Systematic Errors in Alternating Current Hall Effect MeasurementsH. L. McKinzie and D. S. Tannhauser Citation: Journal of Applied Physics 40, 4954 (1969); doi: 10.1063/1.1657320 View online: http://dx.doi.org/10.1063/1.1657320 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/40/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Contactless measurement of alternating current conductance in quantum Hall structures J. Appl. Phys. 116, 154309 (2014); 10.1063/1.4898737 Measuring Systematic Error with Curve Fits Phys. Teach. 49, 54 (2011); 10.1119/1.3527759 Error in Hall Cell Angle Measurement Due to Magnet Edge Effects J. Appl. Phys. 34, 1424 (1963); 10.1063/1.1729593 Alternate Current Apparatus for Measuring the Ordinary Hall Coefficient of Ferromagnetic Metals andSemiconductors Rev. Sci. Instrum. 29, 970 (1958); 10.1063/1.1716070 Sensitive Recording AlternatingCurrent Hall Effect Apparatus Rev. Sci. Instrum. 23, 548 (1952); 10.1063/1.1746081

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JOURNAL OF APPLIED PHYSICS VOLUME 40, NUMBER 12 NOVEMBER 1969

Systematic Errors in Alternating Current Hall Effect Measurements*

H. L. McKINZIE

Department oj Chemistry, Brown University, Providence, Rhode Island 02912

AND

D. S. TANNHAUSER

Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel

(Received 13 June 1969)

This article discusses in the form of a table the spurious signals which can arise in a Hall effect measuring apparatus using alternating current and either an ac or a dc magnetic field. It is pointed out which of these spurious signals can simulate a Hall effect. Checking procedures to distinguish between the real Hall effect and the spurious effects are given.

INTRODUCTION

Measurements of the Hall effect on materials with a small Hall mobility (below about 1 cm2/V· sec) are plagued by a low value of the Hall voltage when com­pared to spurious effects. The most common spurious effects are thermal (Seebeck) or magneto thermal (Ettinghausen, Nernst, Righi-Leduc) .!.2 For this reason highly sensitive Hall effect measurements are usually made by alternating current methods which eliminate these thermal and magnetothermal effects with the exception of the Righi-Leduc effect.!

Alternating current methods can be divided into methods with and without an alternating magnetic field. The use of a dc magnetic field is simpler since it involves standard magnets and power supplies only.

Ac measuring methods introduce their own problems of spurious effects, the discussion of which is scattered through the literature. We have in the present article assembled in Table I a list of effects which can and do plague measurements by ac methods of low Hall mobility values. The table is prepared for the double ac method recently described by Lupu et al.,a where the current I is varying at a much higher frequency (wr= 500 cps) than the magnetic field B (WB= 2 cps). As already mentioned in Ref. 3, this method, while being double ac, uses sequential detection, with two lock-in amplifiers (denoted hereinafter as LIA), first of the current frequency and then of the magnetic field fre­quency. It therefore permits diagnosis of effects which are not in phase with either the current or the magnetic field, in contrast to the conventional double ac method in which the difference OF sum frequency is detected.

At this point we would like to mention that mislead­ing results of the phase with respect of B can be obtained if the Q of the tuned preamplifier in the first LIA is too

* This work was supported by the Advanced Research Projects Agency under Contract SD-86 and by the Aerospace Research Laboratories under Contract No. AF 61 (052)-825 with the European Office of Aerospace Research, U.S. Air Force.

1 E. H. Putiey, The Hall Effect and Related Phenomena, (Butter­worths, London, 1960).

2 H. J. van Daal and A. J. Bosman, Phys. Rev. 158, 736 (1967). 3 N. Z. Lupu, N. M. Tallan, and D. S. Tannhauser, Rev. Sci.

lnstrum. 38, 1658 (1967).

high. This preamplifier will induce a phase delay of the modulation frequency, WB, given by

cp= arctanQ(2wB/wr).

To get a negligible phase delay for WB = 2 and wr= 500 we need Q<5. Some LIA's have a Q normally set at 25 and may need appropriate modification.

