The dielectric constant and quality factor calculation of the microwave dielectric ceramic solid solutions

Author’s Accepted Manuscript The Dielectric Constant and Quality Factor Calculation of the Microwave Dielectric Ceramic Solid Solutions Hetuo Chen, Bin Tang, Chaowei Zhong, Ying Yuan, Yidong Tan, Shuren Zhang www.elsevier.com/locate/ceri

PII: DOI: Reference:

S0272-8842(17)30283-3 http://dx.doi.org/10.1016/j.ceramint.2017.02.092 CERI14708

To appear in: Ceramics International Received date: 27 December 2016 Revised date: 4 February 2017 Accepted date: 20 February 2017 Cite this article as: Hetuo Chen, Bin Tang, Chaowei Zhong, Ying Yuan, Yidong Tan and Shuren Zhang, The Dielectric Constant and Quality Factor Calculation of the Microwave Dielectric Ceramic Solid Solutions, Ceramics International, http://dx.doi.org/10.1016/j.ceramint.2017.02.092 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

The Dielectric Constant and Quality Factor Calculation of the Microwave Dielectric Ceramic Solid Solutions Hetuo Chen,1 Bin Tang,1,* Chaowei Zhong,1 YingYuan,1 Yidong Tan,2 Shuren Zhang.1 1

State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic

Science and Technology of China, Chengdu 610054, China 2

State Key Laboratory of Precision Measurement Technology and Instruments, Department of Prec

ision Instrument, Tsinghua University, Beijing 100084, People’s Republic of China [email protected] [email protected]. *

Corresponding authors. Tel.: +86 28 83206695; fax: +86 28 83202139.

Abstract The dielectric constant of the microwave dielectric ceramic solid solution is usually predicted by the Clausius-Mosotti equation but the quality factor (Q) cannot be precisely calculated. In this paper, it finds that the dielectric constant of the solid solutions also could be well calculated by the Maxwell-Wagner formula, and that the Q of solid solutions can be precisely calculated, by assuming a

solid solution as a two or more materials’ mixture. Keywords: Classical Dispersion Theory; Solid Solutions; Quality Factor. I.

Introduction Due to the high dielectric constant (εr) and high quality factor (Q), microwave dielectric ceramics are

nowadays widely applied in microwave applications, for example, resonators.1-4 The high dielectric constant εr can let the size of components be small and the high quality factor (Q) represents low energy loss.1-4

1

In practice, however, microwave dielectric ceramics are inherently possessing non-zero temperature coefficient at the resonant frequency (TCF).1 One of the most effective methods is to mix the ceramics, with TCF of opposite sign, to form composites or solid solutions, to obtain a zero TCF.2,3,5 The solid solutions’ dielectric constant is usually predicted by the Clausius-Mosotti equation.3-6 However, the quality factor variation trend can be predicted, but cannot be precisely calculated. 3,4,6,7 The quality factor (Q) of the solid solution can be calculated by: n

Q 1  Vi Qi1 ,

(1)

i 1

in which n corresponds to the total number of ceramics, Vi is the volume molar ratio of the i-th n

ceramic,

V i 1

i

 1 , and Qi is the quality factor of the i-th ceramic, if we treat a solid solution as a

two (or more) materials’ mixture.1,4,9-11 Actually, the Clausius-Mosotti equation is also related to two or more materials. Then, the solid solutions’ dielectric constant could be expected by the Maxwell-Wagner formula.9-11 For example, it has been reported that CaTi1-x(Mg1/3Nb2/3)xO3 is a solid solution and the authors predict the dielectric constant based on Ti4+ and (Mg1/3Nb2/3)4+. 9,11 Thus, it is possible to calculate the dielectric constant and quality factor from CaTiO3 and Ca(Mg1/3Nb2/3)O3, by equation (1) and the Maxwell-Wagner formula.9,11 The variant x represents the volume molar ratio (V) of Ca(Mg1/3Nb2/3)O3.10,11 The problem is that equation (1) only considers the starting materials’ Q, so it shows a simple superposition relationship.1 The deviation between equation (1) calculated and the reported results are large.11 In this paper, starting from precursors’ dielectric constants and quality factors, we will show that: 1. the dielectric constant of solid solutions can be calculated by the Maxwell-Wagner formula and a k

2

value will be determined; 2. by introducing the k value, a revised Q calculation formula for the solid solutions is proposed, basing on the classical dispersion theory. II.

