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Potential Calculations in Hall Plates
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ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS VOL . 61
Potential Calculations in Hall Plates GILBERT DE MEY Laboratory of Electronics Ghent State Wniversiry Ghenr. Belgium
I . Introduction ........................................
11. Fundamental Equations for a Hall-Plate Medium
2 3 3 4 5 6
...........................
A . General Equations for a Semiconductor ............ B. Approximations for a Thin Semiconducting Layer ... C . Constitutive Relations with an Externally Applied Magnetic Field .......... D . Boundary Conditions with Externally Applied M q n e Ill . The Van Der Pauw Method ........................................... IV. Influence of the Geometry on Hall-Mobility Measurements ................... V . Conformal Mapping Techniques ..................... A . Basic Ideas ......................................................... B. Approximate Analysis of the Cross-Shaped Geometry .................... C . Exact Analysis of the Cross-Shaped Sample ............................. D . Properties of the Cross-Shaped Hall Plate . . .................... V1. Relaxation Methods .................................... VII . The Boundary-Element Method for Potential Calculations in Hall Plates ........ A . Introduction ...................... ............................... B. Integral Equation for the Potential Distribution in a Hall Plate . C . Numerical Solution of the Integral Equation .......................... D . Application to a Rectangular Hall Plate ..................... E . Zeroth-Order Approximation .............................. F. Numerical Calculation of the Current through a Contact . . . . . . . . . . . . . . . . . . G . Application to the Cross-Shaped Geometry .............................. H . Direct Calculation of the Geometry Correction .................. 1. Application to a Rectangular Hall Generator .................... J . Application to a Cross-Shaped Geometry ............... K . Application to Some Other Geometries ................. VIII . Improvement of the Bounda A . Introduction ................................................. B. Integral Equation ...... .................... C . Calculation of the Funct .......... D . Application to a Rectangular Hall Generator ................... E. Application to a Cross-Shaped Form ................................... F . Application to Some Other Geometries ................................. 1X. Conclusion ............................................................ Appendix 1. The Three-Dimensional Hall Effect .................... Appendix 2. On the Existence of Solutions of Int ations Appendix 3. Green's Theorem ............................................
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9 10 10 11 13 14 15 18 18 18 20 21 22 24 25 21 29 30 33 38 38 39
40 43 45
46 48 49 52 54
1
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Copyright P 1983 by Academic Press Inc. All rights oi reproduction in any form reserved . ISBN n-12-014661-4
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GILBERT DE MEY
Appendix 4. The Hall-Effect PhotovoltaicCell.. ............................ Appendix 5. Contribution of the Hall-PiateCurrent to the Magnetic Field ...... Appendub. Literature .................................................. References .............................................................
57 58 59 59
I. INTRODUCTION
Hall plates are thin semiconducting layers placed in a magnetic field. Owing to the Lorentz force, the current density J and the electric field E are no longer parallel vectors. This means that a current in a given direction automatically generates a potential gradient in the perpendicular direction. With suitable contacts (the so-called Hall contacts), a Hall voltage can then be measured. In a first approximation one can state that the Hall voltage is proportional to the applied magnetic field, the externally supplied current, and the mobility of the charge carriers. Knowledge of the current and of the Hall voltage yields the product pHBof the mobility and the magnetic field B. This indicates two major applications of Hallcffect components. If the mobility is known, magnetic field strengths can be measured. On the other hand, if the magnetic field is known, the mobility can be calculated. The latter is mainly used for the investigation of semiconductors because mobility is a fundamental material parameter. A first series of applications is based on the measurement or detection of magnetic fields. Measurements of the magnetic fields in particle accelerators have been carried out with Hall probes provided with a special geometry in order to ensure a linear characteristic (Haeusler and Lippmann, 1968). Accuracies better than 0.1% have been realized. For alternating magnetic fields, Hall plates can be used at frequencies up to f l0,OOO Hz (Bonfig and Karamalikis, 1972a,b). For higher frequencies emf measurements are recommended to measure magnetic fields. The Hall probe can also be used to detect the presence of a magnetic field. This phenomenon is used in some types of push buttons. On each button a small permanent magnet is provided, and the pushing is sensed by a Hall plate. At this writing Hall plates combined with additional electronic circuitry are available in integrated-circuit form. A Japanese company has produced a cassette recorder in which a Hall probe reads the magnetic tape. The principal advantage here is that dc signals can be read directly from a tape, whereas classical reading heads generate signals proportional to the magnetic flux rate d4/dt. Magnetic bubble memories have also been fitted out with Hall-effect readers (Thompson et al., 1975). A survey of Hallcffect applications can be found in an article written by Bulman (1966) that mentions microwave-power measurements, the use of Hall probes as gyrators, insulators, function generators, ampere meters, etc. Even a brushless dc motor has been constructed using the Hall effect (Kobus
3
POTENTIAL CALCULATIONS IN HALL PLATES
and Quichaud, 1970). Finally, Hall plates can also be applied as transducers for mechanical displacements (Davidson and Gourlay, 1966; Nalecz and Warsza, 1966). A second series of applications, to which this article is mainly devoted, involves the measurement of mobilities. A Hall measurement carried out in a known magnetic field yields the value of p H . This constant is an important parameter for investigating the quality of semiconducting materials. Combined with the resistivity, it also enables us to calculate the carrier concentration. Knowledge of these data is necessary for the construction of components such as diodes, solar cells, and transistors starting from a semiconducting slice. The present article describes the Hall effect and its mathematical representation. The well-known Van Der Pauw method for Hall-mobility measurements is then discussed. The influence of the geometry on the Hall voltage is pointed out using physical considerations. This explains why the potential distribution in a Hall plate should be known in order to evaluate the so-called geometry correction. Then several techniques for potential calculations in Hall plates, such as conformal mapping, finite differences, and the boundary-element method, are outlined and compared. 11. FUNDAMENTAL EQUATIONS FOR
A
HALL-PLATE MEDIUM
A . General Equations f o r a Semiconductor
The fundamental equations for an n-type semiconductor (assuming low injection) are the following: ( a n / a t ) - 4 - 1 V . J , = (an/at)gen - [(P - ~ o ) / ~ p ]
+ q-'
(a~/at)
V *J p = (ap/af)gen- [(P - ~
o ) / ~ p ]
(1)
(substitute [(n - no)/r,,] for a p-type layer); Jn
= nqpnE
V.E
=
+ qDn Vn,
-Vz+
Jp
= PqPpE
= (q/c,C)(p - n
- 4Dp V P
(2)
NA)
(3)
+ND -
where n is the electron concentration, p is the hole concentration, J , is the electron current density, J,, is the hole current density, J = J , J p is the total current density, ND is the donor concentration, N A is the acceptor concentration, E is the electric field, is the electric poential, p, = qD,/kT is the electron mobility, p p = qD,,/kT is the hole mobility, T,, is the electron relaxation time, T~ is the hole relaxation time, no is the equilibrium electron concentration (in the p layer), po is the equilibrium hole concentration
+
+
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GILBERT DE MEY
(in the n layer), L is the dielectric constant, and (a/dt),,, is the generation rate (e.g., due to incident light). Equations (1)-(3) are nonlinear for the unknowns n, p , and 4; however, for Hall generators several reasonable assumptions can be advanced so that the final problem becomes linear.
B. Approximations f o r a Thin Semiconducting Layer For thin-film semiconducting layers with contacts sufficiently distant from each other (order of magnitude in millimeters), one usually assumes that the layer is sufficiently doped to ensure that the contribution of the minority carriers becomes negligible. We shall work later with an n-type semiconductor; however, the same treatment can be carried out for a p-type semiconductor. One also assumes that no space charges are built up in the conductor. It can be shown that an occasional space charge only has an influence over a distance comparable to the Debye length. For an n-type layer, the Debye length is given by (Many et al., 1965) LD
=
[(totkT)/q2ND)]”Z
(4)
Normally, LD varies around 100-1000 A, so that a space charge can only be felt over a distance much smaller than the distance between the electrodes. Practically, space charges can only be realized at junctions or nonohmic contacts. Because Hall generators are provided with ohmic contacts, the space charge is zero everywhere. Hence the right-hand member of (3) should vanish; for an n-type material this gives rise to n = ND
and
p<< n
(5)
The Poisson equation (3) is then reduced to the simpler Laplace equation -V.E = V24= 0
(6)
From Eq. (5), it also follows that V n should be zero. This means that the current density J , only consists of the drift component qp,,nE. The hole current J , can be put at zero because both p and V p are negligible. One obtains for the current density J = J , = NDqp,E = CJE
(7)
A thin semiconducting layer can be seen simply as a sheet with a constant conductivity CJ. For sufficiently high doping concentrations ND, the minority-carrier concentration p can be put equal to its equilibrium value p o , hence p - po = 0. In the time-independent case, Eq. (1) reduces to
POTENTIAL CALCULATIONS IN HALL PLATES
5
if we suppose that the generation term (an/dt),,, vanishes, which will be the case if the layer is not illuminated or irradiated. Equations (6) and (8) constitute the fundamental equations for a semiconducting layer, J and E being related by Eq. (7). From these equations, the boundary conditions can easily be deduced. At a metallic contact the potential 4 should be equal to the applied voltage. At a free boundary the current density must be tangential:
J-U,= 0 (9) where u,, is the normal unit vector. Owing to Eq. (7), the boundary condition (9) is equivalent to (10) v(p*u, = 0 The potential problem in a thin semiconducting laver is reduced to the solution of the Laplace equation in a given geometry, 4 or Vq5.u, being known on each point along the boundary. This is a classical potential problem with mixed boundary conditions.
C. Constitutive Relations with an Externally Applied Magnetic Field The fundamental equations (1) and (2) are still valid in the presence of an externally applied magnetic field. Only the constitutive relations (2) have to be extended in the following way (Smith et al., 1967; Madelung, 1970): j
where the index i denotes the ith component in a rectangular coordinate system. A similar expression can be written for the hole current density Jp,i.The tensorial mobility pij in the presence of a magnetic field B is found by the well-known Jones-Zener expansion, and it turns out that pij has to be replaced by
Applying Eqs. (1 1) and (12) on a flat n-type semiconductor layer, one obtains (De Mey, 1975)
J
=
oE
+ qDVn - opH(E X B) - qC(HD(Vn X B)
(13)
Using the approximation (9,as has also been done in the foregoing section, the diffusion components in Eq. (13) can be dropped, which yields
J = CE - opH(E x B)
(14)
This relation is used later for potential calculations in Hall plates. It has also been assumed that the semiconductor is isotropic, which explains why only one pH coefficient remains in Eq. (14).
6
GILBERT DE MEY
Equation (14) is derived from the Jones-Zener expansion (12) neglecting terms of order pAB2 that describe the physical magnetoresistivity. Hence Eq. (14) can be inverted to E
=
pJ
+ ppH(J x B)
(15)
where p = l/o denotes the resistivity. Equation (15) is also correct to terms of order pHB. The fundamental equations are still V J = 0 and V x E = 0, but for a flat semiconducting Hall plate in a uniform magnetic field B, one can show the following: V * J = C T V . E - I T ~ H V . ( EBX) = a V . E - a p H ( V x E ) . B = o V * E = O V x J = O V x E - opc,V x (E x B) = -opH(B.V)E
-
+ opH(V.E)B = 0
in which (B V)E vanishes since B is directed perpendicular to the Hall plate, whereas E is parallel to it. To solve the problem one can use either V x E = V - E = 0 or V x J = V - J = 0; E can be derived from a potential 9. For reasons which are explained in Section II,D, the current density J can also be derived from a potential function i,b in the following way: J = U, x Vi,b
(16)
where u, is the unity vector directed perpendicular to the Hall plate. From V x J = 0 it can be easily proved that i,b also satisfies the Laplace equation.
D. Boundary Conditions with an Externally Applied Magnetic Field Let us start with the electrostatic potential 4. At a metallic contact 4 is equal to the applied contact voltage. At a free boundary, the current density should be tangential: J - U , = oE'u, - ~/AHBE*u, =0
(17)
or v4.11, = ~ H B V $ . U ,
(18)
where u, = u, x u, is the unit tangential vector along the boundary (Fig. 1). It should be noted that at a free boundary in a Hall plate the electric field can show a nonvanishing normal component. For a p-type semiconductor, the same calculations can be carried out. A minus sign will then appear in the right-hand member of Eq. (18). For Hall plates the current but not the potential through a contact is usually given. It is then easier to introduce the current potential defined by Eq. (16). At a metallic contact E should be perpendicular, or the tangential
POTENTIAL CALCULATIONS IN HALL PLATES
7
FIG. 1. Hall-plate configuration for outlining the stream potential.
component E*u, must vanish. With Eq. (14) this gives rise to ~ E . u=, J . 4
+ pH(J x B).u,= 0
(19)
+ ~ H B V $ * U=, 0
(20)
or V$.U,
At a free boundary, J must be tangential, and due to Eq. (16) $ must be a constant (Fig. 1). At BB' the boundary value can be taken $ = 0. At the opposite side AA', the value $o can be found from the known current 1 injected through the con tact s :
I
=
- JAB' J . u , d l =
JAB
V$*u,dl = $(B') - $ ( A ' ) = -$o
(21)
A similar treatment can be performed if more than two contacts are involved. 111. THEVAN DERPAUW METHOD
Van Der Pauw (1958)has presented an ingenious method for carrying out resistivity and Hall-mobility measurements on thin layers with arbitrary shape. In this article we shall restrict ourselves to Hall-mobility measurements. It should be noted that Van Der Pauw's theory is only valid if the following four conditions are fulfilled: The layer must be perfectly flat (2) The four contacts must be point shaped and placed along the boundary (1)
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GILBERT DE MEY
(3) The layer must be homogeneous (4) The geometry must be that of a singly connected domain Only the second condition (point-shaped contacts) is difficult to meet. This article is therefore mainly devoted to the influence of finite contacts on Hall-mobility measurements. In order to determine the mobility pH,a current I is fed through two opposite contacts A and C (Fig. 2). The voltage across the two other contacts B and D is then measured with and without the magnetic field B. The difference gives us the so-called Hall voltage. If the magnetic field B = 0, the voltage drop V , between the contacts B and D turns out to be
When the magnetic field B is applied, one finds a voltage V2 between B and D given by V2 =
=
pl:J.dr
+ p p H I I ( Jx B).dr
(23)
Because the contacts are assumed to be point shaped, the boundary condition J-u, = 0 holds along the entire boundary. Since the basic equations and boundary conditions for J are unaltered by the magnetic field, one concludes that the current density field J remains unchanged. The J vector in Eq. (22) is thus the same as in Eq. (23). The Hall voltage is then found to be VH
=
V2
- V , = P,UH
f-B
J (J x
B).dr
D
with B = Bu,(u, directed perpendicular to the Hall plate):
FIG.2. Hall generator of arbitrary shape placed in a magnetic field B.
