DISLOCATION CHARGE IN ALKALI HALIDES DETERMINED FROM INTERNAL FRICTION AND PIEZOELECTRIC DEFECT MEASUREMENTS

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DISLOCATION CHARGE IN ALKALI HALIDES

DETERMINED FROM INTERNAL FRICTION AND

PIEZOELECTRIC DEFECT MEASUREMENTS

W. Robinson, J. Tallon, R. Buckley

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C10, supplément a u n 0 1 2 , Tome 46, décembre 1985 page C10-521

DISLOCATION CHARGE I N ALKALI HALIDES DETERMINED FROM INTERNAL FRICTION AND PIEZOELECTRIC DEFECT MEASUREMENTS

W.H. ROBINSON, J.L. TALLON AND R.G. BUCKLEY

Physics

and Engineering

Laboratory, DSIR, Lower Hutt,

New

Zealand

Abstract. By using the ultrasonic composite oscillator operating at 40 kHz and strain amplitudes of 10-lO to ~ X I O - ~ , it is possible to determine the interna1 frictipn, the modulus defect and the direct and converse

piezoelectric defect due to charged dislocations in bent alkali halide single crystals. A mode1 independent equation linking these terms enables the determination of the electrical charge on the dislocations together with the dislocation displacement. The temperature at which the mean dislocation charge vanishes is the isoelectric temperature, Te and from its temperature dependence the vacancy formation parameters have been calculated. We also find that Te is weakly dependent on the strain amplitude.

INTRODUCTION

The ultrasonic composite oscillator /1/ has been used with success at room temperature /2/ and at higher temperatures /3/ to determine the dislocation electric charge and the dislocation displacement. Recently /4,5/ we have investigated in the range 400 to 730°C the temperature dependence of the

dislocation charge for .a number of single crystals of KC1 doped with ca2+ (0.02 to 14 ppm). Above 400°C the charge falls rapidly, passes through an extrinsic

isoelectric temperature, Te where it changes sign and then rises very slowly. From the variation of Te with impurity concentration together with published Schottky parameters, we have determined the following values for the vacancy formation parameters: h+=1.256$.008eV, s+=4.53$.10k, h-=1.334$.008eV and s-=5.08$.10k. The temperature dependence of the equilibrium dislocation charge is successfully modelled by a uniform line charge vibrating in a Debye-Huckel screening cloud of vacancies and divalent impurities. Finally, we discuss the strain dependence of Te. TECHNIQUE

The composite oscillator is shown in Fig. 1. Samples are bent and annealed <110> single crystals of calcium doped KCl mounted on a composite oscillator driven at 40 kHz in the longitudinal mode by a closed loop amplifier. The induced specimen voltage, in phase (vSi) and quadrature (Vsq) to the gauge voltage, Vg is monitored on a lock-in voltmeter and specimen impedance is recorded at regular intervals.

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C10-522 JOURNAL

DE

PHYSIQUE

The divalent impurity concentration is

determined by least squares fitting of Drive-

Arrhenius conductivity plots using the

Lidiard theory 161 for dilute Schottky Gauge-

defects allowing association of divalent impurities and cation vacancies. Al1

Schottky thermodynamic parameters are fixed Quartz

at the values obtained by Chandra and Rolfe

BASIC EQUATIONS

The techniaue owes its success to two analytical developments. Eirstly, the equivalent circuit for the composite

oscillator has been solved

181

allowing the Specimen

various voltages to be related to the strain, €11, mechanical damping, Q-l, modulus defect, ASll/Sll and the

piezoelectric defect, d3i1 of the sample. The relevant equations are

Fig. 1

-

Schematic diagram of the

€ 1 1 = 4 2 mCg ( S 1 1 / ~ 2 d ~ l l ) ~ Xs -1 Vgs (1) composite oscillator. and Q~-' = (U2d311 /si i )g2 8/w2ms~gl Vd/Vg ( 2

where Cg is the capacitance over the gauge crystal, 02 is the electrode width,

As

is the wavelength in the sample, w is the angular frequency, ms is the sample mass and the subscript g refers to gauge crystal. Secondly, the linear coupling between these quantities arising from dislocations with a line charge, q implies the f ollowing model-f ree relationship /1,5/.

where aijk is an orientation tensor, b is the Burger's vector, subscript s refers to sample and Gs and Cs are the conductance and capacitance over the sample. For various practical considerations the charge is best determined by driving well into amplitude dependence. This ensures that the dislocation displacement exceeds the Debye-Huckel screening radius so that the induced

polarisation is unscreened and thereby unable to be cancelled by the highly mobile components in the charge cloud.

ISOELECTRIC TEMPERATURE

Dislocation charge calculated from Our measurements using equation (4) is shown by the data points in Eig. 2 over a range of temperatures and impurity concentrations. A steady increase in Te occurs for increasing impurity concentration. The

dislocation becomes uncharged at Te when the chemical potential of evaporated cation vacancies balances their formation free energy ie when

(4)

Here we neglect the small effect

due to impurity-vacancy 2 0 LOO 500 '°C6'00 700 800 association, so that c is the total ern --W.---

mole fraction of impurity. For O r.

...

