Design of an A.C. micro-gauss sensor

Sensors

and Actuators,

DESIGN

4 (1983) 3 - 9

3

OF AN A.C. MICRO-GAUSS

SENSOR*

T G M KLEINPENNING Department of Electrical (The Netherlands)

Engmeenng,

Emdhoven

Unwerslty

of Technology,

Emdhouen

Abstract

The signal-to-nose ratio (SNR) m Hall elements 1s mvestlgated. The sensltlvlty of Hall elements as magnetic sensors 1s hmlted by two types of noBe (1) voltage noise m the element, and (n) noBe m the electronic system followmg the element The noise m the element IS the sum of Nyqulst nose and l/f nom, the latter having a quadratlc current dependence The SNR 1s proportional to the electnc current m the element Its maxnnum value IS reached either when the l/fnolse m the element exceeds the sum SNR,, of Nyqulst norse and electronic system noise, or when the current reaches its maxnnum level to prevent excessive Joule heatmg Highly-sensltlve elements require (I) mater&s vvlth a hlg;h mobility of free charge carriers,, and (11) high ebctnc power dlsslpatlon A sensltrvlty better than low6 Gauss can be obtamed with elements where the moblllty p > 6 m2/Vs and the power dlsslpatlon P > 0 3 W.

1. Introduction

In the field of magnetic measurements, devices based on the Hall effect have been found useful as magnetic sensors Thin-film InSb Hall elements are widely used for commercial apphcatlons Such elements can be used for magnetic tape swal pick-up (Hall heads), as detectors m magnetic bubble memones, as part of a posltlon sensor, as a Hall sensor for field measurements, as contactless current sensors and so on [l - 31 InSb 1s one of the best mater&s for Hall elements because of its high electron mobrllty (pee,I 6 m2/V s). Hall elements are expected to be supenor to conventional toll heads m thev hgh frequency-independent output voltage. This paper describes the sgnal-to-nose ratio (SNR) m Hall elements mth two types of geometry a Greek cross and a rectangular device. Design rules are gwen for highly-sensltlve elements *Based on a Paper presented at Solid-State Transducers 83, Delft, The Netherlands, May 31- June 3,1983 0260-6874/83/$3

00

0 Elsevler Sequola/Pnnted m The Netherlands

4

2. Calculations of signal-to-noise rat10 The sensitivity of Hall elements as sensors of a.c magnetic fields is hmlted by two types of noBe (1) noise m the element, I e , l/f noise and Nyqulst noise, and (11) noise m the electronic system followmg the element, I e , noise m the first amplifier In this section we present calculations of the SNR m a Greek cross element and m a rectangular element (see Fig 1)

Fig 1 Rectangular Hall element and Greek cross Hall element

2 I Greek cross, theory For a Greek cross the l/f noise across the Hall contacts is given by [ 4 3

SvJf) =

d2(PlQ2

(1)

5 2512tnf

where &(fl is the l/f voltage noise spectral density m V2/Hz at frequency f, p the reslstivity, n the free carrier density, and CYthe Hooge l/f noise parameter This parameter ar 1s dunensionless and of the order of magnitude of 10F3 The Nyqulst noise across the Hall contacts is %,

= 4kTRw s 9 6kTp/t

(2)

with R3+ 2 4 p/t, th e resistance between the contacts The noise contnbutlon of the electronic system is determined mamly by the noise of the preamplifier, which can be described by the relation (3)

h&9 = 4kTRe,(fl

where R,,(f) IS the so-called equivalent noise resistance The noise of an amplifier at frequency f with shorted input is (4kTR,,(f))“” m V/Hz”? For low-noise nanovolt amphfiers we have R,, z 5OC8 at frequencies above lkHz, so (4kTR,,)1’2 z lnV/Hz 1’2 Figure 2 shows a sketch of the noise spectra of the element and the electronic system The Hall voltage is [ 53 l.&=v34=

5!wt

1 [

_

()

045

=twPm ClB

I

s-

I.

wt

where p 1s the free earner mobility There must be a hmrt to the heat dlsslpation, hence

(4)

Fig 2 Typical mic scale)

noise spectra

of a Hall element

I’ = 12R 12= 2 412p/t < Pm,,

Im,, = [Pm,,t/2.4p]

If the bandmdth Af of the electromc f 18 found from eqns. (1) - (4) VH

sNRtn

= (S,Af]“*

1’2

vH

= [S,,Jf)

= pJ3[5 2512tnf/cx]

(logarlth-

(51

system 1s 1 Hz, the SNR at frequency

(‘3)

+ 4kTR,,(f)]1’2

Since the l/f noue D proportional to I2 proportional to I provided I 1s low. At exceeds the sum of the Nyqulst nome reaches its maxmum value (provided that SNR,,(f)

and of a low-noise preamplifier

(eqn. (1)) and V, - I, the SNR 1s high currents where the l/f noise and the amphfler noise, the SNR I < ImaX).