POTENTIAL CALCULATIONS IN HALL PLATES
9
where d represents the thickness of the layer. Equation (25) enables us to determine the mobility pH by measuring the Hall voltage provided that p, B, I, and d are known.
OF THE GEOMETRY ON 1V. INFLUENCE HALL-MOBILITY MEASUREMENTS
Equation (25) of Van Der Pauw is only valid if the contacts are point shaped. Actually, Hall generators always show finite contacts, and this will alter the Hall voltage in a still unknown way. It is shown further on in this article that the so-called geometry correction can be calculated provided that the potential problem in a Hall plate is solved. At this stage it is necessary to emphasize that there are two different kinds of size effects in semiconductor components. The first (and best known) effect is of a purely physical nature. Consider a semiconductor slice that is very thin, in which the mean free path of the charge carriers can become comparable to the thickness. One can easily understand that the mobility will then depend upon the size of the sample (Ghosh, 1961). But in our case the geometry (i.e., finite contacts) has no physical influence on the mobility but will affect the measured Hall voltage. This second kind of size effect is of a purely metrological nature. We shall now review Section 111 for the case of finite contacts on a Hall plate. Equations (22) and (23) are still valid; however, as J-u, = 0 no longer holds along the entire boundary, the current field J will change in the presence of a magnetic field. It is then necessary to replace J by J AJ in Eq. (23), where AJ is the change in the current field caused by the magnetic field B. The Hall voltage VH is then found to be
+
rB
VH
= V , - V, = p ]
AJ-dr - ppHB(l/d)
D
The absolute error ApH introduced by neglecting the influence of the finite contacts is then rs
ApH= (d/Bl)J AJ-dr D
The problem is now to develop a geometry for which the correction (27) is as small as possible in spite of the finiteness of the contacts. One has to find an integration path from D to B in an area where AJ zz 0. This can be done if contacts are placed at the ends of rectangular strips (Fig. 3). From field
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GILBERT DE MEY
FIG.3. Hall generator with contacts placed at the ends of strips.
calculations presented in the next sections, it is shown that disturbances due to the magnetic field (i.e., AJ # 0) only occur near the contacts. These areas are shaded in Fig. 3, and an integration path is drawn which only goes through the shaded areas at D and B. In Hall measurements, however, these contacts are used to measure potentials, and hence the total current through them is zero. Therefore J and, afortiori, AJ are very small in the neighborhoods of D and B. The conclusion is that the correction (27) will be small for the geometry of Fig. 3 where the contacts cannot be considered “point shaped.” Finite contacts with dimensions comparable to those of the Hall plate offer several advantages. The resistance between two contacts is low. Hence the noise induced in the measuring circuitry will be reduced. Small contacts require precise mask positioning and hence cause technological problems which can be avoided by using bigger contacts. The only disadvantage is the introduction of an error ApH; however, in the following section we prove that ApH can be calculated from the potential distribution in a Hall plate. It is then rather convenient to call ApH a geometry correction instead of an error.
V. CONFORMAL MAPPING TECHNIQUES A. Basic Ideas
Conformal mapping is a very useful method for solving the Laplace equation, provided that a geometry can be found for which the problem is solved by inspection. For a Hall-plate problem such a geometry can be found (Fig. 4). Indeed, for the parallelogram of Fig. 4, field lines are perpendicular to the contacts AB and A’B‘. The current lines are parallel to the free sides AA’ and BE’. The homogeneous E field obviously satisfies the Laplace equation (6). One can easily verify that the boundary condition (18) is
POTENTIAL CALCULATIONS IN HALL PLATES
11
A
FIG.4. Homogeneous field and current lines in a Hall plate having thz shape of a parallelogram.
exactly fulfilled if the angles at A, B, A', and B' have the values (n/2) - OH or ( 4 2 ) + as shown in Fig. 4. By mapping a given Hall-plate geometry on the parallelogram of Fig. 4, the potential problem can be solved for a given value of PHB = tan 8,. Wick (1954) was the first to use conformal mapping techniques for the study of the Hall effect. By using the Schwarz-Christoffel transformation formula, the mapping function could be determined. For quite regular Hall plates, however, calculations turned out to be very complicated, even for a simple rectangular shape with two contacts (Lippman and Kuhrt, 1958a,b). A considerable amount of work has been done by Haeusler (1966,1968,1971) and Haeusler and Lippmann (1968), who calculated the Hall voltage for a rectangular Hall plate provided with four finite contacts. The calculations are so complicated that this method cannot be considered a general rule, i.e., applicable to all shapes of Hall plates. We present a semianalytical version of the technique in which the Schwarz-Christoffel transformation formula is still used, but the evaluation of various integrals and constants is done numerically (De Mey, 1973a,b); we shall give an example of this later.
B. Approximate Analysis of the Cross-Shaped Geometry A very useful geometry for Hall-mobility measurements is the crossshaped sample (De Mey, 1973b). It combines broad contacts with a low geometry correction (27), which is now calculated. The cross-shaped form (Fig. 5b) can be mapped onto a circle by the following equations (Haeusler and Lippmann, 1968): y = tan2 26
h/l
=
{2[(k/h) - l]}-'
(28)
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GILBERT DE MEY
b c d e f g H I J K L M N 0 A B C D D , E F G
FIG.5. Conformal mapping of a cross-shaped geometry (b) onto a parallelogram (a).
where F is the hypergeometric function. In the case that c sz 0, i.e., the mapped contacts on the circle are small, one can introduce several approximations in Eqs. (28)-(30) so that one obtains finally (De Mey, 1973b)
2c = 2.58e-"hi'
(31)
In his famous article, Van Der Pauw (1958) has given the following formula for the geometry correction in the case of a Hall sample with circular geometry and one finite contact: ACIHICIH
=
2ch2
(32)
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POTENTIAL CALCULATIONS IN HALL PLATES
A
2010\
P
-
5-
\s-
2 2I=
*0.50
0.5
I.5
I
h/l
2
FIG.6. Geometry correction as a function of h / / for a cross-shaped sample.
Note that Eq. (32) is only valid if 6 is sufficiently small. For four contacts, one can simply multiply Eq. (32) by 4. For the cross-shaped sample, this yields Ap(H/p(H = 4(2€/a2) = 1.0453e-"h/'
(33)
This relationship is shown in Fig. 6. Note that for h = 1, the correction ApH/pH becomes 4.5%, whereas for h = 21 a negligible value of 0.19% is found. A correction of 5% means that experimental pH values have to be increased by 5% in order to obtain the exact value. C . Exact Analysis of the Cross-Shaped Sample
In order to calculate the potential distribution and hence the Hall voltage in a cross-shaped geometry, the Hall plate has to be mapped onto the parallelogram of Fig. 5a (De Mey, 1973b). For this simple geometry, the field distribution is homogeneous. The current density is found to be IJI = I/(IANIcos8(H)
(34)
I
where I is the current supplied through the contacts AN and HG. I AN denotes the length of the contact A N . For unit values of I and p, the Hall voltage is nothing other than the vertical distance between the Hall contacts CDE and JKL (Fig. 5a), or VH = IEI(IACI - ILNI)COS8H
X
(IACI - I L N I ) / ( A N )
(35)
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GILBERT DE MEY
According to Van Der Pauw's theory, V , should be equal to pHB= tan OH, and by comparing it with the value given by Eq. (35), the geometry correction &/pH is found. The conformal mapping of a circle into a semiinfinite plane is well known and is described by the relation x = tan(O/2)
(36)
This half-plane is mapped onto the cross-shaped geometry according t o the Schwarz-Christoffel transformation formula:
From Eq. (37), the values of h and I can be found:
I
=
h=
I-+; (u2
job
-dxx 2 ) 1 ' 2
((c2
dx [(b - x)(x - a)]'"
>"'
(b2 - x2)(f2 - x 2 ) - x2)(e2 - x2)(g2 - x2)
x)(b2 - x 2 ) ( f 2 - x') (x + u)(c2 - x2)(e2- x2)(g2- x 2 )
(
(b
-
These are Gaussian-type integrals and can be easily evaluated numerically (Abramowitz and Stegun, 1965). The conformal mapping of the half-plane onto the parallelogram is performed by dwldz = A ( z - u ) - ~ ~ / ~ (C )z- ( P ~ / ~ ( Z - d ' ) ( z - e)-v2/n ( z - g)-(Pl/n x (z
+
+ C ) - ~ ~ / ~+( Zd " ) ( z + e)-'+'lln(z + g)-'+'2/n
U)-+'~/~(Z
(39)
The unknown constants d' and d" in Eq. (39) are determined by saying that the point E should coincide with C and J with L. From Eq. (39) one can determine the distances I ACI - I L N I and I A N I needed to evaluate the Hall voltage V,. These calculations also lead to Gaussian-type integrals, but further details about the numerical procedure are omitted here. Figure 6 shows several numerically calculated values of ApH/pH. We observe good agreement with the approximate equation (31). These results also prove that conformal mapping can be adequately applied to rather complicated geometries if one starts immediately with numerical integrations once the Schwarz-Christoffel transformation formula has been written. D . Properties oJ the Cross-Shaped Hall Plute
The cross-shaped sample has contacts whose lengths are comparable to the dimensions of the semiconductor layer. This means that the resistance
POTENTIAL CALCULATIONS IN HALL PLATES
15
between two contacts is low, which offers many advantages when low Hall voltages are to be measured. Technological problems caused by small contacts no longer occur. Moreover, the geometry correction ApH/pH is moderate. Even for h = 1, less than 5% is found. This also means that the position of the contacts is not very critical. High-precision mask positioning before evaporation of the contacts is no longer required.
VI. RELAXATION METHODS The relaxation method is the only purely numerical method that has been used to calculate the potential distribution in a Hall-plate medium (Newsome, 1963; Grutzmann, 1966; Chwang et al., 1974; Mimizuka, 1971, 1978, 1979; Mimizuka and Ito, 1972). If one is only concerned with the potential distribution for given voltages at the contacts, the relaxation method gives good results; however, if the current through a contact has to be calculated, the method is far from accurate. Impedance calculations carried out by Mimizuka (1978) show relative errors of 10%. On the other hand, the relaxation method allows us to include nonlinear effects such as the temperature effects inside the Hall plate, a problem which cannot be treated with other techniques (Mimizuka and Ito, 1972; Mimizuka, 1979). We now use the relaxation method to calculate the geometry correction ApH/pH for a cross-shaped plate. Because this involves knowledge of both the Hall voltage and the supply current, we expect moderate accuracy. This is mainly owing to the tangential derivative V 4 u, in the boundary condition ( 1 8), which is difficult to represent numerically. In order to represent the Laplace equation numerically, the cross-shaped form has to be divided into a mesh (Fig. 7). The Laplace equation is then approximated by the well-known five-points formula : 44i.j - 4i+1 . j - A-
1.j
-
4i.j+1
-
4i,j- 1 = 0
(40)
At a contact, the known potential values have to be inserted into Eq. (40). The boundary condition (18) can be written as (Fig. 8) 1 4i,j - 4i- 1 . j = pHBT(4i,j+ 1 - $i,j-
1
-
(41)
At a free boundary, the Neumann boundary conditionV4 u, = 0 is written as (42) 4i.j = 6(4i,j+1 + 4i.j-1 + W i - 1 . j ) which corresponds to putting 4’ = 4i-l . j (Fig. 8). This condition (42) gives
good results in numerical calculations. Note that Eq. (41) cannot be reduced to Eq. (42) when p H B tends to zero. If pHB = 0, Eq. (41) reduces to 4i.j=
16
GILBERT DE MEY
0
0
0
0
0
0
0
0
0
0
0
FIG.7. Grid pattern for the finite-difference approximation in a cross-shaped Hall plate.
FIG. 8. Boundary conditions for the finitedifference approximation.
THEORETICALVALUE= L 51 90
: s : : : : : : : : : : : 10
M f=NI
15
+
FIG.9. Geometry correction as a function of M :p H B= 0.1 ;h/l = 1.
POTENTIAL CALCULATIONS IN HALL PLATES
17
h/I
N
FIG. 10. Geometry correction versus h/l calculated with finite difference approximation: and 13 ( A ) .
