Text-chap4

_______________________________________________4 HIGH PERMITTIVITY DIELECTRICS

The major application of ferroelectrics is for capacitors, utilizing their high dielectric constants around the Curie temperature.

4.1 CERAMIC CAPACITORS There are two classes of ceramic capacitors: one is for thermal compensation of electric circuits, using a TiO 2-based material and the other is a high permittivity capacitor with a BaTiO 3- or Pb(Zr,Ti)O3 -based material. More precisely, there are four primary categories of dielectrics: 1) High-Q, low-dielectric constant K (100) temperature compensating materials (capacitance change ± 30 ppm), 2) Intermediate-K (3000) materials, labeled X7R or BX (± 15%), 3) High K (10,000) formulations, known as Z5U or Z5V (20 - 50%), 4) Nonhomogeneous, barrier layer materials that can have effective K up to 100,000. Figure 4.1 summarizes the various capacitor types, highlighting their sizes and operating frequency ranges.1) Ceramic capacitors with a single parallel plate design are still the most popular, while multilayer ceramic capacitors are 1/20 - 1/30 the size of the single parallel plate type. Semiconductor capacitors exhibit large capacitance using very thin dielectric layers in a semiconductor based ceramic (see Chapter 9, Section 9.3). Micro-chip capacitors are ultra-small capacitors for high frequency applications. The basic specifications required for capacitors are: (a) Small size, large capacitance Materials with a large dielectric constant are desired. (b) High frequency characteristics Ferroelectrics with a high dielectric constant are sometimes associated with dielectric dispersion, which must be taken into account for practical applications. (c) Temperature characteristics We need to design materials to stabilize the temperature characteristics. 105

106

Chapter 4

Satellite Commun. Automobile Commun. UHF TV VHF TV FM Radio

30 GHz 3 GHz 300 MHz 30 MHz

Ceramic Capacitor

3 MHz

Multilayer Ceramic Capacitor

Semiconductor Capacitor

Micro Chip Capacitor

AM Radio 300 kHz 30 kHz

1 cm

Fig. 4.1 Various capacitor types classified according to their sizes and operating frequency ranges.

Example Problem 4.1_________________________________________________ Calculate the wavelength in air (ε = 1) and in a dielectric material with ε = 30 for electromagnetic wave at 10 GHz. Solution _ Taking account of c = 3.0 x 108 m/s in air and v = c / √ε in the dielectric, λ = c / f = 3.0 x 108 / 10 x 109 = 3 x 10-2 [m] = 3 [cm] (in air) __ -2 = 3 x 10 /√30 [m] = 5.5 [mm] ___________________________________________________________________

4.2 CHIP CAPACITORS Multilayer structures have been developed as part of capacitor manufacturing aimed at the integration of electrical circuit components. Figure 4.2 schematically shows a multilayer capacitor chip. Thin sheets made by the tape casting technique, starting from a slurry of the dielectric powder and organic solvents, are coated with Ag-Pd, Ag, or cheaper Ni or Cu paste is used to form the electrodes, then several tens of sheets are stacked together and sintered. Finally, external electrodes, used to connect the chip with the circuit, are painted on. See Chapter 3, Section 3.3(2) for the details of the manufacturing process.

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The layer thickness of multilayer capacitors has been reduced remarkably, with thickness currently on the order of 7 - 10 µm. The electrostatic capacitance of a multilayer capacitor is given by the following formula: C = n ε0ε S / (L/n),

(4.1)

where ε is the relative permittivity of the dielectric material, n the number of layers, S the electrode area, and L the total thickness of the capaciator. Note that the capacitance increases in proportion to the square of the number of layers, when the total size is fixed. Table 4.1 summarizes specifications for several multilayer capacitors.2) The conventional capacitor of 10 µF with a 30 µm layer thickness has a volume of 70 mm3 . By decreasing the layer thickness down to 10 µm, the device volume can be reduced to 7.7 mm3. Note that by reducing the layer thickness by 1/n, the total volume is reduced by a factor of (1/n)2 to sustain the same capacitance.