,.---

,e-s~--ss

-

each value of c, if we take the

-

y

.;Y'-

O

observed Te from Fig. 2, firstly

-

.I

,

,WfJ

-

a(Te) may be calculated, then - 2 0 -

:,'

d'

2'

s

E 1 I ;i'

-

.,

g+(Te) using equations ( 6 ) and (5)

₂

I /r

-

/ /

<;'

respectively. From a plot of - 6 0 - , * , f* E

g+(Te) verses Te we deduce the

-

,4 1.

p

u

Y

individual parameters h+ and s+ ,' f - 2 O

-&O -

listed in the introduction. i f Together with the assumed Schottky

df

parameters these yield the values - B O ₆₀₀ i ₇₀₀, , ₈₀₀, : ' , ₉₀₀, , _1OW

,

, - 3

of h and s- also listed.

T ( K I

DISLOCATION CHARGE

The charge on a dislocation is Fig. 2

-

The temperature dependence or predominantly in the form of a dislocation charge in calcium doped KC1 distributed excess of jogs with for various impurity concentrations; separate charges for which, by virtue left to right: 0.02, 0.23, 1.33, 11.3 of their high core mobility are and 14.4 ppm. Date points: experiment, approximated to a line charge. A solid lines: equations (7) and (8). Debye-Kuckel screening cloud of cation

and anion vacancies as well as less mobile impurities sheathes the core.

The magnitude of the charge is determined by self-consistency between the

thermo-dynamic state of the dislocation and the electrostatic state as determined by Poisson's equation. The former yields /9,3/

where

r

is the density of cation sites along the core, J is the free energy of formation of an unlike pair of well separated jogs and p,(=eV,/kT) is the thermally reduced potential at infinity relative to the core. Poisson's equation provides the simultaneous relation between q and p, necessary for solution, viz /3/

q = (211lrs~lrs~k~/e) tanhp, [t.nXro]-l ( 8 )

rg is the core radius and the parameter

x

is an inverse length related to the Debye-Huckel inverse length, K which recognises that p, may not necessarily be much less than the value one,

X2 = c2 coshp,

,

(9)

and K~ = (8e2/a3tzrtzok~) a = cO2 a (10)

where a is the cation-cation lattice parameter. For each impurity concentration the dashed lines in Fig. 2 give the self consistent solution for q in equations (7) and (8) for the one value of J = 0.45 eV.

STRAIN DEPENDENCE OF THE ISOELECTRIC TEMPERATURE A weak strain dependence was found for the isoelectric temperature, Te. This is more usefully considered in terms of the dislocation displacement dependence of Te. The mean complex displacement is related to the strain, € 1 1 as follows

I l /

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(30-524 JOURNAL

DE

PHYSIQUE

The data points in Fig. 3 show Te as a function of 15

1

for the 1.3ppm doped sample. We interpret the drop in Te at

displacements approaching the ~ebye-Hückel

radius as arising from the elliptical

-

spreading of the charge-cloud. This is u 590 illustrated by eliminating a from equations

0

(5) and (10): Lw

580

Robinson /IO/ found the damping at high strain amplitude to be consistent with a spreading of the ch3rge-cloud radius proportional to 15

1

Z We thus assume the mode1

K ~ =K.- ~ (1kr) ~ -

,

~ (13) Fig. 3

-

Displacement dependence of the isoelectric temperature for the giving (T,/T,~) = (1

+

(k~,~/h+)lln(l+~5)1-~. 1.3 ppm sample. Solid data point:

driving the sample; open data (14) points: driving a quartz crystal.

Dashed curve: predicted behaviour This gives the dashed curve shown in for charge-cloud spreading Fig. 3. The early behaviour is similar to according to equation (13); solid that measured but we observe an inflexion curve: bipolar charge-cloud at higher displacements which is not approximated to two overlapping contained in equation (14). We anticipate cylinders. The Burgers' vector and that the inflexion arises from progressive the ~eb~e-HÜckel radius are

bipolar bunching of the charge-cloud about indicated. the extremes of displacement since the

dislocation spends the greater part of a

vibration period there. Let us estimate this effect by approximating the

charge-cloud to two overlapping cylinders centred on the displacement extremes. Now

and 0 = cosm1 (0.25~~). (16)

This results in the solid line shown in Fig. 3. The initial behaviour closely matches the observed displacement dependence and we assume that the higher strain behaviour will be improved by allowing a distribution of loop lengths.

REFERENCES

111 W H Robinson, Phil. Mag., 25, 355 (1972).

/2/ W H Robinson, A J Glover and A Wolfenden, Phys. Stat. Solidi (a), 48, 155 (1978).

W H Robinson, J L Tallon and P H Sutter, Phil. Mag.,

2,

1405 (1977).

J L Tallon, R G Buckley, M P Staines and W H Robinson, Phil. Mag. (in press).

J L Tallon, R G Buckley, M P Staines, M Tanibayashi, G P Betteridge, and W H Robinson, Phil. Mag. (in press).

A B Lidiard in Handbuch der Physik, ed. by S Flugge, Vol. XX (Berlin, Springer-Verlag, 1957), p.246.

S Chandra and J Rolfe, Can. J. Phys.

48,

412 (1970).

W H Robinson and A Edgar, IEEE Trans. Sonics Ultrason.,

g ,

98 (1974). R M Whitworth, Phil. Mag., l7, 1207 (1968).