Ferroelectrics investigation

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Gervais Newman
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1 Ferroelectrics investigation. Introduction A dielectric is understood as a material where the electric field induces an electric momentum. Let s consider a vacuum capacitor made of two planar metallic electrodes biased with a D voltage, U. The capacitance is given by: Q, () U where Q denotes the electric charge gathered on the capacitor electrodes. The electric capacitance is related to the geometric dimensions of the planar capacitor in the following approimate way: ε S () d - the electrode area, - the distance between the electrodes, - dielectric permittivity of vacuum (ε F/m). The electric field intensity inside the capacitor is given by: U () d When the capacitor is filled with a dielectric, the capacity increases up to: εε rs ε r, () d where ε r is the relative dielectric permittivity lectrodes lektrody Dielectric Dielektryk + Figure The capacitor Part of the charge brought to the capacitor is compensated by charge gathered in the electrodes (Fig.). This charge is called the bound charge. The surface free charge density remains unchanged. That is why when the dielectric is inserted into the biased capacitor the electric field intensity does not change. The electric induction is defined for linear dielectrics as follows: D ε, (5) where ε εε r is called the dielectric permittivity of a medium. The electric field intensity inside the capacitor is connected to the surface free charge density. Whereas the electric induction is equal to the total surface charge density. The electric induction inside the dielectric can be epressed as: D D + P, (6) where P denotes the medium polarization, and D ε. The medium polarization (P) is equal to the surface bound charge density. The polarization is a linear function of the electric field density for most of dielectrics in moderately weak electric fields: P ε χ (7) where χ denotes the medium electric susceptibility. Using the notation mentioned above, qn. (6) may be rewritten in a form:

2 It can come of qn. (7) that: ε + χ and P χ ε ε χ. (8) ε + There are a few mechanisms of a dielectric polarization.. lectron polarization q + L P ε +. (9) ε The electric field deforms electronic shells of atoms so that the gravity centers of the nucleus electric charge and those of the electronic shells do no coincide, forming electric dipoles. The electric dipole diagram is shown in Fig.. Dipole moment Figure An electric dipole µ gl () The electric polarization is the dipole moment of a unit volume P µ i () V i. Atomic (ionic) polarization In ionic materials there is a shift of atoms (or ions) due to the electric field that causes the dipole moment creation. After the electric field is switched off the atomic and ionic polarization decays very fast and this is why this polarization is called elastic.. Dipole polarization. Some dielectrics e.g. water, are built out of particles showing the dipole moments. Thermal motion causes that the dipole orientation is random, and in total the polarization is zero. An electric bias causes that the dipoles in a dielectric are ordered, and the polarization related to the process is called the orientation one.. Polarization connected to the pace charge. q _ Dielectrics can ehibit a non-uniform distribution of the space charge due to the production process or an intentional process after the production. The electric field due to the space charge cause the dielectric polarization called the space-charge polarization. This polarization is used in electrets. The material electric polarization is a sum of all above.. Dielectric in the alternating electric field Let s consider a dielectric placed in an electric field, oscillating according to: where denotes the electric field amplitude, ω πf - - circular frequency ( ) i, t time i ω t ( t ) e ()

3 The induction can be delayed with respect to the electric field in the real dielectrics. The delay reason is a finite time needed for the dipole orientation change. The induction is changed according to the formula: i( ω t ϕ ) D D e () where D denotes the amplitude of the electric induction, is the phase shift between the induction and the field. Based on the dielectric permittivity definition (qn.()) we can rewrite The uler s formula says: therefore where: i( t ϕ ) D D e D iϕ e () ϕ e i e i t cosϕ i sinϕ D D cosϕ i sinϕ 'i" (5) D D ' cosϕ and " sinϕ (6) Quantities and denote the real and imaginary part of the dielectric permittivity. The dielectric permittivity can thus be described as an imaginary number. In practice the tangent of the loss angle (defined below) is used to describe dielectric properties " sinϕ tg tgϕ (7) ' cosϕ. Dielectric permittivity measurement Schering bridge is most frequently used for the dielectric permittivity measurements (Fig ). The bridge is fed with the A. A variable capacitor is placed in one of the bridge arms and parallel to it there is an adjustable resistor. The real capacitor ( *) is substituted by an ideal capacitor ( ) and a parallel resistor ( ). The is responsible for the current flowing through the real capacitor. I * X I Figure Schering bridge diagram and the equivalent circuit of the real capacitor,

