Ferroelectric Switching and the Sievert IntegralP. H. Fang and Irene A. Stegun Citation: Journal of Applied Physics 34, 284 (1963); doi: 10.1063/1.1702599 View online: http://dx.doi.org/10.1063/1.1702599 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Modeling the switching kinetics in ferroelectrics J. Appl. Phys. 110, 114106 (2011); 10.1063/1.3660680 Four switching categories of ferroelectrics J. Appl. Phys. 105, 094112 (2009); 10.1063/1.3117494 Thermal activation of ferroelectric switching J. Appl. Phys. 103, 014101 (2008); 10.1063/1.2822179 Ferroelectric switch for spin injection Appl. Phys. Lett. 87, 222114 (2005); 10.1063/1.2138365 Models for Switching in Ferroelectrics J. Appl. Phys. 32, 1169 (1961); 10.1063/1.1736181

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JOURNAL OF APPLIED PHYSICS VOLUME 34, NUMBER 2 FEBRUARY 1963

Ferroelectric Switching and the Sievert Integral

P. H. FANG* AND IRENE A. STEGUN

National Bureau of Standards, Washington, D. C. (Received 9 July 1962)

A connection between some ferroelectric switching functions and the Sievert integral is established. Switching functions of Landauer et al. and of Franklin are computed and discussed.

I. INTRODUCTION

I N the literature of the ferroelectric switching,1-8 an integral of the following form frequently occurs:

F(a,t) = it exp{ -a/E(t)}dt. (1)

In this integral, a is a constant independent of the applied field E(t) and is called the activation field. The physical implications or consequences of the integral have been discussed only under two special simple forms of E(t), they are: (i) a pulse field (with the rising time much faster than the switching time), where E is practically time-independent; or (ii) E(t) is a linear function of t in the form E=kt. The second form of E(t) can be applied to the case of sine-wave field when the amplitude Eo is much larger than a. However, the general case, where E is a sine wave with arbitrary amplitude, due to the seemingly mathematical complications, has never been investigated theoreti­cally. We would like to point out that this case can actually be evaluated by establishing its connection with the Sievert integral9 :

e S(x,fJ) = 1 exp{ -x secfJ}dfJ. (2)

II. FERROELECTRIC SWITCmNG FUNCTION AND THE SffiVERT INTEGRAL

For an applied sine-wave field,

E=Eosinwt, (3)

the integral connected with the ferroelectric switching is:

11",/ F(ao,wt)=- exp{ -ao/sinfJ}dfJ,

w 0

(4)

where ao=a/ Eo. Bya transformation of the integration variable, we obtain

F(ao,wt)=w-1[S(ao,-7I'/2)-S(ao,1!'/2-wt)]. (5)

The discussions of references 2 and 7 are limited to the case wt~1!'/2. It would be interesting, in order to complete the hysteresis cycle, to extend wt from 1!'/2 to 11'. Thus, for

wt=1I'/2+wt', (6)

F(ao,wt) = w-1[S (aQ,1!'/2)+S (ao,wt')]. (7)

The asymptotic form of F for smallao and wt«1 is

F(ao,wt)rv (ao/w)[(e-A/>-)+Ei( ->-)], (8)

where >- = (\'0 cscwt and

-Ei(-x)= r (e-t/t)dt. (9) • x

For large an, whether wt is small or large,

From the available mathematical table,lO we compute F(ao,wt) = w-1(1I'/2ao)e-"'0{1 + !f[ao(secwt' -1) ]t}, (10) the switching functions of Landauer2 and of Franklin,7 and compare the results. where q> is the probability integral,

* Presently with NASA Goddard Space Flight Center, Greenbelt, Maryland.

I W. J. Merz, Phys. Rev. 95, 690 (1954); J. Appl. Phys. 27, 938 (1956).

