Pressure and temperature dependence of the static dielectric constant of KBr

J. Phys. Chem. Solids, 1974, Vol. 35, pp. 1327-1331, Pergamon Press. Printed in Great Britain

P R E S S U R E A N D T E M P E R A T U R E D E P E N D E N C E OF THE STATIC DIELECTRIC C O N S T A N T OF KBr* P. A. SMITH and D. H. RIEHL'[" Ithaca College, Ithaca, N.Y. 14850, U.S.A. (Received 4 October 1973; in revised form 10 January 1974) Abstract--The static dielectric constant and the temperature and pressure derivatives of the static dielectric constant of KBr have been measured at several temperatures between 4.3°K and room temperature. The sample was prepared in the form of a three-terminal parallel-plate capacitor and the dielectric constant was determined from measurements of the capacitance with a high precision bridge. The dielectric constant decreases with temperature as does the magnitude of the temperature dependence and the magnitude of the pressure dependence. The data were used to calculate the fixed-volume temperature derivative of the dielectric constant. This quantity, (~ In e/aT)v, exhibiting lattice-anharmonicity effects, decreases slightly from its room temperature value as the temperature is lowered, rises to a maximum value at about 33°K and then decreases rapidly at lower temperatures.

1. INTRODUCTION The static dielectric constant of an ionic crystal results from two mechanisms which contribute to the polarization produced by an applied electric field: the " o p t i c a l " part due to the shift of the electronic charge cloud relative to its positive nucleus and the "lattice" part due to the relative shift of the oppositely charged ionic sublattices. These two effects can be separated since at optical frequencies only the electronic mechanism is active and index of refraction measurements determine its magnitude. Many properties of ionic crystals are reasonably well described by harmonic models. H o w e v e r recent measurements suggest that real crystals may be less harmonic than the successes of the quasi-harmonic models at first suggested. Szigeti [1, 2] has considered the effect on the dielectric constant of including anharmonic terms in the crystal-lattice potential. The temperature d e p e n d e n c e of the dielectric constant at fixed volume is a direct anharmonic effect and is thus useful in studies of lattice anharmonicity. This quantity, though not directly measurable, can be obtained from the measured temperature and pressure dependence of the dielectric constant. The dielectric data is particularly useful if it is applicable to a wide temperature range. One of the authors has previously reported on measurements on the static dielectric constants *This work supported by Research Corporation. tPresent Address: Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Ill. 61801, U.S.A.

of the three alkali-halides NaC1, KCI and LiF[3]. The present work on K B r was undertaken to extend the experimental data and thus perhaps add to the understanding of these compounds. 2. EXPERIMENT

Measurements were made on a single crystal of K B r from H a r s h a w Chemical Company. The sample was in the~form of a circular disk of nominal size 20 mm in dia. and 0.5 mm in thickness. The surfaces were polished flat and parallel to within 0.01 mm. Aluminum electrodes were deposited directly on the crystal surfaces by evaporation in vacuum. A thin shadow ring was placed on one side of the sample during this process to achieve the narrow gap between the active center electrode and the guard ring necessary for three-terminal capacitance measurements. The diameter of the center electrode and the width of the gap were measured with a traveling stage and microscope. The width of the gap was just over 0.22 ram. The sample was placed in a sample holder which maintained the shielding of the active leads and then placed directly in the pressure vessel. Two electrical leads were run from the pressure vessel through miniature pressure tubing to a r o o m temperature environment where they were brought out of the pressure system using standard conical leadouts with plastic insulating cones. The pressure vessel itself was used for electrical contact to the guard electrode. Helium gas was used as the pressure fluid with the pressure generated b y an air-operated gas c o m p r e s s o r and measured b y a Bourdon-tube pressure gauge.

1327

P. A. SMITH and D. H. RIEHL

1328

The pressure vessel was installed in a controlledtemperature D e w a r which allowed measurements o v e r a continuous temperature range f r o m about 4-3°K to room temperature. Once the system reached thermal equilibrium no measurable temperature variation at the sample was noted. Temperature measurements were made with a GaAs diode cryogenic sensor and are accurate to 0.2°K. The procedure was to pressurize the system and then allow several hours to achieve thermal equilibrium at the desired temperature. Capacitance measurements were then made at various points of decreasing pressure while the system was maintained at constant temperature. The dielectric constant and its change with pressure and temperature were determined from measurements of the capacitance of the parallel-plate capacitor as a function of pressure and temperature. Since the dielectric constant of an ionic crystal is relatively insensitive to f r e q u e n c y in the low-frequency range the measurements were carried out at the single f r e q u e n c y of 1 kHz. Capacitance measurements were made with a General Radio model 1620A capacitance measuring assembly which yields capacitance measurements to six figures. T h e three-terminal measurement allows capacitance determinations which are independent of lead capacitance and reflect only the variation of the sample capacitance with pressure and temperature. 3. RESULTS Figure 1 shows a raw data plot of capacitance versus pressure. This particular plot for 83°K is typical and shows the nearly linear decrease of capacitance with pressure. The straight line shown on the graph is the tangent to the curve at zero pressure. Since the derivatives desired are those evaluated at zero pressure it is the slope of this line that is used in the calculations. The numerical values of these slopes were found f r o m least-squares parabolic fits to the data. Table 1 shows the results of the measurements in terms of the capacitance and its derivatives. T h e first column is the temperature at which the data was taken and the second column is the zero pressure capacitance at that temperature. The last column, the temperature derivative at zero pressure, was determined from the data in column two. The top twelve slopes come from a least-squares parabolic fit to the top twelve zero pressure capacitance values and the lower three slopes were determined graphically. Table 2 gives the results of this work in terms of

