III.9. THE HYSTERESIS CYCLE OF FERROELECTRIC SUBSTANCES

Transcription

1 III.9. THE HYSTERESIS CYCLE OF FERROELECTRIC SBSTANCES. Work purpoe The analyi of the behaviour of a ferroelectric ubtance placed in an eternal electric field; the dependence of the electrical polariation upon the eternal field, P = P(E).. Theory Ferroelectric ubtance are inulator that have a pontaneou polariation and a domain tructure. In each domain of uch a ubtance, the molecular dipole have the ame orientation, but thi orientation i different for different domain and can be influenced by an eternal electrical field. For ferroelectric ubtance, pontaneou polariation occur only for a certain temperature range. In mot of the cae, ferroelectric crytal may eperience change of their tructure, thu looing their pontaneou polariation. The tructure change of uch crytal, followed by the appearance or diappearance of pontaneou polariation, i called a phae tranition, and the correponding temperature i called tranition temperature or Curie point. The crytal KH PO 4, BaTiO 3, LiTaO 3 and other are ferroelectric if T < T C and paraelectric if T > T C. In the tranition temperature neighbourhood, almot all the propertie of the crytal (electrical, optical, mechanical, thermal etc.) abruptly change and preent everal anomalie.

2 The domain tructure influence everal non-linear propertie of uch ubtance, among which there i the non-linear dependence of the electrical polariation on the eternal field P = P(E). The image below (Fig. ) how the hyterei curve of a ferroelectric ubtance, which prove that the influence of an eternal electrical field caue the rearrangement of domain (piece). The different rate at which the domain and the electrical field change lead to a phae difference between the electric field intenity and the polariation, hence the non-linear variation. Figure. When an electric field influence the crytal, the domain having a polariation parallel to the eternal field increae on the account of thoe domain having a different polariation; thu, the polariation increae and the dependence P = P(E) i hown by the curve a-b-c in Figure. When all the domain have become oriented parallel to the eternal electric field, the polariation reache a aturation level P ; we may ay that the crytal ha became a ingle domain. The value of the pontaneou polariation i obtained by the etrapolation of the linear part in the point c. The value of P that wa obtained i obviouly imilar to the polariation that eited in 79

3 each domain in the initial tate correponding to the point a. Therefore, when we refer to pontaneou polariation, we mean the polariation of each eparate domain, not the total polariation of the whole crytal. When the intenity of the electrical field decreae, o doe the polariation. If the field intenity become zero, there will till be a remnant polariation P r in the crytal, repreented by the curve c-d. In order to detroy thi remnant polariation, a good part of the crytal mut be polaried in the oppoite direction by applying an oppoite electric field. Thi electric field, which i needed to counteract the polariation, i called coercive field; it intenity i quoted -E c and it application i repreented in Figure by the curve d-e. Afterward, the polariation of the ubtance ample change it ene according to the orientation of the domain correponding to the new direction of the electrical field; it will eventually reach it maimum value -P, repreented by the point f in Figure. The ubequent variation of the electrical field toward poitive value will generate a dependence P = P(E) imilar to that already decribed previouly; thi dependence i repreented by the curve f-g-h-c. The ferroelectric ubtance that we will be uing in thi eperiment i a Rochelle alt crytal. The Rochelle alt wa proven to have ferroelectric propertie in 9. It i a alt of odium and potaium reulting from tartric acid. It chemical formula i NaKC 4 H 4 O 6 4H O. The Rochelle alt ha ferroelectric propertie only in the temperature range 8 o C 4 o C. Thi mean that it ha tranition temperature: the lower Curie point at 8 o C, and the upper Curie point at 4 o C. In the temperature range below 8 o C and above 4 o C, the crytal ha an orthorhombic tructure. When it become ferroelectric, the crytal i monoclinic. The Rochelle alt ha only one polar ai and only two 80

4 poible polariation direction, that i, parallel or anti-parallel to the ai. Conequently, the domain tructure of the Rochelle alt i very imple. 3. Eperimental Set-up We will analye the non-linear dependence of the electrical polariation P(E) in an electric field that ocillate a: ( πνt) E = E in, () where E 0 i the amplitude of the field and ν i it frequency. 0 The draft of the eperimental device i repreented in Fig. : The voltage Figure. g adjuted by the elf-tranformer ST, i applied to the tranformer TR, whoe function i to galvanically eparate the generator from the ret of the device. From it econdary, the generated voltage i applied to the circuit. The central piece of thi circuit i the differential amplifier DA (upplied by the ource SD) which generate a voltage f given by: f =. () i the voltage obtained at the jack of the divider P ( ), C, can be computed by thi formula: R V, and = idt = dt + C R V, (3) 8