Of the effects listed in the table, all or part of each of the following can simulate a Hall effect in a routine measurement where phase relations and linearity are determined: Nos. 3, 5, 6, 8, 9, 10.

Table I can easily be reinterpreted for the more common method which uses an alternating current with a constant magnetic field, which is switched to positive and negative values. All effects which are listed as having a frequency WB will change sign with the mag­netic field, the effects which have a frequency 2WB will not change sign. Effect No: 9, which depends on induc­tive pickup of the magnetic field, will give in this case a transient when the magnetic field is switched on or off. For those effects (Nos. 3, 8, 10) which give a phase difference with the magnetic field one gets a time delay after switching which mayor may not be large enough to be detected in the presence of the finite rise time of the magnetic field. Effect No. 11 will of course not appear. All the other effects can appear and will depend on the magnetic field in the manner stated in the table.

REMARKS AND EXPLANATIONS TO TABLE I

Cause of trouble. The combination of circumstances listed has to be present to cause the effect mentioned.

Frequency oj signal in Hall loop. The signal is of the form cosw[t COSWBt, or equivalent for the harmonics. The notation WIXWB is symbolic for this expression.

Diagnosis. This is of course the most difficult part and is the main reason for presenting the table. The method of diagnosis listed is not necessarily the only one, but is considered to be the most convenient method.

Remedy. There may be more than one possible remedy. It is sufficient to eliminate anyone of the contributing factors to the case of trouble.

Effect No. 1. The phase to B will be very close to 4954

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ERRORS IN AC HALL EFFECT MEASUREMENTS 4955

zero because the current leads with their small mass will be able to follow the low-frequency magnetic field without delay. The effect will depend on the square of I since the dc causing the displacement of the current loop will be proportional to the ac through the sample (see appendix).

Effect No.2. We expect the dependence on I to be squared since both the amplitude of vibration and the dc will be proportional to the ac through the sample. The phase to I depends on the mechanical resonance fre­quency of the sample holder (see appendix).

Effect No.3. Here the 1I1R drop changes because of periodic temperature excursions. The phase to B is determined by the thermal relaxation time of the sample.

Effect No.4. The effect will depend on the square of I because both the amplitude of vibration and the flux through the Hall loop depends on the magnitude of the current. The first LIA will stop the induced 2Wl signal (see appendix).

Effect No.5. The effect described here depends basically on the square of the magnetic field since the amplitude of vibration and the flux in the Hall loop are both proportional to B. If B is asymmetric it can be described by B=Bo+Bl sinwBt. B2 will then contain both the first and the second harmonic of WB. The ampli­tude of the first harmonic is BoBl and will be linear in Bl if Bo is constant. The phase of the effect with respect to I depends on the relation between WI and the mechanical resonance frequency of the sample. The 2WB component will be blocked by the second LIA.

This effect will only appear if the magnetic field is nonhomogeneous or else if the sample vibrates in tor­sion, so that the flux changes at the frequency of vibra­tion.4

Effect No.6. This is the usual magnetoresistanceeffect. The dependence on B is the same as in No. S. It should be noted that the component at WB, which appears for an asymmetric magnetic field, can cause a large pseudo Hall effect.

Effect No.7. This again is a magnetoresistance effect. caused, however, by the third harmonic component of the current generated at nonlinear contacts. The signal at 3WI is passed by the first LIA, however, with con­siderable attenuation. Because of this attenuation and because the 3WI component is a small fraction of the current the signal caused by this effect will probably be negligibly small. The expression "six maxima" means that six maxima of the signal appear as one varies the phase of the reference signal of the LIA through 360°, rather than two maxima as for the fundamental.

Effect No.8. This effect, which is a change in the 1I1R drop, will occur typically in high temperature measure­ments where a large dc flows through the furnace. Its phase to B depends on the thermal time constant of the sample.

4 G. L. Guthrie, Rev. Sci. lnstrum. 36, 1177 (1965).

Effect No.9. This is the intermodulation effect de­scribed first by Russell and Wahlig.5 Its phase to B will depend on the impedance at WB in the relevant loop. If the impedance is Ohmic the phase to B will be 90°.

