24
°K
1.3oo b)
135
T,
1,logo2: logo = 3= logo = 4: logo = 5: l o g o 6: logo = 7: logo =
2bo
dE~ ..--"~ dT
Temperature, oK
16o
E#a
Fig. 2. Isoabsorption curves for different values of log a; (a) along the b axis, (b) alongthe a axis.
Temperature,
1
¢-
~ 1.40
C
e'-
~1.45
16o
\
x'\e
\
Eol ~
_~ 1.26
'\
x
1 : l o g o = 3~ 2:logo 2~ 3:logo-2~ 4:logo= 2~ 5:1ogo-2,5 6:logo-2,3
1.50
AE
\,\\ x x \ ,\ x\ x\ x \ ~ x \
E//b
~- 1.28
_~ 1.30
>
I
1.32
136
128
3,0 2,9 2,8 2,7 2,6 2,5 2,4
3bo
",,
,
6
4 t~
Z
¢3
0 Z
©
> t>
rn ,..]
0 Z
Oo
0
Z
<
762
EXPONENTIAL ABSORPTION EDGES IN GeSe
Vol. 19, No. 8
Table 1. Values of parameters which fit the experimental results of the exponential behaviour of absorption in the vicinity of the direct energy gap o f GeSe ellb
100-200°K ella
200-300°K ella
log ao
5.0046
3.6398
3.9908
Eo (eV)
1.382
1.620
1.598
Eo (eV) z2~E(eV)
1.341 0.079
1.54 0.06
-
Tt °K
OEg/3T(eV °K-l)
237.5 -- 5.05 x 10 -4
150 -- 8 x 10 -4
-
Oo
1.009
0.267
0.358
hco (eV)
0.024
0.022
0.013
using light polarized along the a axis are presented in accordance with the treatment applied to the previous results for the b axis, Pertinent values are given in Table 1. In this case though, we have to notice that there are two distinct temperature dependent behaviours, above a certain temperature, in the vicinity of 200°K, and below that temperature. The plots of log a vs hu for temperatures above 200°K result to a focal point (log ao = 3.99, Eo = 1.59 eV) with Oo = 0.3584 and h~o = 0.013 eV (100 + 10 cm-1), while similar plots below 200°K result to a different focal point (log ao = 3.63, Eo = 1.62 eV) with Oo = 0.2671 and hco 0.022 eV (180 -+ 10 c m - ' ) . These results indicate that there is a change in the coupling constant, indicated by the change in Oo, and also a switch-over to a different phonon participating in the interaction. A similar change in the behaviour of the e x c i t o n - p h o n o n interaction has been observed is the case of ZnS 8 which was attributed to the change in the mechanism, since the switch-over was between LA phonons at low temperatures to LO phonons at the higher range of temperatures, in our case far infrared studies 6 indicated the existence of three LO phonons along the a axis situated at 91, 179 and 224 c m - 1 respectively. As it is obvious from a comparison of the phonon values, the switch over mechanism involves two LO phonons, the one with the lower frequency (100 cm -1) participating at the higher temperature range, while the other (180 c m - ' ) at low temperatures. The abrupt change in the coupling parameter Oo at the temperature of the switch-over to lower values, as the temperature decreases, indicates that the exciton localization level becomes deeper, indicating an increase in the ionic character of the bond between
Ge-Se. It may be argued that at the temperature of the switch-over a phase transformation takes place, although the only evidence pointing to that direction is from the results of Asanabe eta/. 9 which indicate an anomaly in the electrical characteristics i.e. electrical resistance and Hall coefficient, occuring at I/T= 5 x 103°K - ' . The behaviour of GeSe is highly anisotropic. Since the deformation from the ideal rocksalt lattice is greatest in the a axis rather than in the b axis, 5 this may account for the following features that may be obtained from a comparison of the tabulated values for the two directions of polarization given in Table 1. First there is a difference in the exciton peak value Eo, which is lower in the b direction. This is accompanied with a larger spread of energy values z2~E.Second the steepness parameter o in the two directions suggests a different nature in the bonding character along these. According to Toyozawa 3 this parameter compares the energy gain of the free exciton in the undeformed lattice with the energy gain of the localized exciton. Therefore a value of o > 1 indicates a more covalent character, while o < 1 requires a strong localization, typical to ionic crystals. The larger distance of the bonds along the a direction may account for the difference that is observed. The results presented here, we believe, that are the first where Urbach's rule found to be applicable in highly anisotropic crystals, therefore the direct application of the reasoning that is found to be sound for isotropic media is a point of concern. In favour of our arguments is the perfect agreement that we have obtained for the values of the participating phonons with the phonons observed by direct experiments.
