Indirect band gap and optical parameters of pure and doped potassium ferrocyanide single crystals

Physica B 292 (2000) 221}232

Indirect band gap and optical parameters of pure and doped potassium ferrocyanide single crystals M.A. Ga!ar*, A. Abu-El Fadl Physics Department, Faculty of Science, Assiut University, Assiut, Egypt Received 7 January 1999; received in revised form 23 September 1999

Abstract The optical transmittance for potassium ferrocyanide (KFCT) single crystals were measured as a function of the wavelength at di!erent temperatures in the range 275}320 K along the two axes (I 0 1 ) and (0 1 0) and hence the absorption coe$cient (a) and optical band gap E were deduced. The type of transition was determined and the validity  of Urbach's rule was checked. The steepness parameter (p) and the exciton energy E were calculated at di!erent  temperatures. The temperature dependence of the energy gap was estimated and the result was con"rmed through the calculation of dE /d¹ from information about p. The refractive index, the extinction coe$cient and both the real and  imaginary parts of the dielectric permittivity were also calculated as a function of photon energy and the results were discussed. The e!ect of doping KFCT with 2% of sodium nitroproced, cobalt sulphate or potassium dichromate on the same physical parameters was also discussed.  2000 Elsevier Science B.V. All rights reserved. PACS: 77.80 Keywords: Ferroelectrics; Potassium ferrocyanide crystals; Optical properties

1. Introduction Potassium ferrocyanide trihydrate K [Fe(CN) ] ) 3H O, hereafter KFCT, is a mem   ber of the family of crystals having the general formula A [B(CN) ] ) 3H O where A is K or NH     while B is Fe, Mn, Ru or Os. In crystalline KFCT the three water molecules are located in layers perpendicular to the (0 1 0) axis and between each layer there are two layers of the Fe(CN)\ groups 

* Corresponding author. Tel.: #20-88-411-438; fax: #2088-312-564. E-mail address: [email protected] (M.A. Ga!ar).

interspersed by K> ions while the Fe> ion is located at the center of the regular octahedra formed by six cyanide ions [1]. This molecular-type ionic crystal undergoes an interesting second-order ferroelectric phase transition below room temperature, at 249 K. This transition is an order}disorder type in which the spontaneous polarization arises along the (1 0 1 ) direction from a collective ordering of the water molecules [2}4]. At the phase transition point the crystal symmetry changes from monoclinic paraelectric, space group C (C /c), to   a monoclinic ferroelectric, space group C (C ) [5].   Extensive investigations have been carried out on the crystal structure and the problem of order}disorder transitions in hydrogen-bonded crystals like

0921-4526/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 4 6 6 - X

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KFCT. Many publications were reported on the study of physical properties of KFCT crystals using di!erent techniques like Raman, infrared, NMR spectroscopy and X-ray absorption near the edge structure [6}9]. Toyoda et al. [10] reported that virgin, as grown, crystals of KFCT have a tetragonal structure which undergoes a monotropic phase transition at about 218 K and transfers into a twined monoclinic phase. Under practical conditions of crystal growth, KFCT often appears with a polytropic structure in which alternating monoclinic and tetragonal layers are found [11]. At 218 K the tetragonal modi"cation of KFCT is changed irreversibly into monoclinic [11]. Savatinova and Anachkova [12] studied the temperature dependence of the two Raman active modes in the vicinity of the phase transition point. The mode bandwidth undergoes a sharp increase which is assigned to the reduction of the barrier height characterizing the water Brownian motion. Using Mason's relaxation theory, they were able to correlate the temperature behavior of the barrier height with the temperature and frequency dependence of the dielectric permittivity of KFCT. Amov et al. [13] assumed that the conductivity in KFCT is electronic in nature which may be produced by a de"nite type of impurity centers and that the increase in the conductivity during the transition might be due to the injection of charges into the crystal from the electrodes. Ga!ar et al. [14] studied the e!ect of thermal recycling on the metastable structure of virgin samples of potassium ferrocyanide single crystals and concluded that the phase transition at 249 K has a `multistagea character. The stable structure of KFCT could be achieved only after thermal recycling treatment. Little attention has been paid to the study of optical properties near the absorption edge of this crystal. In the present work, measurements of the absorption coe$cient at di!erent temperatures from 275 K upto 320 K have been performed. This study is particularly important because it may shed some light on the electronic structure of the optical band gap, in addition it may "ll part of the noticeable gap in the literature about this point. The

e!ect of doping KFCT crystals with di!erent types of metallic molecules on the measured parameters was another goal of this article.