= 7 ( 0 ) .10 ( W),
4i- a boundary condition leading to a reduced accuracy and hence never used. For Hall plates, only Eq. (41) can be applied. This fact can explain the poor results. Figure 9 represents the numerically calculated geometry correction ApH/pH for h = 1 as a function of M (= N).Note that the results show errors of more than SO%, and the convergence is extremely poor when the mesh number M increases. Figure 10 represents the geometry correction A/.iH/pH as a function of h / l . The analytic approximation (33) has also been drawn. Only for low h / l values were acceptable results obtained. For higher h / l values the calculations are meaningless. We conclude that the relaxation method is not suitable for calculating the geometry correction ApH/pH of a Hall plate. Several reasons can be advanced. It turns out that contact current cannot be calculated with sufficient accuracy because it involves numerical differentiations. Also, the numerical representation (41) of the boundary condition (18) is not optimal from a numerical point of view. It should be noted, however, that the high relative errors found here are made on a geometry correction, i.e., the relative difference between the numerical and theoretical Hall voltage, two numbers that are normally close together. The accuracy on the geometry correction will hence be much lower than the accuracy obtained on the Hall voltage.
18
GILBERT DE MEY
VII. THEBOUNDARY-ELEMENT METHODFOR POTENTIAL CALCULATIONS IN HALLPLATES A . Introduction
Several years ago, the boundary-element method (BEM) was a rather unknown technique. Its popularity has since grown considerably, and the method now competes with the finite-element method (FEM). In recent years, some textbooks on the BEM have been published (Jaswon and Symm, 1978; Brebbia, 1978a,b). The BEM replaces the given Laplace equation and the boundary conditions by a single integral equation. This equation involves only the boundary of the given geometry. For a numerical solution the boundary has to be divided into a number of elements. This explains the term boundaryelement method. Because only the boundary is involved, the complexity of the problem is reduced. The method requires less storage allocation and computation time if it is solved numerically. The BEM can also be programmed on small-size computers (e.g., 60K memory) in contrast to the FEM, which requires a large memory. The integral equation technique was very well known to people working in the area of electromagnetic fields. Presumably, the reason is that many problems in electromagnetic theory can be directly formulated in terms of integral equations (Edwards and Van Bladel, 1961; Mei and Van Bladel, 1963a,b). Integral equations have also been used in other fields such as thermal diffusion problems (Shaw, 1974; De Mey, 1976a, 1977a), driftdiffusion problems (De Mey, 1976b, 1977b), elastic problems (Brown and Jaswon, 1971; Symm and Pitfield, 1974),calculation of eigenvalues (De Mey, 1976c, 1977c), semiconductor-component analysis using abrupt depletionlayer approximation (De Visschere and De Mey, 1977; De Mey et al., 1977), and potential distribution in Hall plates (De Mey, 1973c, 1974a, 1976d, 1977d). B. Integral Equation f o r the Potential Distribution in a Hall Plate
We consider the rectangular Hall generator shown in Fig. 11. The potential satisfies the Laplace equation (6) and the boundary conditions
4=
V,
at
A'B',
4 = 0 at A B
V4.u" = p H BV ~ . U , at AA' and BB'
(43)
In order to construct an integral equation, the Green's function G(r I r') of the Laplace equation is used: G(r1r') = (27c)-' lnlr - r'I
(44)
POTENTIAL CALCULATIONS IN HALL PLATES
19
FIG. I I . Rectangular Hall plate.
One can easily verify that G(r I r’) satisfies the Laplace equation provided that a delta function is put in the right-hand member: V:G(rlr’) = d(r - r’)
(45)
The solution of our problem 4(r) is now written as
where p(r) is called the source function and is only defined along the boundary C of the Hall-plate medium. Owing to Eq. (49, the proposed solution (46) automatically satisfies the Laplace equation for every point r inside C. In order to determine the unknown source function p(r), one has to impose the boundary conditions (43) on the proposed solution (46). This gives rise to
- r’(dC’=
&fcp(r’)lnlr
V,,
0,
rEA’B’ rEAB
(47)
where r E AA’ and r E BB’. Equations (47) and (48) constitute an integral equation in the unknown source function p. Once this integral equation has been solved, the potential 4 (and hence all quantities which can be deduced from 4) can be found at an arbitrary point r by evaluating the integral (46). The first term - p ( r ) / 2 ~occurring in Eq. (48) is a so-called fundamental discontinuity. It is caused by the discontinuity of the normal component V$*un at the boundary. The gradient V+ can be calculated at an arbitrary point r by V4(r) = (271.)-’ $C
p(r’)[(r - r’)/(lr - r’l’)]
dc‘
(49)
20
GILBERT DE MEY
C FIG. 12. Coordinate system (t, q ) to explain the fundamental discontinuities around ro.
If r lies on the boundary, the integrand of Eq. (49) becomes infinite because r can coincide with r'. For the normal component the integrand should be treated carefully by taking r close to the boundary at the interior side (Fig. 12). The boundary C can then be approximated by a straight line and p(r) can be considered a constant p. Introducing a (5, q ) Cartesian coordinate system, one has
5
2
By taking the limit 5 + 0 (i.e., r tends to the boundary C), the term -p(r)/2 remains. This problem does not occur for the tangential component (r - r')-u, because this integrand is an odd function of q. There are no fundamental discontinuities for tangential components.
C. Numerical Solution of the Integral Equations
In order to solve the integral equations (47) and (48) numerically, the boundary C has to be divided into n elements ACi. In each of them, the unknown source function p is replaced by an unknown constant p i . Denoting ri as the center point of the element ACi, the expression (46)for the potential can be rewritten as n
(#l(r)=
1 p.G(r(ri)IACjl
j= 1
J
(51)
where IACjI denotes the length of the element ACj. In a similar fashion, the integral equations (47) and (48) can be discretized. However, if r coincides with one of the ris, one obtains G(ri 1 ri) In 0 = co ! In order to avoid this infinity, the self-potential, i.e., the potential at ri due to the source pi itself on
-
POTENTIAL CALCULATIONS IN HALL PLATES
21
Aci, has to be calculated otherwise. Replacing pi by a uniform constant function along Aci, the divergence no longer occurs and the self-potential turns out to be
The integral equations are then replaced by PiIAciIClnOlAciI) 1 + C" -pjIn(ri
j=l27L
11 -
rj(IAcjl =
j#i
= 0,
V,,,
0,
riE A'B' ricAB
(53)
riE AA', BB'
Equations (53) and (54)constitute a linear algebraic set with n equations and n unknowns pi, which can be easily solved numerically by the Gauss pivotal elimination method, for example. The potential can then be calculated at an arbitrary point r by using Eq. (51). D. Application t o a Rectangular Hall Plate
The rectangular Hall plate shown in Fig. 11 has also been treated by conformal mapping techniques (Lippmann and Kuhrt, 1958a,b; De Mey, 1973a). The Hall voltage V , induced between the opposite point contacts P and Q has been calculated accurately. For a square-shaped Hall plate, one has V , = 0.522654 if V , = 1 and 0, = 45" or c(HB = 1. By comparing this value with the numerical results, the accuracy of the BEM can be checked. Figure 13 represents the relative error on the Hall voltage V , as a function of n, where n is the number of unknowns on each side (hence 4n unknowns have to be determined for this problem). One observes that good accuracies are obtained. Precision of better than 1% can be attained with a moderate number of unknowns. The linear behavior of the results shown in Fig. 13 is remarkable. When solving a problem with the BEM the relative accuracy always varies according to a l/n law. In contrast to most other numerical methods, this l / n law does not change when other algorithms for representing the source function are used. It should also be noted that a theoretical explanation for this phenomenon has not yet been published.
22
GILBERT DE MEY
I
I
3
I
1
1
5
1
7
1
1
1
1
1
1
9 11 13 151719
*
n
FIG.13. Relative error on the Hall voltage versus n for a rectangular plate: p H B= I ; V, = 1.
E. Zeroth-Order Approximation
It is a general experience in numerical analysis that higher order approximations yield more accurate results for the same number of unknowns. For example, we can cite that the fact that Simpson's rule for the numerical evaluation of an integral will be more accurate than the trapezoidal rule. In the preceding section a rather crude approximation for the source function p was introduced. Instead of replacing p by a series of constants pi concentrated at the points ri, one can see p as a piecewise constant function. In each element Aci, p equals a constant pi.The expression (51) for the potential now has to be written as
The integral equations (53) and (54) now read
1" p j g1I A c J l n l r -i r ' I d c ' =
j= 1
V,,
0
riEA'B' riEAB
where ri E AA', BB'. This method has also been applied to the square-shaped Hall generator. Figure 14 represents the relative error of the numerically calculated Hall voltage VH. The results obtained in the preceding section
POTENTIAL CALCULATIONS IN HALL PLATES
23
43-
E 4
2-
3 \
c
1-
0.5-
3
-4
5
7
n
9 11 13151719
FIG. 14. Comparison between the direct method and a first-order approximation.
have been redrawn in Fig. 14. One observes a slightly lower error due to the zeroth-order approximation. It is also remarkable that the same l/n law appears again. Similar results have also been found with higher order approximations of the source function (Stevens and De Mey, 1978). These results are in full agreement with the statements made at the end of the foregoing section. A l/n law always occurs, and the relative errors d o not decrease remarkably when higher order approximations are used for the source function. The integrals appearing in Eqs. (55)-(57) can easily be evaluated by complex integrations. By drawing an ( x , y ) coordinate system with the x axis directed tangentially and the origin at ri (Fig. 15), the integral appearing
t’
r z x Cx*yCy
*
Zzxejy
X
FIG. 15. Complex plane used to evaluate some integrals.
24
sAc,
GILBERT DE MEY
in Eq. (55) can be written as In ) r - r‘ I dC’ = Re
s
+ IAcJ1/2
- lAcJl/2
In(z - z’)dz’
(58)
Indeed, the real part of ln(z - z’) is nothing other than lnlr - r’I and dc’ = dz’ along the x axis (Fig. 15). The ( x , y ) plane is considered to be a complex plane, and the points z and z‘ correspond to r and r’. The complex integral appearing in Eq. (58) is easily evaluated analytically:
s
+ lAcj112
-IAcjI/Z
In(z - z’)dz’ = [-(z - z’)ln(z - z’) + (z - z ’ ) ] ~ : ~ ! l ~ ~ l \ ~
If one uses a computer with complex-number facilities, all integrals appearing in Eqs. (55)-(57) can be easily calculated as the real part of an analytic function. F. Numerical Calculation of the Current through a Contact
Using the constitutive relation (14) between the current density J and the electric field E, the total current through a contact AB (Fig. 16) is given by I = C J - u n d C= 0
E.u,dC
(59)
JAB
because the tangential component of E is zero at a metallic contact. From Eq. (46) the electric field in an arbitrary point r is found to be E(r) = -V$
=
-
r
p(r’) VG(r1r’)dC’ 9 C
The current I is then given by I = -Os~dCun.~~p(r’)VG(rIr.)dC.
(61)
If one intends to calculate Eq. (61) numerically, the electric field has to be found at a number of points on the contact AB, and then the summation along AB should be carried out. This procedure will require a lot of computation time, and the accuracy will be moderate because two numerical
FIG. 16. Use of the angle a to calculate current through AB
POTENTIAL CALCULATIONS IN HALL PLATES
25
approximations will be carried out. It is more convenient to interchange the sequence of the integrations in Eq. (61), which gives rise to I = -a~~p(r’)dC~~~’dCu..VG(rlr.)
(62)
The last integral appearing in Eq. (62) is nothing other than the flux of VG through AB. As G(r I r’) depends only upon the distance I r - r‘ 1, the field VG will show a radial symmetry around r’. Hence the flux of VG through AB is given by the angle a normalized to 2n (Fig. 16). The integral (62) reduces to I = -(a/2n)
h
dC’p(r’)a(r’,A, B )
where a(r‘,A, B) denotes the algebraic value of the angle under which AB is seen from the point r’. For the numerical calculations the evaluation of an angle poses no particular problems, and a single summation is sufficient. If r’ coincides with a point on AB, then a = n and the flux is then 4.This is another view of the fundamental discontinuities discussed in Section VII,B. The numerical calculation of contact currents is very important for the study of Hall generators because these devices are normally operated under a constant current through two opposite contacts generating a Hall voltage at the other contacts. G. Applicution to the Cross-Shaped Geometry
The BEM is now applied to the calculation of the geometry correction of the cross-shaped sample. The boundary conditions are shown in Fig. 17.
v0.+IH
q
8V@.
V@
.rn=,+,so@ ..;
--
qj=h/2
IH
IH
--
@ =-vH/2
v@..‘,=p,,[email protected]
.
~#.u;l=~,[email protected];
26
GILBERT DE MEY
FIG. 18. Hall voltage ( VH), supply current (I), and geometry correction (ApH/pH)as a function o f m : h / / = I ; p H B= 0.1.
Normally, the source current I is given and the Hall current is zero. It is difficult to impose the total current through a contact as a boundary condition because this often leads to an ill-conditioned algebraic set. Therefore the potentials at the contacts are given. The current-supplying contacts have potentials 4 = + 1 and - 1. The supply current I is then found from the potential distribution, as outlined in the preceeding section. The calculations are carried out twice, once with V , = 0 and once with V , = pHB.Each time the Hall current I , is also calculated. As V , varies linearly with I,, the correct Hall voltage is then found by putting I, = 0. Figure 18 shows some numerical results obtained for a geometry with h/l = 1. The Hall voltage V,, the supply current I, and the geometry correction ApH/pH have been drawn as a function of m where m is the total number of unknowns used during the numerical procedure. The geometry correction is far from accurate, as can be seen by comparing the numerical value of 12% with the exact value 4.5%. From this consideration one may not conclude that the BEM method is useless. One calculates the Hall voltage numerically and compares it with the theoretical value (25) according to Van Der Pauw's theory. In order to obtain the geometry correction these two values, which are normally close to each other, have to be substracted. This will increase the relative error so that the numerically calculated geometry correction cannot be used. Nevertheless, the potential and field calculations in the Hall plate are still accurate. If one wants to calculate ApH/pH with a 10% accuracy (for example), the Hall voltage and the supply current have to be known with much higher precision. In Section VII,H, a method is presented for finding
POTENTIAL CALCULATIONS IN HALL PLATES
27
t
- 0.86
-
5
21
60 m
120
FIG. 19. Difference between calculated and exact geometry correction versus m. Note that double logarithmic scale is used.
