Eddy Current Method for Low Temperature Resistivity MeasurementsMelvin D. Daybell Citation: Review of Scientific Instruments 38, 1412 (1967); doi: 10.1063/1.1720551 View online: http://dx.doi.org/10.1063/1.1720551 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/38/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Eddy currents: Contactless measurement of electrical resistivity Am. J. Phys. 68, 375 (2000); 10.1119/1.19440 EddyCurrent Method for Measuring Anisotropic Resistivity J. Appl. Phys. 40, 3078 (1969); 10.1063/1.1658143 Analysis of Resistivity Measurements by the Eddy Current Decay Method Rev. Sci. Instrum. 39, 1019 (1968); 10.1063/1.1683554 EddyCurrent Method for Measuring the Resistivity of Metals J. Appl. Phys. 30, 1976 (1959); 10.1063/1.1735100 The Measurement of Specific Resistance by Eddy Current Shielding Rev. Sci. Instrum. 5, 94 (1934); 10.1063/1.1751805

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THE REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 38, NUMBER 10 OCTOBER 1967

Eddy Current Method for Low Temperature Resistivity Measurements*

MELVIN D. DAYBELLt

Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87544

(Received 10 March 1967; and in final form, 8 May 1967)

A modified mutual inductance bridge incorporating a commercial phase-lock amplifier has been used to measure low temperature bulk resistivities in the range from 1 to 10000 nO cm. No electrical contact is required with the spherical metal samples, so that possible heat leakage along electrical leads is eliminated, and measurements can be made below 40 mdeg Kelvin using a dilution refrigerator. The bridge circuit, capable of determining resistivity changes with a precision of better than 0.3%, is described.

INTRODUCTION

RESIDUAL resistivity measurements in metals at low temperatures are commonly made by drawing a fine

wire from the material of interest and using standard bridge methods to determine its resistance. l For recent measurements of the resistivity of dilute magnetic alloys at very low temperatures2 a technique was used which avoids surface contamination and oxidation difficulties3

associated with wire samples as well as eliminating elec­tricalleads to the sample with their attendant heat leaks. The absence of electrical leads allows many samples to be examined in a single low temperature run without over­loading the dilution refrigerator used. Power dissipated in the sample is only a few nanowatts. Direct measurements of true bulk resistivity of magnitudes from 1 to 10000 nQ em can be made with an accuracy of better than 3% and a precision and reproducibility exceeding 0.3% over most of this range. The system is self calibrating. It can be used for the detection of superconductivity,4 and is easily modified for susceptibility measurements in metals and insulators.

MUTUAL INDUCTANCE FUNCTION

Although it can be applied to other sample geometries,5

the technique is based on the fact that the eddy current

* Work performed under the auspices of the U. S. Atomic Energy Commission.

t ARMU visiting scientist on leave from New Mexico State Univ. 1 See, for example, G. L. van den Berg in Progress in Low Temper­

ature Physics, C. J. Gorter, Ed. (North-Holland Pub!. Co., Amster­dam, 1964), Vo!' 4, p. 194.

2 M. D. Daybell and W. A. Steyert, Phys. Rev. Letters 18, 398 (1967).

3 Oxidation and precipitation of the dilute impurity material during sample preparation can remove it from solution in the solid host metal, leading to appreciable errors in the effective concentration of the solid solution alloy. See, for example, D. H. Howling, PhYs. Rev. Letters 17, 253 (1966); also U. Gonser, R. W. Grant, A. H. Muir, and H. Wiedersich, Acta. Met. 14, 259 (1966).

4 William A. Steyert, Jr., Rev. Sci. Instr. 38, 964 (1967). 6 For a cylinder in a long solenoid, see A. G. Warren, Mathematics

Applied to Electrical Engineering (Chapman and Hall Ltd. London, 1958); S. J. Yosim, L. F. Grantham, E. B. Luchsinger, and R. Wike, Rev. Sci. Instr. 34, 994 (1963); W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill Book Co., Inc., New York, 1950), p. 416, prob. 4; H. E. Rorschach and Melvin A. Herlin, Phys. Rev. 81, 467 (1951); and E. W. Johnson and H. H. Johnson, Rev. Sci. Instr. 35, 1510 (1964). The last-named authors use a self-inductance technique;

1412

problem for a spherical conductor in the ac magnetic field of a concentric multiturn circular loop can be solved exactly.6 A solution also exists if the center of the sphere merely lies on the axis of the loop, and this can be used to show that the error introduced by using coils of finite length is negligible for the dimensions chosen.

