The circular hall plate : approximation of the geometrical correction factor for small contacts

(1)

The circular hall plate : approximation of the geometrical

correction factor for small contacts

Citation for published version (APA):

Versnel, W. (1981). The circular hall plate : approximation of the geometrical correction factor for small contacts.

(EUT report. E, Fac. of Electrical Engineering; Vol. 81-E-116). Technische Hogeschool Eindhoven.

Document status and date:

Published: 01/01/1981

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be

important differences between the submitted version and the official published version of record. People

interested in the research are advised to contact the author for the final version of the publication, or visit the

DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page

numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne

Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

(2)

The circular hall plate.

Apprcximation of the geon,etrical correction

factor for small contacts.

by

W. Versnel

A

(3)

Department of Electrical Engineering

Eindhoven The Netherlands

THE CIRCULAR HALL PLATE. APPROXIMATION OF THE GEOMETRICAL

CORRECTION FACTOR FOR SMALL CONTACTS. By

w.

Versnel TH-Report 81-E-116 ISBN 90-6144-116-1 Eindhoven January 1981

(4)

Abstract - The circular Hall plate is considered with four equal finite line contacts symmetrical with respect to two orthogonal axes. A proof is given of an approximation of the geometrical correction factor C(8,m), if

e

tends to zero. The angle

e

corresponds to the length of each

contact. The parameter m depends on the magnetic field, which need not be small.

Versnel, W.

THE CIRCULAR HALL PLATE: Approximation of the geometrical correction factor for small contacts.

Department of Electrical Engineering, Eindhoven University of Technology, 1981.

TH-Report 81-E-116

Address of the author: Dr.ir. W.Versnel,

Electronic Devices Group,

Department of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513,

5600MB EINDHOVEN, The Netherlands

(5)

Contents 1. 2. 3. 4. 5. 6. 7. 8. Introduction Approximation of Approximation of Approximation of Approximation of Approximation of 6.1 Approximation 6.2 Approximation 6.3 Approximation Approximation of Conclusion Acknowledgements References tl and t2 integral J 1 (8,m) integral J 2(8,m) and of p function J 3(8,m) difference J 4(8,m) - J4(8,-m) of of of the J 41 (o,m,a) J 42 (6,m,a) J 44(6,m,a) geometrical + J 41 (o,-m,a) - J 42(6,-m,a) correction factor Page 1 3 4 5 6 6 7 8 8 11

12

12 12

(6)

1. Introduction

Consider a circular Hall plate with four finite contacts of equal length (Fig. 1). It is convenient to take the radius equal to one unit of length. We shall deal with an n-type semiconductor. The current I enters the sample

B

®

Fig. 1

Circular Hall plate with four contacts which are equal in Zength

Radius

=

1

unit of length

at contact 1 and leaves it at contact 3. The Hall electrodes draw no current. The constant magnetic induction B is directed backwards and perpendicular to the plane surfaces of the sample. It is assumed that the sample is homogeneous and has a specific resistivity p and a uniform thickness d. The last-named is

very short with respect to the radius of the sample. As is well known[l] the

method of conformal transformations can then be applied in order to obtain

the potential distribution.

We have analysed the circular structure in [2J. Let V. denote the

J

potential of contact j (see Fig. 1). The geometrical correction factor C(8,m) is defined by the relation

c(S,m) tan

S

(1)

where the Hall voltage V

H V2 -

V

4 ,B

is the Hall angle, m

=

2B/~ and the

angle

e

corresponds to half a contact. We have proved that [2J c (S ,m) 1 sin

S

J 4(S,m) - J4(S,-m) J 3(S,m) The function J 3(8,m) is given by t2

J

1 J 3 (S ,m) sgn p (2 ) (3 )

(7)

with

(Y-p) (y +

~) ~(y

_{+ tl) (y + !J Gl -}

y)r-l

~(y

+ t2) (y + !J(!2 - y)r

Furthermore, the function J

4(e,m) is defined

by

j

tl

-b sgn

P

dy (tl-y)

f

4(y,m)

P with

(y-p )(y +

~)~(t2

- y) (y +

~(y

+ t2)(y +

!J(~2

- yt

b

In these integrals the following parameters tl' t

2, band p occur: and where P Q J

l (8,m)+

J

l (8,-m)

J

2(8,-m)-

J

2(8,m)

The integrals J

1 (8,m) and J2(6,m) are defined by

t

_z

J 1

(8 ,m)

=

f

dy (l_yZ) f (y ,m)

tl

and

t2

J

2(8,m)

=

j

dy y f(y,m)

tl

respectively; where 1+m 2

f(y,m)

~(y-tl)

_{(y + t2) (!2 - y) (y +}

~J

_f-

b

1(y+t

1)

h-

y) (y +

~J

Gl - y)fb-l

(4) (5) (6) (7 ) (8) (9)

(8)

Note that tl and t2 are positive but less than one. The sign in formula (6) has to be chosen such that

Ipl

< 1.