ApH/pH directly, without the intermediate calculation of the Hall voltage
VH or the supply current I .
The same calculations have also been carried out with other h/l values and other numerical approximations for the source function. The results and hence the conclusions remain identical. If the difference between the calculated and exact geometry correction is plotted on a double logarithmic scale as a function of m, one obtains linear behavior with a slope -0.86 (Fig. 19). This agrees with the results shown in Figs. 13 and 14. It also proves that the numerical values converge, although very slowly, to the exact value. H . Direct Calculation o j the Geometry Correction
The geometry correction can be calculated directly, and, as is proved later, this can be easily performed if one introduces the stream potential II/ defined by Eq. (16). The technique is outlined first for a Hall plate with two point-shaped Hall contacts, and a formula is then derived for an arbitrary geometry. Let us consider the rectangular Hall plate with two point-shaped Hall contacts P and Q (Fig. 20). A current 1 is supplied. Figure 20 indicates the boundary conditions if the stream potential t,b is to be used. From Eqs. (15) and (16) one finds the electric field
E = PP,BVt,b
+ P(U,
x V*)
(64)
28
GILBERT DE MEY
FIG.20. Boundary conditions for the stream potential on a rectangular plate.
where uz is the normal unit vector perpendicular to the Hall plate. In order to obtain the Hall voltage VH, the electric field (64) has to be integrated between the Hall contacts P and Q. One obtains VH
= j:E-dr
= pp.sj;V+-dr
= PPHB[$(P) -
(Q)]- P
IQ P
= pPHB(l/d) - P
J;
v$ ’(u:
+ psypC, x v+).dr v$.(U;
X
dr)
dr)
(65)
By Eq. (65),the Hall voltage VHcan be found from the stream potential $. It is remarkable that the first term in the right-hand member of Eq. (65)is nothing other than the theoretical Hall voltage (25) given by Van Der Pauw’s theory. Hence, the remaining term in Eq. (65)is the difference between the theoretical and the actual Hall voltage, or the so-called geometry correction. We remark
FIG.21. Stream potential used on an arbitrary shape.
POTENTIAL CALCULATIONS IN HALL PLATES
29
that the last integral in Eq. (65) is the flux of V+ through a line connecting P and Q. It has also been assumed that the flux V + * (u, x dr) is zero if PHB = 0, which is certainly the case for the symmetrical configuration of Fig. 20. The conclusion is that the geometry correction can be calculated directly as the flux of the V+ field. This can be calculated numerically by the method outlined in Section VII,F. We now consider an arbitrary Hall plate provided with four finite contacts (Fig. 21). The electric field is still given by Eq. (64). On the Hall contacts two points P and Q have been chosen. The integration of the electric field gives us the Hall voltage:
where A$(P) and A+(Q) denote the differences between the II/ values at P and Q, respectively, and the values at the ends of the contacts. For pointshaped contacts these differences are evidently zero and Eq. (66) reduces to Eq. (65).The first term in the right-hand member of Eq. (66) is the theoretical Hall voltage according to Van Der Pauw. The geometry correction is now given by a term containing the stream-potential differences A+(P) and A+(Q) and a second term involving the flux of V+ through PQ. If necessary, the ohmic potential drop occurring when pHB= 0 has to be subtracted from Eq. (66); however, most Hall-plate configurations show sufficient symmetry for this operation to be dropped.
I . Application to a Rectangular Hall Generator
The method outlined in Section VII,H is applied to the rectangular plate shown in Fig. 20. Because the Hall voltage has also been calculated with conformal mapping methods, the accuracy of the numerical values can be controlled. Figure 22 shows the relative error of the geometry correction for various values of PHB obtained by evaluating the last term of Eq. (65). In contrast to previous results, the accuracy is fairly high. It should be noted that a relative accuracy of 10% on the geometry correction means that the Hall voltage is known with a precision much better than 10%. We remark again that the
30
GILBERT DE MEY
20
-t
FIG.22. Relative error on the geometry correction as a function of IM for a square-shaped Hall plate: p H B= 0.1 ( 0 ) , 0 . 3 ( + ) , 0 . 5 ( A ) , a n d I ( 0 ) . IMisthenumberofunknownsperside.
error decreases in inverse proportion to the number of unknowns used in the numerical procedure. We may conclude that the accuracies are sufficient for practical purposes. J . Application to a Cross-Shaped Geometry Equation (66) is now used to analyze the cross-shaped geometry. Two points P and Q have been placed at the centers of the Hall contacts (Fig. 23).
FIG.23. Cross-shaped Hall plate analyzed with the stream potential.
POTENTIAL CALCULATIONS IN HALL PLATES
31
Figure 24 shows the relative error on the numerically calculated geometry correction for h/l = 0.5. For practical purposes these accuracies are more than satisfactory. For high values of pHB(>0.5), the error curve seems to diverge. This does not mean that the method should fail. The reason is that the relative error was calculated taking Eq. (33) as the exact value. However, there is a slight nonlinearity between the Hall voltage and the applied magnetic field B, a secondary effect not included in Eq. (33). Figure 25 shows the geometry correction as a function of h/l calculated for PHB = 0.1. The theoretical relationship (33) has also been drawn. For h/l < 1.5, numerical results are close to the theoretical ones, especially for high values of ZM,which is the number of unknowns for one side (the total number of unknowns is then 12ZM). For h/l 2 1.5, the deviations become remarkable. Nevertheless, the numerical results still converge to the exact values. In order to obtain insight into the practical consequences, let us take h/l = 2. The exact geometry correction is 0.2%, whereas the numerical value obtained with ZM = 9 turns out to be 0.5%. It makes no difference in practice if an experimental value is corrected by 0.2% or OS%, because this correction is much smaller than the accuracy of the Hall-mobility measurements. Only I 20
-
1
+
-ae 10
$
-
20 10
5-
5-
2-
2-
9' 0 z Q
0
: Q b Q
1-
I-
0.5-
0.2-
0.5
+
-
0.2
-
FIG.24. Relative error on the geometry correction versus IM for a cross-shaped Hall plate:
(a) h/l = 0.5; p H B= 0.3 (+).
32
GILBERT DE MEY
!i
+8 z E
9
1-
0.5-
az
\
1
0.5
1
I
1
I h/l
I
1.5
I
I
2
*
FIG.25. Geometry correction as a function of h /lfor a cross-shaped Hall plate: p H B= 0.1 ; I M = 3 (A),5 (OL7 (+), 9 ( 0 ) .
=-
when the geometry correction is greater (e.g., 1%) can it be calculated with sufficient accuracy. The conclusion is that the method can be applied in actual cases. Only when the geometry correction is negligible can it not be calculated with high precision; but in these cases knowledge of it is of no importance. It should be noted that the cross-shaped geometry has four reentrant corners, which makes its use difficult for numerical treatments. Some equipotential lines have been drawn in Fig. 26. Owing to Eq. (16) these lines are also the current lines or streamlines. Only 50% of the Hall plate is shown because the remaining part can be found by symmetry. From Fig. 26 the hypothesis formulated in Section IV can be verified.
POTENTIAL CALCULATIONS I N HALL PLATES
33
FIG.26. Streamlines in a cross-shaped Hall plate.
K . Application to Some Other Geometries
We first consider a rectangular Hall generator provided with four finite contacts (Fig. 27). This type of Hall plate is often used experimentally because monocrystalline semiconductors are usually available in the form of a rectangular bar. It is not always possible to cut a monocrystalline semiconductor slice in the form of a cross; a rectangular Hall plate is then used to carry out Hall-mobility measurements. This geometry has been extensively studied theoretically by Haeusler, using conformal mapping techniques (Haeusler and Lippmann, 1968). Nevertheless, this shape is far from optimal. Even for small contacts, the geometry correction turns out to be 30-40%. A
34
GILBERT DE MEY
h
FIG.27. Rectangular Hall plate provided with four finite contacts.
small shift in the position of the contacts causes a nonnegligible change of the geometry correction and hence an error in the experiments. Figure 28 shows the numerically calculated value of A,uH/PHfor h/b = 2 as a function of a/b. The geometry correction increases rapidly with the contact length a and can attain very high values. For this geometry Haeusler found the following approximate formula valid for sufficiently small contacts (Haeusler and Lippmann, 1968):
Equation (67) is accurate up to 4% for h/b > 1.5 and a/b < 0.18. The first condition is met, but the second one is not fulfilled because the numerical BEM cannot be used if a contact is much smaller than the other sides of the Hall plate. Hence we expect that Eq. (67) will not coincide perfectly with the numerical results. For h/b = 2 and 8, small, Eq. (67) reduces to APH/PH
= 1 - (1 - e-")[l
- (2/a)(a/b)] = 0.0432 + 0.609(a/b) (68)
The linear relationship (68)has also been drawn in Fig. 28, and the agreement is fairly good. One observes a better fit when the number of unknowns per side IM increases. A second particular geometry is the octagon provided with four equal contacts (Fig. 29). Although this shape has not been used in experiments, it is the first geometry that has been studied theoretically by Wick using conformal mapping techniques (Wick, 1954). The reason is that the conformal mapping of the octagon into a circle leads to calculations which can be
POTENTIAL CALCULATIONS IN HALL PLATES
35
< 0.L
0.5
0.6
0.7
0.8
O/b
FIG. 28. Geometry correction for the rectangular Hall plate: h/b = 2; pHB = 0.1 ; IM = 3 (+I, 5 (A), 7 ( x ), 9 (Oh I 1 13 (01, 15 ( 0 ) .
(m),
performed analytically. Figure 30 represents the numerically calculated geometry correction as a function of l / h for various values of the number of unknowns. One observes again that the geometry correction is high. Wick did not calculate any geometry correction, but from several graphs in his article it was possible to deduce approximately ApH/pH. These results are
t1
FIG.29. Octagonal Hall plate.
36
GILBERT DE MEY
+
I
I
I
0.2
0.3
i/h
0.4
I
0.5
*
FIG.30. Geometry correction for the octagonal Hall plate: p H B= 0.1 ; IM = 3 (+), 5 ( A ) , (m),1 3 ( 0 ) , 1 5 ( 0 ) .
7(x).9(0). I1
shown in Fig. 30 and referred to as “theoretical.” One sees good agreement between these theoretical points and the numerical results. Chwang ef al. (1974) analyzed the octagon using finite difference techniques. From the results published in their article, the geometry correction could be derived for small values of l/h. These results are not very accurate because they differ a lot from the results obtained by Wick (conformal mapping) and the BEM. It is also possible to obtain an approximate formula for ApH/pH for the octogon. Van Der Pauw published geometry corrections for a circular shape provided with finite contacts. An octagon can be approximated by a circle with a diameter h and a contact length 1. The geometry correction is then (Van Der Pauw, 1958) AA+/PH
=
4(2/nZHI/h)= (8/n)’(1/h)
(69)
This relation is also drawn in Fig. 30, and it turns out to be a good approximation for l / h % 0.5. For other l/h values the replacement of an octagon by a circle seems to be less adequate.
37
POTENTIAL CALCULATIONS IN HALL PLATES
FIG.31. Unsymmetrical cross-shaped form.
A final interesting geometry is the unsymmetrical cross-shaped form. Thin-film semiconducting Hall plates are usually made by evaporating the materials through a metallic mask. For the contacts on the Hall plate another mask is required, and thus a shift between the mask positionings can never be avoided. This gives rise to an unsymmetrical cross-shaped form. A fortiori, when the contacts are made manually with a conducting ink, for example, a perfect symmetry cannot be guaranteed. We now investigate a cross-shaped form where one arm differs from the others (Fig. 31). The numerical results are shown in Fig. 32, representing the geometry correction ApH/pH as a function of u/l. The geometry correction increases when the arm length shortens, which can easily be understood. These results can also be found theoretically. In Section V,B, we made the assumption that each arm gives its own contribution to the geometry correction. This being the case, the following formula was found for four equal arm lengths:
Applying Eq. (70) to three arms with length h and one arm with length u, one obtains
*
PH = 4L nze x p ( i l n 2 =
For h / l
=
1.04,3[:exp(
- :)[:exp( -n)>
-n:)
+ aexp( -
+ aexp( - n 3 1
31
1, Eq. (71) reduces to ApH/pH = 0.03387
+ 0.2613e-"""
(72)
This relation has also been shown in Fig. 32, and the agreement with the numerical results is very good. This also proves the hypothesis that each
38
GILBERT DE MEY
13
+ 12
A
11
10-
-
s
9
+
8
L
z I
x
d
7
6 5
L
I
0.4
I
0.5
I
0.6
1
0.7
I
0.8
I
0.9
I
I
C
0 4
FIG.32. Geometry correction for the unsymmetrical cross-shaped Hall generator: p H B = IM = 3 (+), 5 (A), 7 ( U ) . 9 (0).
0.1; h / l = I ;
arm generates its own contribution to the geometry correction ApH/pH independent of the other ones.
VIII. IMPROVEMENT OF THE BOUNDARY-ELEMENT METHOD A . Introduction
In this section we present a modified BEM. By investigating the method it was found that the highest errors occurred at the corners of the geometry. Therefore, a method is given here which uses analytic approximations at the corners while the remaining part of the potential is calculated numerically. This technique can be seen as a combination of conformal mapping and the
POTENTIAL CALCULATIONS IN HALL PLATES
39
BEM. The conformal mappings are used to calculate analytic approximations at the corners of the Hall plate. The remaining part of the potential distribution is found by numerical solution of an integral equation. The main advantage of this method is that singularities occurring at a corner can be represented exactly by the analytic approximation (De Mey, 1980).
