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Absorption edge shift: Internal electric field model
Volume 55A, number 5
PHYSICS LETTERS
29 December 1975
ABSORPTION EDGE SHIFT: INTERNAL ELECTRIC FIELD MODEL B.G. YACOBI* Racah Institute of Physics, The Hebrew University, Jerusalem, Israel Received 6 November 1975 The shift of the optical absorption edge caused by defect-induced electric fields through Redfield’s internal FranzKeldysh mechanism is discussed for semiconductors and insulators.
In recent years the electric fields of charged defects have been accepted as a possible reason for the observed behaviour of the optical absorption edge in semiconductors, insulators and amorphous crystals [1—3].The earliest attempt to explain the Urbach rule [4] and the broadening of the absorption edge via the above-mentioned electric fields of charged defects, such as phonons, impurities and dislocations, was made by Redfield [1]. The effect of these fields on a crystal is essentially an internal analog of the Franz-Keldysh effect (FKE) [5, 6]; FKE is the electric field-induced shift of the absorption edge, and for exponential edges this shift ~Eg is: L~Eg=
—
C(a/kT)2F2
(1)
where C = e2i~2/24,i*is a constant, containing /2*, the reduced effective mass, a is the slope parameter which enters the absorption coefficient expression a = ~ exp{a(hr’— hv 0)/kT} and Fis the applied electric field. This is a weak-field approximation (F l0~ 106 V/cm) for direct band-to-band transitions and with a uniform electric field strength in a small (commensurable with the excitonic radius) volume of the crystal. Thus, any charged defect, the electric field of which meets the above requirements of the FKE, can be considered to be responsible for the absorption edge shift. The origin of these internal fields has been discussed in several works [1—3,7]. Experiments on optical absorption edges have also been performed and the results have been explained in the light of the above theoretical models [1—3].Thus the electric fields of charged impurities were found to be responsi—
*
Present address: Department of Metallurgy and Materials Science, Imperial College of Science and Technology, London SW7, UK.
ble for the Urbach slope parameter and the absorption edge shift in covalent semiconductors [8]. In Il-VI compounds phonon-generated microfields were found to have the major influence on temperature dependence of the absorption edge and the Urbach slope parameter [9, 10]. The effect of dislocations on the absorption edge was also detected in silicon [11], and the relative strength of different internal fields was considered as well [12]. Recently, the optical absorption edges in amorphous crystals have also been studied extensively and experiments indicate clearly a similar phenomenon of the absorption edge shift in disordered crystals. Both the Urbach and the by absorption edge energy wereslope foundparameter to be influenced the internal fields in these materials [13]. This similarity in exponential behaviour of the absorption coefficient (the Urbach rule) in a wide variety of crystals has led to the internal electric field (IEF) model for explaining the Urbach rule [2]. Rcently, attempts were made to compute the cornbined influence of different electric fields (external and internal) on the absorption edge [14, lb]. It was also pointed out [15] that eq. (1) is well suited for disordered crystals as well, as it gives reasonable values for the j.f”. The above-mentioned ideas suggest the use of the internal Franz-Keldysh mechanism for the detection of defect-induced electric fields in semiconductors and insulators with exponential absorption edges by using eq. (1). If the defect-induced shift of the absorption edge z~Egiis compared with the shift of the edge ~Ege due to the externally applied field Fe on a crystal without impurities, the internal field will be: 1/2~ =
(2)
Fe(ae/ai)( L~”L~Egi/P: ~Ege)
313
Volume 55A, numberS
PHYSICS LETTERS
Here is the impurity-related effective mass [16], is the reduced effective mass of a “pure” sample, is the slope parameter of the impurity-induced absorption region [17, 18], and Ge is the slope parameter of a “pure” crystal. /2j~can be found from FKE measurements on the impurity-induced absorption region [18]. Comparing z~Egjwith the shift of the edge L\Egei due to the externally applied field Fe on the impurity-induced absorption region, we can write: F~= Fe(L~Egi/~Egei)ltl. It is clear that when several internal fields participate in the absorption process, eq. (2) will be more complicated. Such a complicated influence of LAand LO-phonon-generated fields was shown previously to be responsible for the temperature shift of the absorption edge [9, 10]. Further studies of the IEF model on other defect-induced fields are indicated by the successful application of the above considerations in the case of phonon-generated fields. As has been shown previously [12], the temperature behaviour of various internal fields is expected to be different. This enables us to evaluate their contribution in some cases through temperature-dependent measurements of IEF effect. Therefore, combining the “external” FKE with its “internal” analog can furnish the tool for measuring the internal electric fields of defects in semiconductors and insulators. More experiments are needed before this model can be shown to provide accurate general applicability. The author wishes to thank Dr. Y. Brada for many helpful discussions.