Area S

External electrode Number of layers n

Total thickness L Electric field direction Internal electrode

Fig. 4.2 Structure of a multilayer capacitor. Table 4.1 Dimensions of the multilayer ceramic capacitors. _____________________________________________________________ Capacitance Dimensions Volume Relative Green Sheet at Room (mm) (mm3) Volume Thickness Temp. (µF) L W T (%) (µm) _____________________________________________________________ Present 1 2.0 1.3 0.8 2.1 (100) 10 Ceram. Cap. 10 3.2 1.6 1.5 7.7 100 10 _____________________________________________________________ Conventional 1 3.3 1.7 1.2 6.7 (319) 25 Ceram. Cap. 10 7.0 4.2 2.4 70.0 909 30 _____________________________________________________________ Tantalum 1 3.2 1.6 1.6 8.2 (390) --Electrolytic Cap. 10 4.7 2.6 2.1 25.7 334 --_____________________________________________________________

108

Chapter 4

Fig. 4.3 Cross-sectional view of a monolithic multicomponent ceramic substrate for a voltage control oscillator.

4.3 HYBRID SUBSTRATE Recent technology has introduced a set of capacitors and resistors as well as conducting leads onto a hybrid multilayer substrate, without making dis crete capacitors and resistors. Figure 4.3 shows a sectional view of a monolithic multicomponents ceramic (MMC) substrate for a voltage controlled oscillator, 3) where resistors and capacitors are included in the substrate using a tape casting technique. Using this MMC substrate, the voltage controlled oscillator has been reduced by 1/10 in size (volume).

4.4 RELAXOR FERROELECTRICS Relaxor ferroelectrics such as Pb(Mg1/3Nb 2/3)O3 and Pb(Zn1/3Nb 2/3)O 3 have been utilized for very compact chip capacitors. The reasons why these complex perovskites have been investigated intensively for capacitor applications are:

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(1) their very high permittivity, and (2) their temperature - insensitive characteristics (i.e., diffuse phase transition) in comparison with the normal ferroelectric perovskite solid solutions. However, the relaxors exhibit a problem, namely, dielectric relaxation, a characteristic highlighted by their name. (1)

High Permittivity

An intuitive crystallographic model (rattling ion model) has been proposed to explain the giant permittivity of these disordered perovskites.4) Figures 4.4(a) and 4.4(b) show the ordered and disordered structures for an A(BI1/2BII1/2)O3 perovskite crystal. Assuming a rigid ion model, a large "rattling" space is expected for the smaller B ions in the disordered structure [Fig. 4.4(b)] because the large B ions prop open the lattice framework. Much less "rattling" space is expected in the ordered arrangement [Fig. 4.4(a)] where neighboring atoms collapse systematically around the small B ions. When an electric field is applied to a disordered perovskite, the B ions (usually high valence ions) with a large rattling space can shift easily without distorting the oxygen framework. Larger polarization can be expected for unit magnitude of electric field; in other words, larger dielectric constants and larger Curie-Weiss constants should be typical in this case. On the other hand, in ordered perovskites with a very small rattling space, the B ions cannot move easily without distorting the octahedron. A smaller permittivity and a Curie-Weiss constant are expected.

(a) (b) Fig. 4.4 Crystal structure models of the A(BI1/2BII1/2)O3 type perovskite: (a) the ordered structure with a small rattling space and (b) the disordered structure with a large rattling space [open circle = BI (lower valence cation) and solid circle = BII (higher valence cation)].

110 (2)

Chapter 4 Diffuse Phase Transition

The exact reason why the phase transition is diffuse in the relaxor ferroelectrics has not yet been clarified. We introduce here the "microscopic composition fluctuation" model which is one of the most widely accepted models for the relaxor ferroelectrics.5-7) Within a single Kaenzig region [the minimum polar region size in which cooperative polarization (ferroelectricity) can occur], typically on the order of 10 - 100 nm, the model applied to a Pb(BI 1/3BII 2/3)O 3 relaxor assumes a local fluctuation of the B I2+ and BII 5+ ions. Figure 4.5 shows a computer simulation of the composition fluctuation in an A(BI1/2BII1/2)O 3-type crystal calculated for various degrees of ionic ordering. The fluctuation of the BI/BII fraction x obeys a Gaussian error distribution. H. B. Krause has reported the short-range ionic ordering of Pb(Mg1/3Nb 2/3)O3 observed by electron microscopy.8) The high resolution image in Fig. 4.6 reveals somewhat ordered islands in the range of 2 - 5 nm, each of which may have a slightly different transition temperature.