B. Integral Equation We consider a Hall plate as having the shape of an arbitrary polygon (Fig. 33). Along each side zi the following boundary condition holds: a4
+ BV4.U" + yV4.q
= fo
(73)
Forametalliccontactatpotential Vo,onehasa = 1,p = y = O,andf, = Vo. For a free boundary a = 0, /?= 1, y = -pHB, and fo = 0. All types of boundary conditions along a Hall-plate medium can be represented by Eq. (73). It makes no difference in the subsequent analysis whether one is interested in the electrostatic potential or the stream potential II/. We assume now that in the neighborhood of each corner an analytic approximation for the potential is known. At the ith corner one has
4(r)
AiPi(r)
(74)
where cpi(r) is a known function satisfying the Laplace equation; Ai is a still unknown proportionality constant. The constants Ai will be determined together with the solution of the integral equation. The potential inside the Hall plate is now written as
4(r) =
Aicpi(r) i
+ f C p(r')G(rlr') dC'
(75)
The function cpi(r) includes all the singularities at the corners. Note that cpi(r) is not only defined at the ith corner, but in the entire region. This means that cpi(r) will not necessarely be zero at the other corners but that the functions cpi(r) will be chosen in such a way that cpi(r) is a smooth potential
\
z,-i
FIG. 33. Boundary condition at one side of an arbitrary polygon: aiQ yIvQ.u,=jO.
+ fii V+.u. +
40
GILBERT DE MEY
distribution at the other corners. The basic idea is that 1 Aiqi(r)is not a first-order approximation but a particular solution involving all singularities, especially those occurring in the gradient field. The remaining part will then be a perfectly smooth surface and is represented by the last term in Eq. (75). Applying the boundary condition (73) to the proposed solution (75) gives rise to C A i [ a q i + PVqi.un + ~ v q i . ~ ] i
- P+[p(r)l
+ (jcP(rWNG(rIr7
+ P V G ( r l r ’ ) - u , + yVG(rIr’)*u,]dC’ = fb(r)
(76)
Equation (76) constitutes an integral equation in the unknown function p. We remark that the constants Ai are also unknown and have to be determined simultaneously with the numerical calculation of p. C. Culculation of the Functions qi(r)
The purpose of this section is to determine the q i , potential functions being good approximations at corners, and to include all singularities: especially those occurring in the gradient fields Vq,.The method is outlined for the corner A of the Hall plate shown in Fig. 20. The corner can be replaced by two infinite sides. The function qi is then a potential satisfying the boundary conditions indicated in Fig. 20 along two infinite sides. This can be easily done, as it is always possible to map one corner onto another at which the potential is reduced to a constant and homogeneous field. This conformal
FIG.34. Approximation of a corner by two circular arcs.
POTENTIAL CALCULATIONS IN HALL PLATES
41
mapping is done by the simple analytic function 2'. Some problems may arise, however. Consider again Fig. 20 for the particular case p H B = 0. In each corner one side has a constant potential and the other side requires that the normal gradient should be zero. The field is then homogeneous because the corners are rectangular. The potential pi then varies linearly with distance. The same situation is valid for all corners, however, which means that the four functions cpl, cpz, cp3, and cp4 will be linearly dependent. In this case it is not possible to determine the constants A , , A z , A 3 , and A, numerically. Linearly dependent functions should always be avoided. In order to eliminate this problem, a corner will not be approximated by two infinite sides but rather by two circular arcs, as shown in Fig. 34, representing a corner with angle cp. One must now solve the Laplace equation V Z q i= 0 with the condition cpi = 0 at one circular arc and Vcpi-u,+ p H BVqi u, = 0 at the other. In this case it is almost impossible to obtain linearly dependent solutions at two corners. Because the circles are tangent at the sides of the corners, qi is still a good approximation for the potential. The solution of this potential problem is carried out with conformal mapping techniques. The geometry bounded by two circular arcs can be mapped onto a semiinfinite plane using (Fig. 39,
-
2' =
A"z/(z -
Zo)]n'e
(77)
where A' is a complex constant. Let us consider now the particular case where c2 is a free boundary and c1 a metallic contact held at constant potential. We
FIG.
35. Conformal mapping on a corner with angle ( 4 2 ) - OH
42
GILBERT DE MEY
then have to transform into a corner with an angle 4 2 - OH ( w plane in Fig. 3 9 , because for this geometry the potential can be written immediately. The transformation from the z to the w plane is then w = A ’ ( 1 / 2 ) - ( @ H / 2[) z / ( z - zO)](n/2o)-(@H/9) = A [ z / ( z - ~ ~ ) ] ( n / 2 1 ) - ( @ H / O ) (78) For a point z located on c1 (Fig. 35), one has
- zo) = ( z / ( z - zo)l e-jqlz
z/(z
(79)
then, w = A 1 z/(z - z o )1 ( ~ / W - ( @ H / V ) exp( - j i n
+ jl-0 2 H)
(80)
A point z on c1 is mapped onto a point w situated on the real u axis. Hence Eq. (80) has to be real, from which A can be determined:
A = exp(jin - j + O H )
(81)
Because the function cpi is multiplied by a constant Ai in the proposed solution (75), the absolute value of A has no influence on our problem. From the transformation formula (78) one obtains dw/dz = w [ ( $ n - &)/~][-zO/z(z - ZO)]
(82)
Equation (82) is used later to calculate the transformation of the electric fields. In the w plane, the potential cp corresponds to a homogeneous field directed perpendicular to the u axis (Fig. 35) because the angle was taken to be $n - 0,. Hence pi can be written as Vi(U,O)
=
(83)
IJ
The potential function can be derived from the complex potential W : W = -jw
[cpi
(84)
= Re( W ) ]
with d W / d w = -j
The potential in the z plane can be easily determined because the complex potential W is invariant. The electric field components in the z plane are determined by dW - - - - E , + j E. dz
y
acpi ax
= - - j - = -acpi -= ay
dwdw dw dz
-j-dw
dz
(86)
Inserting Eq. (82) into Eq. (86) directly gives us the field components in the original z plane. It has been proved that the potential cpi in a corner can be determined by a simple conformal mapping.
POTENTIAL CALCULATIONS IN HALL PLATES
43
It should be noted that cpi has only been determined up to a proportionality constant. Because the functions cpi are preceded by an unknown constant Ai in Eqs. (74) and (75), this causes no problems. For other situations at the corners, e.g., two adjacent free boundaries at a corner, the procedure to determine cpi remains identical; however, some formulas may be appropriately changed. D. Application to a Rectangular Hall Generator
Figure 36 shows a rectangular Hall plate with the circles for the approximation of the potential in the corners. First, the influence of the radius R was investigated. Figure 37 indicates the relative error of the Hall voltage and the current I as a function of the radius R for various values of I M , which is the number of unknowns per side. One observes that the error is almost independent of R. As for the current, it seems to be an optimum around R = 1; however, this is owing to the fact that the numerical value of the current minus the exact value changes its sign when R x 1, so one cannot consider it to be optimal. Figure 38 shows the relative error of the Hall voltage and the supply current as a function of IM. The numerical results obtained without the analytic approximation C Aicpi have also been drawn in Fig. 38. The accuracy of the Hall voltage remains practically unchanged, but the accuracy on the current calculation increases by almost two decades. Because both quantities are needed for practical applications, an overall improvement has been achieved. The geometry correction has also been investigated. Figure 39 shows the relative error on the geometry correction as a function of IM. The same
FIG.36. Square-shaped Hall plate with approximate arcs at the corners.
5-
-z
2-
6
1-
2
a5-
E c
2 a20.1-
M5-l
-., t
a1
R
FIG. 37. Relative error as a function of radius R : p H B = 1 ; AVH/VH ( 0 .
(0.0. A).
A); A I / l
4
3
5
9
1M
15
29
FIG.38. Relative error on the Hall voltage and the supply current: R = 1 ; pHB = I , with ( + ) and without ( 0 )analytic approximation.
POTENTIAL CALCULATIONS IN HALL PLATES
45
t
i
5
rb
IM
.?a
50
FIG.39. Relative error on the geometry correction as a function of IM:p,J = 0.I (+, x ); R = I ; with analytic approximation (+, A, 0 , 0); 0.3 (A, A); 0.5 ( 0 ,0 ) ; I (W, 0); without analytic approximation ( x , A , 0, m).
results without the analytic approximation are also drawn in Fig. 39. Generally, the introduction of analytic approximations yields better results, especially for low values of p H B ,when the Hall voltage is low and difficult to detect. E. Application to a Cross-Shaped Form
The cross-shaped form has also been analyzed using the same method. In this case, there are 12 corners and hence 12 qifunctions have to be evaluated. The geometry correction was calculated using the method outlined in Section VII,H. The results are shown in Fig. 40.The data obtained in Section VII,J are also drawn in Fig. 40. One now observes, contrary to the results found in the preceeding section for a square-shaped form, that the introduction of analytic approximations at the corners yields no improvement in the numerical results. Noting the fact that the calculation of the cpi functions involves computation time, one must conclude that the method of Section VII,H should now be preferred. These results indicate that the present method is only useful for a limited number of corners. The cross-shaped sample has also been analyzed using an analytic function at a reduced number ( c12) of corners, but this did not lead to any improvement.
46
GlLBERT DE MEY
A
1
a5
I
1
1
1
I
1.5
I
1)
2
h/ I
FIG.40. Geometry correction versus h / l for a cross-shaped sample; pHB = 0.1 ; with analytic approximation, IM = 3 (A), 9 ( 0 )without ; analytic approximation, IM = 3 (A),9 ).(
F . Application to Some Other Geometries
Because the number of corners seems to influence the numerical accuracy, two geometries with eight corners are now investigated. The rectangular and octagonal Hall plates treated in Section VII,K are now analyzed. It should be noted that the rectangular Hall plate provided with four finite contacts should be viewed as having eight corners. Because the electric field at both ends of each Hall contact shows a singular behavior, the introduction of analytic functions may be useful. Figure 41 represents the geometry correction as a function of a/b. Some results of Figure 28 have been redrawn. One sees that more accurate results are now found.
60
-
2
-A
M-
*5 z
Q
LO-
30
I
I
0.5
0.4
I
I
0.7
d.6
0.6
o/b
FIG.41. Geometry correction for a rectangular Hall plate: hlb = 2; pHB= 0.1 ; without analyticapproximation, IM = 3 ( A ) , 15 ( 0 ) withanalyticapproximation, ; IM = 3 ( A), 15 (0).
0.2
0.6
0.3
as
I/h
FIG. 42. Geometry correction for an octagonal Hall plate: pHB= 0.1, without analytic approximation, IM = 3 ( A ) , 15 ( 0 ) ;with analytic approximation, IM = 3 ( A ) , 15 ( 0 ) .
48
GILBERT DE MEY
Similar data for the octagonal shape are shown in Fig. 42. The same conclusion still holds because one observes that the numerical results with analytic approximations are closer to the theoretical curve. As a general conclusion of this section, one can state that the introduction of analytic approximations that include the singularities at the corners is only useful if the number of corners is limited to eight.
1X. CONCLUSION In this article two objectives concerning Hall-effect devices have been outlined. First, a review was given concerning several methods for calculating the potential distribution in a Hall plate. Second, an attempt was made to answer the question: What can be done with such a potential distribution? It turns out that the Hall voltage depends upon the geometry of the plate, an effect which can only be calculated if the potential problem is solved. Because many applications of Hall effects involve measurement of the Hall voltage, it is important to know all the influences on this voltage. When several methods for potential calculations were tested, not only did we draw streamlines or equipotential lines, but we emphasized the evaluation of the geometry correction to the Hall voltage. The first objective of this article was to check several methods for potential calculations in Hall plates. The potential satisfies the Laplace equation, but the boundary conditions are rather unusual, so that common techniques such as separation of variables or eigenfunction expansion cannot be applied. Three methods were found to give acceptable results: conformal mapping, finite difference approximation, and the BEM, the last two being purely numerical. Conformal mapping was the first method used to investigate the Hall effect. Some geometries can be analyzed with a completely analytical treatment; however, the calculations are very lengthly and complicated. Conformal mapping has also been applied successfully in a semianalytical approach : after the Schwarz-Christoffel transformation formula is written down further steps are performed numerically. In this way the high accuracy associated with conformal mapping techniques can be achieved with only moderate computational effort. The finite difference method is the most obvious numerical method, but complications arise owing to the special boundary conditions. This method can be used for a potential distribution, but if current and impedances are calculated the finite difference approach gives unacceptable results. In the BEM the special boundary condition presents no particular difficulties. Potential calculations can be done accurately. With some modifications, even small geometry corrections may be
POTENTIAL CALCULATIONS IN HALL PLATES
49
calculated. Finally, we can say that either theconformal mapping or the BEM can be used. Conformal mapping yields high accuracies, but for every new geometry, calculations have to be done again. The BEM can be applied to all geometries but its accuracy is lower. One of the two methods will be appropriate for each individual case. In the last section a combination of these two methods was presented. The potential was still calculated by the BEM, but at the corners analytical approximations obtained with conformal mapping were introduced. I t turns out that this combined method only constitutes an improvement for geometries with a limited number of corners. The second objective of this article was to show the influence of the geometry on the Hall voltage, i.e., the geometry correction. Van Der Pauw’s theory gives a formula for the Hall voltage, but owing to geometic effects (i.e.,finitecontacts) a lower value will be measured in practice. Thiscorrection has been calculated for several geometries, and the conclusion is that the cross-shaped form requires the smallest correction. This may be useful because some geometric parameters are not always known with sufficient accuracy. The cross-shaped form is also fitted with contacts comparable to the dimensions of the Hall plate, so that the resistance between two contacts and hence the noise in the measuring circuitry will be minimal. APPENDIX 1. THETHREE-DIMENSIONAL HALLEFFECT The Hall generators studied in this article are essentially two dimensional. Van Der Pauw’s theory, as outlined in Section 111, cannot be extended to a three-dimensional semiconducting volume having four contacts on its surface and placed in a magnetic field. Even when all contacts are point shaped, a general formula for the Hall voltage cannot be given. For every geometry one has to solve the potential problem from which the Hall voltage is found. Neglecting terms of order &B2, it turns out that the potential still satisfies the Laplace equation (De Mey, 1974b). At a metallic contact the potential is given, but on a free surface one has where u, is the unit vector perpendicular to the semiconductor surface and u, is the unit vector in the direction of the magnetic field. Note that u, and u, need to be perpendicular, as in the two-dimensional case. Because u, and u, can include all possible angles, the right-hand side of Eq. (87) turns out to be a complicated boundary condition. A perturbation method is therefore used to solve the problem. Treating PHB as a small quantity, the potential 4 can be
50
GILBERT DE MEY
written as a zeroth-order approximation 4oand a first-order perturbation
4
=
40
+ P(HB4I
The equations and boundary conditions for do and
v24, = 0 v40 'U, = 0 4o = applied potential v24, = 0 V 4 , 'u, = (V40 x uz)*u, 41
=o
:
(88)
4, are then (89) (90)
on a free surface on a metallic contact
(91) (92)
on a free surface
(93)
on a metallic contact
(94)
These equations are to be solved for a cylindrical semiconducting bar, as shown in Fig. 43. If a potential difference V is applied, the zeroth-order potential is easily found: 40
=
(95)
(V/a)x
The zeroth-order current I, through the contacts is given by I0
(96)
= (V/p)(nR2/a)
For the first-order perturbation, an eigenfunction expansion is used. In a circular section, the Dirichlet eigenfunctions are
where N,, is the normalization constant, Jn is the Bessel function of the first kind of order n and xnpis the pth root of the transcendental equation The first-order potential
41 is then written
where
,
the other coefficients being zero. The normalization constant N, is given by
3
FIG.43. Cylindrical Hall medium placed in a magnetic field.
