Effect of the ceria dopant on the structural and dielectric properties of zno semiconductors

Journal of Science: Advanced Materials and Devices (2018) 433e439 Contents lists available at ScienceDirect Journal of Science: Advanced Materials and Devices journal homepage: www.elsevier.com/locate/jsamd Original Article Effect of the ceria dopant on the structural and dielectric properties of ZnO semiconductors B Rajesh Kumar a, *, B Hymavathi b, T Subba Rao c a Department of Physics, Gandhi Institute of Technology and Management (GITAM-Deemed to be University), Visakhapatnam, 530045, Andhra Pradesh, India b Department of Physics, Anil Neerukonda Institute of Technology and Sciences (Autonomous), Sangivalasa, Visakhapatnam, 531162, Andhra Pradesh, India c Department of Physics, Materials Research Lab, Sri Krishnadevaraya University, Anantapur, 515003, Andhra Pradesh, India a r t i c l e i n f o a b s t r a c t Article history: Received June 2018 Received in revised form 31 August 2018 Accepted September 2018 Available online 11 September 2018 ZnO doped with different concentrations (2, 4, 6, and 10%) of ceria was synthesized by the conventional solidestate reaction method X-ray diffraction spectra confirm that all the samples have a hexagonal structure The structural properties of the samples were studied from X-ray diffraction data The surface morphology and elemental composition of the ceria doped ZnO samples were characterized by scanning electron microscopy and energy dispersive X-ray spectroscopy The variation of the dielectric constant, the dielectric loss and the ac conductivity as functions of frequency in the range of 100 Hze1 MHz for the as-prepared material were studied at room temperature by using impedance spectroscopy All the samples exhibit a normal dielectric dispersion behavior, i.e it decreases with increasing frequency due to the MaxwelleWagner type of interfacial polarization The ac conductivity data was used to evaluate the maximum barrier height, the minimum hoping distance, and the density of localized states at Fermi level © 2018 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Keywords: Solid state reaction X-ray diffraction Structural properties Dielectric constant AC conductivity Introduction ZnO is a wide bandgap semiconductor with 3.37 eV and a large exciton energy of 60 meV It crystallizes in the hexagonal wurtzite structure Due to its unique chemical and physical properties, ZnO had found widespread applicationst in UV light emitters, surface acoustic wave (SAW) devices, solar cells and gas sensors [1] Doping of ZnO can improve the structural properties of ZnO for various applications [2] Rare earth metal-oxide nanoparticles have good electronic, optical luminescence and magnetic properties due to their unique electronic structure [3,4] They are widely used in magnetic, electronic and functional materials owing to their special characteristics Among the rare earth elements, Cerium oxide (Ceria, CeO2) has received much attention due to its peculiar optical and catalytic properties because of the availability of the shielded 4f levels with only one electron in the 4f state, Ce3ỵ [5e7] CeO2 can be obtained in crystalline or amorphous forms at low temperatures with a bandgap close to 3.1 eV (about 400 nm) Furthermore, the * Corresponding author Fax: ỵ91 8554 255710 E-mail address: rajphyind@gmail.com (B Rajesh Kumar) Peer review under responsibility of Vietnam National University, Hanoi refractive index of CeO2 in the visible region is 2.1e2.2, almost the same as that of ZnO (2.0e2.1) This makes it a very attractive material [8] However, there are very few reports on the crystal structure and dielectric properties of the CeO2 doped nanocomposite ZnO semiconductor In the present work, we have synthesized the ZnO doped with different concentrations of CeO2 by the solidestate reaction method When compared with other conventional methods, the solidestate reaction has an advantage because of its low cost, high yield, and ability to achieve high purity in making oxide nano powders [9] A systematic investigation on structural and frequency dependent dielectric properties of CeO2 doped ZnO is reported Experimental ZnO doped with 2, 4, 6, and 10% of CeO2 were synthesized by the conventional solidestate reaction method The appropriate ratio of the constituent oxides i.e ZnO and CeO2 (99.99% Aldrich