Fig. 4.5 Computer simulation of the composition fluctuation in an A(BI1/2BII1/2)O 3 - type crystal calculated for various degrees of ionic ordering (Kaenzig region size: 4 x 4).

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Fig. 4.6

High resolution electron-microscope image of a Pb(Mg1/3Nb 2/3)O3 single crystal (110). Note ion-ordered islands in the range of 2 - 5 nm. In the case of the permittivity, for example, by superimposing the Curie-Weiss law for each cluster with a different Curie temperature, we obtain a rather broad permittivity peak, which provides more stable temperature change. Thus, sometimes the "Curie range" is specified rather than the "Curie point." The permittivity of relaxor ferroelectrics in the paraelectric region obeys the following quadratic relation: 1 / ε = 1 / ε0 + (T - T C) 2/ C* ,

(4.2)

rather than the normal law 1 / ε = (T - T C) / C .

(4.3)

In order to improve the temperature coefficient of permittivity, by promoting a more diffused phase transition, the following techniques are applied:7) (a) Ion-disordered crystals produced by (a-1) adding a nonferroelectric component (e.g. (Pb,Ba)(Zr,Ti)O3 where BaZrO3 is a non-polar material). (a-2) the generation of lattice vacancies (e.g. (Pb,La,¨)(Zr,Ti)O3 ). (b) Short-range ordering within the crystal due to the generation of cation-ordered clusters (e.g. Pb(Mg1/3Nb 2/3)O3 ,Ti)O 3 and Pb(Mg1/2W 1/2,Ti)O3 (PMW-PT)).

112

Chapter 4

Improvement of the temperature coefficient of permittivity by means of (b) is exemplified by the solid solution PMN-PT incorporating PMW or Ba(Zn 1/3Nb 2/3)O3 (BZN). The addition of PMW tends to generate microclusters of the 1:1 ordered-type, and BZN clusters of the 1:2 ordered type.

(3)

Dielectric Relaxation

Another significant characteristic of these "relaxor" ferroelectrics is dielectric relaxation (frequency dependence of the permittivity) from which their name is derived. The temperature dependence of the permittivity for Pb(Mg1/3Nb 2/3)O 3 is plotted in Fig. 4.7 at various measuring frequencies.9)

Fig. 4.7 The temperature dependence of the permittivity and tan δ in Pb(Mg1/3Nb 2/3)O3 for the various measuring frequencies (kHz): (1) 0.4, (2) 1, (3) 45, (4) 450, (5) 1500, (6) 4500.

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(a) Skanavi - type relaxor

(b) Ferroelectric relaxor

Fig. 4.8 Multi-potential-well model for (a) the Skanavi-type and (b) the ferroelectric relaxors: Note the difference in the cooperative phenomenon. With increasing frequency, the permittivity in the low-temperature (ferroelectric) phase decreases and the peak temperature near 0o C shifts towards higher temperature; this is contrasted with the behavior of normal ferroelectrics such as BaTiO 3, where the peak temperature changes little with the frequency. This is understood generally to be caused by shallow multipotential-wells associated with the locally distorted perovskite cell due to the disordered ionic arrangement (Skanavi-type dielectric relaxation),10) in addition to a ferroelectric phase transition phenomenon. Figure 4.8 illustrates the model. The Skanavi-type provides local dipoles and exhibits an "electret" like property. When a long-range cooperative phenomenon (ferroelectricity) is superposed (pictured by the springs connecting the constituent ions), the net polarization appears. Example Problem 4.2_________________________________________________ Consider an order-disorder type ferroelectric with an ion trapped in a doubleminimum potential with a relatively low barrier between the two minima (Fig. 4.9). Under a quasi-dc field, the ion follows the electric field alternating between the positive and negative potentials. However, with increasing drive frequency the ionic motion exhibits a delay with respect to the electric field due to the potential barrier ∆U. This is an intuitive explanation for the dielectric relaxation. (1) Using a mathematical representation, derive the Debye dispersion relation for a monodispersive case: ε(ω) = εs / (1 + j ωτ) .