2. Experimental Single crystals of KFCT were grown using the slow evaporation technique from a KFCT aqueous solutions at 316 K. The addition of some materials such as KOH or K CrO (about 2 g/l KFCT) was   found to be useful to have a perfect crystals morphology [15]. Over a period of about seven weeks, large (4;3.5;3 cm), transparent, well-de"ned planes and yellow colored single crystals could be obtained. Samples of KFCT in the form of slabs having dimensions 0.3;0.5;0.1 cm with its smallest length parallel to one of the two axes (1 0 1 ) (which is the polar axis) and [0 1 0] were cut using a wet thread saw. The samples used were clear, transparent and free from any noticeable defects, i.e. optically perfect. Crystals of KFCT were also grown in the presence of 2 mol% of either of sodium nitroprocid, Na [Fe(CN) NO] ) 2H O, cobalt sulphate    CoSO ) 7H O or potassium dichromate   K Cr O (here after SNP, Co and Cr, respective   ly). The additives were chosen to be with di!erent physico-chemical activities. SNP has a unit cell symmetry and dimensions close to that of KFCT crystals. Cr> is believed to substitute Fe> in the lattice while Co> may replace K> or exist in interstitial position in the lattice. The optical transmittance and re#ectance of the samples were recorded using Shimadzu UV-VIS2101 PC dual beam scanning spectrophotometer in the wavelength ranged from 350 to 900 nm. The surrounding medium was air and all the measurements were carried out between 275 and 320 K. The system comprises photometer unit, a microcomputer of the IBM PC/PS2 type and UV-2101 PC personal spectroscopy software package. The re#ectance measurements were made using specular re#ectance attachment. The relative specular re#ectance was measured at an incident angle of 53 while the sample was placed horizontally on the stage facing downward and illuminated from the bottom. The temperature of the sample was

M.A. Gawar, A. Abu-El Fadl / Physica B 292 (2000) 221}232

controlled between 275 and 320 K using ultra thermostat [mgw Lauda type K 2R]. The temperature of the sample was measured with an accuracy of $0.1 K using a chromel}alumel thermocouple and a DC microvoltmeter (Philips type PM 2434). The measurements were performed at each temperature after being sure of thermal stabilization. The overall accuracy of our measurements was about 3%. The absorption coe$cient a was calculated in the present work using the formula:




1 1 a"! ln , ¹ d

(1)

where d is the thickness of the sample and ¹ is the transmittance.

3. Results and discussions 3.1. Absorption coezcient The results of the absorption coe$cient a against photon energy u for the un-doped KFCT crystals along the two crystallographic directions (0 1 0) and (1 0 1 ) and for KFCT crystals doped with 2% of SNP, Cr or Co along the (1 0 1 )-axis at di!erent temperatures are presented in Fig. 1. The general trend of the dependence is continuous increasing of a with increasing u. Near the absorption edge a increases more rapidly with u. As it is clear, doping KFCT a!ects both the magnitude of a and the shape of the a- u dependence at all energies and temperatures. Several weak absorption bands due to Cr>centers were observed in the visible region at 1.60 eV (j"775 nm) and between 2.15 and 2.35 eV (j between 577 and 528 nm) in the case of Cr-doped KFCT crystals. Similar absorption bands due to Fe> centers in K SeO   single crystals were also observed by Pacesova et al. [16]. The absorption coe$cient a of a crystalline solid obeys the following relationship [17]: a uJ( u!E )L, (2)  where E is the optical gap, u is the photon energy  and n being an index which characterizes the op-