Effect No. 10. The effect is caused by periodic changes in the 1I1R drop. Its phase to B depends on the thermal time constant of the sample.

E.tfect No. 11. The signal at 3WB will be passed by the second LIA with considerable attenuation. If B is severely distorted the signal may be large enough to give a measurable component at 90° to B and cause trouble for the experimenter. The expression "six maxima" is explained in effect No.7.

EXPERIMENTAL

We have checked some of these effects experi­mentally, using a constant magnetic field. The impor­tance of the rectifying effects was found by adding a dc component to the ac current, and then attempting to make a measurement on a platinum sample at room temperature in a constant magnetic field. Using an alternating current of 100 rnA at 400 cps and varying the dc from 0 to 10 mA, it was found that an error in the Hall voltage as large as 100% could result from the dc component, even though the leads were taped to the supporting rod as firmly as possible. Since the sample was at room temperature, the effect which was observed must have been No.2 of Table 1.

The magnitude of the vibration effects was deter­mined by using several different lead configurations on a platinum sample which was glued onto an insulating strip and fIrmly mounting the strip on a !-in. aluminum rod 18 in. in length. The current through the sample was 100 mA at 400 cps. The leads were multiple-stranded 30 gauge copper wire which was Teflon coated. The magnetic field was again constant with a maximum value of 6 kG.

Measurements were made as follows:

(a) A voltage lead touching mechanically to a cur­rent lead (leads are insulated) with about 5 cm of the leads in the magnetic field not secured.

(b) The leads randomly placed but not touching. (c) The same as b, except care was taken to minimize

the current loop. (d) The same as c, except the sample and the leads

were immersed in heavy oil. (e) The leads were taped to the supporting rod as

firmly as possible, with the current loop minimized.

The results of the measurements are given in Table II. It should be noted that the measured Hall voltage was not affected, in spite of the widely varying magnitude of the vibration induced signal.

The difference between measurements "b, c, d" on the

5 B. R. Russell and C. Wahlig, Rev. Sci. Instrum.21, 1028 (1950).

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4956 H. L. McKINZIE AND D. S. TANNHAUSER

TABLE I. Effects which plague measurements

Cause of trouble

Currents Sample Sample in I asym- Inon- B asym- B non- leads holder Furnace 2 cps temperature metric sinusoidal metric sinusoidal loose loose loose pick up gradient Effect

X X 1. Rectifying contacts cause dc in current leads; current leads move at WB; inductive coupling of current loop to Hall loop changes at WB.

X X 2. Rectifying contacts cause dc in current loop; current loop vibrates at WI in mag-netic field and induces signal at WI in Hall loop.

X X X 3. Rectifying contacts cause dc in current leads; sample moves at WB in tempera-ture gradient; AR changes atwB·

X 4. Current loop vibrates in magnetic field at WI. Cur-rent induces signal in voltage loop.

X X 5. Sample vibrates in mag-netic field at frequency WI.

Asymmetric magnetic field induces signal in Hall loop.

X 6. Asymmetric magnetic field modulates sample re-sistance or electrode re-sistance.

X X 7. Nonlinear current contacts cause 3w[ current in sam-ple; asymmetric magnetic field modulates this current by modulating contact resistance.

X X 8. Furnace moves in magnetic field at WB; temperature distribution on samble changes at WB; AR c anges atwB.

X 9. Magnetic field induces cur-rent at WB in sample voltage or current loop. This intermodulates with sample current through nonlinear element in sample or electronics.

X 10. Temperature gradient in sample. Magnetic field modulates temperature distribution in sample through Righi-Leduc effect. AR changes at WB.

X 11. Third Harmonic of mag-netic field causes Hall effect.

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ERRORS IN AC HALL

by ac methods of low Hall mobility values.