I0~00~

83"K

10'500%~ Z O

O

IOAO0

0

I0I000 PRESSURE

i 20000

(psi)

Fig. 1. A raw data plot of sample capacitance vs pressure. Data is for 83°K. The line shown is the tangent to the curve at zero pressure as determined by a least-squares parabolic fit. Table 1. Results of the capacitance-temperature-pressure measurements. All quantities are evaluated at zero pressure

T (*K) 295.4 211.3 197-0 177-7 157.3 135-2 117.8 97.3 83.3 75.1 68-9 47.8 26-0 13.8 4.3

C1,.o (pF) 11.2952 10-9749 10-9282 10.8667 10.8047 10-7418 10.6871 10-6252 10.5851 10-5611 10.5443 10-4853 10.4305 10-4114 10-4097

(oClOP)T

(OCIOT),,.o

(10-2 pF/kbar)

(10-3 pF/°K)

-

16.097 14.657 14.317 14.247 13-646 13.343 13-108 12.819 12-577 12.546 12.345 12.107 11-740 11.735 11.427

3.912 3.454 3.376 3.271 3.160 3.040 2-946 2.834 2-758 2.713 2-679 2.565 1.906 0.786 -0.428

the static dielectric constant and its derivatives. Thermal expansion and elastic data are needed over the entire temperature range in order to reduce the measured quantities[4-10]. T h e value of the dielectric constant was obtained f r o m the zero pressure capacitance value and the dimensions of the capacitor. Corrections for the gap b e t w e e n the center electrode and the guard ring were taken into account by adding the gap-width to the diameter of the guarded electrode [11]. T h e value of the dielectric constant ~ at the temperature T is determined from the relation

,ct

E -- ~

exp

(1)

Pressure and temperature dependence

1329

Table 2. Results of the dielectric constant-temperature-pressure measurements expressed in several forms. All quantities are evaluated at zero pressure T (°K)

~

295.4 211.3 197.0 177.7 t57,3 135,2 1t7-8 97.3 83-3 75.1 68,9 47,8 26.0 13.8 4,3

4,909 4.755 4-732 4.702 4-672 4.64t 4-615 4-586 4-567 4-555 4-547 4.520 4,496 4,487 4.486

(0 In elOP)r (Oe/OP)r (8 In e/OT)p (10-2/kbar) (t0-~/kbar) (10-'PK) - 1.1922 - 1.1156 - 1.0924 - 1-0963 - 1,0509 - 1.0330 - 1,0200 - 1-0026 - 0.9862 - 0-9870 - 0-9708 - 0,9573 - 0.9300 -0.9319 - 0-9027

- 5.853 - 5.304 -5.169 - 5,155 - 4.910 -4-795 - 4-708 - 4,598 - 4-504 - 4-496 - 4-414 - 4-327 - 4-181 -4.182 - 4.050

w h e r e C is the zero pressure capacitance at temperature T, t. is the thickness of the crystal at r o o m temperature, d, is the corrected diameter of the center electrode, and a is the linear coefficient of thermal expansion. T h e exact exponential f o r m has been used for the effect of the thermal expansion on sample dimension rather than the more c o m m o n first order formulation since the temperature range involved is substantial [t2]. The relative change of ¢ with pressure at constant temperature is calculated from the expression (0 In E/OP)r = (0 In C/OP)r + 1/(3Br)

(2)

w h e r e Br is the isothermal bulk modulus. The relative change of ~ with temperature results f r o m the relation (O In elOT)e = (a In C/aT), - / 3 / 3

(3)

w h e r e / 3 is the v o l u m e coefficient of thermal expansion. T h e last column in Table 2 represents the ratio of the fractional change in dielectric constant to the fractional change in crystal dimensions at constant temperature. These numbers result from the expression (0 In ~/0 In r)r = - 3Br(0 In e/OP)r,

