Urbach tail for ferroelectric materials with an order-disorder-type phase transition Ken-ichi Noba and Yosuke Kayanuma College of Engineering, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan Received 4 November 1998 We present a model describing the Urbach tail of exciton absorption spectra in ferroelectric materials with order-disorder-type phase transition. In addition to the thermal effect of exciton-lattice interactions, the internal Stark effect by the local electric field due to the displacement of protons is taken into account. The random distribution of the direction of local electric field induces a random polarization of the lowest exciton state. This results in an additional randomness of the exciton band in the disordered phase. The anomalous tempera- ture dependence of the steepness parameter for the Urbach tail observed in PbHPO 4 is reproduced by a numerical simulation within the present model. S0163-18299903128-8 In many insulators, the low-energy tail of fundamental absorption edge depends exponentially on the energy of pho- tons. Since the first report by Urbach, 1 this rule has been widely observed in many kinds of materials. The tail of the spectra called the Urbach tail is empirically given as a func- tion of photon energy E, F  E  =A exp  - E 0 -E k B T  , 1 where the convergence energy E 0 and the steepness param- eter  are almost constant at sufficiently high temperatures. 2 In order to explain the origin of the Urbach tail, a number of theoretical studies have been carried out. 3–12 Among them, two types of theories seem to be worth mentioning at present. One is the theory based on the model of the internal Stark effect. 4,7,11 In this theory, it is asserted that the Franz- Keldysh effect by the microfield due to lattice vibrations is responsible for the appearance of the exponential tail in the absorption edge. Dow and Redfield 11 have made a compre- hensive survey of Urbach’s rule from the viewpoint of this microfield model. On the other hand, Toyozawa and co-workers 3,8–10 developed a theory of Urbach’s rule on the basis of the model of short-range interaction of phonons with the center-of-mass motion of excitons. According to the theory of Sumi and Toyozawa, 9 thermal fluctuations of lat- tice systems result in an adiabatic modulation of site ener- gies, which can be described by a Gaussian distribution of random potentials. It was shown that the optical transitions to the exciton state momentarily trapped by this local poten- tial gives rise to the exponential tail at the absorption edge. By applying Einstein’s relation to the absorption and emis- sion spectra, they succeeded in deriving a criterion described in terms of the steepness parameter  , which determines whether the exciton is self-trapped or not. Furthermore, nu- merical simulations by Schreiber and Toyozawa 13 have clearly demonstrated that the model of momentarily trapped exciton reproduces well the line shape given by Eq. 1, in- cluding its temperature dependence. At least for insulating materials with relatively strong electron-phonon coupling, this model gives the most reasonable explanation of the Ur- bach tail. It should be noted that the Urbach tail is widely observed also in the absorption spectra of amorphous materials: 14–16 The steepness of the exponential tail is essen- tially independent of temperatures in this case. It can be said, therefore, that we detect various kinds of randomness of crystals through the measurement of the Urbach tail. PbHPO 4 is a ferroelectric material containing hydrogen bonds. The ferroelectric transition occurs at T c =310 K. This transition is a second-order one, and the spontaneous polar- ization P s slowly rises below T c . 17–19 It is known that the one-dimensional ordering of protons between the two stable configurations of O—H—O bonds is a primary cause of this phase transition. 18 Although displacements of heavy ions are also observed, its temperature dependence is not directly cor- related with that of P s . Recently, Kida et al. 20 studied ex- perimentally the optical-absorption spectra of PbHPO 4 . The first exciton peak has been assigned as due to the cationic excitation of 6 s ˜6 p in Pb 2 + ions. They found that the Ur- bach tail of the absorption spectra exhibits an anomalous temperature dependence. It was pointed out that, as the tem- perature is lowered below T c , the steepness parameter  increases from the value above T c , and this increment be- haves similarly to P s as a function of temperature. This result strongly suggests that the ordering of protons has a close relation with the anomalous behavior of  . In the present study, we propose a simple model for the Urbach tail of exciton absorption in ferroelectric materials with order-disorder-type phase transition. Our theory is es- sentially based on the model of the internal Stark effect. It is quite natural to expect that strong electric fields are acting on the internal motion of excitons in ferroelectric materials. Generally it is difficult to make a precise evaluation of the strength of microscopic fields in crystals. As a very rough estimation for PbHPO 4 , we take the electric field E =2.4 10 7 V cm -1 at low temperatures, which is calculated from the saturated value of P s =2.1  C cm -2 . 17 This is an order of magnitude larger than those estimated in Ref. 11 for the field strength induced by the LO phonons in alkali halides at room temperature. Such an electric field will strongly polar- ize the excitons as shown in Ref. 11. It should be noted, however, that the conventional microfield model of the Ur- bach tail will lead to the prediction that the steepness param- eter  becomes smaller below T c since the effective-field strength increases below T c . This is in contradiction with the experimental observation, in which  becomes larger below PHYSICAL REVIEW B 15 AUGUST 1999-I VOLUME 60, NUMBER 7 PRB 60 0163-1829/99/607/44184/$15.00 4418 ©1999 The American Physical Society