1’2

(7)

If the l/f noise does not exceed the other two types of nome at I = Imar, then with the help of eqns (4) - (6) we obtam SNR,,(f)=

’ k 2Rwqnt [ kT

Usmg an amphfler mth B,,(fl

l/2

l-1 + %,VVR,11’2

1

= 4 SyI’Ty’;;;;2 w

l 1,2 34

< Rs4, eqn. (8) reduces to

From eqns (l), (2) and (5), It follows that the l/f noise does not exceed the Nyqulst noBe provided that P,,,jkT

< 5.25 X 4

8212tnf/az

12012tnf/cx

(10)

From eqns (l), (2) and (6) It xs obvious that the h@est value for SNR can be obtamed if Rep < RM and inequality (10) prevruls So the highest attamable value for SNR IS @ven by eqn. (9) and can be obtamed If the free earner density obeys the mequality

6

with ~max the heat dlsslpatlon per unit volume For pmax = 100 W/cm3 and

f = 100 Hz, we have n > 1016 cm- 3 So the detection hmlt for the magnetic mductlon 1s given by

Here we take SNR = I, and R,,(f) < R34 The sensitivity can be mcreased by usmg a material with a high carrier mobility cc, and a rather high carrrer density n according to eqn (11) Good thermal contact between sample and surroundings is necessary m order to avoid excessive heatmg The value of R34 should not be too low (Rs4 > R,, > 100 a), which can be realized by takmg a thm sample Hughvalues for P maL can be obtained by choosmg a large Greek cross A suitable matenal1s n-InSb, where p, = 6 m*/V s, n z 1016 cmw3 and p z IO-* a cm Taking t < 2 4 pm, we obtam Rs4 > 102 32 For such thm samples the maximum heat dlsslpatlon IS about 0 1 W/mm* [ 1, Z] For I = 1 mm we have P,,, = 0 5 W and thus B,,, = 7 X 10-l’ T = 0 7 PG

2 2 Rectangular element, theory For rectangular elements the results are as follows The l/f noise across the Hall contacts (see Fig 1) ISgiven by 16 3

%&@I =

atiT*ln(w/a)

(13)

2nfrlt

where E IS the electric field strength m the longitudmal dlrectlon Nyqulst norse across the Hall contacts is

The

RN s (2p/;rrt) ln(w/a) %4 = 4kTRW, The Hall voltage IS [ 71

(14)

VH = v, [email protected][l--exp(-

x’~~~)][l-

“rtr]

(15)

For 1> w and a Q w, eqn (15) can be approximated by v, = /.lBEw

(16)

The heat dissipation has to be lower than Pm_ P = I=pl/wt = E*lwt/p < Pm_

(17)

In the same way as described m Section 2 1, we obtam for the highest attamable value of SNR

This value can be obtamed provrded that %#9

< %

(19)

and n > arp,,,l(16kTf)

(20)

Here, too, we fmd that the sensltlvlty can be mcreased by usmg a matenal with a high cmer moblllty and a rather high earner density Furthermore, the thermal contact has to be optnnum, the element thm, and the area large An element wrth p = 6 m2/V s, l/w = 2, w/a = 4, and PmBx = 0.5 W has a detection lnnlt (SNR = 1) for magnetic mductlons of Bmin = 4 X lo-l1 T = 041.tG. 2.3 Rectangular element, experrmen t Kotera et al [ 1, 21 have mvestlgated thin-film Hall elements of n-InSb The results of a typical sample (J 375 - C2) are aven m Fig 3. The dunenslons of the sample are w = 200~m,I=l000~m,t=16~m,anda=50~m Other data are c(~ = 6 m2/V s, RH = 430 cm3/C at low heat dlsslpatlon (I< 10 mA; P< 0 03 W), and Af = f,,, - fmin= lo4 - lo2 Y lo4 Hz The devlatlons from the straight lmes above I = 10 mA are due to Joule heatmg. Since P,,, = 0 03 W, we find mth eqn (18) for this sample B,,, = 2.6 X 10-l’ T or 2 6 X 10m6 G At I = 10 mA, Kotera et al found Bmin = 1 7 X 10m3 G The difference of a factor 650 1s mamly due to (1) the bandwidth Af = lo4 Hz, which explams a factor 100, and (n) the amphfler noue, No=