Vol. 19, No. 8
EXPONENTIAL ABSORPTION EDGES IN GeSe
763
REFERENCES 1.
VLACHOS S.V., LAMBROS A.P., THANAILAKIS A. & ECONOMOU N.A. (to be published).
2.
LAMBROS A.P., GERALEAS E. & ECONOMOU N.A., J. Phys. Chem. Solids 35,537 (1974); ARAI T., ONOMICHI M. & KUKO I., Proc. Int. Conf. Phys. Chem. Semicond. Hetero]unctions and Layer Structures, Vol. IV, p. 51. Akademial Kiado, Budapest (1971).
3.
SUMI H. & TOYOZAWA Y., J. Phys. Soc. Japan 31, (1971).
4.
LAMBROS A.P. & ECONOMOU N.A.,ScL Ann. Fac. Phys. and Mathem. Univ. Thessaloniki13, 85 (1973).
5.
OKAZAKI S.A., J. Phys. Soc. Japan 13, 1151 (1958);ZACHARIANSENW.H.,Phys. Rev. 40, 9t7 (1932).
6.
SIAPKAS D., KYRIAKOS D.S. & ECONOMOU N.A. Solid State Commun. 19,765 (1976).
7.
ROBERTS G.G., TUTIHASI S. & KEEZER R.C., Phys. Rev. 166,637 (1968).
8.
BRADA V., YACOBI B.G. & PELED A., Solid State Commun. 17, 193 (1975).
9.
ASANABE S.A. & OKAZAKI A., J. Phys. Soc. Japan 15,989 (1960).
Pergamon Press.
Printed in Great Britain
EXPONENTIAL ABSORPTION EDGES IN GeSe S.V. Vlachos, A.P. Lambros and N.A. Economou Department of Physics, University of Thessaloniki, Greece
(Received 16 January 1976 by M. Balkanski) The exponential absorption edges of GeSe were analysed and the pertinent parameters calculated as a function of temperature, using polarized light with the plane of polarization along the main axis of the (001) plane. The behaviour is anisotropic, and it was found that different LO phonons participate in the exciton-phonon interaction. Along the e II a direction at a temperature ~ 200°K a switch-over mechanism from a LO phonon of 0.022 eV to a LO phonon 0.013 eV is operating. The difference in the value of the slope parameter of the exponential rise along the two directions of polarization indicates that the binding of the atoms is different in character.
GERMANIUM SELENIDE has an indirect energy gap 1 similar with the behaviour of the isomorphic compounds SnS and SnSe. 2 At higher photon energies the absorption of GeSe follows a semilogarithmic dependence on energy, indicating that the absorption of this material in the vicinity of the direct absorption edge is best described by Urbach's rule a = ao exp
o(hp -- Eo) kT
(1)
where a is the absorption coefficient, h, u, k T have their usual meaning, ao and Eo are constants and the factor o is expressed by the relation
2kT
hw
o = Oo h w t a n h 2 k T .