Chemical, USA) were milled in a planetary ball mill (Retsch PM 200) with tungsten carbide balls (10 mm diameter) at a ball-to-powder weight ratio of 10:1 with a speed of 350 rpm for 24 h These mixed powders were calcined in a programmable SiC furnace at 900  C for https://doi.org/10.1016/j.jsamd.2018.09.001 2468-2179/© 2018 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) 434 B Rajesh Kumar et al / Journal of Science: Advanced Materials and Devices (2018) 433e439 10 h and then pressed into pellets of mm thickness and 10 mm in diameter using PVA (polyvinyl alcohol) as a binder at a pressure of 400 kg/cm2 Finally, the CeO2 doped ZnO pellets were sintered at 1100  C for h Structural characterization of the samples was carried out by using a Bruker D8 Advance X-ray Diffractometer with a CuKa radiation (l ¼ 0.154 nm) source operated at 40 kV and 40 mA XRD measurements were performed in the wide range of Bragg angles 2q (10 < 2q < 80 ) with a scan speed of 2 minÀ1 The surface morphology of the samples has been studied using a scanning electron microscope (SEM) (Model No Evo 18, Carl Zeiss Germany) and the elemental composition of the samples was determined by the energy dispersive X-ray spectroscopy (EDX; Oxford Instruments, X-max) attached with SEM For electrical measurements, the powdered samples have been pressed into pellets of uniform diameter and thickness The pellets were coated with silver paste on opposite faces in order to established a parallel plate capacitor geometry and then sintered for h at 100  C Dielectric measurements of the samples were taken at room temperature using an LCR HI-Tester (HIOKI 3532-50, Japan) in the frequency range of 100 Hze1 MHz Results and discussion 3.1 Structural analysis X-ray diffraction (XRD) patterns of ZnO doped with different concentrations of CeO2 are shown in Fig 1(a) The narrow and sharp XRD peaks confirm that the sample is of high quality with good crystallinity The diffraction peaks at 2q ¼ 31.65 , 34.36 , 36.21, 47.44 , 56.46 , 62.80 , 66.18 , 67.74 and 68.87 are identified with the Miller indices (1 0), (0 2), (1 1), (1 2), (1 0), (1 3), (2 0), (1 2) and (2 1), respectively, corresponding to ZnO (space group p63mc, JCPDS no 36e1451) phase having wurtzite structure In the range of 2q ¼ 28 e33 , two weaker diffraction peaks are attributed to (1 1) and (2 0) planes which indicate that the face-centered cubic (FCC) crystalline structure CeO2 exists in the samples according to the standard JCPDS (No.75-0390) card It is noticed from Fig 1(b) that the diffraction peaks are obviously shifted toward lower angles in the range of 2q ¼ 31e37, indicating that cerium ions were doped into the ZnO lattice This is due to the fact that the ionic radius of Ce3ỵ (0.103 nm) is much larger than that of Zn2ỵ (0.074 nm), which cause an expansion of the lattice parameter in the cerium doped ZnO crystallites [10] The lattice parameters ‘a’ and ‘c’ of the prepared samples are calculated using the equations as reported in our previous work [11] The lattice parameter ‘a’ increases from 0.3254 to 0.3268 nm, whereas ‘c’ increases from 0.5211 to 0.5268 nm with the increase of doping concentrations of CeO2 from to 10% in ZnO The average crystallite size (D) of CeO2 doped ZnO samples was calculated using Scherer formula [12] D ¼ kl=b cosq (1) where D is the crystallite size, k (¼ 0.94) is the shape factor and l (¼0.154 nm) is the wavelength of Cu-Ka radiation The decrease in crystallite size from 33 to 27 nm with the increase of the Ceria dopant concentration in ZnO is due to the distortion of host ZnO lattice by Ce3ỵ ions, which reduces the nucleation and subsequent the growth rate of ZnO This similar behavior was also observed in the previous literature works [13e16] The specific surface area of the crystallites of the samples was determined from XRD data using the Sauter formula [17]   D:r S ¼  103 (2) where S is a specific surface area, D is the crystallite size and r is the density of ZnO which equals to 5.606 g/cm3 The specific surface area of the samples increases with the increase of CeO2 concentration in ZnO The increase in the specific surface area is due to the presence of pores which leads to the decrease in particle size The presence of dislocations strongly influences the material properties generally, a larger dislocation density implies a higher hardness The dislocation density (d) in the sample was determined by using expression [18], d ¼ 15b cosq=4aD (3) where d is dislocation density, b is the broadening of diffraction line measured at half of its maximum intensity (in radian), q is the Bragg's diffraction angle (in degree), a is the lattice constant (in nm) and D is the crystallite size (in nm) The dislocation density (d) was found increase from 1.16  1015 to 1.51  1015 mÀ2 with the increase of the concentration of Ceria from to 10% in ZnO for the (0 2) orientation The structural parameters of CeO2 doped ZnO calculated from the X-ray diffraction data are given in Table The wurtzite structure of ZnO has two types of first-neighbour Zn-O bond distances: Zn-Oc along the c-axis (one bond), Zn-Ob in the basal plane (three bonds) and two bond angles a and b [19] Fig X-ray diffraction patterns of ZnO doped with (a) 2%, (b) 4%, (c) 6%, (d) 8%, and (e) 10% CeO2 Zn À Ob ¼ uc (4) B Rajesh Kumar et al / Journal of Science: Advanced Materials and Devices (2018) 433e439 435 Table Structural parameters of CeO2 doped ZnO nanocomposite semiconductor Concentration of ceria (%) Lattice parameter a (Å) c (Å) Volume (Å3) c/a Scherrer method, D (nm) Specific surface area, S (m2/g) Dislocation density, d (0 2) (  1015 mÀ2) 10 3.254 3.260 3.262 3.264 3.268 47.78 48.17 48.4 48.51 48.72 1.601 1.606 1.610 1.611 1.612 33 30 23 25 27 19 19.5 20.1 24.7 25.1 1.16 1.19 1.24 1.33 1.51 h 5.211 5.234 5.252 5.258 5.268 2 Zn Ob1 ẳ 1=3ịa ỵ ðð1=2Þ À uÞ c i1=2 composition The presence of O Ka peak at 0.56 keV, Zn La peak at 1.01, Zn Ka peak 8.68 keV, Ce M peak at 0.88 keV and Ce La peak at 4.84 keV was identified No other peaks related to impurities were detected in the spectrum confirming that CeO2 is successfully doped in ZnO (5) The wurtzite ZnO structure in its basal plane had the lattice constants ‘a’ and ‘c’ in the basal direction, ‘u’ parameter is expressed as the nearest-neighbor distance or the bond length b divided by c (0.375 in an ideal crystal), and a and b (109.47 in the ideal crystal) as the bond angles are shown in the figure as reported in the earlier work [20] n o1=2 À1  a ẳ p=2ị ỵ arc cos ỵ 3c=aị2 1=2 uị2 b ẳ 2arc sin n 4=3ị ỵ 4c=aị2 1=2 À uÞ2 o1=2 À1  3.3 Room temperature frequency dependent dielectric properties (6) The dielectric constant, dielectric loss tangent and ac electrical conductivity of CeO2 doped ZnO were studied as a function of frequency from 100 Hz to MHz using the impedance spectroscopy at room temperature The dielectric constant (εr) was calculated using the following formula [21] (7) εr ¼ Cp d=εo A where Cp is the capacitance of the specimen, d is the thickness of the pellet, A is the cross-sectional area of the sample and εo is permittivity of free space 8.85  10À12 F.mÀ1 Dielectric loss or the imaginary dielectric constant (ε”) was calculated using the relation where u denotes the cell internal parameter given by   u ẳ ẵ1=3ị a2 =c2 ỵ 1=4ị (8) In addition three types of second-neighbour cationeanion dis0 0 tance connecting the cation M to the anions O b1, b2, b3 Zn Ob1 ẳ uị c ðone neighbour along the c À axisÞ Zn À Ob2 ẳ ẵa2 ỵucị2 1=2 six neighboursị (12) ¼ εr tand (13) where tand is the dielectric loss tangent proportional to the loss of energy from the applied field into the sample Fig 3(a) and (b) shows the variation of the real and the imaginary part of dielectric constant (εr and ε”) as a function of frequencies for CeO2 doped ZnO The variation of the real and the imaginary dielectric constant values with the frequency shows a similar behavior The dielectric constant values are found to be decreased with the increase of frequency At low frequency, the dielectric constant is high due to the accumulation of charges at the grain boundary, and at the interface of the electrode sample and the electrode which is also called space-charge polarization As the frequency increases, the dielectric constant decreases due to the diminishing of the spacecharge polarization, indicating the electronic and atomic contribution domination According to MaxwelleWagner interfacial model, a dielectric medium consists of double layers having wellconducting grains which are separated by poorly conducting grain boundaries [22,23] The charge carriers can easily migrate to the grains by an external electric field, but they are accumulated at the grain boundaries by resulting a large polarization with a high