The volume of converted material therefore follows a simplelaw:

dV_

~d7 Twhere k is a constant which depends weakly on the bombard-ment ion and its energy, G is the number of ions striking theCMT per second and NA is the concentration of mercuryvacancies initially present.

Applying this to a planar experiment with bombardmentover a large area A, we obtain a junction depth d of

A - JL9L~NAA

For bombardment through a small aperture on the surfacedefined by resist, or using a focused beam, then an approx-imately spherical interface develops, of radius

r = (3kGt/2nNAy3

unless the thickness d of the material is so small that it isentirely removed by the bombardment, in which case cylin-drical diodes develop2 where

r = (kGt/n dNA)1/2

Fig. 1 shows plots of the product VNA (i.e. the number ofannihilated vacancies) as a function of dose of argon ions,

10

S 10a

1010' 1012 10' 101

dose Gt 1807/H

Fig. 1 Variation of number of annihilated vacancies as a function ofintegrated dose Gt of argon ions at an energy of 1 keV

Dependence is close to linear with k ^ 2 x 10~4 (see text). For aplanar geometry, volume V is proportional to junction depth

from which it can be deduced that fc ~ 2 x 10"4. Fig. 2 showsan EBIC scan of a section through a diode in relatively thickmaterial showing the approximate spherical form, togetherwith evidence for removal of material through a resistwindow. Measurements of the converted volume in suchthree-dimensional geometries are consistent with thoseobtained in the one-dimensional case.

Fig. 2 EBIC scan through sectioned diodes which have been formed bybombardment through resist windows of different apertures

Ruled horizontal line indicates material surface. Overall junctionradius of largest diode at surface is 16-5/im

Conclusion: The conversion of vacancy-doped mercurycadmium telluride as a result of ion bombardment occursduring the bombardment process and not during any sub-sequent anneal. The rate limiting process is therefore the inputflux and not the duration or temperature of later processing.Diffusion of mercury interstitials is therefore deduced to bevery fast even at ambient temperatures. For production of pnjunctions the nature of the bombarding or implant ion is quiteirrelevant.

M. V. BLACKMAN 27th July 1987D. E. CHARLTONM. D. JENNERD. R. PURDYJ. T. M. WOTHERSPOON

Mullard LimitedSouthampton, Hants. SO9 7BH, United Kingdom

C. T. ELLIOTTA. M. WHITE

Royal Signals & Radar EstablishmentSt. Andrews RoadMalvern, Worcs. WR14 3PS, United Kingdom

Copyright © Controller HMSO, London, 1987

References

1 WOTHERSPOON, j . T. M.: UK patent GB 2095898, 19812 BAKER, I. M., JENNER, M. D., PARSONS, J., BALLINGALL, R. A., BLENKIN-

SOP, i. D., and FIRKINS, J. H.: IEE Conf. Publ. 228, Conf. onadvanced infra-red detectors and systems, 1983, p. 12

3 BLACKMAN, M. v., et al.\ NATO workshop on MCT and relatednarrow gap materials, Freiburg, March 1987

PHYSICAL MODEL FOR THE COMPLEXRESISTIVITY OF THE EARTH

Indexing term: Dielectrics

We show that the empirical Cole-Cole model used to inter-pret low-frequency complex resistivity data does have a phys-ical basis. Such a configuration is a homogeneous conductorloaded by a sparse distribution of spheroidal metallic par-ticles with a polarisable surface layer which can be character-ised by an interface impedance.

It is quite remarkable how effective the Cole-Cole model hasbeen in characterising the frequency dependence of thecomplex resistivity of geological materials in situ. This modelin the context of electrical geophysics is written as1

= Po\ 1 " ' " o f 1 -1

1 +(1)

where pe(jco) is the complex resistivity appropriate for a timefactor exp (jot), p0 is the low-frequency or DC limit, x is thecharacteristic relaxation time, m0 is the chargeability and k isa dispersion index. The four parameters p0, T, m0 and k arechosen to fit observed data for pe over a frequency rangetypically from 01 to 103Hz. Clearly, the Cole-Cole model isempirical and a physical or chemical basis is only consideredafter the fact. Sometimes an equivalent circuit is postulatedwhere one of the elements is an impedance which varies as afraction inverse power of frequency. Certainly we do not wishto find fault with any of these empirical approaches. However,physical insight is often lacking if the basis for the model isnot established.

It has been pointed out to us by E. O. McAlister and M.Halverson in a personal communication that an earlier math-

ELECTRONICS LETTERS 10th September 1987 Vol. 23 No. 19 979

ematical model2 of mine could be cast into the Cole-Coleform. In this model the disseminated mineral particles wereidealised as spheres and the host medium was an electrolyticconductor. The electrochemical double layer at the surface ofthe particle was characterised by an interface impedance z{jco),which accounted for the discontinuity in potential across theinterface between electrolyte and metal. At the time we putforward this coated spherical particle model2 the interfaceimpedance or coating layer was represented as a lossy con-denser to facilitate the analysis. As we have indicated else-where,3 the interface impedance could assume other morecomplicated forms that might be more realistic. The principalconclusion remained intact, namely that the effective relax-ation time of the ensemble increased monotonically with thesize of the particles, all other things being equal.

Here we would like to shed some further light on thesubject by treating a somewhat more general particle modelinvolving spheroidal shapes. Thus needle- and disc-shapedparticles could be included along with the limiting sphericalshape.