(P4.2.1)

(2) Also discuss how the above dispersion obeys the so-called Cole-Cole relation (i.e., the real and imaginary parts of permittivity trace a half circle on a complex permittivity plane).

114

Chapter 4 αα+

-

+ ∆U

µF

µF F

Fig. 4.9 Ion in a double-minimum potential. Solution When an external electric field E is applied, the local field F in the crystal is described by F=E+γP.

(P4.2.2)

The transition probability for an ion from the - to the + in Fig. 4.9, α +, and the opposite transition probability α-, are expressed as α+ = Γ exp[- (∆U - µF)/kT] ,

(P4.2.3)

α- = Γ exp[- (∆U + µF)/kT] .

(P4.2.4)

Here, ∆U is the barrier height between the two potential minima, µ the dipole moment, and Γ is a constant. If we introduce the number of + (or -) direction dipoles per unit volume N+ (or N-), the total dipole number is given by N = N+ + N-, and the polarization (per unit volume) is represented as P = (N+ - N- ) µ .

(P4.2.5)

The time dependence will be expressed as

Then,

dN+/dt = N- α+ - N + α -

,

(P4.2.6)

dN-/dt = N+ α- - N - α +

,

(P4.2.7)

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dP/dt = µ (dN+/dt - dN -/dt) = 2 µ (N- α + - N + α- ) .

(P4.2.8)

N+ = (1/2) (N + P/µ), N- = (1/2) (N − P/µ),

(P4.2.9) (P4.2.10)

Suppose that the external electric field E = E0 e jωt is small and that the polarization is given by P = PS + ε0 ε E0 e jωt

(P4.2.11)

,

From Eq.(P4.2.8), ε0ε E0 (jω) e jωt = 2 µ (N- Γ exp[- (∆U - µF)/kT] - N + Γ exp[- (∆U + µF)/kT]) = 2 µ (N- Γ exp[- (∆U - µ(E + γ P))/kT] - N + Γ exp[- (∆U + µ(E + γ P))/kT]) = 2 µ (N- Γ exp[- (∆U - µ(E + γ(PS + ε0 εE)))/kT] - N + Γ exp[- (∆U + µ(E + γ(P S + ε0εE)))/kT]) =2 µ (N- Γ exp(- ∆U/kT) exp(µγPS/kT) (1 + µ(1 + γε 0 ε)E/kT) - N + Γ exp(- ∆U/kT) exp(- µγP S)/kT) (1 - µ(1 + γε 0 ε)E/kT)) =2Γexp(-∆U/kT)µ[(1/2)(N−P/µ)exp(µγP S/kT)(1+µ(1+γε0 ε)E/kT) - (1/2)(N+P/µ) exp(- µγPS )/kT) (1 - µ(1 + γε 0 ε)E/kT)) =2Γ exp(- ∆U/kT) [µN(sinh(µγPS/kT) + µ(1+γε 0ε)(E/kT)cosh(µγPS/kT)) - P(cosh(µγPS/kT) +µ(1+γε0 ε)(E/kT)sinh(µγP S/kT))] (P4.2.12) Consequently, we obtain ε(ω) = εs / (1 + j ωτ) ,

(P4.2.13)

where τ = (1+γε 0 εs ) τ0 /cosh(µγPS/kT) τ0 = 1 / 2Γ exp(- ∆U/kT) .

,

(P4.2.14) (P4.2.15)

The subscript s stands for a static value (ω = 0), and in the paraelectric phase εs = C / (T - T C).

(P4.2.16)

116

Chapter 4

ε”

ω= ∞

0

ω=0

εs /2

εs

ε’

Fig. 4.10 Cole-Cole plot for a double-minimum potential model. Equation (P4.2.13) can be rewritten as ε(ω) = ε'(ω) + j ε"(ω) , ε'(ω) = εs / (1 + (ωτ) 2) , ε"(ω) = = ωτ εs / (1 + (ωτ) 2) .