dielectric constant value The larger value of the dielectric constant at low frequencies is due to the grain structure and the porosity It is also noticed that the dielectric constant decreases with the increase (9) (10) Zn Ob3 ẳ ẵ4=3ị a2 þð1=2 À uÞ2 c2 Š1=2 ðthree neighboursÞ (11) The calculated unit cell internal parameters and cationeanion distances between the nearest and the second-nearest neighbours (given in Å) as well as the bond angles (given in degrees) for ceria doped ZnO are given in Table 3.2 Surface morphology and elemental analysis Figure 2(a) e (e) shows the SEM images of CeO2 doped ZnO The surface morphologies of the samples appear smooth, but with a lot of pores It is also observed that the surface of the sample is dense up to 6% doped CeO2 in ZnO, and as the CeO2 content is increased to 10%, the surface becomes bumpy and rough Fig 2(f) shows the EDX spectra of the 6% CeO2 doped ZnO The spectra reveals the presence of the O, Zn and Ce elements along with their atomic percentage Table Calculated cationeanion distances between nearest and second-nearest neighbours (given in Å) as well as bond angles (given in degrees) for CeO2 doped ZnO 0 Concentration of ceria (%) Zn-O b Zn-O b1 Zn-O b1 Zn-O b2 Zn-O b3 a ( ) b ( ) 10 1.980 1.985 1.988 1.990 1.993 1.980 1.985 1.988 1.990 1.993 3.231 3.249 3.264 3.268 3.275 3.809 3.817 3.820 3.823 3.828 3.809 3.817 3.820 3.823 3.828 108.41 108.55 108.72 108.74 108.77 110.50 110.37 110.22 110.18 110.17 436 B Rajesh Kumar et al / Journal of Science: Advanced Materials and Devices (2018) 433e439 Fig SEM images of ZnO doped with (a) 2%, (b) 4%, (c) 6%, (d) 8%, and (e) 10% CeO2 (f) EDS of ZnO doped with 6% CeO2 of the concentration of ceria upto 6% in ZnO because of the high density of defects in ZnO But the further increase of ceria concentration from to 10% in ZnO, the values of dielectric constant is found to be increased This increase in the dielectric constant may be due to the less electronegativeness of Ce (0.89) than that of Zn (1.78) that makes the ionic bonds of Zn-O-Ce weaker than Zn-O-Zn bonds [24] The dielectric loss tangent (tand) represents the energy dissipation in the dielectric system The variation of tand with frequency for CeO2 doped ZnO is shown in Fig The dielectric loss is high in the lower frequency region of 100 Hze1 KHz and sharply decreased with the increasing frequency compared with the dielectric constant The dielectric loss decreases with the increase of the frequency up to 10 kHz and then increases at higher frequencies Accordingly, the dielectric loss at low and moderate frequencies is characterized by high values of the dielectric loss due to the contribution of ion jump and conduction loss of ion migration However, at higher frequencies the ion vibrations may be the source causing the dielectric loss [25] As the CeO2 dopant concentration increases from to 6% in ZnO the dielectric loss decreases, whereas the dielectric loss increases with the increase of the ceria dopant concentration from to 10% A shift of the peak towards lower frequencies is observed The decrease of the dielectric loss tangent with the increase in frequency as seen in 2, 4, 6% of CeO2 doped ZnO is attributed as due to the space charge B Rajesh Kumar et al / Journal of Science: Advanced Materials and Devices (2018) 433e439 437 The ac conductivity (sac) of the samples was calculated from the relation [26] sac ¼ uεr εο tand (15) where εr is the relative permittivity (or the real dielectric constant), εo (¼8.85  10À12 F/m) is the permittivity of the free space and u ¼ 2pf is the angular frequency The dependence of the ac conductivity (sac) on the frequency for the ceria doped ZnO samples is shown in Fig The ac conductivity of CeO2 doped ZnO samples increases with the increase of the frequency showing the semiconducting behavior This increase in the electrical conductivity of the samples is related to the increase in the drift mobility of the electrons and holes by the hopping conduction [27] That the ac conductivity decreases with the increase of the concentration of CeO2 from to 6% in ZnO may be attributed as due to the fact that the dopants can introduce defects in the ZnO lattice These defects lead to segregation at the grain boundaries due to the diffusion process resulting from cooling and sintering processes With the further increase of the CeO2 concentration from to 10%, the