Theory

In the classical dispersion theory, the dielectric constant of a ceramic is expressed as:

   '  i '' ,

(2)

in which, ε’ and ε’’ are real and imaginary part of the dielectric constant.1 According to reference [1], the dielectric loss of the ceramic is defined as:

tan  

 '' . '

(3)

In the following, we will write the i-th real part of the complex dielectric constant, εi’ as εri, and write the dielectric constant of the solid solution as εr, for simplicity. When a solid solution is formed, if we

treat it as a two or more materials’ mixture, the εr can also be related to the two starting materials. The εr could be predicted by the Maxwell-Wagner formula:6,12 n

 r  (Vi rik )1/ k

(4)

i 1

where εri is the dielectric constant of the i-th ceramic, and k can be determined by fitting the experimental results to this equation. In the following, we will compare the reported experimental data of solid solutions and equation (4) calculated results, to confirm the effectiveness of it. We assume that the dielectric loss of the solid solution is determined by the total ε’ and ε’’. From equation (3), ref [1] claims that the tanδ is linearly frequency dependent for a ceramic and the claim has been widely accepted.1 In other words, the imaginary part of the dielectric constant of the solid solution also will obey equation (4). Therefore, the dielectric loss of the solid solution can be obtained from equation (3), by putting (4) into (3). After some algebra steps, we can get the total dielectric loss of the solid

3

solution: n

tan k  

V 

k i ri

i 1

tan k  i (5)

n

V 

k i ri

i 1

in which tanδi is the dielectric loss of the i-th ceramic. In practice, people would like to use quality factor, Q-1 = tanδ,1 so we rewrite equation (5) in the terms of Q: n

Q

k

V Q



i 1



k k i ri

i

(6)

n

V 

k i ri

i 1

From this equation, the quality factor is actually not only depending on the quality factor, but also the dielectric constant, εri, of the starting ceramics and k. When k = 1 and εri = constant, equation (6) is the same as equation (1).

Usually, the target is to modify the properties of the first material, so we write equation (6) as: n

RQk 

V R i

i 1

k Qi1

Rkr i1 .

n

V R

(7)

k

i 1

i

r i1

in which, RQ 

R r i1 

 ri is  r1

Q Q1 is the relative quality factor; RQi1  1 is the quality factor ratio; Qi Q

the dielectric constant ratio. Once the dielectric constant data of the solid solution is

determined, the relation between the relative quality factor and the volume molar ratio could be predicted. Then by Q1, the total quality factor, Q, could be obtained. Based on current reports, most situations are actually two-phases-mixing and each combination corresponds to one k value.13-15 Equation (7) could be write as:

4

(

Q k )  Q1

 r 2 k Q2  k ) ( ) V  r1 Q1  1  V  ( r 2 )k V  r1

1V  (

.

(8)

In figure (1), we take k = 1 for example to show the importance of dielectric constant ratio and quality factor ratio on the quality factor calculation of the solid solutions. Other k values obey the same rule of equation (5) – (8). The quality factor ratio, RQ 21 , and dielectric constant ratio, R r 21 , between the two starting materials are randomly set as 5, 1, 1/5 and 3, 1, 1/3 respectively to show how these parameters influence the Q variation. Figure (1) shows the relation between the relative quality factor RQ, quality factor ratio RQ 21 and dielectric constant ratio R r 21 , versus the volume molar ratio V, for k = -1. When 1, RQ will increase as the volume molar ratio increases. If

RQ 21 is higher than

R r 21 is also higher than 1, RQ shows a

concave increase, meanwhile, it firstly shows a slow increase and then rapidly increases to 5, versus V; as

R r 21 decreases, the concaveness of RQ becomes more inconspicuous to linear and until R r 21 <<

1, RQ will show a convex increase versus V. If RQ 21 = 1, the RQ will be unity, no matter what value

R r 21 is. These two situations correspond to RQ ≥1. Otherwise, when RQ 21 is smaller than 1, the situation essentially is the same as

RQ 21 >1 by exchanging Q1 and Q2.

III. Results and Discussions After showing the theoretical prediction, in the following, we will confirm the effectiveness of the equation (8) by giving relative deviation between the reported and calculated results. In recent decades, plenty of researches concerning solid solutions have been reported, however, the two starting ceramics are usually chosen from MgTiO3, LnAlO3, Ba6-3xLn8+2xTi18O54, and CaTiO3 (Ln represents rare earth elements) et al. based ceramics.4,5,7,8,12,14,15 We will firstly get the k values from the reported dielectric constant data according to equation (4), in figure 2(a). The relative quality factor of calculated, 5

RQ calculated 

Qi  reported Qi  calculated , and reported, RQ  reported  , results of 6 typical ceramics Q1 reported Q1 reported

based solid solutions will be depicted in figure 3(a): 1. Mg1-xZnxTiO3, εr ~ 20; 2.