A mutual inductor was formed by placing a coplanar multi turn circular loop inside the (primary) loop mentioned above, with a spherical sample of radius a at their common center. The complex mutual inductance m7:(x) of this mutual, which depends on the resistivity of the sample, can be defined as a function of frequency f (angular fre­quency w) and resistivity p by

E.;;jwm7:(x)Ip, [j= (-1)!, x=p/j]

which expresses the complex secondary emf E. in terms of the complex primary current I p. Using the results of Ref. 6, m7:(x) can be written in mks units as

7rJ.LaNM ( a2 )n

m7:(x)=M.+ L kn -2 n odd BnCn

where M B[ =m7:( 00 ) ] is the mutual inductance of the empty inductor, J.L is the permeability of free space, N and M are the number of primary and secondary turns, respec­tively, and

kn = [ 1·3·5···n J2 2

2·4·6··· (n-1) n(n+1)'

Bn -n is the average of b-n over the interval bmin to bmax ,

where bmin and b max are the minimum and maximum radii of the secondary loop. Cn-n is a similar parameter for the primary loop. The argument of the modified Bessel func­tions J n+i(V), etc., defined by Smythe6 is given by v= (j)! X (J.Lw/p)!a, which can be expressed as a function of x, vex).

they also give extensive references to earlier work. The problem of a thin disk in a uniform field has also been treated. CW. R. Smythe, op. cit., p. 418, prob. 24.)

6 W. R. Smythe, op. cit., p. 416, prob. 2.

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LOW TEMPERATURE RESISTIVITY 1413

A more useful form for ~(x) can be obtained by the following steps:

Let

(2)

where

K =7rp.a3NM /2B1C1, hn=kna2n-2(BICI) (BnCn)-n

'::::=:.k n (a2/ BC) n-l,

in which Band C are some sort of mean secondary and primary radii. Since the first term in brackets vanishes as ;\: goes to zero [so that ~(O) (=M 0) is real], solving (2) for K at x=O yields

K= (M.-Mo)/'E.h".

1.o..---- -----------,-----,

0.8

0.2

0.4

-; i 0.2

E

0.1

O·O~--------~S------~IO~----~IS

x (aplf)

(3)

(a)

(b)

FIG. 1. (a) Real part of a typical mutual inductancefunction W (x). (b) Imaginary part of W (x). The parameter x is the sample resistivity divided by the bridge frequency.

REFERENCE CURRENT AMPLIFIER

A C REFERENCE SOURCE

JjFREOUENCYI COUNTER

FIG. 2. Simplified schematic diagram of the eddy current resistivity bridge. Spherical sample at center of mutual inductor :ms. Bucking mutual Ms' is optional. If Ms' is omitted, Mr is made equal to the value of M, with the sample removed. S' balances the real part of :ms, S balances its imaginary part.

Defining m=Ms-Mo and using (3) in (2),

(4)

and finally, calling the term in brackets W(x), ~(X)=M8 -mW(x).

The parameters m and M 8 are easily measured experi­mentally, while only the relative dimensions of the sample and coils appear in the hn, and then only the higher order terms where errors in determining these dimensions are of reduced significance. W (x) is readily calculated on a digital computer for given a and hn • It is convenient to choose x (in nQ cm/Hz) as the dependent variable, but to calculate using v, which in these units (with a in centi­meters) is v(x)= h(1+j)a(x)-!. For small v, the Bessel functions are evaluated using a power series in v2. At large v, an exponential expression7 is used to evaluate

In both cases, a recursion formula for (2n+1)/In+!(V)/ vI n+t(V) is used to find the low order terms after the highest order term required has been computed. The results of such a calculation are shown in Fig. 1 for a=O.S cm, (BnCn)-i= 1.0 cm (all n).