314
29 December 1975
References [1] D. Redfield, Phys. Rev. 130 (1963) 914,916. [2] J.D. Dow and D. Redfield, Phys. Rev. B5 (1972) 594. [3] J. Tauc, Mater. Res. Bull. 5 (1970) 721. [4] F. Urbach, Phys. Rev. 92 (1953) 1324; F. Moser and F. Urbach, Phys. Rev. 102 (1956) 1519. [5] W. Franz, Z. Naturforsch. A13 (1958) 484. [6] L.V. Keldysh, Soviet Phys.-JETP 7 (1958) 788. [7] J.D. Dow, D.L. Smith and F.L. Lederman, Phys. Rev. (1973) 4612. [81 B8 D. Redfield and M.A. Afromowitz, Appl. Phys. Lett. 11(1967)138. [9] Y. Brada and B.G. Yacobi, in Proc. 12th Intern. Conf. on the Physics of semiconductors, ed. M.H. Pilkuhn (Teubner, Stuttgart, 1974), p. 1212. [10] B.G. Y. Brada, Rev. Yacobi, Bli (1975) 2990. U. Lachish and C. Hirsch, Phys. [11] S.Kh. Mil’shtein and B.G. Yacobi, Phys. Lett. A, to be published.
[121G.H. Jensen,
in Proc. 12th Intern. Conf. on the Physics of semiconductors, ed. M.H. Pilkuhn (Teubner, Stutt1974), p.and 1217. [13] gart, K.L. Chopra S.K. Bahi, Thin Solid Films 11(1972) [14] J.G. Krieg, Z. PhysIk 205 (1967) 425. [15] B. Esser, Phys. Status Solidi (b) 51(1972) 735. [16] S.T. Pantelides, in Proc. 12th Intern. Conf. on the Physics of1974), semiconductors, ed. M.H. Pilkuhn (Teubner, Stuttgart, p. 396. [17] Y. Brada, B.G. Yacobi and A. Peled, Solid State Commun. 17 (1975) 193. [18] B.G. Yacobi and Y. Brada, Solid State Commun., to be published.
PHYSICS LETTERS
29 December 1975
ABSORPTION EDGE SHIFT: INTERNAL ELECTRIC FIELD MODEL B.G. YACOBI* Racah Institute of Physics, The Hebrew University, Jerusalem, Israel Received 6 November 1975 The shift of the optical absorption edge caused by defect-induced electric fields through Redfield’s internal FranzKeldysh mechanism is discussed for semiconductors and insulators.
In recent years the electric fields of charged defects have been accepted as a possible reason for the observed behaviour of the optical absorption edge in semiconductors, insulators and amorphous crystals [1—3].The earliest attempt to explain the Urbach rule [4] and the broadening of the absorption edge via the above-mentioned electric fields of charged defects, such as phonons, impurities and dislocations, was made by Redfield [1]. The effect of these fields on a crystal is essentially an internal analog of the Franz-Keldysh effect (FKE) [5, 6]; FKE is the electric field-induced shift of the absorption edge, and for exponential edges this shift ~Eg is: L~Eg=
—
C(a/kT)2F2
(1)
where C = e2i~2/24,i*is a constant, containing /2*, the reduced effective mass, a is the slope parameter which enters the absorption coefficient expression a = ~ exp{a(hr’— hv 0)/kT} and Fis the applied electric field. This is a weak-field approximation (F l0~ 106 V/cm) for direct band-to-band transitions and with a uniform electric field strength in a small (commensurable with the excitonic radius) volume of the crystal. Thus, any charged defect, the electric field of which meets the above requirements of the FKE, can be considered to be responsible for the absorption edge shift. The origin of these internal fields has been discussed in several works [1—3,7]. Experiments on optical absorption edges have also been performed and the results have been explained in the light of the above theoretical models [1—3].Thus the electric fields of charged impurities were found to be responsi—
*
Present address: Department of Metallurgy and Materials Science, Imperial College of Science and Technology, London SW7, UK.
