Surface Conductivities of a Thin Sample Four-Point Method for Measuring the Volume

Communication de l'Institut fed~ral pour l'etude de la neige et des avalanches

Four-Point Method for Measuring the Volume and Surface Conductivities of a Thin Sample

By C. ]ACCARD

Weissfluhjoch-Davos, mai 1966

Tire a

part du

JOURNAL DE MATHEMATIQUES ET DE PHYSIQUE APPLIQUEES (ZAMP) Vol. 17, Fasc. 6 (1966) BUlKHl usn Vl:RLAG BASEL Pages 657-663

Reprint from

JOURNAL OF PPLIED MATHEMATICS A D PHYSICS (ZAMP)

Vol. 17, Fasc. 6 (1966) 8JRKHAU$ER VERLAG 8AS£1, Pages 657-663

Four-Point Method for Measuring the Volume and Surface Conductivities of a Thin Sample

By C.

J

ACCARD, Federal Institute for Snow and valanche Research, 7260 WeissfluhjochfDavos, witzerland

1. Introduction

A four-point method for measuring the ani otropy of resi tivity has been de cribed by P. ^HNABEL (Philips Re . Repts., vol.19, pp. 43-52, 1964). On a thin ample with two flat parallel surfaces, two electrode A and B are brought on one side and two other ones C and D ju t opposite on the other ide. An electric current flow first betwe n A and C and the potential difference between Band Dis measured; then, the current flow between A and Band the potential i mea :ured between C and D.

If the material i isotropic, the resi tance R1

=

V8vlJAc and R₂

=

VcvllAB allow to determine any two of the following physical parameters: volume conductivity, electrode pacing or ample thickne ; in the case of electric ani otropy, two compo- nents of the conductivity tensor can be obtained.

We attempted to apply thi method to measure the DC conductivity tensor of ice, but the obtained re ults were incompatible with known propertie of the material.

The di crepancy has been attributed with ucce to a large urface conductivity.

chnabel' method can be extended to obtain from the two mea ured re istances the bulk and the surface conductivities imultaneou ly. The purpo e of thi paper i to de cribe such a method. A correction term i introduced to account for the urface current and is then included into the general formula relating the mea ured re i tance with the conductivities. The sen itivity of the method and the influence of a finite

ample urface are evaluated.

The urface conductivity of ice cry tal ha been determined according to thi method (C. ^JACCARD: Electrical conductivity of the urface Layer of Ice, Proceedings of the International Conference on now and Ice, apporo 1966, to be publi hed).

The re ult have been compar d with direct mea urement on thin amples, extra- polated to zero thickness. The agreement i well within the experimental error and this prove the us fulness of the method de cribed here.

2. Potential with Surface Current

Let u consider a system of cylindrical coordinate fl, 1/J, z with the uppei- half pace z

>

0 filled with an homogeneou and i otropic material having a bulk electric conductivity a. and a urface conductivity a, in the plane z

=

0. At the origin (z = 0, fl= 0) a current

J

is injected into the material. there are no oth r cuiTent ources and as the bulk conductivity i a con tant, the lectric potential <p must ati fy in the

ZAMP 17/42

65 ^C.JACCARO upper half pace to the Laplac equation:

Liq:,= 0 and must vanish at infinity:

o=

" ^z

(z

>

⁰⁾

7.AM'r

(]) (2) On th urface, the bulk currentj ha a normal componentj, produced by the diver- gence of the urface curr nt js:

div_, js

+

j, = C Introducing the characteri tic length

(z = 0) .

thi condition can be expre ed for the potential by J, iJ (o iJ<p )

+

iJrp = 0

fl iJe ~ iJe iJz (z

=

0)

(3)

(4)

(5) as the symmetry i obviou ly cylindrical. The olution of the y tern (1, 2, 5) take then the form :

(6) The first term corre pond to the applied current source and the second one to a current dipole with a trength J }, with the ource out ide (z = - 0) and the sink in- side (z =

+

0) the material. The dipole field account for the finite urface conduct- ivity and vanishes with it, as it is proportional to the length k