Zn1-xMgxTa2(1-x)Nb2xO6, εr ~ 35; 3. Ca(Mg1/3Nb2/3)1-xTixO3 , εr ~ 50; 4. (1-x)BaNd2Ti4O12-xNdAlO3, εr ~ 60 - 70; 5. (1-x)CaTiO3-xNdAlO3, εr ~ 80 - 100; and 6. (1-x)Ca0.8Sr0.2TiO3-xNdAlO3 εr ~ 100 -

140, versus x.6,7,8,13-16 In addition, the relative deviation of dielectric constant and relative deviation of quality factor, defined as:

R r deviation | RQ deviation |

R r calculated  R r reported R r calculated RQ calculated  RQ reported RQ calculated

| 100%

| 100% ,

(9)

and

(10)

will be given to confirm the effectiveness of equation (8), in figure 2(b) and 3(b) respectively. The experimental dielectric constant and quality factor data depicted in figure 2(a) and figure 3(a) are cited from literatures.6,7,8,13-16 In figure 2(a) and figure 3(a), C represents “Calculated results”, M means “Measured data” and 1 - 6 are solid solution group numbers in this paragraph. In addition, in the above mentioned literatures, only Qf, not Q, values are reported. The resonant frequency f is inversely proportional to the dielectric constant.1,17,18 We extract the Q value from Qf values, basing the dielectric constant and assuming constant radii. In figure 2(a), the reported dielectric constant fits well with the calculation, and the k values are 2, 10, -1.2, 4, -1.5 and 5. In figure 2(b), the relative deviation of dielectric constant, around ±10%, is given. Therefore, the effectiveness of the Maxwell-Wagner formula is confirmed for the solid solution. By applying these k values into equation (8), in figure 3(a), the reported quality factor also fits well with the calculation, with the relative deviation around ±10% shown in figure 3(b). The fact that some data show larger deviation is probably because of the assumption of constant radii. As a result, with the 6

properties of the two starting materials, we can precisely predict the dielectric constant and quality factor of the solid solutions, without considering the microstructures, after checking the scanning electron microscopy (SEM) results from these references.6,7,8,13-16 In other words, from equation (8), if the ceramic solid solutions are well sintered, what the change of the microstructure actually influences is the dielectric constant. The variation of the quality factor is independent of the microstructure. III.

Conclusions

In this paper, we calculate the dielectric constant of the microwave ceramic solid solutions by the Maxwell-Wagner formula. Once the k value of a solid solution is determined from the formula, Q could be precisely calculated. It is found that the quality factors of solid solutions are not only determined by each component’s quality factor, but also related to their dielectric constant and the k value. We also found that, if sintered at proper temperatures, quality factor of the ceramics will be independent of the microstructures. IV.

Acknowledgments

The support provided by China Scholarship Council (CSC, 201506070064) during a visit of Hetuo Chen to California Institute of Technology is acknowledged. This work is supported by the Open Foundation of National Engineering Research Center of Electromagnetic Radiation Control Materials (ZYGX2014K003-6) and the National Natural Science Foundation of China (Grant No. 51672038 and 51402039). V.

References

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VII. Figure Captions Figure 1. The example of relative quality factor versus dielectric constant ratio, quality factor ratio and volume molar ratio for k = -1. Figure 2(a). The calculated and reported dielectric constant results of: 1. Mg1-xZnxTiO3; 2.

Zn1-xMgxTa2(1-x)Nb2xO6;

3.

Ca(Mg1/3Nb2/3)1-xTixO3;

4.

(1-x)BaNd2Ti4O12-xNdAlO3;

5.

(1-x)CaTiO3-xNdAlO3; and 6. (1-x)Ca0.8Sr0.2TiO3-xNdAlO3, versus x. C and M represents “Calculated data” and “Measured data”, respectively; the number 1 – 6 correspond to the six group materials. Figure 2(b). The relative deviation between the reported dielectric constant data and calculated

data from equation (4). 9

Figure 3. The calculated and reported dielectric constant results of: 1. Mg1-xZnxTiO3; 2.

Zn1-xMgxTa2(1-x)Nb2xO6;

3.

Ca(Mg1/3Nb2/3)1-xTixO3;

4.

(1-x)BaNd2Ti4O12-xNdAlO3;

5.

(1-x)CaTiO3-xNdAlO3; and 6. (1-x)Ca0.8Sr0.2TiO3-xNdAlO3, versus x. C and M represents “Calculated data” and “Measured data”, respectively; the number 1 – 6 correspond to the six group materials. Figure 3(b). The relative deviation between the reported quality factor data and calculated data

from equation (8).

10

11