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The Urbach rule in ZnS crystals
Solid State CominunicationsYol. 17, pp. 193—196, 1975.
Pergamon Press.
Printed in Great Britain
THE URBACH RULE IN ZnS CRYSTALS* Y. Brada, B.G. Yacobi and A. Peled Racah Institute of Physics, The Hebrew University, Jerusalem, Israel (Received 28 January 1975 by W. Low)
The slope parameters of the exponential absorption edges of 3C cubic, 4H polytypic and 2H hexagonal ZnS crystals were measured as a function of temperature. Their behavior show that different phonons interact with the electron transitions in different te~peratureranges, an LA piezoelectric phonon below~pproximately100 Kand an LO phonon above this tempera. ture~upto 300 K. In cubic ZnS the slope parameter was measured up to 750 K and the same LO phonon was found to be involved at these temperatures. Recently Dow and Redfield9 presented a unified model pertaining both to the exponential form of absorption edges and their temperature dependence. This theory applies both to ionic and covalent materials and is essentially a unification of two previous models: Redfield’s internal Franz—Keldysh mechanism’°and
ZnS POLYTYPES, like many ionic crystals, are known to have an exponential absorption edge.12 The photon energy and temperature dependence of the absorption coefficient in the spectral range of this exponential edge is characterized by the Urbach rule :~ a
=
a 0 exp {a(hv—hvo)IkT}
(1)
Dexter s Stark-shifted exciton model. 11 ,
where a is the absorption coefficient, h, v, k and T have the usual meaning; a0 and ~oare constants of the material, The slope parameter ci itself is a function of ternperature and in some ionic crystals it was found to 4 depend on a phonon energy: a = ao(2kT/hwo) tanh (hwo/2kT) (2) where hc~ was defined as an “effective” phonon energy and 0 depends on the material. The theoretical treatment of this slope parameter was first developed by Toyozawa and Mahr for exciton—phonon interaction in ionic crystals.4”6 In covalent crystals the slope parameter was found to depend on the concentrations and the electrical 7’8 charges of impurities. * Supported in part by the Central Research Fund of the Hebrew University. ____________
193
.
.
This unified theory ascribes exponential edges to electric field ionization of the exciton, i.e. the fieldinduced tunneling of the electron away from the hole. The dominant sources of the ionizing electric microfields may be imperfections as phonons, impurities andcharged dislocations. In alkali such halides and il—VI compounds the form of the exponential absorption edges has been correlated with the LO phonon generated 9 In some cases electrical fields due to microfields. phonons were also found to be involved.’2 piezoelectric It has been shown previously’2 that in cubic crystals h~ 0= hc~LA= 13.6 meV below approximately l00°Kand hc~0= hWLO = 43.5 meV up to 300°K. In the present work we extend this research to higher temperatures in the cubic case also of check on the behavior of slope parameter in and the case hexagonal 2H and polytypic 4H ZnS crystals, where two directions of polarization of the incident light have to be considered.
THE URBACH RULE IN ZnS CRYSTALS
194
Vol. 17, No.2
7.(. Zn S 6.5
measure the absorption spectra. The system has been described elsewhere.’4 Figure 1 shows a typical absorption spectrum, which
3C
,
/
T~ 728°K -
/
served to evaluate the experimental results. We observe
I’ 6.0
two well differentiated regions. A straight line of steep slope in the in a vs hv plot near the edge and a straight line of lower slope extending to the longer wavelengths. This latter part of the curve is most probably due to charged impurities in the crystal7’8”5 and will be dealt with later.
‘
‘
/ 55
-
5.0 3.26
1ev)
PHOTON ENEkGY
The slope at higher densities was influenced by the sample temperature and was in addition identical for different samples of the same crystallografic
3.38
structure. This is considered to be the phonon shifted absorption edge and all the data published here and
FIG. 1. Natural logarithms of the absorption coe~. ficient a vs photon energy. Cubic ZnS, T = 728 K.
2HJ.LE 2.0
ZnS
50
2.
2H.EII~
343/12
100
,~..._)“b
150
200
250
4H,EIC ZnS 248/PR
//
20 05
300
.—“‘5”b .—‘
o~~0
2.0
50
343/12
100
ISO
200
250
300
2~0
~
4H,EUc ZnS 248/58 —I.