as
0
1
1
1
1
1
5
FIG.44. Hall voltage for a cylindrical semiconductor.
.-
52
GILBERT DE MEY
From Eqs. (96), (99), and (lOO), the Hall voltage V , between the points P and Q can be found for a given current I , : VH = -4/&BpI,(nR)-'c ( x t p - l)-'[l - cosh(x,,/R)(a/2)]-'
(102)
P
Figure 44 shows the Hall voltage normalized to pHBpIo/R as a function of u/R. If u/R + co, i.e., the semiconducting bar becomes infinitely long, the normalized Hall voltage equals 0.6324. This means that the cylinder generates the same Hall voltage as a two-dimensional Hall plate with the same width 2R and a thickness R/0.6324.
APPENDIX 2. ON THE EXISTENCE OF SOLUTIONS OF INTEGRAL EQUATIONS If the Laplace equation for a given geometry with suitable boundary conditions is replaced by an equivalent integral equation, generally a Fredholm equation of the first kind will be obtained. On the other hand, the existence theorems for integral equations deal only with equations of the second kind (Hochstadt, 1973).For these equations it is possible to construct an iteration procedure, which can be used to prove the existence theorems. With regard to Fredholm integral equations of the first kind, the literature is quite contradictory. Some authors claim that the existence has been proved (Symm, 1963), whereas other authors allege the reverse (Tottenham, 1978). A particular geometry is now treated in order to investigate the existence of a solution. The conclusion cannot be extended to arbitrary geometries, but it will provide us with a deeper insight into the problem. Also, the contradictions occurring in the literature will be better understood. As integral equations for potential problems are mainly used as numerical approximations, the treatment in this appendix is also done from a numerical point of view. According to Fredholm theory (Courant and Hilbert, 1968) an integral equation can be regarded as the limit of a set of algebraic equations whose number of unknowns goes to infinity. We shall proceed now in a similar way. In order to investigate the integral equation for the circular geometry shown in Fig. 45, the boundary is divided into n equal parts Acj. The integral equation fcp(r')G(rlrc)dC'= V(r),
is then replaced by the finite algebraic set
reC
(103)
POTENTIAL CALCULATIONS IN HALL PLATES
53
I'
FIG.45. Circular geometry used to study the existence of a solution of the integral equation.
This set will have a unique solution if the determinant of the matrix [A,,] is not zero: (105) det[A,,] # 0 Similarly, the integral equation will have a unique solution if the Fredholm determinant does not equal zero. In our problem, the easiest way to check if Eq. (105) is fulfilled is to calculate all the eigenvalues of [A,,]. When all the eigenvalues are other than zero, the determinant will also be nonzero and vice versa. For the particular geometry of Fig. 45, one can easily verify that =
A(l-J)mOdn
(106)
if the boundary is divided into equal parts AC,. Equation (106) means that the matrix [A,] is circulant. It can be proved that the eigenvalues are then given by 11
i.,
=
'm
=
C
k= 1 n- I
1A l . k + l
k= 0
(107)
A,,
e - 2 n jmkln
,
m = l , 2 ,..., n - 1
(108)
If the matrix dimension increases, the matrix still remains circulant. In the limit for n + 00, the expressions (107)and (108)are then replaced by integrals.
54
GILBERT DE MEY
The eigenvalues of the integral equation (103) are then found to be r
i o= (2n)-'fClnlr - r,IdC
i,, = (2n)-
'
In I r - r, Ie-JmedC
where r, can be chosen arbitrarily on the boundary. FOPa circular geometry, the integrals (109) and (1 10) can be easily evaluated: j . , =
(R/n)
i,, = (R/n)
1; [i
ln(2R sin +6)d6 = R In R
(111)
ln(2R sin +6)cos m6 d6 = - R/2m
(1 12)
If R = 1, the eigenvalue I, turns out to be zero. The integral equation will then have no solution at all because the Fredholm determinant is zero. This fact has been investigated by Jaswon and Symm (1978) and Symm (1964). By using an appropriate scaling factor, this problem can be easily eliminated. It has been verified experimentally that eigenvalues of the matrix [ A i j ] (with dimension n) show behavior similar to the first n values l o ,R,, . . .,i n given by Eqs. (111) and (112). Especially, it was found that the smallest eigenvalue is inversely proportional to the matrix dimension n. This implies that I , , - , tends to zero when n + 00. One therefore concludes that the Fredholm determinant of the integral equation (103) will always be zero, meaning that Eq. (103) will have no solution at all. O n the other hand, as long as n remains finite, all the eigenvalues are other than zero and the algebraic set has a unique solution. It can thus be stated that the integral equation only has a solution if one considers it to be the limit solution of an algebraic set. If the solution of the set converges for n + co,one can define it as the solution of the integral equation. Many properties of Fredholm integral equations of the first kind have not been studied or explained because this matter turns out to be a difficult area of functional analysis. This explains why some numerical results, such as the l/n law for the relative error versus the number of unknowns, have not been declared yet. A theoretical background will indicate to us in which direction research should be continued in order to improve the BEM.
APPENDIX 3. GREEN'S THEOREM The method outlined in Section VIII,B is rather unusual for constructing an integral equation for the BEM. Normally, Green's theorem is used,
55
POTENTIAL CALCULATIONS IN HALL PLATES
leading to an integral equation where the unknown source function p has a physical meaning. If the potential is given on a part of the boundary, the unknown function turns out to be the normal component of the gradient and vice versa. However, this can only be done if the so-called natural boundary conditions are given, i.e., the boundary conditions only involve 4 or V 4 sun. The boundary condition (18) is not a natural one. An attempt has been made to integrate the tangential component along the boundary in order to obtain the potential 4 rather than V 4 mu,. The procedure is outlined later. Consider the Hall plate shown in Fig. 11. The potential satisfies
v24(r)= 0
(113)
and the Green's function satisfies V2G(r(r')= 6(r - r ' )
(1 14)
Multiplying Eq. (1 13) by G and Eq. (1 14) by 4 and substracting the resulting equations from each other, one obtains after applying Green's theorem fc [4(r)VG(rlr')*un- G(rlr')V4-un]dC = 4(r')
(1 15)
Equation (1 15) holds for each point r' inside C and can be transformed into an integral equation if r' is placed on the boundary. When 4 is given (e.g., on a metallic contact), then V 4 * u, is the unknown function. However, along the side BB', Eq. (18) is valid and V4.u" is not given, so 4 cannot be treated directly as the unknown function. Some intermediate steps are required:
The last integral in Eq. (1 16) can be transformed by partial integration: pHB
jBB'
G V$
*
U, dC
jBB'4
= pHB[G(r1r')4(r)]:r:~,- pHB
VG -u, dC
(1 17)
Equation (1 16) now reads jBB'[4VG-un- GV4.un]dC
IBB'
= pHBG(rB,lr')vo-I-
[4VG*U, - p~BvG'll,]dC
(118)
The tangential component V 4 -u, has been eliminated, and only 4 appears as an unknown function in the intergrand. Hence it is possible now to consider 4
56
GILBERT DE MEY
the unknown function along BB’. The same procedure can be carried out along AA’, so that Eq. (1 15) can be rewritten as n
n
- pHBG(rB lr’)Vo +
[ V o V G . u , - G V d * u , ] dC s.,A.
d[VG.U, - p H B V G ’ U , ] dC + JAA,
+ pHBG(rA,lr’)Vo,
CE
c
(119)
I f r’ E C , Eq. (1 19) is an integral equation that can be solved numerically; however, in Eq. (119) terms of the form VoG(rB,lr’) occur. Since G is a logarithmic function, these terms will be singular at the corner points A ’ and B’. Generally, it can be stated that singularities will occur at the end points of every contact staying at a nonzero potential. These difficulties d o not happen with the technique presented in Section VII,B, because the Green’s function then always appears under the integral sign. The integral equation (119) with r’E C has been solved numerically. In the term G(rA,lr’), r‘ was put equal to ri, which is the center point of the ith boundary element. Singularities were hence eliminated. Figure 46 shows the relative error on the Hall voltage. These results are compared with those
t
\
5
APPROXIMATION
I
1
3
1
1
5
I
I
9
7 n
1
I1
I
I
13 15
I
I
n 19
+
FIG.46. Relative error on the Hall voltage calculated by using Green’s theorem.
POTENTIAL CALCULATIONS IN HALL PLATES
57
obtained in Section V11. One observes that the error is very high, making the method unusable. For p H B = 0, accurate results were found because the singular terms in Eq. (1 19) vanish. These results explain why Green’s theorem is not used to construct an integral equation for the field problem in Hall plates. The methods outlined in Section VII should be preferred in this case. APPENDIX 4. THEHALL-EFFECT PHOTOVOLTAIC CELL The Hall effect can also be used to convert solar energy to electric power. The basic principle of all photovoltaic cells is the generation of electrons and holes in semiconductors owing to the absorbed light. If electrons and holes are accelerated in opposite directions a net current will be delivered to an attached load. In junction solar cells the separation of electrons and holes is done by the junction field. The same effect can be achieved with the Hall phenomenon because electrons and holes are deflected in different directions by the Lorentz force. A possible configuration of a Hall-effect photovoltaic cell is shown in Fig. 47. The incident light generates electron-hole pairs in the layer. Owing to the exponential decay of the light intensity, a concentration gradient for the charge carriers is built up. Hence electrons and holes diffuse in the y direction. Owing to the Lorentz force, charge carriers will be deviated along the x axis in such a way as to give a net current through the load resistor R,. A theoretical analysis has been published (De Mey, 1979).Assuming that one type of charge carrier has a much higher mobility than the other (which
gs
58
GILBERT DE MEY
is the case for InSb, InAs, etc.), the maximum attainable efficiency was found to be where p is the mobility, B the magnetic induction, E , the band gap of the semiconductor, L the diffusion length, and u the light absorption coefficient. For E , = 1 eV, and giving GILits optimum value, Eq. (120) reduces to = 0.00625(puB)2
(121)
For p B = 1, the efficiency turns out to be 0.6%, a low value compared with that of junction solar cells, which have efficiencies better than 10%. Note that p B = 1 is a rather high value because B = 1 Wb/m2 is difficult to attain, and there are only two semiconductors having p > 1 : InSb ( p = 7) and InAs ( p = 3). Because Eq. (120)was developed for monochromatic light, the efficiencywill be reduced at least by 0.44 in order to take the solar spectrum into account, and because both InSb and InAs have small band gaps, these conductors are not matched to the solar spectrum, which results in a much lower spectrum factor (< 0.44). The conclusion is that the Hall-effect solar cell is not suitable for energy production because of its low conversion efficiency.
APPENDIX 5. CONTRIBUTION OF THE HALL-PLATE CURRENT TO THE MAGNETIC FIELD A current must be supplied through two of the contacts of a Hall plate, which gives rise to a Hall voltage (in combination with an externally applied magnetic field). However, the current in the Hall place also generates its own magnetic field, which can also influence the Hall effect. The mathematical analysis of this secondary influence is rather complicated because the magnetic field caused by a current distribution in a plate turns out to be a three-dimensional problem. However, a crude approximate analysis indicates that the contribution of the currents in the Hall plate to the applied magnetic field is negligible. The situation changes if an alternating magnetic field is applied. Eddy currents are then generated in the Hall plate, creating an additional magnetic field and influencing the original Hall voltage. It is even possible to get a Hall voltage for a zero supply current, the eddy currents only being responsible for the Hall effect. If the frequency of the alternating magnetic field is high, the reaction of the eddy currents on the magnetic field can be important. A theoretical analysis carried out for a circular Hall plate in an ac magnetic field indicates that the parameter Q = 2n f o p o d / R should be considered (f,
POTENTIAL CALCULATIONS IN HALL PLATES
59
frequency; 0 , conductivity; po , permeability; d, thickness; and R, radius) (De Mey, 1976d). If R < 0.1, the contribution of the eddy currents to the magnetic field is negligible. Eddy currents are calculated directly from the applied field, and the Hall voltage is found by integrating the electric field between the Hall contacts. For high values of R, the calculation of the eddy-current pattern constitutes a complicated field problem. For practical purposes (e.g., magnetic field measurements in electrical machines), one is interested in working under the condition R < 0.1. The quantity d/R can then be replaced by the quotient of the thickness and a typical dimension of the Hall plate if the shape is not circular.