5 where R i the variable reitance and C V i an adjutable air capacitor; both element are conidered ideal, without any electric charge. The capacitor C 0 i the circuit element that collect the electrical charge of the ample; thi capacitor mut atify the condition C >> C, C, o that 0 V mot of the voltage will be ditributed on C and C V, repectively. The ferroelectric ample, repreented by the parallel ytem C R, i placed in the divider C, R,. The obtained voltage ( ) will be: where Q = C 0 = C 0 i dt = C Q f i the ample ferroelectric charge, non-ferroelectric capacitance. By adjuting R = R 0 and V dt + C R + Q f, (4) R it reitance and C it C = C, the voltage f i obtained at the terminal of the differential amplifier AD; thi voltage control the ocillocope OSC on the vertical, and i proportional only to the ferroelectric charge of the ample: where A i the ample area and P it polariation. Q AP f f = = =, (5) A ignal proportional to the electric field intenity E in the ample i applied to the horizontal plate of the ocillocope, E being related to a: E =, (6) d where d i the thickne of the ample, and i the voltage meaured by the voltmeter V (For a correct fitting onto the creen, the voltage i applied to the plate through the potentiometer P ). 8

6 4. Working Procedure The following tak will be eecuted in thi eact order:. Adjut the elf-tranformer ST and the ource SD to a voltage equal to 0 V.. Plug in the ource SD and adjut the voltage of the two channel to 0 V (thi value hould be marked red on the diplay). 3. Plug in the elf-tranformer and the ocillocope. Turn on the ocillocope and the ource SD. 4. Eamine the curve P = P(E) on the creen of the ocillocope and centre them, if neceary. In order to obtain the larget poible image that till fit the creen, adjut P for the appropriate horizontal dimenion of the cycle. 5. Adjut P and C V, in order to obtain a correct hyterei curve. In Figure 3 below we have repreented everal poible dependencie P = P(E). In order to obtain a correct curve (that ha horizontal and pointed aturation ide, a een in fig. 3.e), you mut adjut P to correct ditortion 3a and 3b, and C V to correct ditortion 3c and 3d. Figure 3. 83

7 5. Eperimental Data Proceing The adjutment uggeted previouly mut be done for each value of the voltage applied to the ample: 00, 50, 00, 50, and 300 V. For each voltage, you mut read on the creen poition n, n, n 3, and n 4 correponding to the maimum electric intenity E, the coercive electric field intenity E c, the remnant polariation P r and the maimum polariation P m, repectively. The collected data are written in Table : Table (V) n n n 3 n 4 E E c P r P m The value of E are computed uing the formula (6). Since horizontal diplacement are proportional to the electric field, it follow that: From equation (6), we obtain: where (/ range). Pm n E c = n f n = = K yc 4 0, S S n E =. (7) n d Pr n = K yc 3 0, (8) S K y i the vertical enitivity of the ocillocope at the terminal X3 The value of the parameter that are ued in thi eperiment are d = 4 mm, K y = V/div, C 0 = 0 7 F, A = cm. 84

8 Plot the dependencie Pr ( E), Pm ( E) and E c ( E). Oberve the tendency toward aturation of thee quantitie a the intenity of the electrical field increae and etimate their limit value. When computing a quantity that i not directly meaured, one mut apply the error propagation method. According to thi method, any quantity (,y,z... ) X = f can be determined by applying the following equation: where (,y,z,... ) X = X ± S X, (9) X = f, = N and the diperion or the mean quare root deviation i: with X f = = f + y N y= y y ( i ) i= N( N ) N i, (0) i= f + z z= z z +..., () =, () For thi particular eperiment, you mut determine the value of the coercive field intenity E c, the remnant polariation P r and the maimum polariation P m by applying the above method. For the firt quantity: X = E ; = n ; y n. In order to compute the average value and the c i i i = i mean quare root deviation, you mut make 0 independent meaurement for each voltage value. The final reult will have the form E = E ± S. c c E c Similarly, you will obtain P = P ± S, r r P r P = P ± S. m m P m 85