ac conductivity is found to increase as a consequence of the electron hopping i.e., the number of the hopping charge carriers increases at the grain boundaries and correspondingly the conductivity increases rapidly According to the Jonscher's power law, the frequency dependence of the conductivity is given by the expression [28] sac ẳ sdc ỵ Aus Fig Frequency dependent (a) real dielectric constant and (b) imaginary dielectric constant for ZnO doped with CeO2 Fig Frequency dependent tand with different concentrations of CeO2 doped ZnO (16) where sdc indicates the dc conductivity when s ¼ 0, the electrical conduction is frequency independent (dc conduction) and the second term is the frequency dependent ac conductivity Here, A is known as the strength of polarizability and s is the temperature dependent parameter The parameters A and s can be obtained from the plot of lnsac versus lnu and the equation is known as a Jonscher's power law The values of the exponent s, sdc and A obtained by fitting the equation (5) are given in Table The fit quality is usually evaluated by comparing the squared value of the linear correlation coefficient (R2) (see Table 3) Different models have been studied to explain the conduction mechanism on the basis of the parameter s Among these models, the quantum mechanical tunneling (QMT), the correlated barrier hopping (CBH), the small polaron hopping and the overlapping large e polaron (OLP) are the most applicable ones [29e32] If s is temperature independent, quantum mechanical tunneling is expected The CBH is usually associated with a decrease in s with the temperature Small polaron (SP) conduction is predominant if s increases with the temperature In the OLP conduction mechanism, s decreases with the temperature reaching a minimum value and then increases again In our polarization This behavior, as well as the low loss factor compared to that of the undoped ZnO makes the prepared samples suitable for high-frequency device applications With the further increase of doping concentrations of CeO2 in ZnO, the dielectric loss is found to increase Peaks in the dielectric loss as observed indicate the characteristic feature of the Debye-type relaxation process It is noticed that the peak shifts toward lower frequencies with the increase of CeO2 concentration in ZnO The curves observed in Fig are called the Debye curves and um is the maximum angular frequency, um ¼ 2pfmax, fmax is the relaxation frequency which is given by f max ¼ 1=2pt (14) where t is the relaxation time The relaxation time for the 2, 4, 6, and 10% ceria doped ZnO is found to be 2.44  10À6, 2.65  10À6, 3.88  10À6, 3.98  10À6 and 2.56  10À6 s, respectively Fig Variation of ac conductivity with frequency for ZnO doped with CeO2 438 B Rajesh Kumar et al / Journal of Science: Advanced Materials and Devices (2018) 433e439 Table Fitting parameters obtained from experimental data of ac conductivity as a function of frequency using the Jonscher's power law and calculated values of Wm and Rmin at 10 KHz Concentration of ceria (%) sdc (U.cm)À1 10 4.62 3.78 2.24 2.82 3.37      10À6 10À6 10À6 10À6 10À6 A 3.96 3.47 1.77 1.12 2.19      10À8 10À8 10À6 10À7 10À8 s R2 Wm (eV) Rmin (nm) 0.63 0.58 0.24 0.36 0.42 0.996 0.995 0.998 0.996 0.997 0.425 0.374 0.271 0.246 0.207 1.36 2.43 5.91 6.77 7.20 system, the decreasing trend of s with increasing ceria dopant in ZnO implies the CBH mechanism of conduction In this model, the conduction of carriers takes place through the barriers separating the localized sites In case of the present samples, the value of s decreases from 0.63 to 0.24 with the increase of concentrations of CeO2 from to 10% in ZnO, suggesting that the conduction phenomenon in the material under study is due to the correlated barrier hopping Therefore, the conduction in the system may be considered as due to the short-range translational type hopping of charge carriers This indicates that the conduction process is a thermally activated one A similar trend in the ac conductivity has been observed in many nanocrystallized semiconductor materials [33,34] The ac conductivity and frequency exponent expressions of CBH model are given by the following equations [35,36] s ẳ ẵ6kB T=Wm kB Tln1=uto ịị (17) where kB is the Boltzmann's constant, T is the temperature, to is the characteristics relaxation time of the carriers and Wm is the binding