In this paper we shall derive an approximative expression for the geometrical correction factor C(8,m) if 8 tends to zero. It has already been given without proof in [2J.

In the next section an approximation will be obtained for t l and t

2, i f

e

approaches zero. In Section 3 the integral J

1 (8,m) will be approximated. Section 4 contains an analysis which provides approximations of the integral

J

2(8,m) and of p. We are then in a position to ascertain the behaviour of

J

3(8,m) and of the difference J4(8,m) - J4(8,-m) i f 8 tends to zero. This is done in Sections 5 and 6. Finally, we shall derive an approximative formula for the geometrical correction factor C(8,m) in Section 7_

2. Approximation of tl and t 2.

Let

6 =

e

where

6

is a small positive number. Define E

=

t

2

- t

1 -

If

0

tends to zero, then E also approaches zero. I t is convenient to approximate C(o,m) by a simple function of £. Later on, the relationship between

0

and

E will be used in order to obtain an approximation in

o.

It is easily seen from (5) that o Then we find Furthermore, 1 - 1

+

h

-2 1 - 1 +

12

+

2

0 + o

+

h

2 3

16

0 +0(0 ) (0-+0) . (0-+0) (0-+0) The integral J

1 (8 , m) will be considered in the next section.

( 10)

(11 )

(9)

3. Approximation of integral J 1(B,m) Let t2 L(c,m,k)

f

k-b b-l

=

dy (y-t) (t -y) 1 2 tl

where k is zero or a natural number. Introducing a new variable v

=

(y-t

1)/" we obtain

L(c,m,k) , k B(k+l-b,b)

where B(x,y) is the beta function

B(x,y)

=

f

l t x-I (1-t)y-l dt

(R{x}>O, R{y}>O)

°

The integral J

1 (8,m) can be written in the form

where

g(y,m) =

(1_

y

2)

~(y

+ t

_{2) G2 -}

y) (y +

!J (

l(y + t 1) (y +

!J

Ul - y)

f

b-l

We now develop the function g(y,m) in a series in y-t

1

Using (11) and (12), an elementary calculation yields

A(m) = 2

+

12

16

[

(

12 12)

,2 { 1 + ,

"2 -

b

4

+

16

5 +

12 -

b (19 2 0(, ) (13) (14) (15) (16)

(10)

From

(14), (15)

and

(16)

one finds

1 £ { - }

=

i/2 B(1 - b,b) +

16

(2 + 12 - b) B(1 - b,b) - (3+2/2) B(2-b,b) if E tends to zero. 4. Approximation of integral J 2(8,m) and of p The integral J 2(8,m) can be rewritten as where t2 J 2 (o,m)

=

f

dy

tl

h(y,m)

_{y 1(y+t}

2 ) (!2 -

y)

~

+

! J ( b

l(y+t

₁

)(y +

! J (!1 -

Y)!

b-l

The function h(y,m) is developed in a series in y -

tl:

h (y

,m)

Using (11) and (12), after some straightforward calculations we obtain

I 1 2 12 3 = - 1

+

v 2 -

"2

£

+

£

16 +

0(£ )

~(-5

+ 12) + 0(£) 8 (18) (19)

The value of A(m) was already obtained in the preceding section~ From (14),

(18)

and

(19)

it is easily seen that

J

2(o,m)

=

A(m) {corm) B(I-b,b)

+

8 C1 (m) B(2-b,b)

+

(20)

i f £ tends to zero.

(11)

From (6), (7), (17) and (20) we can derive an expression for p (note that

Ipl

< 1). Extensive but simple calculations yield

p - - 8 -2+12 (l-2b) 8

+

0(8 ) 3 (8->{» 5. Approximation of function J

3(6,m)

From (3), (8) and (9) we have

From (17), (20), (21) and (22) We find

sgn p 1-12

J 3 (o,m)

=

8(1-2b) - 2 - B(l-b,b) + 0(£) (8->{) ) It should be noted that J

3(8,m) is invariant for reversal of the magnetic

field: J

3(8,m)

=

J3(8,-m).