(4)

where r is the lattice constant. T h e dielectric constant of K B r has been measured at room temperature by numerous investigators and we find no serious discrepancy with

3-070 2.784 2.731 2-660 2-586 2-506 2-446 2,380 2-340 2.322 2.309 2.293 1.783 0.748 - 0-411

(10-~PK)

(Oe/oT)F

(0 In e/O in r)r r = lattice constant

1.507 1.324 1,292 1,251 1,208 t,163 I' I29 1-092 1-069 1-058 1,050 1.036 0.802 0-336 - 0,185

5.118 5-074 5.017 5-101 4-956 4-940 4-939 4-918 4-885 4-912 4-855 4-851 4-754 4.775 4-625

their values. In addition measurements of e as a function of temperature have been made by Robinson and Hollis Hallett[13] and by L o w n d e s and Martini14]. Our values agree with the former to better than 1 per cent and with the latter to a slightly less degree o v e r the entire temperature range. Fontanella, Andeen and Schuele[15] give a room temperature value for (0 In e/OP)r which differs from the present value by about three percent when corrected for the temperature difference. T h e value given by Jones[16] for this quantity at room temperature agrees with our results to better than one percent. 4, DISCUSSION

Pressure and temperature measurements on the dielectric constants of three of the alkali-halides, NaCI, K C L and LiF, as well as MgO were made previously by one of the authors and have been reported earlier [3]. Discussion of those results was rather extensive and wilt not be repeated here for the K B r work except to point out the significant differences b e t w e e n those results and the present work. (a) Low-temperature minimum in E While the evidence is by no means conclusive the data appears to show a minimum in the dielectric constant of K B r at our lowest available temperatures. This effect has been discussed in the earlier paper in relation to the measurements on NaCI and is consistent with the effect of impurity ions with permanent electric dipole moments. This accounts f o r the negative value of (0 in E/OT)p and hence the negative value of (0 In e/OT)v at the lowest temperature reported,

P. A. SMITH and D. H. RIEHL

1330

Co) Constant-volume temperature dependence The constant-volume temperature d e p e n d e n c e of an ionic crystal's dielectric constant is of interest since it is directly related to lattice anharmonicity. A t constant volume the electronic polarizability should not depend on temperature and, if the interionic forces are harmonic, neither should the lattice polarizability. The value of (a6/OT)v can be determined from the relation (a In ~lOT)p = (a In elaT)v - ~Br(a In 6faP)r,

Table 3. Constitution of the temperature dependence of the dielectric constant. Units are 10-'/°K T(°K) (O In 6/c3T)p = (a In ~lOT)v -/3Br(a In e/aP)r 3-070 2.784 2-731 2.660 2.586 2-506 2.446 2.380 2-340 2.322 2-309 2-293 1.783 0.748 -0.411

1-059 0-937 0-930 0-870 0.906 0.906 0-915 0.969 1.046 1-109 1-t83 1-551 1.574 0-714 -0-411

C

T V

21.O

0

(5)

where ~ is the volume coefficient of thermal expansion and B r is the isothermal bulk modulus. Table 3 gives the relative size of the three terms in this relation over the entire temperature range. The first and third columns are derived from measured quantities and the middle column is determined using equation (5).

295.4 211.3 197.0 177.7 157.3 135.2 117.8 97.3 83-3 75.1 68.9 47.8 26.0 13-8 4.3

-~,o%'K

2-011 1-847 1.801 1.790 1.680 1.600 1.53t 1-411 1.294 1-213 1-126 0-742 0-209 0-334 0

Figure 2 is a plot of (a In E/aT)v vs reduced D e b y e temperature T/Oo for the present w o r k on K B r as well as for NaC1, KCI and L i F from the earlier work[3]. A t room temperature, except for L i F , this quantity is positive; it then in all cases increases toward a maximum positive value at lower temperatures, and finally, in the case of L i F and KC1 approaches zero near absolute zero. The negative (a In E/aT)v for NaCI and K B r at very low temperatures is the impurity effect discussed earlier. Robinson and Hollis Hallett[13] calculated this quantity for NaC1, KCI and K B r and their results are similar to those shown in Fig. 2. H o w e v e r they did not have available the quantity ( a E / a P ) r over the vchole temperature range and so they estimated