EDDY CURRENT BRIDGE

A simplified schematic of the bridge used to determine W(x) is given in Fig. 2. A current I p flowing in the primary circuit at the left causes an emf

to appear across the secondary circuit where it is detected by a low noise amplifier locked to the phase of the primary

7 Smythe, op. cit., p. 197, Eq. 8.

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1414 MELVIN B. DAYBELL

FRO'" PAR~' III REFERENCE ,~ OUTPUT I

MAIN CHASSIS I

PRIMARY SECONDARY TO REFERENCE MUTUAL.

FIG. 3. Bridge circuit diagram. The phase amplifier is designated PAR. All resistors precision noninductive type. Current amplifier G capable of yielding 1oo-mA short circuit output for 1 V input. Switches low-loss with silver-plated contacts. Coax cables insulated from chassis. Case of ratiotran connected to chassis but not to ratio­tran common. Case of divider connected to chassis but not to divider common.

current.8 The phase selector on the amplifier is used with the voltage divider 5 to null the in-phase component of the secondary emf, and with 5' to null the out of phase component. The total emf is then zero, so Re~(x)=M.s', and Im~(x)= -RS, where 5 and 5' are the divider ratios at null. Thus from (4),

M.( Mr) [Mr ] Re W(x)=~ 1- M.S" -;;(1-S') if Mr=M. (5a)

and

1m W(x)=RSlwm=H(5If) where H=RI27rm. (5b)

The value of the instrumental constant H can be found by varying f with an arbitrary (fixed) value of p, and de­termining 51f as a function of 11f. Comparison of the height of the peak in this function with that of the peak in 1m W (x) at x = x m yields the value of H. The circuit can then be used as a fixed frequency bridge. As p varies, the null value of 5 determines 1m W(x) and hence x and p.

The value of 5' indicates whether x is above or below x". and removes the ambiguity arising from the fact that 1m W(x) is double valued.

In a refinement of the above circuit, a compensating M.'(""-'-M.) is connected in series with M. and is located near it in the Dewar while the reference mutual M r is re­placed by a smaller mutual Mr', ~-m. In this case, Re W(x)= (M.-IM.'I-Mr5')/~5', while the expres­sion for 1m W(x) is unaffected.

Figure 3 is a detailed schematic of the bridge. Chassis

8 Princeton Applied Research Corp., Box 565, Princeton, N. J. 08540, model HR-8.

grounds of the individual parts are common, but it is mandatory that all circuit grounds be kept isolated from the chassis and brought separately to a common physical point forming the sole contact with chassis ground. Com­ponent values shown were used with a sample mutual having N=M=300 turns of No. 38 Formvar-coated copper wire, bmin = 9 mm, and a coil length of 3 mm. Its mutual inductance was 2.0 mHo The reference mutual was similar, except that N was 305 turns. A resistive divider was used for 5' to avoid the low frequency phase shift present in a ratio transformer. A one-ohm rheostat in series with the divider provided fine adjustment. For the value of Mr used an error in 5 of lXlo-6 at 1 kHz is introduced by loading of this mutual by the 5' divider; this can usually be ignored. This error is even smaller if M.' is used, since Mr will then be smaller. The inductive divider used for 5 caused a small but unimportant error in 5'.

SAMPLE MOUNTING

Samples were cast in an epoxy9 rod mounted below a 3He-4He dilution refrigerator. Copper coil foil clamped to the samples with a brass screw formed the heat link to the refrigerator. The coil foil-refrigerator joint was made with a Cd-Bi eutectic solder, which has recently been shown to have a relatively high thermal conductivity at very low temperatures.4

Space was pro~ided in the refrigerator volume for up to four samples, as well as for a cerium magnesium nitrate thermometer. The sample and reference mutuals were mounted outside this volume in a 10 K helium bath.