6. Approximation of difference J

4(8,m) - J4(8,-m)

Eqn. (4) can be rewritten as

where t1 sgn p

f

dy { / - 1 +

(~-

p)

Y}

(t 1-y)-b .. p (21) (22) (23) (24 )

Because of the singularity of the integrand in (24) i t is found to be

necessary to split the interval of integration into four parts. Introducing

a

positive number a less than one, we have

with J 41 (o,m,a) sgn p

{J

41 (o,m,a)+

(i -

p)

J

42(o,m,a) + - J 43(o,m) + J44(O,m,al} (25) (26)

(12)

(27) p J

43

(8,m) =

fdY

{y2_1

+

(~_

p)

Y}

o

and J

44

(8,m,a) (28) respectively. Since J

43(6,m) O(E), if E tends to zero, we find

sgn p

{J

41 (8,m,a) +

J

41 (8,-m,a)} +

+

sgn p

(~-

p)

{J

42(8,m,a) -

J

42(8,-m,a)}

+

0(0) (0-+0)

Successively, the termson the right-hand side of (29) will be determined.

The value which the number a should have in order to obtain an accurate approximation will be clear later on.

6.1 Approximation of J

41 (o,m,a) + J41 (o,-m,a)

From (26) it follows that the constant term in the approximation of

J

41 (a,m,a) + J41 (o,-m,a) is

J

41 (O,m,a)

+

J41 (O,-m,a)

, l'" '(

t-l--Y')~(-:":':'l )-(-;-Y-+--;-~ l')-(-;;;~-l---;Y

)

in which tl

= -

1 +

12.

By elementary integration we obtain the result

(independent of m) J 41 (O,m,a)

+

J41 (O,-m,a) h (l-a) (l-a(3-2h)}

In

4 (1+a) {l+a (3-2/2)}

The linear term in the Taylor series is evidently of the order E.

(29)

(13)

6.2 Approximation of J

42(o,m,a) - J42(c,-m,a)

From its definition (27) it is easy to see that J

42(O,m,a) is independent of the magnetic field. Hence, the constant term in the Taylor series of J

42(o,m,a) - J42(6,-m,a) is zero. In order to obtain its linear term we need the derivation of J

42(o,m,a):

aJ

42 (0 ,m,a)

a

t

2

By differentiating" (27), we find, for example

2 a t1 ---;;---=---;;---:.-

+

2 2 4 (1-a ) (1-a t j ) in which t

=

1 - I

+

/2. The variables tl and t2 are represented

and (12) respectively. Carrying out laborious but

as functions

of E in ( 11 ) straightforward

integrations, we obtain the first-order approximation

where tl .. [ atl 2

2

4

16(a tl -I) - j +

h.

6.3 Approximation of J 44(o,m,a)

The problem of deriving an approximation of the function J

44(8,m,a) for small values of 8 is very complicated.

First, we develop the function {y2 -1

+

(~-

p) y} h4 (y ,m) in a Taylor

series in the variable y - tl:

(14)

D(m) {EO (m) + E1 (m) (y-t_j ) +

Using (11), (12) and (21), after some straightforward calculations we find

that D(m) A(-m) EO(m)

2-212

+ {4 - 312 +

t

(2

- 12)}

_{E (2b-1)}4 E1 (m) 2 + 12 +

{s -

212 + E

₄

1 12

_-S

5 , 1 } 4 Eb2 _E(2b-1) 2 E 2(m)

12-712

3+612

312 20 (2b-1) - - - 4 - + 2(2b-1)

We refer to Section 3 for an expression for A(-m). Second, let

K(o,m,a,k) =

J

dy at

1

where k is zero or a natural number. By introducing a new variable

1

k k 1 - b i k-b b-1

K(o,m,a,k)

=

(I-a) t1 x dv (I-v) (I-xv)

o

in which

1

- =

x

1

+

t j (1-a) E

It is easily seen that

k k l-b K(o,m,a,k)

=

(I-a) t1 x

*

k

.. L:(-l)n

(~)

B(n+1, I-b) F(1-b, n+1; 2-b+n; x) n=O

where B(x,y) is the beta function and F(a,bicix) is the hypergeometric function F(a,b;CiX) f(c) f(b) Hc-b) (32) (33 ) (34)

(15)

if b > 0, c-b > 0 and \x\ < 1. It is known that [3] with F(p,q;p+q;X) 1 B(p,q) i=O (P)i(q)i

(il>

2 i (I-x) f. (p,q,x) ~

(j

l-x\ < 1)

f. (p,q,x) = 2 .(i+l) - .(p+i) - .(q+i) - In(l-x)

~

,

.(z) =

r

(z)/f(x)

(P)i = r(p+i)/r(p) for i=I,2,3 . . . ; (p)O=I.