T/eo Fig. 2. Fixed-volume temperature dependence of the dielectric constant plotted as a function of reduced Debye temperature. Values are shown for KCI, NaCI, LiF as well as KBr. the second term on the right side of equation (5) using the room temperature value for the product of the bulk-modulus and the pressure derivative of the dielectric constant. They show the maximum in (aElOT)v at a value of T/OD = 0-2. In addition they show a minimum for all three samples at T/OD = 0.8. As reported earlier we saw no evidence of this minimum in NaCI or KCI. Our measurements show that in a temperature drop of one hundred degrees the quantity Br(ae/OP)r decreases about ten percent in NaCI and about five percent in KC1 and this discrepancy is enough to account for the minimum that they show. In the case of K B r h o w e v e r we do find this minimum. The lower D e b y e t e m p e r a t u r e of K B r (175 °) allows measurements at r o o m temperature as high as 1.7 in T/OD which is not the case for NaCI or KC1 and this may account for the fact that the minimum is not seen in the earlier work. Szigeti[1,2] has calculated the static dielectric constant for ionic crystals, representing the anharmonic terms by a power series in (kT/to2). The result of this analysis yields the following expression

e~, - eop= ~ + G

(6)

where = (40, + 2"~2 4.trN(Ze*) \--~/ Mtoo~--~ , ~s, and ~0~ are the static and optic dielectric constants respectively, N is the number of cationanion pairs per unit volume, J~/is the reduced mass of these cation-anion pairs, (Ze*) is the effective charge on the ion and too the f r e q u e n c y associated

Pressure and temperature dependence with the long-wavelength limit of the optical branch of the crystal's phonon spectrum. "0 represents the contribution of the linear dipole moment and harmonic potential, and G represents, in the lowest order, the effect of both the mechanical and electrical anharmonicity. Szigeti argues that above the D e b y e temperature G can be written as G=AT

(T>Oo)

(7)

where. A is a constant. In addition since the quantities defining "0 depend on the temperature only because they are unique functions of the volume, which varies with temperature, (O'o/OT)v = 0 and thus ( OO / OT)v = ( a ( e,, - eo, )/ OT)v = (oe,,/aT)v - (aEo.loT)v

(8)

at any temperature. H e argues further that (Oeop/OT)v can be regarded as negligible c o m p a r e d with (Oe,,/OT)v and thus A = ( o G / a T ) v = (aE,,laT),, = constant.

(9)

The measurements do extend above the D e b y e temperature for K B r and although there is not an abundance of data in this region ( a e / a T ) v appears to be increasing slightly suggesting that the higher order terms in the expansion are not negligible. Experimental measurements to higher values of T/Oo would be of sbme value in this regard. In any case further calculations are required to establish the detailed behavior of ( a e / a T ) v over the whole temperature range. Following the work of Szigeti, Fuchs[17] considered both anharmonic terms in the lattice potential and higher order electric dipole effects due to ionic overlap. The result of this analysis shows that in addition to the term r/ on the right hand side of

1331

equation (6) there are two competing anharmonic terms, one making a positive and the other a negative contribution to (a In e / a T ) v . T h e sign of (a In e / a T ) v is then determined by which of the two terms dominate. While F u c h s did not actually calculate the relative magnitudes of these two terms he did show that the negative term is proportional to e,, and will thus be relatively larger for materials with a large dielectric constant. Figure 2 shows this to be experimentally verified with the value of (0 In e / a T ) v for the different alkali halides increasing in order with the decreasing value of E at 0o. REFERENCES

I. Szigeti B., Proc. R. Soc. Lond. A252, 217 (1959). 2. Szigeti B., Proc. R. Soc. Lond. A261, 274 (1961). 3. Bartels R. A. and Smith P. A., Phys. Rev. B7, 3885 (1973). 4. Yates B. and Panter C. H., Proc. Phys. Soc. 80, 373 (1962). 5. Meincke P. P. M. and Graham G. M., Can. J. Phys. 43, 1853 (1965). 6. White G. K., Proc. R. Soc. Lond. A286, 204 (1965). 7. Sharko A. V. and Botaki A. A., Soviet Phys. Solid State 12, 1796 (1971). 8. Gait J. K., Phys. Rev. 73, 1460 (1948). 9. Clusius K., Goldmann J. and Perlick A., Z. Naturforsch. 4a, 424 (1949). 10. Berg W. T. and Morrison J. A., Proc. R. Soc. Lond. A242, 467 (1957). 11. Harris F. K., In: Electrical Measurements, p. 683. Wiley, New York (1952). 12. Bartels R. A., Am. J. Phys. 41, 78 (1973). 13. Robinson M. C. and Hollis Hallett A. C., Can. J. Phys. 44, 2211 (1966). 14. Lowndes R. P. and Martin D. H., Proc. R. Soc. Lond. A316, 351 (1970). 15. Fontanella J., Andeen C. and Schuele D., Phys. Rev. B6, 582 (1972). 16. Jones B. W., Phil. Mag. 16, 1085 (1967). 17. Fuchs R., Technical Report No. 167, MIT Laboratory for Insulation Research, Massachusetts Institute of Technology, Cambridge, Mass., 1961 (unpublished).