3. Potentials and Currents in a Flat Sample

Let u con ider now an infinite sampJe with parallel flat . urface and a thicknes w ituated in the original coordinate system between z = 0 and z = - w. Four point electrodes are brought on the surface at the point A (z = 0, 1P = 0,

e

= 0), B (z = 0, 'f/J

=

0,

e

= s), C (z = - ^w, 'f/J = 0,

e

= 0), and D (z = - ^w, 'f/J = 0,

e =

s). In a fir t configuration, the current i injected into the sample at A and flows out at C, and the potential is measured between Band D. To account for the boundary conditions we superpose an infinity of virtual current ources on the z-axis with the potential given by Equation (6), the positive ones at the ordinate z

= ±

^{2 n w} and the negative ones at z =

±

(2 n

+

1) w with n = 0, I, 2, 3 . . . . The sources and sinks mu t have a strength of

±

2

J:

at point A for example, half of the current flows into the real plate but the other half flowing into the uperposed virtual plate is not taken into account in the current measurement. The current dipoles have to be replicated in an analogous way. Formula (6) applied to thi sy tern yields then the following value for the potential difference:

with

<X.= s

w

(7)

(8)

VOL 17, 1966 4-Point Method for Measuring the Volume and Surface Conductivities 659

The function G(a) has already b en given by SCHNABEL and has the value

00

G(ct.) = a-l _ 2

J; ( -

1)11+1 (ct.2

+

^n2)-112 (9)

11.-1

and the function G₁(a) i

00

G₁(a) = 2

J; ( -

^{J )"+l n (a2}

+

^n2)-a12. (JO)

ti•l

In the second configuration, the current i injected at A and flows out of the sample at B, and the potential is mea urcd between C and D. The virtual current sources are then taken at z =

±

2 n w, with Q = 0 for the positive ones, {! = s for the negative ones and '!fl = 0 for both, with a corresponding distribution of the dipoles.

The potential difference is then

Ve - VD= (

!~

⁸

av)

^{[H(a) - ( ~) H1(ct.)], (11)}

where H(ct.) is al o one of chnabel's functions:

00

H(a) = 2

J;

{(2 n - 1)-^{1 - [a 2}

+

(2 n - 1) -^112} (12}

n•l

and

00

H₁(a) = 2

J;

^{{(2 n - 1)-2 - (2 n - l} [a2}

+

(2 n - 1)~~-3'2} . (13)

u-1

The re istance as defined in ection 1 are then given by R1nwa.= G(ct.)

+ (~)

^G1(a),

R2n w a.= H(a.) - (:) H1(a).

Solving these expression yield the following formulae for the conductivitie

with

R F,

n w a. ⁱ = [F2

+

(R.2/RilJ ' R F3[F4 - (R.2/R.1)]

n a, ⁱ = [F2

+

(R2/Ri)J2 , .l. _

(Fa)

^{[F~ -} (R.2/R.1)]

w - Fi [F2

+

(R.2/Ri)J

Fi(a)

=

(_G H^{1 ;}

1

G¹ H)

F2(a)

= :: ,

Fa(a) = ^(G Hi +G~¹ H) G

l

(14) (15)

(16) (17)

(1}

(19) (20) (21) (22)

660 C. JACCARLJ

Xumerical values of G and H ha\·e been tabulated b r CHXABEL for a = 0,05 to 2,0 in 0,05 teps. The two other fun tion can be obtained easily from the function Q(a}

00

Q(oc) =

J;

11 (a2

+

112)-3/2 . (23)

n- 1

F r oc

<

^{2, the term for ·n}

>

6 can be developed in a^{2 ,} ith a fast convergence.

The formula

5 00

Q(a)

=};

¹¹ (o:²

+

n²)-3'² -r . . J (- 1)"' (;

)2"'

^{(2 m}

⁺

^{1) ! (m !)-2}

J;

^11,-2(m+1)

,,. .. 1 111 0 n-o

(24)

+

0.00115 (

~ t-

0.000]2 (

~ )6 +

0.00001 ; ⁸

give Q with an accurate 4th place after the decimal point. Th function G₁ and H₁ are related with Q by

G₁(oc) = ² Q(o:) - Q ( ~) ,

H1(a) =

!

^{Q(0) - 2 Q(oc)}

^{+ ~}

^{Q (;).}

From the function (for oc

<

²⁾

00

J[(a) = 2

J;

[n-1 - (n2 + a:2)-1/2]

,i-J

5 9

= 2

J;

^{[11-1 - (n2}

+

^ix2)-112

+

0.06558 ( ~ ^)-

n-1

- 0.00319 (;

)4 +

0.00023 (

~ )6 -

0.00002 (;

)8

the function G and H can be calculated according to the expression G(ix) =

~

- 2 ln 2

+

M(a) - M (;) ,

H(ix) = M(a) -

~

M ( ~) .