223

tical absorption process. For direct allowed transition n", for direct forbidden transition n",   for indirect allowed transition n"2 and, "nally, for indirect forbidden transition n"3. For indirect allowed transitions, the measured absorption coe$cient a can be expressed as: a "a #a  where symbol a and a being, respectively, the  contributions due to emission and absorption of phonons. According to Philipp and Taft [18] and Tandon [19] the temperature and energy dependence of the absorption coe$cient takes the form ( u!E #E ) ( u!E !E )   #   a ( u)J . [exp(E /k¹)!1] [1!exp(!E /k¹)]   (3) The "rst term corresponds to photon absorption with absorption of phonon with energy E and is to  be taken zero for u(E !E , while the second   term represents the contribution of photon absorption with emission of phonon with energy E and  must be taken to be zero for u(E #E .  

 

A ( u!E #E )   , u'E !E ,  

u [exp(E /k¹)!1] a "  0,

u(E !E ,   A ( u!E !E )   , a " u [1!exp(!E /k¹)]  0,

u'E #E ,  

u(E #E ,  

where A is the constant of proportionality between the absorption coe$cient and photon energy. For simplicity, at certain constant temperature, let us consider A B(¹)" . (4) [exp (E /k¹)!1]  Hence, the contribution to the absorption coe$cient due to phonon absorption may be represented by B(¹)( u!E #E )   , u'E !E ,  

u a " 0,

u(E !E ,  



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M.A. Gawar, A. Abu-El Fadl / Physica B 292 (2000) 221}232

Fig. 1. (a) Dependence of the absorption coe$cient a on the photon energy at di!erent temperatures for KFCT crystals along the (0 1 0) axis. (b) Dependence of the absorption coe$cient a on the photon energy at di!erent temperatures for KFCT crystals along the (1 0 1 ) axis. (c) Dependence of the absorption coe$cient a on the photon energy at di!erent temperatures for Co-ions doped KFCT crystals along the polar axis. (d) Dependence of the absorption coe$cient a on the photon energy at di!erent temperatures for Cr-ions doped KFCT single crystals along the polar axis. (e) Dependence of the absorption coe$cient a on the photon energy at di!erent temperatures for SNP-doped KFCT single crystals along the polar axis.

M.A. Gawar, A. Abu-El Fadl / Physica B 292 (2000) 221}232

Fig. 1. (Continued ).

The corresponding relation for phonon emission is given by



B(¹)( u!E !E )   , u'E #E ,  

u a "  0,

u(E #E ,   Where E is the indirect energy gap and B(¹) is  a constant similar to B(¹) and both are nearly independent of photon energy and known as the disorder parameter [20]. It was found that this parameter is sensitive for phase transitions in ferroelectrics [21]. An analysis of absorption data was carried out to determine the predominant optical transition. According to the power law of Eq. (2), Fig. 2 represents a plot between (a u) and u. The points lie close to one straight line for E !E ( u(   E #E to a steeper straight line for u'   E #E . This behaviour is just what one expects if   the absorption in this range is due to indirect transition in which there is a marked change in momentum between the initial and "nal states and

225

in which direct transition do not play a role. Accordingly, the value of n is 2 and hence the type of transition is the allowed indirect. The values of the optical band gap E and the constant B could be  determined from Fig. 2 by the least-squares "ttings of the data. The values of E obtained for un-doped  and doped KFCT crystals at di!erent temperatures are presented in Fig. 3 while the corresponding values of the disorder parameter B are graphically represented in Fig. 4. From Fig. 4 it is clear that pure crystals along the (0 1 0) as well as crystals doped with Co or SNP have the same attitude while pure crystals along the polar axis has a maximum B at 307 K. Crystals doped with Cr has a plateau between 289 and 305 K. From the overall behaviour one can say that there are anomalous changes in the value of B around 307 K. The graph representing (a u) against u (Fig. 2) may be resolved into two straight-line portions. The straight line obtained at lower photon energies corresponds to phonon absorption process and cuts the energy axis at E !E . The other line   which represents the dependence in the high-energy range corresponds to a phonon emission process and cuts the energy axis at E #E . From the   intercept of the two straight lines, the values of E and E at room temperature were calculated   and they are listed in Table 1. From Fig. 3 and Table 1 it is clear that doping KFCT crystals with Co>, Cr> or SNP decreases the value of E . The maximum decrease is observed  in case of doping with Co> while SNP has less e!ect on E which may be due to the similarity in  the crystal lattice between KFCT and SNP. The phonon energy changes from one dopant to another, its value is too high to be considered as lattice phonon. So, we believe that the structure observed in the (a u) plot is not due to emission and absorption of phonon. The structure may be ascribed to the presence of two valence bands which originate due to the spin}orbit interaction and crystal-"eld splitting. 3.2. Temperature dependence of the absorption edge The equation relates the absorption coe$cient (a) with the photon energy u at di!erent