Fig 3 Expenmental data of a thm-fdm rectangular Hall element of n-InSb [ 1, 2 ] VH IS the Hall voltage, No the r m s value of the nose voltage of the resistance of the element and the amphfler m a bandvndth Af I: lo4 Hz, IV, the r m s value of the l/f nolee voltage, RN = llnq 18the Hall coefficient, S/N 18the signal-to-noise ratlo at B = 20 G and B, 1s the sensitivity for detecting magnetic mductlons The arrows mdlcate the current where No = N,

8

[4kT(RM + R,,)Af]l’* = 0 4 pV, thus RM + R,, = 970 St Usmg R, = 40 a “* = 5 Therefore we conclude that we can explam a factor [ 1 + R,,/RJ the experunental data of Kotera et al agree well mth the calculated data The sensltlvlty of their elements can be improved (1) by using a low-noise nanovolt amphfler with R,, “Y 50 SZ (such amphfrers are commercially available), (n) by taking a smaller bandwldth of the electronic system, (1~) by lmprovmg the thermal contact and (iv) by using an element with a lower Z/w ratio and a larger area 3. Remarks and conclusions Several remarks can be made here Frost, the mmlmum detectable magnetic mductlon can be decreased by attachmg a femte core to the element [ 23 Secondly, besides the Hall voltage an a c magnetic field gives an mduction voltage Vina = AdB/dt, where AB IS the magnetic flux m the clrcult If B = B, an wt, using eqn (16) for a rectangular element we fmd VInd/V~ = Aw/l,cEw For typical values such as A = 0 1 cm*, w = IO4 rad/s, fi = 6 X lo4 cm*/V s, E = 10 V/cm and w = 1 mm, we fmd VIna/VH s lo-* Generally V Ind will be negligible Thirdly , the sensltlvlty of a Hall sensor 1s often defined by [ 3, 81 G = V,/IB = l/(qnt),so the sensltlvlty increases with decreasmg earner density However, from our mvestlgatlons it follows that for a c measurements this defmltlon 1s mcorrect (cf eqns (11), (20)) Fourthly, m order to measure very low Hall mobllltles, one usually performs a c Hall effect measurements The detection hmlt is then @ven by prnin= B,,-1[kZ’/P,,]1’2 1: lo-’ m*/Vs for B = 0 1 T and P,,,,r = 1 W On the basis of our mvestlgatlons, we conclude that highly sensltwe sensors have to satisfy the followmg condltlons (1) the carrier mobllrty p m the matenal has to be as high as possible, suitable materials are n-InSb, n-InP, n-InAs, (11)the carrier den&y n has to be high enough to avoid detenoratlon as a result of l/f noise, (in) the element has to be so thm that the resistance between the Hall contacts exceeds the equwalent noue resistance of the amphfler followmg the element, (iv) the electnc power dlsslpatlon has to be as large as possible, which can be achieved by good thermal contact between element and surroundlllgs, and by usmg an element with a large area, (v) the bandmdth of the electronic system followmg the Hall element has to be as narrow as possible References 1 N Kotera, J Shlgeta, T 01, M Nakashuna, and K Sate, Transverse l/f no=e In InSb thm films and the wgnal-to-nolse ratlo of related Hall elements, J Appl Phys , 49 (1978) 6990 - 5996

9 2 N Kotera, J Shlgeta, K Narlta, T 01, K Hayashr and K Sate, A low-noise InSb thm film Hall element fabncatlon, device modeling, and audio apphcatlon, ZEEE Trans Mugn, MAG-16 (1979) 1946 - 1966 3 J H J Flultman, A survey of solid state magnetic held sensors, Summer course

1982 Solid-state June I982

sensor

and

transducers,

Katholzeke

Unwerslte#t

Leuven,

Belgrum,

4 L

K J Vandamme and J Kedzla, Concentration, moblhty and l/f noise of electrons and holes m thm bismuth films, Thm Solrd AZms, 65 (1980) 283 - 292 5 W Versnel, Analysxs of symmetrical Hall plates with fmlte contacts, J Appi Phys , 52 (1981) 4669 - 4666 6 T G M Klempennmg and L K J Vandamme, Comment on ‘Transverse l/f nolSe m InSb thm films and the SNR of related Hall elements’ J Appl Phys , 50 (1979)

5547 7 J Haeusler and H J Llppmann, Hallgeneratoren Solrd-State Electron, 11 (1968) 173 - 182 8 P Danul and E Cohen, (1982) 8257 - 8269

Low

field

Hall effect

mlt klemem magnetometry,

Lmearlserungsfehler,

J Appl

Phys , 53