I
“.~
15
2~
3~
~
TEMPERATURE.°K
TEMPERATURE.
°~(
FiG. 2. The slope parameter a vs temperature for 2H and 4H structures of ZnS, for both perpendicular and parallel directions of the electric vector E of the polarized light. Lines theoretical, according to equation 2 and Table 1; points experimental. —
—
Vapor gro~n~3 single crystals of a thickness varying from 7—50gm were used in this work. A Xenon 500W high pressure lamp served as the light source and a Jarell—Ash 0.5 m monochromator was used to
previouslylLi4 pertain to this absorption region, which in our crystals corresponds to an absorption coefficient a from about 102 to several times ~
Vol. 17, No.2
THE URBACH RULE IN ZnS CRYSTALS
195
Table 1. The slope pammetera0 and values ofphonon energies which fit the experimental results ZnS structure
Polarization of light
Temperature range (°K) Oo (a) 10— 90 (b) 90—300 (a) 10—100 (b) 100—300 (a) 10—140 (b)140—300
1.2 2.5 1.2 2.6 1.5 2.5
13.6 43.5 9.9 43.4 11.2 43.4
LA LO (LA) LO (LA) LO
E C
(a) 10—140 (b) 140—300 (a) 10—130
1.5 2.5
E II C
(b) 130—300
2.6
12.4 43.4 13.6 43.4
(LA) LO (LA) LO
3C E IC 4H EJIC
2H
___________________________ ‘
2.0
,._
1.5-
/
-
ZnS
,
in our case the piezoelectric LA phonons, while at the
higher temperatures the fields are due to the LO 2 14
the value of A.axis phonon with a wave-vector small enough to provide a quasi.constant electric field over the exciton radius.9 The same A.axis phonon was also
/ ~ / 0.5
100
1.5
phonons.’ The LA phonon energy involved, 13.6 meV, fits
1.0.
0
Phonon energy h~o(meV) Identification
200
300
400
TEMPERATURE
500
600
700
800
,
FiG. 3. The slope parameter a vs temperature for 3C cu Figures 2 and 3 show the value of the slope para’ meter as a function of temperature between 10 and 300°Kin the 2H and 4H structures and up to 750 K in the cubic case. Two clearly distinct regions have been observed in all the above samples: one below approximately 1 00°Kand the other above it. The lines are theoretical, using equation (2) and the parameters tabulated in Table 1. Both a 0 and hw0 of equation (2) change their values in the vicinity of lOO°K.The change 5 in a0 is due to the change in the coupling constant between the electrons and phonons, while the change in hc~., 0is due to the difference 4”6 in the phonon energy of the interacting phonons. As the interactions are due to electrical fields,9”0 we can postulate that at low temperatures the highest mean-square fields (F2 ) are due to lower energy phonons,
observed in second order Raman scattering.’6 The coupling factor entering is the polaron coupling constant17 aLa in the LO case and a similar parameter ciLA for the piezoelectric polaron18 in the LA case. The ratio ao(LO)/ao(LA) gives an idea on the relative strength of these different types of coupling in ZnS.~ The long-wave impurity induced tails (Fig. 1) were also influenced by temperature, like the higher density region. The slope parameter of this part of tail rose again from a small value at low temperature, reaching the value of a 1.0 at approximately 350°Kand then remained constant at higher temperatures, up to 750°K. The temperature shift of these tails will be discussed elsewhere 20 In conclusion, we have shown that in the investigated structures of ZnS different types of phonons influence the absorption edge slope in different tern: perature intervals. This should be also true of other Il—VI compounds, have a piezoelectric of lower energy thanwhich the LO phonon. A similarphonon behavior was indeed observed recently in ZnO.21 At high enough temperatures, above about half the Debye temperature, the LO phonon created fields are high enough to have a major influence and the slope parameter will change
196
THE URBACH RULE IN ZnS CRYSTALS
in accordance. No transition to multiphonon processes was observed at higher temperatures, and the slope
Vol. 17, No.2
parameter a remains practically constant above the Debye temperature.
REFERENCES 1.