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Fig. 2. (a) Dependence of (a u) on the photon energy at di!erent temperatures for KFCT crystals along the (0 1 0) axis. (b) Dependence of (a u) on the photon energy at di!erent temperatures for KFCT single crystals along (1 0 1 ) axis. (c) Dependence of (a u) on the photon energy at di!erent temperatures for Co-doped KFCT single crystals along the polar axis. (d) Dependence of (a u) on the photon energy at di!erent temperatures for Cr-ions doped KFCT single crystals along the polar axis. (e) Dependence of (a u) on the photon energy at di!erent temperatures for SNP-doped KFCT single crystals along the polar axis.

M.A. Gawar, A. Abu-El Fadl / Physica B 292 (2000) 221}232

227

Fig. 4. Temperature dependence of the disorder parameter (B) for pure and doped KFCT single crystals.

Table 1 Values of E , E and ¹ for undoped and doped KFCT crystals    at room temperature Fig. 2. (Continued ).

temperatures according to Martienssen [22], who modi"ed the Urbach's rule, takes the form





p(¹) a"a exp ( u!E ) ,   k¹

Crystal

E (eV) 

E (eV) 

¹ (K) 

Undoped (0 1 0) Undoped (1 0 1 ) KFCT#Co KFCT#Cr KFCT#SNP

2.40$0.02 2.66$0.02 1.67$0.02 1.85$0.02 2.02$0.02

0.19$0.01 0.21$0.01 0.28$0.01 0.19$0.01 0.34$0.01

2200$20 2430$20 3250$20 2200$20 3940$20

(5)

where E is a parameter characterizing the energy  gap, a is the pre-exponent constant and p(¹) is the  parameter characterizing the broadening of the absorption edge known as the steepness parameter

Fig. 3. The temperature dependence of the optical energy gap for pure and doped KFCT single crystals.

and it is temperature dependent. So, a relation between ln a and u at certain constant temperature should yield a straight line. In fact, a family of straight lines could be obtained at di!erent temperatures between 275 and 320 K indicating the exponential dependence of the absorption coe$cient a on the energy of the incoming photon u in the 2.4}3.8 eV region as required by Urbach's rule (Fig. 5). The straight lines, treated by the least-squares method, converge to a point with coordinates a and E . Table 2 contains the values of a and    E for the undoped and doped KFCT crystals.  From the table it is clear that, doping KFCT with Cr slightly increases the values of both a and E .   In the other two cases, the value of a strongly  decreases while the value of E considerably in creases. To verify the values obtained for E , iso-absorp tion curves were constructed. Fig. 6 shows the temperature dependence of the photon energies u

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M.A. Gawar, A. Abu-El Fadl / Physica B 292 (2000) 221}232

corresponding to constant values of the absorption coe$cient a. Again, all the lines converge to the same point which is identical with the value ob-

tained from Fig. 5. The same veri"cation proved equally successful for the three samples containing SNP, Co SO or K Cr O . The iso-absorption     

Fig. 5. (a) Relation between ln(a) and photon energy for KFCT single crystals at di!erent temperatures along the (0 1 0) axis. (b) Relation between ln(a) and photon energy for KFCT single crystals at di!erent temperatures along (1 0 1 ) axis. (c) Relation between ln(a) and photon energy for Co-doped KFCT single crystals at di!erent temperatures along the polar axis. (d) Relation between ln(a) and photon energy for Cr-doped KFCT single crystals at di!erent temperatures along the polar axis. (e) Relation between ln(a) and photon energy for SNP-doped KFCT single crystals at di!erent temperatures along the polar axis.