2 R. Landauer, D. R. Young, and M. E. Drougard, J. Appl. Phys. 27, 752 (1956).

3 H. H. Wieder, J. Appl. Phys. 27, 413 (1956); 31, 180 (1960). 4 M. Prutton, Proc. Phys. Soc. (London) B70, 64 (1957); 72,

307 (1958). 5 C. F. Pulvari and W. Kuebler, J. Appl. Phys. 29, 1315 (1958). 6 E. Fatuzzo and M. J. Merz, Phys. Rev. 116,61 (1959). 1 A. D. Franklin, Progress in Dielectrics, edited by J. B. Birks

(John Wiley & Sons, Inc., New York, 1960), Vol. 1, p. 171. 8 R. C. Miller and G. Wienre~h, Phys. Rev. 117, 1460 (1960). 9 R. M. Sievert, Acta Radiol. 11,249 (1930). 10 National Bureau of Standards, Table of the Sievert Integral,

Appl. Math. Series (U. S. Government Printing Office, Washington, D: C., 1962).

q>(s) = (2/11'1) i 8

e- t2dt. (11)

III. TWO SWITCHING FUNCTIONS

The switching function of Landauer et al.,2 when extended to the case of sine-wave field, is

P(t)/P.=1-2 exp{-~ wt""

x [S(ao,1I'/2)-S(ao, 1I'/2-wt)]}. (12)

This is obtained as the solution of the differential

284

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FER ROE LEe T RIC S WIT CHI N G AND THE S t EVE R TIN T E G R A L 285

p

1 +1

~+----~-/-r-7 I I I '

I-T I

/ / / I

I I +.5 I I .,/ .. i ".c

, I I

....}-I.O I I I

I I

I I

I Ii

J , Eo

/ I I I

I I / I

) / .I -'------4-~~-~

I I I I

I ) ./ -

o

-.5

-I

FIG. 1. Hysteresis loop calculated from the switching function of Landauer et al.

equation: dP(t)/ dt= (t.)-l(Ps- P), (13)

where PCt) is the transient polarization, P s is the spon­taneous polarization, and

(14)

with t", corresponding to the switching time at the "infinite field." The initial condition is taken such that

P(t)= -Ps at t=O. (15)

It might be interesting to point out that a differential equation somewhat analogous to (13) was studied by Gurevich and Gribov in connection with the dielectric loss in ionic dielectrics at strong fields.u

On the other hand, the switching function of Franklin7

is obtained from the differential equation,

d P(t)+Ps -log'---dt 2P.

1 { P(t)+Ps}2 log .

t. 2P. (16)

The notations have the same meanings as those of Eq. (17). With the same initial condition:

P(t)/ p.= 2 exp{ -w(,jS(ao,n/2) -S(ao,n/2-wt)}-1. (17)

From (12) or (17) we can calculate the reversal polarization as a function of the transient field. In

1] L. E. Gurevich and V. N. Gribov, J. Exptl. Theoret. Phys. USSR 29, 629 (1959) [translation: Soviet Phys.-JETP 2, 565 (1956)].

+1 r-

.---+-I I . / I / I f

I / I

-~ I . I ! / / / .

=4/ =8/ I I

+.5

I, E o

f I / I I I / i

-.5

I .

~~--.) -I

FIG. 2. Hysteresis loop calculated from the switching function of Franklin.

order to complete the hysteresis cycle, we make the assumption that the extension of (12) and (17) from wt= n/2 to 7r, through (7), is valid for both switching functions under discussion. To extend the field from wt=7r to 27r, one can either change the initial condition (15) or simply impose the symmetry argument.

To make an explicit calculation, we take the case of BaTi03 which has a value of a=6.1 kV/cm and t",=4X1O-7 sec. l2 With a 60-cycle ac field, we calculate the cases ao=a/ E= 2, 4, and 8, respectively. Figure 1 is obtained from Landauer's function and Fig. 2 from Franklin's.

IV. DISCUSSION

By making comparisons of Figs. 1 and 2, we observe that when the applied field is very high, i.e., Eo>a/4, both switching functions give a reasonable square hysteresis loop as in qualitative agreement with that observed in actual experience. On the other hand, the switching function of Landauer et al. still gives a satu­ration polarization at a low field (Eo"-'O.1a) where the switching function of Franklin predicts a "minor hysteresis loop." In this region Franklin's function would be closer to actual observation.