(2)
The h6o term represents the energy of the phonon participating in the interaction, while Oo is describing the state of the exciton. Thus o is proportional to the energy gain due to the transfer of the exciton over the undeformed lattice and is inversely proportional to the energy gain of the localized exciton due to the interaction with lattice vibrations. 3 A value for Oo smaller than unity indicates the existence of excitonic states lying lower than free excitons in the undeformed lattice, where excitons become self trapped by a local lattice deformation induced by itself. Therefore Oo expresses in a way the ionic character of the material. The experimental procedure for measuring the absorption coefficient was described previously. 4 GeSe has an orthorhombic structure with lattice parameters a = 4.40 A, b = 3.85 A and c = 10.82 A, ~ therefore in examining its properties the anisotropy should be considered. Due to difficulties that arise from the easy shear along {001} the measurements of the optical 759
properties were confined using polarized light with the plane of polarization along the a and b axis. In Fig. 1 the relationship between log a vs hu is shown for various temperatures. Figure l(a) represents the results obtained with light polarized along the b axis. The linear relationship extending over large ranges of a and T confirms that we are actually dealing with Urbach's rule. Extrapolating the straight lines, each of which corresponds to results at a different temperature, they converge to a focal point (log ao, Eo), with values log ao = 5.004 and Eo = 1.38 eV. In the inset the steepness coefficient o is plotted vs temperature, while the solid line represents the expression of equation (2), which was obtained using the values 1.009 and 0.024 eV for the parameters Oo and hco. The value 0.024 eV (or 201 + I0 cm -~) found for the phonon participating in the exciton phonon interaction should be compared with the one phonon observed along the b direction in the far infrared spectrum of GeSe 6 which has a value equal to 208 cm-l. It is known 7 that the isoabsorption curves show a linear behaviour for temperatures above a certain temperature Tt, obeying the relation Eo = Eo -- A E - - Tt 0E~ 3T where Eo is the focal point energy, z5~7the spread of energy values that result from the curvature of the isoabsorption curves for T < Tt, Eo the exciton peak energy and OEg/OT the variation of tire energy gap with temperature. In Fig. 2(a) a plot of the results according to the above scheme is presented. The pertinent values obtained from this plot are given in Table I. In Figs. l(b) and 2(b) the results obtained by
(3)
2.8
3.2
3fi
/
2~1.21
~
,Go 2~o T (OK) b
/
//~/
/// ~
1.~3
1.~
Photon
1;,7
, . , ~ , . ~ ~
o.o
3~o
/
f/ / "
/
34
energy,
1.2e hv (eV)
1.31 -
1.33
1 : 300°K
1.3~
1.37
10, 131°K
9: 151°K
8: 177°K
7, 197°K
5 232°K
2=273°K 3: 257°K
2°~130 b)
2~,
2~
3.0
3.2
•
1.34
2~ 3/ 4/
,/
0%
5
'
!.38
/
1oo
6
/
/
,,
1~,2
/
260
/
•
/"
1~,6
)10
/
36o
Photon energy,
/
T (OK)
~~ ~ ~
1.50 hv {eV)
£11a
"~
/
//
/
/
5: 222°K
1 : 300°K 2 : 273°K 3 : 25q°K
10:
~"
1.~8
113°K
9 : 136°K
7' 177 OK 8: 155°K
/
1.~,4
/
,/
/
1.62
(a02,E02)
(aovEo)
Fig. I. The linear relationship in GeSe of log a vs hu at different temperatures (a) along the b axis, (b) along the a axis. (Inset; the steepness coefficient vs temperature.)
a)
C)
t
4~3
4.4
48
lo//
a2
(a0'E0)
30
Ellb
0.6
~.o
4.O
Z 0 oo
70
< o
Cb
bl
Z rtl
0
>
Y~
Z bl Z,-]
--3 0
\
x
x
\
xx\
26o
T#
\ ~, 360
dT
>
1.55
1.60 Eo~
O_
a)
1.20
1.22
1.24
°K
1.3oo b)
135
T,
1,logo2: logo = 3= logo = 4: logo = 5: l o g o 6: logo = 7: logo =
2bo
dE~ ..--"~ dT
Temperature, oK
16o
E#a
Fig. 2. Isoabsorption curves for different values of log a; (a) along the b axis, (b) alongthe a axis.
Temperature,
1
¢-
~ 1.40
C
e'-
~1.45
16o
\
x'\e
\
Eol ~
_~ 1.26
'\
x
1 : l o g o = 3~ 2:logo 2~ 3:logo-2~ 4:logo= 2~ 5:1ogo-2,5 6:logo-2,3
1.50
AE
\,\\ x x \ ,\ x\ x\ x \ ~ x \
E//b
~- 1.28
_~ 1.30
>
I
1.32
136
128
3,0 2,9 2,8 2,7 2,6 2,5 2,4
3bo
",,
,
6
4 t~
Z
¢3
0 Z
©
> t>
rn ,..]