PIPER W.W.,Thys. Rev. 92,23 (1953). 2. SHACHAR G., Ph.D. thesis, The Hebrew University, Jerusalem (1969). 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
URBACH F., Phys. Rev. 92, 1324 (1953); MOSER F. and URBACH F., Phys. Rev. 102, 1519 (1956). MAHRH.,ThyL Rev. 125,1510(1962); 132, 1880(1963). TOYOZAWA Y.,frogr~Theoret. Phys. 22,455 (1959). TOYOZAWA Y., Suppi. Progr. Theorer. Phys. (Japan) 12, 111(1959). DIXON J.R. and ELUS J.M.,Thys. Rev. 123, 1560 (1961). REDFIELD D. and AFROMOWITZ M.A.,Appl. Phys. Lett. 11, 138 (1967). DOW J.D. and REDFIELD D.,Thys. Rev. B5, 594 (1972). REDFIELD D., Trans. New York Acad. ScL 26, 590 (1964). DEXTER D.L.,Phys. Rev. Lert. 19, 1383 (1967). BRADA Y. and YACOBI B.G., in F~oc.12th mt. Conf Phys. Semicond. (Edited by PILKUHN M.H.), p. 1212, Teubner, Stuttgart, (1974). ALEXANDER E., L~LMANZ.H., MARDIX S. and STEINBERGER I.T., Phil Mag. 21, 1237 (1970). YACOBI BG., BRADA Y., LACHISH U. and HIRSH C. (to be published in Phys. Rev. B! 5).
16.
JENSEN GH.,froc. 12th mt. Conf Phys. Semicond. (Edited by PILKUHN M.H.), p. 1217. Teubner, Stuttgart (1974). NILSEN W.G.,Thys. Rev. 182, 838 (1969).
17.
FROHLICHH.,Adv. Phys. 3,325(1954).
18.
DOW J.D., SMITH Di. and LEDERMAN F.L.,Phys. Rev. B8, 4612 (1973).
19. 20.
TOYOZAWA Y., Techn. Rept. ofISSP, Series A, 119, (1964). BRADA Y. and YACOBI B.G. (to be published). JENSEN G.H. and SKETTRUP T., Phys. Status Solidi (b) 60, 169 (1973).
15.
21.
Pergamon Press.
Printed in Great Britain
THE URBACH RULE IN ZnS CRYSTALS* Y. Brada, B.G. Yacobi and A. Peled Racah Institute of Physics, The Hebrew University, Jerusalem, Israel (Received 28 January 1975 by W. Low)
The slope parameters of the exponential absorption edges of 3C cubic, 4H polytypic and 2H hexagonal ZnS crystals were measured as a function of temperature. Their behavior show that different phonons interact with the electron transitions in different te~peratureranges, an LA piezoelectric phonon below~pproximately100 Kand an LO phonon above this tempera. ture~upto 300 K. In cubic ZnS the slope parameter was measured up to 750 K and the same LO phonon was found to be involved at these temperatures. Recently Dow and Redfield9 presented a unified model pertaining both to the exponential form of absorption edges and their temperature dependence. This theory applies both to ionic and covalent materials and is essentially a unification of two previous models: Redfield’s internal Franz—Keldysh mechanism’°and
ZnS POLYTYPES, like many ionic crystals, are known to have an exponential absorption edge.12 The photon energy and temperature dependence of the absorption coefficient in the spectral range of this exponential edge is characterized by the Urbach rule :~ a
=
a 0 exp {a(hv—hvo)IkT}
(1)
Dexter s Stark-shifted exciton model. 11 ,
where a is the absorption coefficient, h, v, k and T have the usual meaning; a0 and ~oare constants of the material, The slope parameter ci itself is a function of ternperature and in some ionic crystals it was found to 4 depend on a phonon energy: a = ao(2kT/hwo) tanh (hwo/2kT) (2) where hc~ was defined as an “effective” phonon energy and 0 depends on the material. The theoretical treatment of this slope parameter was first developed by Toyozawa and Mahr for exciton—phonon interaction in ionic crystals.4”6 In covalent crystals the slope parameter was found to depend on the concentrations and the electrical 7’8 charges of impurities. * Supported in part by the Central Research Fund of the Hebrew University. ____________
193
.
.