We are well aware of the vast literature on the effectivemedium properties of disperse particle systems (see, forexample, the review by van Beek4). Attempts to analyse non-spherical coated particles are relatively few. A recent analysis5

adopted a spheroidal model for the particle shape, and theinterlace impedance was assumed to have a confocal form. Inother words, the equivalent coating was bounded by confocalsurfaces. The more realistic case of a uniform coating led tocomplicated mode matching and crosscoupling of the harmo-nic solutions.6 On the basis of numerical calculations it wasindicated that the confocal assumption was not crucial. In anycase we will use the confocal model for the present discussion.We also assume that the relative volume of the particles issmall compared with the total volume of the ensemble.

The spheroidal shaped particles are taken to have a resis-tivity px, a semi-minor axis a and a semi-major axis b. Theprolate spheroidal geometry corresponds to taking b > awhile the oblate spheroidal geometry corresponds to b < a.

The background medium has a resistivity p which we willregard as being real, but this assumption is not necessary. Therelevant interface impedance parameter6 is defined by

(2)

where z0 is the actual interface impedance at the waist of thespheroidal particle with dimensions Qm2. Then, on using anextension of the Maxwell-Wagner formalism,3'5 we can showthat for all particles aligned with axes parallel the effectiveresistivity in the axial direction is obtained from

and

P - PeO'«)

p,/p)<x(3)

where t; is the volume of the particles relative to the totalvolume and a is a polarisability coefficient, which equals 2 inthe case of spherical particles. It can also be writtena = — 1 + 17l in terms of the more common depolarisabilityfactor L. For the same distribution of particles, the formulafor the effective resistivity in the transverse direction hasexactly the same form as eqn. 3, except that a is replaced by a,the appropriate polarisability coefficient for excitation of thespheroid by an electric field transverse to its axis.

With a minor amount of algebraic manipulation, we canwrite eqn. 3, for the case v <̂ 1, in the form

where

Po = PI 1 +

P (1 + °Q2

mo = vPo a

(4)

(5)

(6)

Pi

P(7)

Now, to obtain eqn. 4 in the Cole-Cole form, we would firstneed to assume that pjp <^ 1, which is certainly justified formetallic particles. Then we see that if

Q ~ Q = (j(OT)~k/a (8)

we have eqn. 1 precisely. Thus we are saying that if the inter-face impedance can be approximated by the power-law form

= A/(jco)k (9)

where A is a constant, then the corresponding relaxation timeis given by

(10)

This simple result is telling us that, for k < 1, the time con-stant decreases with diminishing particle size, as has oftenbeen observed in laboratory studies.7 In fact it is often statedthat T varies as the square of the mean particle size, whichmeans that k = 1/2. This particular value of k suggests thatthe corresponding form of eqn. 9 is the Warburg impedanceassociated with diffusion-dominated electrochemical processesat the metal/electrolyte interface,8 but values as low as 0-2 arepossible.

As mentioned, the case of transverse excitation of the spher-oidal particles is handled in the same fashion. Eqns. 3 and 4are still applicable, but now we must replace <* by a. Here it issignificant that the expression for T is changed from the casefor axial excitation. Thus we can say that the relxation time isdependent on particle orientation at least for the model wehave assumed.

A limitation in the interface impedance model exists whenthe radius of curvature of the particle becomes very small (e.g.less than 10/xm). Then the electrochemical double layercannot be adequately simulated by an interface impedance.Possibly the formulation of Wong and Strangway9 may berelevant here, in spite of their unconvincing method of solvingthe coupled wave equations for the spatial and frequencydependence of the ion densities in the electrolyte. This point isdiscussed elsewhere.5'6

Another limitation of the interface impedance model iswhen the local current density at the particle surface becomeslarge (e.g. greater than 1 /iA/cm2), in which case the linearityassumption would be violated. A modified formulation isneeded to handle this case.

J. R. WAIT 24th July 1987

Department of Electrical & Computer EngineeringUniversity of ArizonaTucson, AZ 85721, USA

References

1 PELTON, w. H.: 'Interpretation of complex resistivity data',Geophys. Trans., 1983, 29, pp. 297-330

2 WAIT, J. R. : 'A phenomenological theory of induced polarization',Can. J. Phys., 1958, 36, pp. 1634-1644

3 WAIT, J. R.: 'Geoelectromagnetism' (Academic Press, 1982)4 VAN BEEK, L. K. H.: 'Dielectric behaviour of heterogeneous systems',

Prog. Dielectrics, 1967, 7, pp. 69-114 (Heywood Books, London)5 WAIT, J. R.: 'Complex conductivity of disseminated spheroidal ore

grains', Gerlands Beitr. Geophys., 1983, 92, pp. 49-696 FLANAGAN, p. w., and WAIT, j . R. : 'Induced polarization response of

disseminated mineralization for spheroidal geometries', Radio Sci.,1985,20, pp. 147-148 (Summary of microfiche paper)

7 OLHOEFT, c R.: 'Low frequency electrical properties', Geophysics,1985,50, pp. 3001-3030

8 BOCKRIS, j . O'M., and REDDY, A. K. N.: 'Modern electrochemistry'(Plenum Press, 1970)

9 WONG, J., and STRANGWAY, D. W. : 'Induced polarization in dissemi-nated elongated mineralization', Geophysics, 1981, 46, pp. 1258-1268

980 ELECTRONICS LETTERS 10th September 1987 Vol. 23 No. 19