4. Sensitivity of the Method

(25) (26)

(27)

(28) (29)

The sensitivity depends upon the relative importance of the surface and of the volume contribution to the overall sample conductance. If th urface conductivity plays a minor role; i.e. if 2/w $ 0,1 (for i x ~ 1), the bulk conductivity can be obtained by any of the expre sions (14) and (15). The relative error /lavfav is then the mallcst of the um w/w

+

.JR1/R1 or ,1wfw

+

LIR2/R2 . In this case, the ratio R₂/R₁ i near the value of~ and a, or ). have a correspondingly small absolute value according to Equation (J 7) or (18). The absolute error i. . mall, but the relative one is large, and

Vol. 17, 1966 ·I-Point Method for ;\fcasuring the \'{)lumc and SurfacP Conductivities 661 only the order of magnitude can be determined with certainty. The most favorable situation occurs when both contributions to the conductance are of the ame order.

The sensitivity is then roughly the same for a" and fora, or}., about w/w +LI R₁,2/R₁,2 •

If the smiace conductivity play th major role, e.g. if J../w ~ 1 (for a '.'.::'. 1), the ratio R_2/ R₁ i near the value - F₂ and a,, being the inverse of a mall number, cannot be determined accurately, and neither does av.

Another electrode configuration gives better results for a,. A very thin ample

.bears on one of its sides four collinear electrodes: A and D (separation s ~ w) for the

current, and Band ^{C (} eparation s'

<

s) for the potential, with equal di tance AB and CD. The re i. tance R' = Vec/J AD is then given by the expre ion

2 n a, R' = ln

u: ^~ : : n

⁽³⁰⁾

and the relative error on a, is of the order of Js/s

+

s'/s'

+ ..

IR'/R'.

5. Error Produced by a Finite Sample Radius

The calculations of Section 3 apply trictly only to a ample with an infinite ex ten ion perpendicular to the z-axis. Let us examine what happen when the ample

cir ular with a finite radius R. The boundary condition i

O<p

be = 0 for f2 = R . ⁽³¹⁾

the potential produced by the dipole resulting from the ··urface conductivity have a ma.Her range than tho e of the main current source and ink , they are affected at a le er degree by condition (24). Therefore, as we need only the order of magnitude of the error, we may di regard the urface conductivity.

We take a source and a sink on the upper urface of the di c haped ample (z = 0, f2 = s/2, '1/J = 0 and n re pectively). The unperturbed potential (up to a multiplicative con tant)

<pl =

f {[

^{[22 +}

^~

^{+ (z - 2 n w)2 + (1 s co '1/J}

J-

^1'2

l

11• -oo -

[ e2 + ~

^{+ (z-} 2nu•)2-(2SCO

'!J'rl/ 2}

⁽³²⁾

ha to be correct d b, an harmonic term <p_{2 ,} periodic in lp, with the general form

"" (µnz) (µ"(!)

cp2 = "-' "-' A,.,, co (v '1/J) co ^w

I,.

w v-0 1• l

(33)

where the I, are modified Bes ·el function . The coefficients A,.µ are determined by condition (24) which, with

B,.,, ⁽³⁴⁾

takes the form:

}; }; B o. (vv,) co ( 1-nr Z) = - [ : <p₁(z, I!, lJl)] = F(z, R, lJ)) • (35)

v-0 /t l vµ W (! e-R

62 ^{C. J CCARD}

The B,₁^, an be calculated exact! , by a Fourier inversion, but to estimat their order of magnitude, only an upper limit for Fi neces ary. This limit i obtained in replacing co 1/J by - 1, z by + w, in the fir t term of <p_{1 ,} and co tp by+ l, z by - w in the econd term. The urns, which are in fact integral over stair like function , are replaced by integrals over the mooth functions enveloping the stairs, from above or from below according to the ign. With the hypothe is R - s/2

>

w, the integral limits can be replaced by

±

J and the end re ult i

I

F

I < F.\t

= Rw 17 . ⁽³⁶⁾

The larger of the Bv,. i obviou ly B₁₁ which i estimated to be

'+:C +w

( 1 ) / / (. 1t z ) 68

B 11 = ^{; i} w F -co 1P cos w d'I/J dz

<

4 F;,1 = R w- (37)

and therefore

(38)

In the region of interest,

I

<p₁ l can be approximated by 1/w and the relative correction i then

6 (w/R) l1 (n s/2 w)

< - . ·-

- I~ (;i R/w) (39)

If thi e timation i done for the configuration in which the ource and the ink are on oppo ite ide of the ample, the ame order of magnitude is found.