APPENDIX 6. LITERATURE Most books on the Hall effect describe the physical aspects of Hall mobility. Putley’s well-known book (1960) describes the galvanomagnetic properties of a large number of semiconductors. The book also provides many references. An excellent work has also been published by Wieder (1979), which describes both the physical nature and the measuring techniques related to the study of galvanomagnetic properties.
ACKNOWLEDGMENTS I wish to thank Professors M. Vanwormhoudt and H. Pauwels for their continuous interest in this work. I am also grateful to my collegues B. Jacobs, K.Stevens, and S. De Wolf, who collaborated on several topics treated in this article. I want also to thank Ms.H. Baele-Riems for careful typing of the manuscript and Mr. J. Bekaert for drawing the figures.
REFERENCES Abramowitz, M., and Stegun, 1. (1965). “Handbook of Mathematical Functions,” pp. 888-890. Dover, New York. Bonfig, K. W., and Karamalikis, A. (1972a). Grundlagen des Halleffektes. Teil 1. Arch. Tech. Mess. 2, 115-118. Bonfig, K. W., and Karamalikis, A. (1972b). Grundlagen des Hall-effektes. Teil 11. Arch. Tech. 3, 137-140. Brebbia, C. (1978a). “The Boundary Element Method for Engineers.” Pentech Press, London. Brebbia, C., ed. (1978b). “Recent Advances in Boundary Element Methods.” Pentech Press, London. Brown, I. C., and Jaswon, E. (1971). “The Clamped Elliptic Plate under a Concentrated Transverse Load,” Res. Memo. City University, London. Bulman, W. E. (1966). Applications of the Hall effect. Solid-State Electron. 9,361-372. Chwang, R., Smith, B., and Crowell, C. (1974). Contact size effects on the Van Der Pauw method for resistivity and Hall coefficient measurements. Solid-Stare Electron. 17, 12171227.
60
GILBERT D E MEY
Courant, R.,and Hilbert, D. (1968). “Methoden der Mathematischen Physik I,” pp. 121-124. Springer-Verlag, Berlin and New York. Davidson, R . S., and Gourlay, R. D. (1966). Applying the Hall effect to angular transducers. Solid-State Electron. 9,47 1-484. De Mey, G. (1973a). Field calculations in Hall samples. Solid-Slate Electron. 16,955-957. De Mey, G. (1973b). Influence of sample geometry on Hall mobility measurements. Arch. Elektron. Uebertragungsrech. 27, 309-3 13. De Mey, G .(1973~).Integral equation for the potential distribution in a Hall generator. Electron. Lelt. 9, 264-266. De Mey, G . (1974a). Determination of the electric field in a Hall generator under influence of an alternating magnetic field. Solid-State Electron. 17,977-979. De Mey, G. (1974b). An expansion method for calculation of low frequency Hall etTect and magnetoresistance. Radio Electron. Eng. 44,321-325. De Mey, G. (1975). Carrier concentration in a Hall generator under influence of a varying magnetic field. Phys. Status Solidi A 29, 175- 180. De Mey, G. (1976a). An integral equation approach to A.C. diffusion. In:. J. Heat Mass Transjer 19,702-704. De Mey, G . (1976b). An integral equation method for the numerical calculation of ion drift and diffusion in evaporated dielectrics. Computing 17, 169- 176. De Mey, G. (1976~).Calculation of eigenvalues of the Helmholtz equation by an integral equation. In:. J. Numer. Methods Eng. 10, 59-66. De Mey, G. (1976d). Eddy currents and Hall effect in a circular disc. Arch. Elektron. Uebertragungstech. 30, 312-315. De Mey, G . (1977a). A comment on an integral equation method for diffusion. In!. J. Heat Mass Transjer 20, I8 1 - 182. De Mey, G . (1977b). Numerical solution of a drift-diffusion problem with special boundary conditions by integral equations. Comput. Phys. Commun. 13,81-88. De Mey, G. (1977~).A simplified integral equation method for the calculation of the eigenvalues of Helmholtz equation. In:. J. Numer. Methods Eng. 11, 1340-1342. De Mey, G. (1977d). Hall effect in a nonhomogeneous magnetic field. Solid-State Electron. 20, 139-142. De Mey, G . (1979). Theoretical analysis of the Hall effect photovoltaic cell. IEE Trans. Solid Slate Electron Decices 3,69-71. De Mey, G . (1980). Improved boundary element method for solving the Laplace’ equation in two dimensions. Proc. In/. Semin. Recent Ado. Bound. Elem. Methods, I980 pp. 90- 100. De Mey, G., Jacobs, B., and Fransen, F. (1977). Influence of junction roughness on solar cell characteristics. Electron. Lert. 13, 657-658. De Visschere. P., and De Mey, G. (1977). Integral equation approach to the abrupt depletion approximation in semiconductor components. Eleclron. Lelt. 13, 104- 106. Edwards, T. W., and Van Bladel, J. (1961). Electrostatic dipole moment of a dielectric cube. Appl. Sci. Res. 9, 151-155. Ghosh. S. (1961). Variation of field effect mobility and Hall effect mobility with the thickness of deposited films of tellurium. J. Phys. Chem. Solids 19,61-65. Gray, R. M. (1971). “Toeplitz and Circulant Matrices: A Review,” Tech. Rep. No. 6502-1, pp. 16- 19. Information Systems Laboratory, Stanford University, Stanford, California. Grutzmann, S. (1966). The application of the relaxation method to the calculation of the potential distribution of Hall plates. Solid-State Electron. 9,401-416. Haeusler, J. (1966). Exakte Losungen von Potentialproblemen beim Halleffekt durch konforme Abbildung. Solid-Stare Eleclron. 9,417-441.
POTENTlAL CALCULATlONS 1N HALL PLATES
61
Haeusler, J. (1968). Zum Halleffekt Reaktanzkonverter mit vier Elektroden. Arch. Elektr. Uebertragung 22,258-259. Haeusler, J. ( I 971). Randpotentiale von Hallgeneratoren. Arch. Elekrrotech. (Berlin) 54, 77-81, Haeusler, J., and Lippmann, H. (1968). Hall-generatoren mit kleinem Linearisierungsfehler. Solid-Stare Electron. 11, 173- 182. Hochstadt, H. (1973). “Integral Equations,” Chapters 2 and 6. Wiley, New York. Jaswon, M. A., and Symm, G. T. (1978). “Integral Equation Methods in Potential Theory and Electrostatics.” Academic Press, New York. Kobus, A., and Quichaud, G. (1970). Etude d’un moteur a courant continu sans collecteur a commutation par un generateur a effet Hall en anneau. RGE, Rev. Gen. Electr. 79,235-242. Lippmann, H.,and Kuhrt, F.( l958a). Der Geometrieeinflussaufden transversalen magnetischen Widerstandseffekt bei rechteckformigen Halbleiterplatten. 2.Naturforsch. 13,462-474. Lippmann, H., and Kuhrt, F. (1958b). Der Geometrieeinfluss auf den Hall-effekt bei rechteckformigen Halbleiterplatten. Z. Naturjorsch. 13,474-483. Madelung, 0. (1970). “Grundlagen der Halbleiterphysik,” Chapters 37-39. Springer-Verlag, Berlin and New York. Many, A., Goldstein, Y., and Grover, N. B. (1965). “Semiconductor Surfaces,” p. 138. NorthHolland Publ., Amsterdam. Mei, K.,and Van Bladel, J. (1963a). Low frequency scattering by rectangular cylinders. IEEE Trans. Antennas Propagation AP-I 1,52-56. Mei, K., and Van Bladel, J. (1963b). Scattering by perfectly conducting rectangular cylinders. IEEE Trans. Antennas Propag. AP-11, 185-192. Mimizuka, T. (1971). Improvement of relaxation method for Hall plates. Solid-Stare Electron. 14, 107-110. Mimizuka, T. (1978). The accuracy of the relaxation solution for the potential problem of a Hall plate with finite Hall electrodes. Solid-State Electron. 21, 1195-1 197. Mimizuka, T. (1979). Temperature and potential distribution determination method for Hall plates considering the effect of temperature dependent conductivity and Hall coefficient. Solid-State Electron. 22, 157- 161. Mimizuka, T., and Ito, S. (1972). Determination of the temperature distribution of Hall plates by a relaxation method. Solid-State Electron. IS, 1197-1208. Nalecz, W., and Warsza, Z. L. (1966). Hall effect transducers for measurement of mechanical displacements. Solid-State Electron. 9,485-495. Newsome, J. P. (1963). Determination of the electrical characteristics of Hall plates. Proc. Inst. Electr. Eng. 110, 653-659. Putley, E. H. (1960). “The Hall Effect and Related Phenomena.” Butterworth, London. Shaw, R. (1974). An integral equation approach to diffusion. In:. J . Hear Mass Transjer 17, 693-699. Smith, A,, Janak, J., and Adler, R. (1967). “Electronic Conduction in Solids.” Chapters 7-9. McGraw-Hill, New York. Stevens, K., and De Mey, G . (1978). Higher order approximations for integral equations in potential theory. In:. J. Elecrron. 45,443-446. Symm, G. (1963). Integral equation methods in potential theory. Proc. R. Soc. London 275, 33-46. Symm, G. (1964). “Integral Equations Methods in Elasticity and Potential Theory,” Res. Rep. Natl. Phys. Lab., Mathematics Division, Teddington, U.K. Symm, G., and Pitfield, R. A. (1974). “Solution of Laplace Equation in Two Dimensions,” NPL Rep. NAC44. Natl. Phys. Lab., Teddington, U.K.
62
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Thompson, D. A., Romankiw, L. T., and Mayadas, A. F. (1975). Thin film magnetoresistors in memory, storage and related applications. IEEE Trans. Magn. MAG-11,1039-1050. Tottenham, H. (1978). "Finite Element Type Solutions of Boundary Integral Equations." Winter School on Integral Equation Methods, City University, London. Van Der Pauw, L. J. (1958). A method of measuring specific resistivity and Hall effect of discs of arbitrary shape. Philips Res. Rep. 13, 1-9. Wick, R. F. (1954). Solution of the field problem of the Germanium gyrator. J . Appl. Phys. 25,741-756.
Wieder, H. H. (1979). "Laboratory Notes on Electrical and Galvanomagnetic Measurements." Elsevier. Amsterdam.
ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS VOL . 61
Potential Calculations in Hall Plates GILBERT DE MEY Laboratory of Electronics Ghent State Wniversiry Ghenr. Belgium
I . Introduction ........................................
11. Fundamental Equations for a Hall-Plate Medium
2 3 3 4 5 6
...........................
A . General Equations for a Semiconductor ............ B. Approximations for a Thin Semiconducting Layer ... C . Constitutive Relations with an Externally Applied Magnetic Field .......... D . Boundary Conditions with Externally Applied M q n e Ill . The Van Der Pauw Method ........................................... IV. Influence of the Geometry on Hall-Mobility Measurements ................... V . Conformal Mapping Techniques ..................... A . Basic Ideas ......................................................... B. Approximate Analysis of the Cross-Shaped Geometry .................... C . Exact Analysis of the Cross-Shaped Sample ............................. D . Properties of the Cross-Shaped Hall Plate . . .................... V1. Relaxation Methods .................................... VII . The Boundary-Element Method for Potential Calculations in Hall Plates ........ A . Introduction ...................... ............................... B. Integral Equation for the Potential Distribution in a Hall Plate . C . Numerical Solution of the Integral Equation .......................... D . Application to a Rectangular Hall Plate ..................... E . Zeroth-Order Approximation .............................. F. Numerical Calculation of the Current through a Contact . . . . . . . . . . . . . . . . . . G . Application to the Cross-Shaped Geometry .............................. H . Direct Calculation of the Geometry Correction .................. 1. Application to a Rectangular Hall Generator .................... J . Application to a Cross-Shaped Geometry ............... K . Application to Some Other Geometries ................. VIII . Improvement of the Bounda A . Introduction ................................................. B. Integral Equation ...... .................... C . Calculation of the Funct .......... D . Application to a Rectangular Hall Generator ................... E. Application to a Cross-Shaped Form ................................... F . Application to Some Other Geometries ................................. 1X. Conclusion ............................................................ Appendix 1. The Three-Dimensional Hall Effect .................... Appendix 2. On the Existence of Solutions of Int ations Appendix 3. Green's Theorem ............................................
7
9 10 10 11 13 14 15 18 18 18 20 21 22 24 25 21 29 30 33 38 38 39
40 43 45
46 48 49 52 54
1
.
Copyright P 1983 by Academic Press Inc. All rights oi reproduction in any form reserved . ISBN n-12-014661-4
2
GILBERT DE MEY
Appendix 4. The Hall-Effect PhotovoltaicCell.. ............................ Appendix 5. Contribution of the Hall-PiateCurrent to the Magnetic Field ...... Appendub. Literature .................................................. References .............................................................