energy or the maximum barrier height, which is defined as the energy required to remove an electron completely from one site to another site For a small value of T,Wm >> kBT ln (1/ut0) and, therefore, the equation (3) can be written as s ẳ ẵ1 À ð6kB T=Wm ފ (18) where Wm is the maximum barrier height or binding energy The values of Wm were calculated by putting values of s and T (¼303 K) in the above equation Using the values of the binding energy, minimum hopping distance Rmin was calculated with the equation Rmin ¼ 2e2 =Pεεo Wm (19) where εo is the permittivity of free space and ε is the dielectric constant Fig 6(a) represents the variation of Rmin with frequency for ZnO doped with different concentrations of CeO2 It is noticed that Rmin increases with the increase of ceria dopant in ZnO A low value of Rmin in the lower frequency region was obtained (~10À10 m) A continuous dispersion with the increase in frequency has been observed This can be attributed to the conduction phenomenon originating from the short-range mobility of charge carriers A sigmoidal increase in the value of Rmin with the frequency approaches to a saturation value These observations are related to a lack of a restoring force which governs the mobility of charge carriers with the action of an induced electric field [37] The Rmin values at 10 KHz are found to be increased from 1.36 to 7.20 nm with the increase of doping concentrations of CeO2 from to 10% in ZnO According to the AustineMott formula based on the CBH model, the ac conductivity sac (u) is expressed in terms of the hopping of electrons between a pair of localized states N(EF) at the Fermi level The ac conductivity sac is related to the number of sites per unit energy per unit volume N(EF) at the Fermi level and is given by the equation [38] Fig Variation of (a) the minimum hopping distance, Rmin and (b) the density of states at Fermi level N(EF) with frequency for ZnO doped with CeO2 sac uị ẳ p=3ị e2 ukB TððNðEF ÞÞ2 aÀ5 ðln f o =uÞ4 (20) where e is the electronic charge, fo the photon frequency and a is the localized wave function The density of the localized states N(EF) was calculated according to equation (20) assuming fo ¼ 1013 Hz, a ¼ 1010 mÀ1 Fig 6(b) illustrates the variation of N(EF) with frequency for the ceria doped ZnO samples, where the values of N(EF) decreases with the increasing frequency It is also observed that the obtained value of N(EF) increases with the increasing ceria doping amount The high N(EF) values suggest that the hopping between the pairs of sites dominates the mechanism of the charge transport [39] Conclusion The CeO2 doped ZnO samples were prepared by the conventional solidestate reaction method The XRD patterns showed that these samples had a hexagonal wurtzite structure Since rare earth elements have larger ionic radii as compared with zinc, the incorporation of trivalent cerium ions in the ZnO host lattice can cause a significant distortion therein The dielectric properties of the samples reveal that the dielectric constant and the dielectric loss of the samples decreases with the increase of frequency, whereas the electric conductivity increases with frequency The decrease of the dielectric loss tangent with an increase in frequency seen in the B Rajesh Kumar et al / Journal of Science: Advanced Materials and Devices (2018) 433e439 ceria doped ZnO samples is attributed to the space charge polarization A Debye-like relaxation in the dielectric loss was observed for the ceria doped ZnO samples with a peak at a maximum frequency um, which is shifted to the lower frequency with increasing the doping concentration of ceria in the Debye curves As CeO2 doping increased up to 6% in ZnO, the loss factor decreased to a large 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the concentration of ceria upto 6% in ZnO because of the high density of defects in ZnO But the further increase of ceria concentration from to 10% in ZnO, the values of dielectric constant is found... to the increase in the drift mobility of the electrons and holes by the hopping conduction [27] That the ac conductivity decreases with the increase of the concentration of CeO2 from to 6% in ZnO. .. the incorporation of trivalent cerium ions in the ZnO host lattice can cause a significant distortion therein The dielectric properties of the samples reveal that the dielectric constant and the