(P4.2.17)

So-called Cole-Cole relation is obtained from Equ. (P4.2.17) (see Fig. 4.10): (ε'(ω) - εs /2)2 + ε"(ω)2 = (εs /2) 2. (P4.2.18) ___________________________________________________________________

Another explanation for the dielectric relaxation has been proposed by Mulvihill et al. for Pb(Zn 1/3Nb 2/3)O 3 single crystals.11) Figures 4.11(a) and 4.11(b) show the dielectric constant and loss versus temperature for an unpoled and a poled PZN sample, respectively. The domain configurations are also pictured. The macroscopic domains were not observed in an unpoled sample even at room temperature [Fig. 4.11(a) right], in which state only large dielectric relaxation and loss were observed below the Curie temperature range. Once the macrodomains were induced by an external electric field [Fig. 4.11(b) right], the dielectric dispersion disappeared and the loss became very small (that is, the dielectric behavior became rather normal!) below 100o C. As the temperature is increased, the macroscopic domains disappear in the poled sample at 100o C, then immediately above this temperature both large dielectric dispersion and loss appear [Fig. 4.11(b) left]. Therefore, the dielectric relaxation appears to be associated with the microdomains generated in this material. Mathematical treatment of the relaxor behavior originating from the presence of microdomains has not yet been conducted.

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(a) A depoled 100%PZN single crystal measured on the <111>

(b) A poled 100%PZN single crystal measured on the <111>

Fig. 4.11 Dielectric constant and loss versus temperature for (a) an unpoled and (b) a poled PZN sample. The domain configurations are also pictured. Once the macrodomains were induced by an external electric field, the dielectric dispersion disappeared and the loss became very small.

CHAPTER ESSENTIALS_________________________________ 1.

Basic specifications required for capacitors: (a) Small size, large capacitance (b) High frequency characteristics (c) Temperature characteristics

2.

Electrostatic capacitance of a multilayer capacitor (layer number = n): C = n 2 (ε0 ε S / L)

118 3.

Chapter 4 Characteristics of relaxor ferroelectrics: (a) high permittivity (b) temperature - insensitive characteristics (i.e., diffuse phase transition) (c) dielectric relaxation

4.

Modified Curie-Weiss law for relaxor ferroelectrics: 1 / ε = 1 / ε0 + (T - T C) 2/ C* 5. Dielectric relaxation in some relaxor ferroelectrics is attributed to the presence of microdomains. Once macrodomains are induced by an external electric field, the dielectric dispersion disappears and the loss becomes very small. ___________________________________________________________________

CHAPTER PROBLEMS 4.1

A multilayer capacitor (50 layers) is made from a 10 µm thick sheet with a dielectric material ε = 3000. Assuming a 90% ratio of overlapped electrode area over the chip surface area, calculate the chip area to obtain a total capacitance of 10 µF.

4.2

When the relaxation time is distributed, the permittivity dispersion follows as ε(ω) = εs / (1 + (j ωτ)β), where β < 1. Discuss the Cole-Cole plot change in comparison with the case β = 1.

REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)

Murata Catalog: Miracle Stones. K. Utsumi: Private communication at 4th US-Japan Seminar on Dielectrics & Piezoelectric Ceramics (1989). K. Utsumi, Y. Shimada, T. Ikeda and H. Takamizawa: Ferroelectrics 68, 157 (1986). K. Uchino, L. E. Cross, R. E. Newnham and S. Nomura: J. Phase Transition 1, 333 (1980). W. Kanzig: Helv. Phys. Acta 24, 175 (1951). B. N. Rolov: Fiz.Tverdogo Tela 6, 2128 (1963). K. Uchino, J. Kuwata, S. Nomura, L. E. Cross and R. E. Newnham: Jpn. J. Appl. Phys. 20, Suppl. 20-4, 171 (1981). H. B. Krause, J. M. Cowley and J. Wheatley: Acta Cryst. A35, 1015 (1979). G. A. Smolensky, V. A. Isupov, A. I. Agranovskaya and S. N. Popov: Sov. Phys.Solid State 2, 2584 (1961). G. I. Skanavi, I. M. Ksendzov, V. A. Trigubenko and V. G. Prokhvatilov: Sov. Phys.-JETP 6, 250 (1958). M. L. Mulvihill, L. E. Cross and K. Uchino: Proc. 8th European Mtg. Ferroelectricity, Nijmegen (1995).