This unified theory ascribes exponential edges to electric field ionization of the exciton, i.e. the fieldinduced tunneling of the electron away from the hole. The dominant sources of the ionizing electric microfields may be imperfections as phonons, impurities andcharged dislocations. In alkali such halides and il—VI compounds the form of the exponential absorption edges has been correlated with the LO phonon generated 9 In some cases electrical fields due to microfields. phonons were also found to be involved.’2 piezoelectric It has been shown previously’2 that in cubic crystals h~ 0= hc~LA= 13.6 meV below approximately l00°Kand hc~0= hWLO = 43.5 meV up to 300°K. In the present work we extend this research to higher temperatures in the cubic case also of check on the behavior of slope parameter in and the case hexagonal 2H and polytypic 4H ZnS crystals, where two directions of polarization of the incident light have to be considered.
THE URBACH RULE IN ZnS CRYSTALS
194
Vol. 17, No.2
7.(. Zn S 6.5
measure the absorption spectra. The system has been described elsewhere.’4 Figure 1 shows a typical absorption spectrum, which
3C
,
/
T~ 728°K -
/
served to evaluate the experimental results. We observe
I’ 6.0
two well differentiated regions. A straight line of steep slope in the in a vs hv plot near the edge and a straight line of lower slope extending to the longer wavelengths. This latter part of the curve is most probably due to charged impurities in the crystal7’8”5 and will be dealt with later.
‘
‘
/ 55
-
5.0 3.26
1ev)
PHOTON ENEkGY
The slope at higher densities was influenced by the sample temperature and was in addition identical for different samples of the same crystallografic
3.38
structure. This is considered to be the phonon shifted absorption edge and all the data published here and
FIG. 1. Natural logarithms of the absorption coe~. ficient a vs photon energy. Cubic ZnS, T = 728 K.
2HJ.LE 2.0
ZnS
50
2.
2H.EII~
343/12
100
,~..._)“b
150
200
250
4H,EIC ZnS 248/PR
//
20 05
300
.—“‘5”b .—‘
o~~0
2.0
50
343/12
100
ISO
200
250
300
2~0
~
4H,EUc ZnS 248/58 —I.
I
“.~
15
2~
3~
~
TEMPERATURE.°K
TEMPERATURE.
°~(
FiG. 2. The slope parameter a vs temperature for 2H and 4H structures of ZnS, for both perpendicular and parallel directions of the electric vector E of the polarized light. Lines theoretical, according to equation 2 and Table 1; points experimental. —
—
Vapor gro~n~3 single crystals of a thickness varying from 7—50gm were used in this work. A Xenon 500W high pressure lamp served as the light source and a Jarell—Ash 0.5 m monochromator was used to
previouslylLi4 pertain to this absorption region, which in our crystals corresponds to an absorption coefficient a from about 102 to several times ~
Vol. 17, No.2
THE URBACH RULE IN ZnS CRYSTALS
195
Table 1. The slope pammetera0 and values ofphonon energies which fit the experimental results ZnS structure
Polarization of light
Temperature range (°K) Oo (a) 10— 90 (b) 90—300 (a) 10—100 (b) 100—300 (a) 10—140 (b)140—300
1.2 2.5 1.2 2.6 1.5 2.5
13.6 43.5 9.9 43.4 11.2 43.4
LA LO (LA) LO (LA) LO
E C
(a) 10—140 (b) 140—300 (a) 10—130
1.5 2.5
E II C
(b) 130—300
2.6
12.4 43.4 13.6 43.4
(LA) LO (LA) LO
3C E IC 4H EJIC
2H
___________________________ ‘
2.0
,._
1.5-
/
-
ZnS
,
in our case the piezoelectric LA phonons, while at the
higher temperatures the fields are due to the LO 2 14
the value of A.axis phonon with a wave-vector small enough to provide a quasi.constant electric field over the exciton radius.9 The same A.axis phonon was also
/ ~ / 0.5
100
1.5
phonons.’ The LA phonon energy involved, 13.6 meV, fits
1.0.