(35 )

Third, using the above results, we shall now determine an approximation for

J

44(o,m,a). From (28), (32) and (33) we obtain

It is assumed here that (i-a) is a small positive number. Applying (35) in (34), we find after some straightforward calculations that

1+12

K(o,m,a,O) =

.(1) -

.(I-b) - In I-a - In £ +

Note that we used the relation

.(z+l) - .(z) 1

z

(36)

and that the term with index i in (35) is of the order of O(£i).In the same way the functions K(6,m,a,1) and K(o,m,a,2) have been approximated, viz.

K(o,m,a,1) 2

+

0(£ In£) l-b I-a / { l-b (1-a) (-1+ 2) 1

+

-I-a (1+12)£ In £ + (1 +12)

{-.(2)

+

.(2-b)

+

In - -

1+12}J}

+

I-a

(16)

and

K(o,m,a,2)

2

(I-a) (3-212)

2

Let a ::::: 0.9. Then, from (36) we obtain after straightforward calculations J 44(6,m,0.9) 0.20711 2b-l In E

- - +

E + {O. 79077 + 0.20711 Iji (l-b)} -,'1:-,-.,...,- + E (2b-l) 0.14017 + 2.64276 - l-b + 1 { 0.14017} + 2b-l - 2.39956 + l-b + +O(EInE) (E+O)

Note that for a <0.9 the approximation of J

44(o,m,a) by three terms (see

(36» may not be accurate enough (see (34».

7. Approximation of the geometrical correction factor

In this section an approximation for the geometrical correction factor

C(8,m) will be derived. From (37) i t is easily seen that

From (30) we obtain 0.20711 E(2b-l) + 5.28552 {Iji(l-b) - Iji(b)} + (E+O) J 41 (6 ,m, 0.9) + J 41 (6 ,-m, 0.9) = - 1.15108 + OrE) (E+O)

From (21) and (31) i t is shown that

Furthermore, from (29), (38), (39) and (40) i t is found that

(37 ) ( 38) (39) (40) [ 0.20711 - J_{4 (6,-m)}

=

sgn p e(I-2b) + O. 70687J (41 )

(17)

C (8 ,m) 1 + 3.41304 (1-2b)E ljJ (b) - ljJ (l-b)

Since ljJ(b) - ljJ(l-b)

=

TI tan(S) and £ 8(4-2/2) + 0(03), if 0 tends to zero,we finally obtain

C (0 ,m) 1 - 0.8103 8 cotan (8) 0 (8"* 0)

Herewith we have found an important formula for the geometrical correction factor C(6,m} if 8 tends to zero. Data on the accuracy of formula (42) have been given in Section 5 of [2J.

8. Conclusion

(42)

It has been shown that the geometrical correction factor C(8,m) can be approximated by a simple analytical expression if the angle e tends to zero. For the accuracy of the approximation reference is made to [2J.

Acknowledgements. The work has been carried out in the Electronic DeviceS

Group (Prof.dr.H.Groendijk).

REFERENCES

1. R.F. Wick, Solution of the field problem of the germanium gyrator,

J.Appl. Phys. ~ (1954) 741-756.

2. W.Versnel, Analysis of a circular Hall plate with equal finite contacts,

Solid-State Electron. 24 (1981) 63-68.

3. Higher transcendental functions, Vol. 1. Based, in part, on notes left

by H. Bateman, Ed. by A.Erd(lyi et al. New York, McGraw-Hill, 1953.

(18)

DEPARTMENT OF ELECTRICAL ENGINEERING

Reports:

93) Duin, CaA. van

DIPOLE SCATTERING OF ELECTROMAGNETIC WAVES PROPAGATION THROUGH A RAIN

MEDIUM. TH-Report 79-E-93. 1979. ISBN 90-6144-093-9

94) Kuijper, A.H. de and L.K.J. Vandamme

CHARTS OF SPATIAL NOISE DISTRIBUTION IN PLANAR RESISTORS WITH FINITE

CONTACTS. TH-Report 79-E-94. 1979. ISBN 90-6144-094-7

95) Hajdasinski, A.K. and A.A.H. Damen

96)

REALIZATION OF THE MARKOV PARAMETER SEQUENCES USING THE SINGULAR VALUE

DECOMPOSITION OF THE HANKEL MATRIX. TH-Report 79-E-95. 1979.

ISBN 90-6144-095-5

Stefanov,

B.