In mo t of cases, the eparation s i about the ame a the ample thickne w, so that I₁(n s/2 w) '.::'. J₁(n/2) '.::'. 1. But if the sample radiu i at least equal to four times the thickness, the argument of I~ in the denominator i at least 12, and the

Table I. T'alues of the functions Q, G, H, G_{1 ,} H 1, F 1, ^F2 , F₃, F< of the parameter x

X Q G H G1 H1 Fl F2 Fa F4

0,1 1,629 ,623 0,01044 1,617 0,03007 0,1708 0,01 60 0,9109 0,001211 0,2 1,583 3,649 0,04091 1,537 0,1160 0,3163 0,07549 0,7501 0,01121 0,3 1,513 2,023 0,0 89 1,416 0,2464 0,4409 0,1740 0,6297 0,04399 0,4 1,426 1,242 0,1512 1,270 0,4061 0,5483 0,319 0,5360 0,1218 0,5 1,332 0, 015 0,2239 J,113 0,5796 0,6414 0,5209 0,4620 0,2793 0,6 1,235 0,5325 0,3032 0,9568 0,7542 0,7229 0,78 2 0,4023 0,5695 0,7 1,141 0,3601 0,3861 0,8111 0,9207 0,7949 1,135 0,3529 1,072 0,8 1,053 0,2465 0,4701 0,6 04 1,074 0,8591 1,57 0,3112 1,907 0,9 0,9730 0,1701 0,5533 0,5665 1,211 0,9170 2,138 0,2754 3,252 1,0 0,9005 0,1182 0,6345 0,4695 1,332 0,9697 2,837 0,2440 5,369 1,1 0,8356 0,08248 0,712 0,3883 1,43 1,018 3,703 0,2163 8,642 1,2 0,7778 0,05781 0,78 0 0,3209 1,529 1,163 4,766 0,1916 13,63 1,3 0,7264 0,04064 0,8597 0,2654 1,608 1,106 6,060 0,1693 21,15 l,4 0,6806 0,02866 0,9279 0,2200 1,677 1,146 7,622 0,1493 32,38 1,5 0,6397 0,02025 0,9927 0,1829 1,736 1,185 9,493 0,1312 49,02 1,6 0,6030 0,01434 l,054 0,] 527 1,788 1,222 11,71 0,ll48 73,51 1,7 0,5701 0,01018 l,1J3 0,1280 1,833 1,259 14,33 O,l 001 109,4 l,8 0,5404 0,007231 1,169 0,1078 1,873 1,294 17,38 0,086 2 161,6 1,9 0,5135 0,005146 1,222 0,09128 1,908 1,329 20,90 0,07493 237,4 2,0 0,4891 0,003666 1,272 0,07769 1,939 1,364 24,96 0,06435 347,0

\'ol.17, 1966 4-l'oint Method for Measuring the Volumf' and Surface Conductivities 663

{unction itself at least 20000, because it grows rapidly, like an exponential. The correction is then at most 0,02% and is quite negligible. If a correction smaller than l

%

is required, the sample radius has to be larger than three times the thickness, which is easy to obtain practically.

This favorabl situation is due to the presence of I~ in the denominator in the formula for A_{11 ,} and this results from the boundary condition (24). If we assume that the sample is enclosed in a highly conducting ring, we have to require that the potential itself vani he on the boundary, not its differential with respect to the radius. This does not let appear the function

t

and the corrective term ha to be considered for much larger values of the ratio Rfw. Thu the sample circumference mu t not be grounded.

Resume

n echantillon mince d'epaisseur w, avec une conductivite electrique de volume et de surface finie, a sur un de ses c6tes deux electrodes ponctuelles A et B (avec une separations) ct vis

a

vis sur l'autre c6te, deux autres electrodes C et D. Un courant passe d'abord entre A et C, ensuite entre A et B; une difference de potentiel est mesuree entre Bet D, ensuite entre C et D. vec les resistances VsolJAc et Vco/JAD, on peut calculer la valeur de la conductivitc de surface et de volume

a

!'aide de fonctions universelles du parametre s/w.

(Received: May 31, 1966.)