M.A. Gawar, A. Abu-El Fadl / Physica B 292 (2000) 221}232

229

Fig. 5. (Continued ).

curves in the case of Cr-doped KFCT crystals exhibit peculiar character. The iso-absorption curves taken over all the absorption range do not intersect in one point as it is expected. The absorption range was then divided into two parts. The "rst represents the low-absorption range for which it was found that the lines merge in a single point with the same E as that deduced from Fig. 5. The problem arises  in the high-absorption range where the straight lines also converge to a single point but with E lower by 13% than that in the low-absorption  range. The reason for this behaviour is not clear to us at present and we believe that this point needs more study. It is worthy knowing that the temperature range through which Urbach's rule is valid changes from one sample to the other. While Urbach's rule is valid for the Cr-doped and the undoped samples along the (0 1 0) axis almost in the whole temperature range used in this study, it is valid only in the range 304}314 K for the pure sample along the polar axis and in the range 301}317 K in case of Co- and SNP-doped KFCT crystals. The exponential behaviour of the absorption edge may be explained as a consequence of random thermal #uctuations in the band gap energy as

Fig. 6. (a) Isoabsorption curves as a function of temperature for pure KFCT crystal along (1 0 1 ) axis. (b) Isoabsorption curves at di!erent temperatures for KFCT crystals doped with Cr>: (i) low absorption range; (ii) high absorption range.

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M.A. Gawar, A. Abu-El Fadl / Physica B 292 (2000) 221}232

Table 2 Values of a and E for undoped and doped KFCT crystals   Crystal

a (cm\) 

E (eV) 

Undoped (0 1 0) Undoped (1 0 1 ) KFCT#Co KFCT#Cr KFCT#SNP

175.9$0.1 120.3$0.1 66.7$0.1 134.3$0.1 66.7$0.1

3.315$0.005 3.150$0.005 3.950$0.005 3.200$0.005 3.400$0.005




u 2k¹  , tanh p(¹)"p  u 2k¹  where p is the electron}optical}phonon coupling  constant and u is the energy of the optical  phonon most strongly bound with the electron (exciton). Developing tanh( u /2k¹) in a power  series and taking the "rst two terms we have

proposed by Skettrup [23]. In this model the steepness parameter p is related to the temperature dependence of the energy gap through the relation dE 3k ! " . p d¹

the steepness parameter p(¹) is the photon energy and temperature dependent:

(6)

The validity of this relationship was checked by calculating the temperature dependence of the energy gap and then comparing the obtained results with the corresponding values estimated from the linear parts of the lines shown in Fig. 3. The obtained results are listed in Table 2. From Table 3, it is clear that the values of b are more or less typical for the crystals. In addition good agreement is found between the values of the b estimated from Fig. 3 and those calculated using Eq. (6). 3.3. Broadening of the absorption edge




1 u   . p(¹)"p 1!  3 2k¹

(7)

So, from a graph between p(¹) and 1/¹ (Fig. 7), the values of p and u can be obtained. Fig.   7 shows such dependence. Values of u and p de  duced from straight-line portions at high temperatures are listed in Table 4. 3.4. Extinction coezcient and refractive index The re#ectivity R, the extinction coe$cient K and the refractive index n of crystalline solids are related through the two equations: aj K" , 4p

(n!1)#K R" . (n#1)#K

(8)

Using these relations the values of K and n could be determined from measurements of R and ¹. The refractive index (n) of virgin KFCT crystals as well as for crystals doped with either Co, Cr or

According to the following equation, proposed by Mahr [24] on the basis of Toyozawa's model,

Table 3 Calculated values of b"(!dE /d¹) for pure and doped KFCT  single crystals Crystal

b (eV K\) Eq. (6)

Pure KFCT (0 1 0) (1.67$0.01) 10\ Pure KFCT (1 0 1 ) (1.85$0.01) 10\ Co-doped KFCT (12.95$0.05) 10\ Cr-doped KFCT (4.71$0.02) 10\ SNP-doped KFCT (4.31$0.02) 10\

Fig. 3 (1.80$0.01) (2.03$0.01) (12.65$0.05) (4.76$0.02) (4.65$0.02)

10\ 10\ 10\ 10\ 10\

Fig. 7. Dependence of the Steepness parameter (p) on temperature for pure and doped KFCT single crystals.