0
Phonon energy h~o(meV) Identification
200
300
400
TEMPERATURE
500
600
700
800
,
FiG. 3. The slope parameter a vs temperature for 3C cu Figures 2 and 3 show the value of the slope para’ meter as a function of temperature between 10 and 300°Kin the 2H and 4H structures and up to 750 K in the cubic case. Two clearly distinct regions have been observed in all the above samples: one below approximately 1 00°Kand the other above it. The lines are theoretical, using equation (2) and the parameters tabulated in Table 1. Both a 0 and hw0 of equation (2) change their values in the vicinity of lOO°K.The change 5 in a0 is due to the change in the coupling constant between the electrons and phonons, while the change in hc~., 0is due to the difference 4”6 in the phonon energy of the interacting phonons. As the interactions are due to electrical fields,9”0 we can postulate that at low temperatures the highest mean-square fields (F2 ) are due to lower energy phonons,
observed in second order Raman scattering.’6 The coupling factor entering is the polaron coupling constant17 aLa in the LO case and a similar parameter ciLA for the piezoelectric polaron18 in the LA case. The ratio ao(LO)/ao(LA) gives an idea on the relative strength of these different types of coupling in ZnS.~ The long-wave impurity induced tails (Fig. 1) were also influenced by temperature, like the higher density region. The slope parameter of this part of tail rose again from a small value at low temperature, reaching the value of a 1.0 at approximately 350°Kand then remained constant at higher temperatures, up to 750°K. The temperature shift of these tails will be discussed elsewhere 20 In conclusion, we have shown that in the investigated structures of ZnS different types of phonons influence the absorption edge slope in different tern: perature intervals. This should be also true of other Il—VI compounds, have a piezoelectric of lower energy thanwhich the LO phonon. A similarphonon behavior was indeed observed recently in ZnO.21 At high enough temperatures, above about half the Debye temperature, the LO phonon created fields are high enough to have a major influence and the slope parameter will change
196
THE URBACH RULE IN ZnS CRYSTALS
in accordance. No transition to multiphonon processes was observed at higher temperatures, and the slope
Vol. 17, No.2
parameter a remains practically constant above the Debye temperature.
REFERENCES 1.
PIPER W.W.,Thys. Rev. 92,23 (1953). 2. SHACHAR G., Ph.D. thesis, The Hebrew University, Jerusalem (1969). 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
URBACH F., Phys. Rev. 92, 1324 (1953); MOSER F. and URBACH F., Phys. Rev. 102, 1519 (1956). MAHRH.,ThyL Rev. 125,1510(1962); 132, 1880(1963). TOYOZAWA Y.,frogr~Theoret. Phys. 22,455 (1959). TOYOZAWA Y., Suppi. Progr. Theorer. Phys. (Japan) 12, 111(1959). DIXON J.R. and ELUS J.M.,Thys. Rev. 123, 1560 (1961). REDFIELD D. and AFROMOWITZ M.A.,Appl. Phys. Lett. 11, 138 (1967). DOW J.D. and REDFIELD D.,Thys. Rev. B5, 594 (1972). REDFIELD D., Trans. New York Acad. ScL 26, 590 (1964). DEXTER D.L.,Phys. Rev. Lert. 19, 1383 (1967). BRADA Y. and YACOBI B.G., in F~oc.12th mt. Conf Phys. Semicond. (Edited by PILKUHN M.H.), p. 1212, Teubner, Stuttgart, (1974). ALEXANDER E., L~LMANZ.H., MARDIX S. and STEINBERGER I.T., Phil Mag. 21, 1237 (1970). YACOBI BG., BRADA Y., LACHISH U. and HIRSH C. (to be published in Phys. Rev. B! 5).
16.
JENSEN GH.,froc. 12th mt. Conf Phys. Semicond. (Edited by PILKUHN M.H.), p. 1217. Teubner, Stuttgart (1974). NILSEN W.G.,Thys. Rev. 182, 838 (1969).
17.
FROHLICHH.,Adv. Phys. 3,325(1954).
18.
DOW J.D., SMITH Di. and LEDERMAN F.L.,Phys. Rev. B8, 4612 (1973).
19. 20.
TOYOZAWA Y., Techn. Rept. ofISSP, Series A, 119, (1964). BRADA Y. and YACOBI B.G. (to be published). JENSEN G.H. and SKETTRUP T., Phys. Status Solidi (b) 60, 169 (1973).
15.
21.