Since both theories do not give a coercive field, that is, a field below which there is no change of polarization, one would expect at the initial portion (wt"-'O) of the

12 P. H. Fang and E. Fatuzzo, J. Phys. Soc. Japan 17, 238 (1962).

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286 P. H. FANG AND I. A. STEGUN

sine field, a continuous increase of the absolute value of the polarization. This is not shown by both switching functions.

When the sine-wave sweeps from wt=7r/2 to 71", both switching functions show that the polarization is either a constant (strong field) or increasing (weak field) from the value at 1»1=71"/2. In the experiment, however, one actually observes a decrease in the value of po­larization. Therefore, in order to extend the switching function to the weak field case, one has to introduce a

depolarization term into the switching equation. A depolarization term which depends on dE/dl seems to be able to account for this effect.13

ACKNOWLEDGMENT

We would like to acknowledge the help of Dr. F. Oliver who derived the asymptotic formula for the case of small x and ao.

)3 E. Fatuzzo (private communication).

JOURNAL OF APPLIED PHYSICS VOLUME 34, NUMBER 2 FEBRUARY 1963

Electrical Resistivity of Scandium

R. V. COLVIN AND SIGURDS ARAJs

Edgar C. Bain Laboratory For Fundamental Research, United States Steel Corporation Research Center, Monroeville, Pennsylvania

(Received 18 July 1962)

Electrical resistivity of polycrystalline scandium has been studied between 1.38° and 358.8°K. The meas­urements show a step-type anomaly in the resistivity between 4 and 5 OK and p3ssibly a very small minimum at 9.5°K. Considerable hysteresis in the electrical resistivity exists at low temperatures, depending upon the previous thermal history of the sample. It is very likely that these anomalies may be associated with a super­conductive substructure resulting from the presence of small amounts of tantalum as an impurity. However, the possibility that the anomalous behavior is due to the occurrence of some cooperative magnetic states at low temperatures cannot be completely eliminated at the present time. Between about 80° and 3600 K the intrinsic resistivity of scandium is describable by the Bluch-Griineisen formula with Debye (J of 27S°K. At lower temperatures the temperature dependence of the intrinsic resistivity is consistent with expectations derived from the interband electron-phonon scattering. Ziman's reduced electrical resistivity at 3()()OK is calculated to be 50.8.

INTRODUCTION

SCANDIUM is the first member of the first series of transition elements. The physical properties of this

metal are quite unexplored, primarily due to its limited availability. The recent interest in the electronic struc­tures of the transition elements has stimulated us to investigate the electrical resistivity of scandium from liquid helium to room temperatures. Such measure­ments have not been made before. These are of interest, first, because of the suggested possibilityl that scandium becomes antiferromagnetic at low temperatures and, secondly, to gain additional information on the behavior of transport properties of transition metals. This paper presents the results of the electrical resistivity measure­ments of scandium and a short discussion of their significance.

EXPERIMENTAL CONSIDERATIONS

Polycrystalline scandium in the form of a rod2

(diameter 0.635 cm, length 7.62 cm) was compared by spectroscopic analysis with a sample of SC203 in powder

) H. Montgomery and G. P. Pells, Proc .. Phys. Soc. (London) 78, 623 (1961).

2 Obtained from St. Eloi Corporation.

form.3 Scandium oxide of 99.9% purity, according to the supplier, contained less than 0.1% yttrium and spectrographic traces of silicon and zirconium. The analysis indicated that the major impurity in the metal which was not present in SC203 was tantalum (pick-Up from casting in a tantalum crucible). The presence of other contaminants such as calcium and iron were also noted but these were in very small concentrations. The presence of tantalum in scandium was confirmed by obtaining an insoluble residue from a portion of the scandium sample after a treatment with HCI and HN03. The residue, ignited at lOOO°C, was identified by x-ray diffraction and fluorescence techniques to be Ta205.

A section of the rod was examined metallographically and a few dark inclusions and possibly some small amount of structural imperfections were found. No preferred orientation was observed. The size of grains was found to be in the range from 0.14 to 0.22 mm in approximate diameter.

A cylindrical sample for the electrical resistivity measurements having a diameter of 0.486 cm and length of 6.35 cm was machined from the original rod. This

3 Obtained from the Research Chemicals Division of Nuclear Corporation of America.

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