M.A. Gawar, A. Abu-El Fadl / Physica B 292 (2000) 221}232 Table 4 Calculated values of u and p for pure and doped KFCT   single crystals Crystal

u (meV) 

p 

Pure KFCT (0 1 0) Pure KFCT (1 0 1 ) Co-doped KFCT Cr-doped KFCT SNP-doped KFCT

60$2 82$2 83$2 64$2 84$2

0.205$0.005 0.710$0.005 0.090$0.005 0.125$0.005 0.380$0.005

231

for the undoped crystal is higher, by about 10%, than that for the refractive index along the (0 1 0) axis. The refractive index decreased after doping, the maximum e!ect appear in the case of doping with SNP. The extinction coe$cient as a function of the photon energy is presented in Fig. 8b. Doping KFCT crystals with Cr not only increase the value of the extinction coe$cient in the hole energy range but also in#uenced the two absorption bands at 1.60 and 2.25 eV. 3.5. Dielectric permittivity The real (e ) and imaginary (e ) parts of the   dielectric constant were calculated as a function of the photon energy for virgin and doped KFCT crystals using the formula e "n!K,  e "2nK. (9)  The energy dependence of the real part of the dielectric permittivity (e ) preserve the same shape  as that for the [n! u] dependence while the shape of the [e ! u] dependence is  quite similar to the [K! u] dependence (Figs. not included). The reason for this behaviour is the small values of the extinction coe$cient in comparison with the values of the refractive index for KFCT crystals.

4. Conclusions

Fig. 8. (a) Refractive index (n) versus photon energy for pure and doped KFCT single crystals. (b) Variation of the extinction coe$cient with photon energy for pure and doped KFCT single crystals.

SNP along the (1 0 1 ) axis was calculated in the energy range 1.4}3.1 eV Fig. 8a shows the energy dependence of (n). As it is clear from the "gure, the values of the refractive index along the polar axis

E The absorption process at the absorption edge of KFCT crystals in the temperature range from 275 up to 325 K and the e!ect of doping with Co, Cr or SNP were studied. The undoped sample exhibits exponential behaviour of the absorption edge following Urbach's rule. In the case of doped samples, the validity of Urbach's rule existed in a narrower temperature range than that in case of pure KFCT crystals. The type of transition is the allowed indirect. The Urbach's parameters were determined and their temperature dependence was investigated.

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M.A. Gawar, A. Abu-El Fadl / Physica B 292 (2000) 221}232

E The steepness parameter p is always less than one and hence the condition for self-trapping is satis"ed. E The most acceptable model in the case of KFCT crystals is that due to Skettrup [23]. E Pronounced variations are detected in the values of a , E , p and B after doping KFCT crystals   with either SNP, CoSO or K Cr O .     References [1] V.A. Pospelov, G.S. Zhdanov, Zh. Fiz. Khim. 21 (1947) 879. [2] D.E. O'Reilly, G.E. Schacher, J. Chem. Phys. 43 (1965) 4222. [3] A.Ya. Krasnikova, V.A. Koptsik, B.A. Strukov, D. Wang Ming, Sov. Phys. Solid State 9 (1967) 85. [4] S. Waku, K. Masuno, T. Tanaka, H. Iwasaki, J. Phys. Soc. Japan 15 (1960) 1185. [5] R. Kiritama, H. Kiriyama, T. Wada, N. Niizeki, H. Hirabayashi, J. Phys. Soc. Japan 19 (1964) 540. [6] J.C. Taylor, M.H. Mueller, R.L. Hitterman, Acta Crystallogr. A 26 (1970) 559. [7] I. Savatinova, E. Anachkova, E.V. Chisler, Sov. Phys. Solid State 23 (1981) 1568.

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