ELECTRON MOMENTUM TRANSFER CROSS-SECTION IN CESIUM AND RELATED CALCULATIONS

OF THE LOCAL PARAMETERS OF Cs

+

Ar MHD PLASMAS. TH-Report 79-E-96. 1979.

ISBN 90-6144-096-3

97) Worm, S.C.J.

RADIATION PATTERNS OF CIRCULAR APERTURES WITH PRESCRIBED SIDELOBE LEVELS.

TH-Report 79-E-97. 1979. ISBN 90-6144-097-1

98) Kroezen, P.H.C.

A SERIES REPRESENTATION METHOD FOR THE FAR FIELD OF AN OFFSET REFLECTOR

ANTENNA. TH-Report 79-E-98. 1979. ISBN 90-6144-098-X

99) Koonen, A.M.J.

ERROR PROBABILITY IN DIGITAL FIBER OPTIC COMMUNICATION SYSTEMS.

TH-Report 79-E-99. 1979. ISBN 90-6144-099-8

100) Naidu, M.S.

STUDIES ON THE DECAY OF SURFACE CHARGES. ON DIELECTRICS.

TH-Report 79-E-l00. 1979. ISBN 90-6144-100-5

101) Verstappen, H.L.

A SHAPED CYLINDRICAL DOUBLE-REFLECTOR SYSTEM FOR A BROADCAST-SATELLITE

ANTENNA. TH-Report 79-E-l0l.

1979. ISBN 90-6144-101-3

102) Etten, W.C. van

THE THEORY OF NONLINEAR DISCRETE-TIME SYSTEMS AND ITS APPLICATION TO

THE EQUALIZATION OF NONLINEAR DIGITAL COMMUNICATION CHANNELS.

TH-Report 79-E-l02. 1979. ISBN 90-6144-102-1

103) Roer, Th.G. van de

ANALYTICAL THEORY OF PUNCH-THROUGH DIODES.

TH-Report 79-E-l03. 1979. ISBN 90-6144-103-x

104) Herben, M.H.A.J.

DESIGNING A CONTOURED BEAM ANTENNA.

(19)

/.

,

_,.

\

_§,

~

DEPARTMENT OF ELECTRICAL ENGINEERING

Reports:

lOS) Videc, M.F.

STRALINGSVERSCHIJNSELEN IN PLASMA'S EN BEWEGENDE MEDIA: Een

geometrisch-optische en een golfzonebenadering.

TH-Report 80-E-IOS. 1980. ISBN 90-6144-IOS-6

106) Hajdasifiski, A.K.

LINEAR MULTIVARIABLE SYSTEMS: Preliminary problems in mathematical

description, modelling and identification.

TH-Report 80-E-I06. 1980. ISBN 90-6144-106-4

107) Heuvel, W.M.C. van den

CURRENT CHOPPING IN SF6.

TH-Report 80-E-l07. 1980. ISBN 90-6144-107-2 108) Etten, W.C. van and T.M. Lammers

TRANSMISSION OF FM-MODULATED AUDIOSIGNALS IN THE 87.5 - 108 MHz BROADCAST BAND OVER A FIBER OPTIC SYSTEM.

TH-Report 80-E-l08. 1980. ISBN 90-6144-108-0 109) Krause, J.C.

SHORT-CURRENT LIMITERS: Literature survey 1973-1979. TH-Report 80-E-109. 1980. ISBN 90-6144-109-9

lio) Matacz, J.S.

UNTERSUCHUNGEN AN GYRATORFILTERSCHALTUNGEN. TH-Report 80-E-ll0. 1980. ISBN 90-6144-110-2 111) Otten, R.H.J.M.

STRUCTURED LAYOUT DESIGN.

TH-Report 80-E-ll1. 1980. ]SBN 90-6144-111-0 (in preparation) 112) Worm, S.C.J.

OPTIMIZATION OF SOME APERTURE ANTENNA PERFORMANCE INDICES WITH AND WITHOUT PATTERN CONSTRAINTS.

TH-Report 80-E-112. 1980. ISBN 90-6144-112-9 113) Theeuwen, J.F.M. en J.A.G. Jess

EEN INTERACTIEF FUNCTIONEEL ONTWERPSYSTEEM VOOR ELEKTRONISCHE SCHAKELINGEN.

TH-Report 80-E-113. 1980. ISBN 90-6144-113-7

114) Lammers, T.M. en J.L. Manders

~%

l·.\t~.

EEN DIGITAAL AUDIO-DISTRIBUTIESYSTEEM VOOR 31 STEREOKANALEN VIA GLASVEZEL.

TH-Report 80-E-114. 1980. ISBN 90-6144-114-5