0 Z
Oo
0
Z
<
762
EXPONENTIAL ABSORPTION EDGES IN GeSe
Vol. 19, No. 8
Table 1. Values of parameters which fit the experimental results of the exponential behaviour of absorption in the vicinity of the direct energy gap o f GeSe ellb
100-200°K ella
200-300°K ella
log ao
5.0046
3.6398
3.9908
Eo (eV)
1.382
1.620
1.598
Eo (eV) z2~E(eV)
1.341 0.079
1.54 0.06
-
Tt °K
OEg/3T(eV °K-l)
237.5 -- 5.05 x 10 -4
150 -- 8 x 10 -4
-
Oo
1.009
0.267
0.358
hco (eV)
0.024
0.022
0.013
using light polarized along the a axis are presented in accordance with the treatment applied to the previous results for the b axis, Pertinent values are given in Table 1. In this case though, we have to notice that there are two distinct temperature dependent behaviours, above a certain temperature, in the vicinity of 200°K, and below that temperature. The plots of log a vs hu for temperatures above 200°K result to a focal point (log ao = 3.99, Eo = 1.59 eV) with Oo = 0.3584 and h~o = 0.013 eV (100 + 10 cm-1), while similar plots below 200°K result to a different focal point (log ao = 3.63, Eo = 1.62 eV) with Oo = 0.2671 and hco 0.022 eV (180 -+ 10 c m - ' ) . These results indicate that there is a change in the coupling constant, indicated by the change in Oo, and also a switch-over to a different phonon participating in the interaction. A similar change in the behaviour of the e x c i t o n - p h o n o n interaction has been observed is the case of ZnS 8 which was attributed to the change in the mechanism, since the switch-over was between LA phonons at low temperatures to LO phonons at the higher range of temperatures, in our case far infrared studies 6 indicated the existence of three LO phonons along the a axis situated at 91, 179 and 224 c m - 1 respectively. As it is obvious from a comparison of the phonon values, the switch over mechanism involves two LO phonons, the one with the lower frequency (100 cm -1) participating at the higher temperature range, while the other (180 c m - ' ) at low temperatures. The abrupt change in the coupling parameter Oo at the temperature of the switch-over to lower values, as the temperature decreases, indicates that the exciton localization level becomes deeper, indicating an increase in the ionic character of the bond between
Ge-Se. It may be argued that at the temperature of the switch-over a phase transformation takes place, although the only evidence pointing to that direction is from the results of Asanabe eta/. 9 which indicate an anomaly in the electrical characteristics i.e. electrical resistance and Hall coefficient, occuring at I/T= 5 x 103°K - ' . The behaviour of GeSe is highly anisotropic. Since the deformation from the ideal rocksalt lattice is greatest in the a axis rather than in the b axis, 5 this may account for the following features that may be obtained from a comparison of the tabulated values for the two directions of polarization given in Table 1. First there is a difference in the exciton peak value Eo, which is lower in the b direction. This is accompanied with a larger spread of energy values z2~E.Second the steepness parameter o in the two directions suggests a different nature in the bonding character along these. According to Toyozawa 3 this parameter compares the energy gain of the free exciton in the undeformed lattice with the energy gain of the localized exciton. Therefore a value of o > 1 indicates a more covalent character, while o < 1 requires a strong localization, typical to ionic crystals. The larger distance of the bonds along the a direction may account for the difference that is observed. The results presented here, we believe, that are the first where Urbach's rule found to be applicable in highly anisotropic crystals, therefore the direct application of the reasoning that is found to be sound for isotropic media is a point of concern. In favour of our arguments is the perfect agreement that we have obtained for the values of the participating phonons with the phonons observed by direct experiments.
Vol. 19, No. 8
EXPONENTIAL ABSORPTION EDGES IN GeSe
763
REFERENCES 1.
VLACHOS S.V., LAMBROS A.P., THANAILAKIS A. & ECONOMOU N.A. (to be published).
2.
LAMBROS A.P., GERALEAS E. & ECONOMOU N.A., J. Phys. Chem. Solids 35,537 (1974); ARAI T., ONOMICHI M. & KUKO I., Proc. Int. Conf. Phys. Chem. Semicond. Hetero]unctions and Layer Structures, Vol. IV, p. 51. Akademial Kiado, Budapest (1971).
3.
SUMI H. & TOYOZAWA Y., J. Phys. Soc. Japan 31, (1971).
4.
LAMBROS A.P. & ECONOMOU N.A.,ScL Ann. Fac. Phys. and Mathem. Univ. Thessaloniki13, 85 (1973).
5.
OKAZAKI S.A., J. Phys. Soc. Japan 13, 1151 (1958);ZACHARIANSENW.H.,Phys. Rev. 40, 9t7 (1932).
6.
SIAPKAS D., KYRIAKOS D.S. & ECONOMOU N.A. Solid State Commun. 19,765 (1976).
7.
ROBERTS G.G., TUTIHASI S. & KEEZER R.C., Phys. Rev. 166,637 (1968).
8.
BRADA V., YACOBI B.G. & PELED A., Solid State Commun. 17, 193 (1975).
9.
ASANABE S.A. & OKAZAKI A., J. Phys. Soc. Japan 15,989 (1960).