4 Admittance of the capacitor in question is: Similarly Y + (8) + i i i The bridge undergoes the condition of equilibrium when: Y + (9) I I I I where denotes the impedance of the respective arm of the bridge. Dividing the equations by sides we can obtain: thus () () () Substituting qn.() into (8) and (9) we can obtain: + i + i qn. () is fulfilled when the respective parts: real and imaginary are equal to each other, so: and () () Based on the comple form of the dielectric permittivity the capacitor impedance can be written as: or admittance Therefore From qn.(7) (5) i ( ) i'i" i' + " (6) ' and " (7) ' " (8)

5 Dividing by sides of qns. (8) and based on qn. (7) we can obtain: " ' tg (9) We can easily calculate, and thus, when we know geometric dimensions of the capacitor: ' () Most of automatic bridges make it possible to read and tg. The measurements should consider of the wiring capacitance, d. This capacitance should be subtracted from the measured one..basic terms related to ferroelectrics Some crystals possess certain polarization even under absence of the electric field. The polarization is called the spontaneous polarization and labeled as P s. It is found that the spontaneous polarization can only occur in crystals with polar aes of symmetry. The crystals are called pyroelectrics. A temperature dependence of the polarization is linear for pyroelectrics. dp s (T) dt () Factor is called the pyroelectric coefficient. When the polarization direction can be changed by an eternal electric field the crystal is called a ferroelectric (analogous to a ferromagnetic). A temperature of the polar symmetry ais decay (so the spontaneous polarization decay) is called the phase transition temperature. This is a transition from the ferroelectric phase to the paraelectric one. The phase transition is a process of the crystal structure change related to its symmetry reconstruction. There also are phase transitions between phases with polar symmetry aes which does not cause a decay of the spontaneous polarization. Both phases keep the spontaneous polarization but its direction or its value may change. There are some other phase transitions, but we shall restrict ourselves to transitions from the paraelectric (non-polar) phase to the ferroelectric one. 5. Ferroelectric phase transitions the thermodynamic description The crystal state can be described with the Gibbs free energy which is defined as: U Y X ST () where: U the crystal internal energy, Y K the electric field intensity or the mechanical strain, X K the electric induction, the polarization or the deformation, S the entropy, T the absolute temperature. K Let us assume that the mechanical strain is constant. The free energy is thus a function of the temperature and the electric field intensity. K 5

6 ( ) U P ST T, () The energies of the both phases are equal to each other at the phase transition temperature where: F P () F - the free energy of the ferroelectric phase, P - the free energy of the paraelectric phase. A phase transition is of the n-th order when the (n-)-order partial derivatives of the free energy are equal each other, and the n-th order partial derivatives are not. The first-order transition but F F P F P P (5) T T Φ P and S T There is a jump in polarization and entropy of the crystal at the phase transition temperature. The second-order transition and but F (6) F P F P P (7) T T T F T S p T T T where p the specific heat. P F P The ferroelectric phase polarization and entropy are equal to those of the paraelectric phase (a continuous change). The specific heat p and the electric susceptibility change in a jumplike style. Let us consider a transition from the ferroelectric phase to the paraelectric phase in more detail. Additionally let us assume that the spontaneous polarization occurs in one direction only. Devonshire, based on the basic theory of phase transitions derived by Landau, suggested to develop the free energy into a power series with respect to the polarization. Since the crystal energy does not change when the spontaneous polarization direction changes, even powers of P s should be considered only P (8) 6 + AP + BP + P +... P (9) 6 omponent P stands for the eternal electric field interaction energy, - the free energy of the paraelectric phase, A, B, are the development coefficients. oefficient A is a linear temperature function ( T ) where T c is the urie-weiss temperature. A () T c 6

7 The relation comes from the eperiment (urie-weiss law). oefficients B and weakly depend on temperature, so we omit these relations. In the case of the second-order phase transitions we can omit P 6 and higher epressions in qn.(9). But it cannot be, however, omitted in the case of the first-order phase transitions. Our considerations will be restricted to the second-order phase transitions only because: ) triglycine sulphate (TGS) will be a subject of the investigation (the crystal shows a second-order phase transition), ) the considerations are more educational. The crystal stable state corresponds to a minimum of the free energy, therefore From qn.() AP + BP P AP + () BP () Figure Ferroelectric hysteresis loop A relation between the polarization and the electric field intensity is shown in a figure according to qn.(). P s denotes the spontaneous polarization (the polarization at ), and c is the coercion field, i.e. an electric field intensity required for the polarization orientation change. The curve shown in the figure is called a hysteresis loop. A ferroelectric crystal consists of domains, i.e. regions with various polarization directions. The crystal partitioning into domains is caused by a high depolarization energy, the energy related to a uniform surface charge distribution. A rise of the intermediate layer energy, called as domain walls, counteracts this partitioning process (by analogy to a surface tension in liquids). An equilibrium is established and this state corresponds to the energy minimum. At equilibrium the macroscopic polarization equals zero. A change in the macroscopic polarization consists in an increase of a domain volume with one polarization direction at epense of domains with an opposite orientation. In the etreme case the entire crystal constitutes a single domain (a uniform polarization). Let us come back to the thermodynamic considerations. When we assume in qn. (), then we get: Hence: AP + BP () 7