ble for the Urbach slope parameter and the absorption edge shift in covalent semiconductors [8]. In Il-VI compounds phonon-generated microfields were found to have the major influence on temperature dependence of the absorption edge and the Urbach slope parameter [9, 10]. The effect of dislocations on the absorption edge was also detected in silicon [11], and the relative strength of different internal fields was considered as well [12]. Recently, the optical absorption edges in amorphous crystals have also been studied extensively and experiments indicate clearly a similar phenomenon of the absorption edge shift in disordered crystals. Both the Urbach and the by absorption edge energy wereslope foundparameter to be influenced the internal fields in these materials [13]. This similarity in exponential behaviour of the absorption coefficient (the Urbach rule) in a wide variety of crystals has led to the internal electric field (IEF) model for explaining the Urbach rule [2]. Rcently, attempts were made to compute the cornbined influence of different electric fields (external and internal) on the absorption edge [14, lb]. It was also pointed out [15] that eq. (1) is well suited for disordered crystals as well, as it gives reasonable values for the j.f”. The above-mentioned ideas suggest the use of the internal Franz-Keldysh mechanism for the detection of defect-induced electric fields in semiconductors and insulators with exponential absorption edges by using eq. (1). If the defect-induced shift of the absorption edge z~Egiis compared with the shift of the edge ~Ege due to the externally applied field Fe on a crystal without impurities, the internal field will be: 1/2~ =
(2)
Fe(ae/ai)( L~”L~Egi/P: ~Ege)
313
Volume 55A, numberS
PHYSICS LETTERS
Here is the impurity-related effective mass [16], is the reduced effective mass of a “pure” sample, is the slope parameter of the impurity-induced absorption region [17, 18], and Ge is the slope parameter of a “pure” crystal. /2j~can be found from FKE measurements on the impurity-induced absorption region [18]. Comparing z~Egjwith the shift of the edge L\Egei due to the externally applied field Fe on the impurity-induced absorption region, we can write: F~= Fe(L~Egi/~Egei)ltl. It is clear that when several internal fields participate in the absorption process, eq. (2) will be more complicated. Such a complicated influence of LAand LO-phonon-generated fields was shown previously to be responsible for the temperature shift of the absorption edge [9, 10]. Further studies of the IEF model on other defect-induced fields are indicated by the successful application of the above considerations in the case of phonon-generated fields. As has been shown previously [12], the temperature behaviour of various internal fields is expected to be different. This enables us to evaluate their contribution in some cases through temperature-dependent measurements of IEF effect. Therefore, combining the “external” FKE with its “internal” analog can furnish the tool for measuring the internal electric fields of defects in semiconductors and insulators. More experiments are needed before this model can be shown to provide accurate general applicability. The author wishes to thank Dr. Y. Brada for many helpful discussions.
314
29 December 1975
References [1] D. Redfield, Phys. Rev. 130 (1963) 914,916. [2] J.D. Dow and D. Redfield, Phys. Rev. B5 (1972) 594. [3] J. Tauc, Mater. Res. Bull. 5 (1970) 721. [4] F. Urbach, Phys. Rev. 92 (1953) 1324; F. Moser and F. Urbach, Phys. Rev. 102 (1956) 1519. [5] W. Franz, Z. Naturforsch. A13 (1958) 484. [6] L.V. Keldysh, Soviet Phys.-JETP 7 (1958) 788. [7] J.D. Dow, D.L. Smith and F.L. Lederman, Phys. Rev. (1973) 4612. [81 B8 D. Redfield and M.A. Afromowitz, Appl. Phys. Lett. 11(1967)138. [9] Y. Brada and B.G. Yacobi, in Proc. 12th Intern. Conf. on the Physics of semiconductors, ed. M.H. Pilkuhn (Teubner, Stuttgart, 1974), p. 1212. [10] B.G. Y. Brada, Rev. Yacobi, Bli (1975) 2990. U. Lachish and C. Hirsch, Phys. [11] S.Kh. Mil’shtein and B.G. Yacobi, Phys. Lett. A, to be published.
[121G.H. Jensen,
in Proc. 12th Intern. Conf. on the Physics of semiconductors, ed. M.H. Pilkuhn (Teubner, Stutt1974), p.and 1217. [13] gart, K.L. Chopra S.K. Bahi, Thin Solid Films 11(1972) [14] J.G. Krieg, Z. PhysIk 205 (1967) 425. [15] B. Esser, Phys. Status Solidi (b) 51(1972) 735. [16] S.T. Pantelides, in Proc. 12th Intern. Conf. on the Physics of1974), semiconductors, ed. M.H. Pilkuhn (Teubner, Stuttgart, p. 396. [17] Y. Brada, B.G. Yacobi and A. Peled, Solid State Commun. 17 (1975) 193. [18] B.G. Yacobi and Y. Brada, Solid State Commun., to be published.