Signal in Hall loop

Depend-Frequency ence on I

Dependence onB Phase to I

WIXWB Square Linear 90°

WIXWB Square Linear Any

WIXWB Linear Linear 00

2WIXWB Square Linear Not applicable

wIX2wB Linear and

Square for 2WB, linear for WB if

Any

WIXWB asymmetry is constant

WrX2WB Linear Square for 2WB, 0° and linear for WB if WIXWB asymmetry is

constant

3wIX2wB Linear Square for 2WB, Not applicable and linear for WB if (six maxima) 3wrXWB asymmetry is

constant

WrXWB Linear Linear 0°

WrXWB Linear Linear 0°

WrXWB Linear Linear

WrX3WB Linear Linear

EFFECT

Phase toB

Close to 0°

0°

Any

0°

0° forwB

0° forwB

0° forwB

Any

Any

Any

Not applicable (six maxima)

MEASUREMENTS

Diagnosis

Intentional dc in cur-rent loop changes signal

Intentional dc in cur-rent loop changes signal

Intentional dc in cur-rent loop changes signal

Through dependence onI

Behaves like mag-netoresistance but is not in phase with I

Adding constant value to magnetic field changes signal

Changing Q of first LIA affects signal

Reversing dc through furnace changes signal

Changing amount of pickup changes signal

Changing temperature distribution affects signal

Changing Q of second LIA affects signal

4957

Remedy

Stiffen sample holder

Stiffen sample holder

Stiffen sample holder

Stiffen sample holder

Stiffen sample holder

Improve contacts, symmetrize B

Symmetrize B

Stiffen furnace holder

Compensate care-fully signal at WB

Minimize tempera­ture gradients

Eliminate harmonics from magnetic field

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4958 H. L. McKINZIE AND D. S. TANNHAUSER

TABLE II. The effect of various electrode configurations on the signal measured in the Hall voltage loop.

Signal in phase Signal 90° out of with current. phase with current.

Field Field Field forward reversed forward

(a) lOJ'V 10 J'V +20J'V

(b) -100nV -145 nV +100nV (c) -70nV -120nV +70nV (d) -215 nV -265 nV +300nV

(e) +lOnV +60nV +20nV

one hand and measurement "e" on the other must be due to effect No.5 of Table I, i.e., the pseudomagneto­resistance caused by vibration.

ACKNOWLEDGMENT

The authors would like to thank Mr. H. N. S. Lee for many helpful discussions and suggestions in the prep­aration of this paper.

APPENDIX: ANALYSIS OF RELATIVE MOVE­MENT AND VIBRATION OF CURRENT LOOP

AND VOLTAGE LOOP

The instantaneous flux caused by the current loop in the stationary voltage loop is given by

<I>(x, 1) =J(x)I.

Here x is the instantaneous distance between current loop and voltage loop (or an equivalent geometrical parameter which depends on the Lorentz forces on the current loop), I is the current through the sample. The dependence ofJ(x) on x will be linear for small changes of x around its equilibrium value xo(x=xo for 1=0 or B=O)

J(x) =axo+Mx.

ax will depend on the current through the sample, 1= 10+ II COSWIt and the slowly varying external mag-

Field Hall reversed voltage Comments

+20J'V Very noisy, Hall voltage cannot be determined.

+100nV 22nV Much more stable. +70nV 25nV Further decreased noise.

+300nV 25 nV Mechanically coupled through the oil, more noisy.

+20nV 25nV Very stable.

netic field B as follows:

Ax= eoIoB+edlB cos (WIt+4» .

Here the first term is the constant displacement due to the dc part of I. The second term oscillates at WI but is not in phase with the current because of the mechanical resonance properties of the current loop.

Substituting we get

4>(x, 1) = (axo+Mx)I

= [axo+b(eoIoB+edIB cos (WIt+4» ]

X (10+11 coswIt)

= axofo+axoh coswIt+beoI02B

+beoIoIIB cosw1t

+bedollB cos(wII+4»

+bed12B cos(wII+4» COSWIt.

The first three terms in this expression do not con­tribute anything of interest since they are either in­dependent of the magnetic field or independent of the alternating current. The remaining three terms give effects Nos. 1, 2, and 4 of Table I. If we assume that 10 is proportional to II then all three effects are propor­tional to 112, as mentioned in the table. The phases of these three effects with respect to I follow directly if one calculates the voltage in the Hall signal loop by V = -04>/01.

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