8 A P or P ± () B The case with P corresponds to the paraelectric phase, whereas P to the ferroelectric one. Substituting qn.() into qn.() we get: ( T T ) c P ± (5) B In order that qn.(5) has a physical meaning, the following conditions must be fulfilled: α >, B > (T-T < for the ferroelectric phase). A temperature dependence derived from qn.(5) is shown in the figure 5. Figure 5 Temperature dependence of the spontaneous polarization for ferroelectrics with second-order phase transition. Let us calculate a derivative from qn.() P for the paraelectric phase, so: P A + BP ( T ) >> for ferroelectrics, so we can assume qn. (8) and: hence: T c (6) A (7) (8) ( T ) (9) This is the urie-weiss law. qn.(9) makes it possible to determine a temperature relation of coefficient A. The urie-weiss law is also written in a form: where: the urie constant, T c the urie-weiss temperature. In the ferroelectric phase: T c T c ' (5) T 8

9 A A + B A ( T T c ) (5) B The urie constant can be calculated from a temperature relation of /ε in the paraelectric phase. And knowing α we can determine coefficient B from a temperature dependence of P s. Figure 6 Temperature dependence: of the relative dielectric permittivity for ferroelectrics with the second-order phase transitions (a), and of the reciprocal dielectric permittivity (b). elations of ε and /ε obtained from the considerations conducted above are shown in the figure 6. The urie temperature T can be determined from the temperature dependence of /ε. 6. Sawyer-Tower polarization measurement There are a few known methods of the spontaneous polarization measurements in ferroelectrics. A method elaborated by Sawyer and Tower in 9 is one of the most frequently used. The Sawyer-Tower circuit is shown in the figure, where * denotes the measured sample, and n the reference capacitor. The current intensity in the circuit is as follows: where S denotes the electrode surface area of the sample. harge gathered in capacitor n Voltage on capacitor n dp I S (5) dt t t P dp Q Idt S dt S dp (5) dt P Q S U n dp (5) n n Polarization P can be calculated when we know the electrode surface area S, the capacity n, and when we measure the voltage U n. For simplicity the capacitor n is chosen so that the condition of n >> is satisfied, and then U n << U. Voltage on capacitor is practically equal to the feed voltage. We use an oscilloscope to measure voltages U and U n (see Fig. 7). The oscilloscope does not overload the measurement circuit as voltmeters could do. A vertical 9

10 spot deflection is proportional to the voltage on capacitor n, thus to the polarization. Whereas a horizontal spot deflection is proportional to the sample bias, therefore to the electric field intensity U /d The hysteresis loop as a relation of the polarization upon the electric field intensity is displayed on the oscilloscope screen. Often the hysteresis saturation voltage is too high for the oscilloscope circuitry, then a voltage divider D n is employed to reduce the voltage. In order to compensate the phase shift due to the ferroelectric losses a shunt resistor is connected to capacitor n. The measurement diagram is shown in the figure 7. Figure 7 Diagram of Sawyer - Tower ircuit The spontaneous polarization and the coercion field can be determined from the hysteresis loop dimensions (see figure ). The spontaneous polarization can be calculated from the formula: where U y is the vertical spot deflection voltage for, S denotes the electrode surface area of the sample, n is the reference capacitor capacity. The coercion field: U P y n s S (55) U c kd (56) where k is the voltage divider factor, U denotes the voltage of the shift between the hysteresis and the OX ais, and d is the sample thichness.

11 7. Measurement task. Determination of the sample capacity and the sample tg with respect to temperature.. Determination of the oscilloscope voltage sensitivity along the X-direction.. Determination of the temperature relation of the spontaneous polarization by means of the Sawyer-Tower circuit. 8. esults in detail. alculate and plot relations of ε and /ε versus temperature.. ead the urie temperature and calculate the urie-weiss constant (from the paraelectric phase!) from /ε plot.. Plot a temperature relation of tg.. Plot a temperature dependence of the spontaneous polarization P s. 5. Plot a temperature dependence of P s. 6. Determine coefficient B from P s (T) relation.

Experiment 1: Laboratory Experiments on Ferroelectricity

Experiment 1: Laboratory Experiments on Ferroelectricity 1. Task: 1. Set up a Sawyer-Tower circuit to measure ferroelectric hysteresis curves. 2. Check the D(E) curves for a capacitor, a resistor and an