LIMITATIONS

Unless difference measurements can be made with the sample in and out of position, capacitive effects for f above 1 kHz impose an upper limit of the order of 10000 nQ cm on the resistivities accessible to the bridge. A lower re­sistivity limit of about 1 nQ cm is imposed by the loss of sensitivity at low frequency (near the 1.5 Hz low frequency cutoff of the amplifier), and by the decrease of 1m W(x) at small x. Obtainable precision is only about 10% at 1 nn cm. Extremely careful grounding and a more powerful current source might push the lower limit down an order of magnitude, but lower resistivities will require methods based on the lifetime of persistent currents.10

OTHER APPLICATIONS

This bridge has also been used for susceptibility meas­urements in laminated metal alloy samples2 and in para-

9 Epibond 121 obtained from Furane Plastics, 4516 Brazil St., Los Angeles, Calif. This is a low susceptibility material. [G. L. Salinger and J. C. Wheatley, Rev. Sci. Instr. 32, 872 (1961).J

10 See C. P. Bean, R. W. DeBlois, and L. B. Nesbitt, J. App!. Phys. 30, 1976 (1959) for a possible approach.

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LOW TEMPERATURE RESISTIVITY 1415

magnetic salts. A bucking coil was used, and the resistive divider and ratio transformer were interchanged. A 1000: 1 fixed resistive divider was placed across the output of the adjustable S divider since S was quite small; this fixed divider also presented a low impedance to the bridge secondary circuit and permitted a 100: 1 stepup trans­former to be used at the detector amplifier input to in-

THE REVIEW OF SCIENTIFIC INSTRUMENTS

crease sensitivity. For the metal samples, S' was measured as a function of (f)2 and extrapolated to zero frequency to correct for small eddy current effects.

The author thanks Dr. William Steyert for suggesting the possible utility of the eddy current technique, and the Los Alamos Scientific Laboratory for the use of their facilities.

VOLUME 38. NUMBER 10 OCTOBER 1967

500 MHz Ring Counter

z. C. TAN

Physics Department, University of Auckland, New Zealand

(Received 29 May 1967; and in final form, 23 June 1967)

This paper describes a new tunnel diode-transistor ring counter employing current mode operation. With suitably shaped input pulses it is capable of operating reliably at input repetition rates in excess of 500 MHz.

INTRODUCTION

A RING counter is essentially a number of flipflops connected in a ring and driven from a common input,

with coupling arranged to prime and reset adjacent flip­flops. Initially one flipflop is in a state different from the rest and successive input pulses propagate this state around the ring. A ring counter made up of n flipflops thus has a scaling capacity of n.

A ring counter employing tunnel diodes as the basic flipflops is described in this paper. Coupling between them is by means of transistor current switches which perform the functions of priming and resetting. The maximum speed of operation depends essentially on the delay in priming the next tunnel diode.

PRINCIPLE OF OPERATION

Figure 1 shows the basic arrangement of a quinary ring counter. Each tunnel diode is biased by a constant current Ib and can be in either a 0 or 1 state, as shown on the static characteristic diagram of Fig. 2. Switches SI to S5 are con­trolled by tunnel diodes TD1 to TD5 respectively. Each switch is normally open but closes when the associated tunnel diode switches to a 1 state.

FIG. 1. Basic arrangement of ring counter.

Initially one of the tunnel diodes (say TD5) is switched into a 1 state.1 Switch S5 is therefore closed and current Ion is extracted from TD4 and supplied to TDI. With reference to Fig. 2, TD4 is in a On state and TD1 is in a Op state, i.e., it is primed. Input current pulses of ampli­tude lin, where Ip-h-Ion<lin<Ip-h, are applied simultaneously to all the tunnel diodes. The first pulse switches TD1 to a 1 state and closes switch SI. TD5 is reset to a On state and TD2 becomes primed, i.e., in a Op state. Following the reverse switching of TDS switch S5 opens and TD4 returns to a ~ state. The 1 state and its adjacent On and Op states have been thereby shifted one stage around the ring. Further pulses will propagate these states around the ring one stage at a time.

Figure 3 shows the theoretical waveforms of the tunnel diodes in response to a number of successive input pulses.

Ib - Ion I

-0 states-: · · :-1 states-· FIG. 2. Static characteristic diagram showing 0 and 1 states.

1 A current of magnitude greater than (I p-h) supplied momen­tarily to this tunnel diode will achieve this.

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