57 58 59 59
I. INTRODUCTION
Hall plates are thin semiconducting layers placed in a magnetic field. Owing to the Lorentz force, the current density J and the electric field E are no longer parallel vectors. This means that a current in a given direction automatically generates a potential gradient in the perpendicular direction. With suitable contacts (the so-called Hall contacts), a Hall voltage can then be measured. In a first approximation one can state that the Hall voltage is proportional to the applied magnetic field, the externally supplied current, and the mobility of the charge carriers. Knowledge of the current and of the Hall voltage yields the product pHBof the mobility and the magnetic field B. This indicates two major applications of Hallcffect components. If the mobility is known, magnetic field strengths can be measured. On the other hand, if the magnetic field is known, the mobility can be calculated. The latter is mainly used for the investigation of semiconductors because mobility is a fundamental material parameter. A first series of applications is based on the measurement or detection of magnetic fields. Measurements of the magnetic fields in particle accelerators have been carried out with Hall probes provided with a special geometry in order to ensure a linear characteristic (Haeusler and Lippmann, 1968). Accuracies better than 0.1% have been realized. For alternating magnetic fields, Hall plates can be used at frequencies up to f l0,OOO Hz (Bonfig and Karamalikis, 1972a,b). For higher frequencies emf measurements are recommended to measure magnetic fields. The Hall probe can also be used to detect the presence of a magnetic field. This phenomenon is used in some types of push buttons. On each button a small permanent magnet is provided, and the pushing is sensed by a Hall plate. At this writing Hall plates combined with additional electronic circuitry are available in integrated-circuit form. A Japanese company has produced a cassette recorder in which a Hall probe reads the magnetic tape. The principal advantage here is that dc signals can be read directly from a tape, whereas classical reading heads generate signals proportional to the magnetic flux rate d4/dt. Magnetic bubble memories have also been fitted out with Hall-effect readers (Thompson et al., 1975). A survey of Hallcffect applications can be found in an article written by Bulman (1966) that mentions microwave-power measurements, the use of Hall probes as gyrators, insulators, function generators, ampere meters, etc. Even a brushless dc motor has been constructed using the Hall effect (Kobus
3
POTENTIAL CALCULATIONS IN HALL PLATES
and Quichaud, 1970). Finally, Hall plates can also be applied as transducers for mechanical displacements (Davidson and Gourlay, 1966; Nalecz and Warsza, 1966). A second series of applications, to which this article is mainly devoted, involves the measurement of mobilities. A Hall measurement carried out in a known magnetic field yields the value of p H . This constant is an important parameter for investigating the quality of semiconducting materials. Combined with the resistivity, it also enables us to calculate the carrier concentration. Knowledge of these data is necessary for the construction of components such as diodes, solar cells, and transistors starting from a semiconducting slice. The present article describes the Hall effect and its mathematical representation. The well-known Van Der Pauw method for Hall-mobility measurements is then discussed. The influence of the geometry on the Hall voltage is pointed out using physical considerations. This explains why the potential distribution in a Hall plate should be known in order to evaluate the so-called geometry correction. Then several techniques for potential calculations in Hall plates, such as conformal mapping, finite differences, and the boundary-element method, are outlined and compared. 11. FUNDAMENTAL EQUATIONS FOR
A
HALL-PLATE MEDIUM
A . General Equations f o r a Semiconductor
The fundamental equations for an n-type semiconductor (assuming low injection) are the following: ( a n / a t ) - 4 - 1 V . J , = (an/at)gen - [(P - ~ o ) / ~ p ]
+ q-'
(a~/at)
V *J p = (ap/af)gen- [(P - ~
o ) / ~ p ]
(1)
(substitute [(n - no)/r,,] for a p-type layer); Jn
= nqpnE
V.E
=
+ qDn Vn,
-Vz+
Jp
= PqPpE
= (q/c,C)(p - n
- 4Dp V P
(2)
NA)
(3)
+ND -
where n is the electron concentration, p is the hole concentration, J , is the electron current density, J,, is the hole current density, J = J , J p is the total current density, ND is the donor concentration, N A is the acceptor concentration, E is the electric field, is the electric poential, p, = qD,/kT is the electron mobility, p p = qD,,/kT is the hole mobility, T,, is the electron relaxation time, T~ is the hole relaxation time, no is the equilibrium electron concentration (in the p layer), po is the equilibrium hole concentration
+
+
4
GILBERT DE MEY
(in the n layer), L is the dielectric constant, and (a/dt),,, is the generation rate (e.g., due to incident light). Equations (1)-(3) are nonlinear for the unknowns n, p , and 4; however, for Hall generators several reasonable assumptions can be advanced so that the final problem becomes linear.
B. Approximations f o r a Thin Semiconducting Layer For thin-film semiconducting layers with contacts sufficiently distant from each other (order of magnitude in millimeters), one usually assumes that the layer is sufficiently doped to ensure that the contribution of the minority carriers becomes negligible. We shall work later with an n-type semiconductor; however, the same treatment can be carried out for a p-type semiconductor. One also assumes that no space charges are built up in the conductor. It can be shown that an occasional space charge only has an influence over a distance comparable to the Debye length. For an n-type layer, the Debye length is given by (Many et al., 1965) LD
=
[(totkT)/q2ND)]”Z
(4)
Normally, LD varies around 100-1000 A, so that a space charge can only be felt over a distance much smaller than the distance between the electrodes. Practically, space charges can only be realized at junctions or nonohmic contacts. Because Hall generators are provided with ohmic contacts, the space charge is zero everywhere. Hence the right-hand member of (3) should vanish; for an n-type material this gives rise to n = ND
and
p<< n
(5)
The Poisson equation (3) is then reduced to the simpler Laplace equation -V.E = V24= 0
(6)
From Eq. (5), it also follows that V n should be zero. This means that the current density J , only consists of the drift component qp,,nE. The hole current J , can be put at zero because both p and V p are negligible. One obtains for the current density J = J , = NDqp,E = CJE
(7)
A thin semiconducting layer can be seen simply as a sheet with a constant conductivity CJ. For sufficiently high doping concentrations ND, the minority-carrier concentration p can be put equal to its equilibrium value p o , hence p - po = 0. In the time-independent case, Eq. (1) reduces to
POTENTIAL CALCULATIONS IN HALL PLATES
5
if we suppose that the generation term (an/dt),,, vanishes, which will be the case if the layer is not illuminated or irradiated. Equations (6) and (8) constitute the fundamental equations for a semiconducting layer, J and E being related by Eq. (7). From these equations, the boundary conditions can easily be deduced. At a metallic contact the potential 4 should be equal to the applied voltage. At a free boundary the current density must be tangential:
J-U,= 0 (9) where u,, is the normal unit vector. Owing to Eq. (7), the boundary condition (9) is equivalent to (10) v(p*u, = 0 The potential problem in a thin semiconducting laver is reduced to the solution of the Laplace equation in a given geometry, 4 or Vq5.u, being known on each point along the boundary. This is a classical potential problem with mixed boundary conditions.
C. Constitutive Relations with an Externally Applied Magnetic Field The fundamental equations (1) and (2) are still valid in the presence of an externally applied magnetic field. Only the constitutive relations (2) have to be extended in the following way (Smith et al., 1967; Madelung, 1970): j
where the index i denotes the ith component in a rectangular coordinate system. A similar expression can be written for the hole current density Jp,i.The tensorial mobility pij in the presence of a magnetic field B is found by the well-known Jones-Zener expansion, and it turns out that pij has to be replaced by
Applying Eqs. (1 1) and (12) on a flat n-type semiconductor layer, one obtains (De Mey, 1975)
J
=
oE
+ qDVn - opH(E X B) - qC(HD(Vn X B)
(13)
Using the approximation (9,as has also been done in the foregoing section, the diffusion components in Eq. (13) can be dropped, which yields
J = CE - opH(E x B)
(14)
This relation is used later for potential calculations in Hall plates. It has also been assumed that the semiconductor is isotropic, which explains why only one pH coefficient remains in Eq. (14).
6
GILBERT DE MEY
Equation (14) is derived from the Jones-Zener expansion (12) neglecting terms of order pAB2 that describe the physical magnetoresistivity. Hence Eq. (14) can be inverted to E
=
pJ
+ ppH(J x B)
(15)
where p = l/o denotes the resistivity. Equation (15) is also correct to terms of order pHB. The fundamental equations are still V J = 0 and V x E = 0, but for a flat semiconducting Hall plate in a uniform magnetic field B, one can show the following: V * J = C T V . E - I T ~ H V . ( EBX) = a V . E - a p H ( V x E ) . B = o V * E = O V x J = O V x E - opc,V x (E x B) = -opH(B.V)E
-
+ opH(V.E)B = 0
in which (B V)E vanishes since B is directed perpendicular to the Hall plate, whereas E is parallel to it. To solve the problem one can use either V x E = V - E = 0 or V x J = V - J = 0; E can be derived from a potential 9. For reasons which are explained in Section II,D, the current density J can also be derived from a potential function i,b in the following way: J = U, x Vi,b
(16)
where u, is the unity vector directed perpendicular to the Hall plate. From V x J = 0 it can be easily proved that i,b also satisfies the Laplace equation.
D. Boundary Conditions with an Externally Applied Magnetic Field Let us start with the electrostatic potential 4. At a metallic contact 4 is equal to the applied contact voltage. At a free boundary, the current density should be tangential: J - U , = oE'u, - ~/AHBE*u, =0
(17)
or v4.11, = ~ H B V $ . U ,
(18)
where u, = u, x u, is the unit tangential vector along the boundary (Fig. 1). It should be noted that at a free boundary in a Hall plate the electric field can show a nonvanishing normal component. For a p-type semiconductor, the same calculations can be carried out. A minus sign will then appear in the right-hand member of Eq. (18). For Hall plates the current but not the potential through a contact is usually given. It is then easier to introduce the current potential defined by Eq. (16). At a metallic contact E should be perpendicular, or the tangential
POTENTIAL CALCULATIONS IN HALL PLATES
7
FIG. 1. Hall-plate configuration for outlining the stream potential.
component E*u, must vanish. With Eq. (14) this gives rise to ~ E . u=, J . 4
+ pH(J x B).u,= 0
(19)
+ ~ H B V $ * U=, 0
(20)
or V$.U,
At a free boundary, J must be tangential, and due to Eq. (16) $ must be a constant (Fig. 1). At BB' the boundary value can be taken $ = 0. At the opposite side AA', the value $o can be found from the known current 1 injected through the con tact s :
I
=
- JAB' J . u , d l =
JAB
V$*u,dl = $(B') - $ ( A ' ) = -$o
(21)
A similar treatment can be performed if more than two contacts are involved. 111. THEVAN DERPAUW METHOD
Van Der Pauw (1958)has presented an ingenious method for carrying out resistivity and Hall-mobility measurements on thin layers with arbitrary shape. In this article we shall restrict ourselves to Hall-mobility measurements. It should be noted that Van Der Pauw's theory is only valid if the following four conditions are fulfilled: The layer must be perfectly flat (2) The four contacts must be point shaped and placed along the boundary (1)
8
GILBERT DE MEY
(3) The layer must be homogeneous (4) The geometry must be that of a singly connected domain Only the second condition (point-shaped contacts) is difficult to meet. This article is therefore mainly devoted to the influence of finite contacts on Hall-mobility measurements. In order to determine the mobility pH,a current I is fed through two opposite contacts A and C (Fig. 2). The voltage across the two other contacts B and D is then measured with and without the magnetic field B. The difference gives us the so-called Hall voltage. If the magnetic field B = 0, the voltage drop V , between the contacts B and D turns out to be
When the magnetic field B is applied, one finds a voltage V2 between B and D given by V2 =
=
pl:J.dr
+ p p H I I ( Jx B).dr
(23)
Because the contacts are assumed to be point shaped, the boundary condition J-u, = 0 holds along the entire boundary. Since the basic equations and boundary conditions for J are unaltered by the magnetic field, one concludes that the current density field J remains unchanged. The J vector in Eq. (22) is thus the same as in Eq. (23). The Hall voltage is then found to be VH
=
V2
- V , = P,UH
f-B
J (J x
B).dr
D
with B = Bu,(u, directed perpendicular to the Hall plate):
FIG.2. Hall generator of arbitrary shape placed in a magnetic field B.
POTENTIAL CALCULATIONS IN HALL PLATES
9
where d represents the thickness of the layer. Equation (25) enables us to determine the mobility pH by measuring the Hall voltage provided that p, B, I, and d are known.
OF THE GEOMETRY ON 1V. INFLUENCE HALL-MOBILITY MEASUREMENTS
Equation (25) of Van Der Pauw is only valid if the contacts are point shaped. Actually, Hall generators always show finite contacts, and this will alter the Hall voltage in a still unknown way. It is shown further on in this article that the so-called geometry correction can be calculated provided that the potential problem in a Hall plate is solved. At this stage it is necessary to emphasize that there are two different kinds of size effects in semiconductor components. The first (and best known) effect is of a purely physical nature. Consider a semiconductor slice that is very thin, in which the mean free path of the charge carriers can become comparable to the thickness. One can easily understand that the mobility will then depend upon the size of the sample (Ghosh, 1961). But in our case the geometry (i.e., finite contacts) has no physical influence on the mobility but will affect the measured Hall voltage. This second kind of size effect is of a purely metrological nature. We shall now review Section 111 for the case of finite contacts on a Hall plate. Equations (22) and (23) are still valid; however, as J-u, = 0 no longer holds along the entire boundary, the current field J will change in the presence of a magnetic field. It is then necessary to replace J by J AJ in Eq. (23), where AJ is the change in the current field caused by the magnetic field B. The Hall voltage VH is then found to be
+
rB
VH
= V , - V, = p ]
AJ-dr - ppHB(l/d)
D
The absolute error ApH introduced by neglecting the influence of the finite contacts is then rs
ApH= (d/Bl)J AJ-dr D
The problem is now to develop a geometry for which the correction (27) is as small as possible in spite of the finiteness of the contacts. One has to find an integration path from D to B in an area where AJ zz 0. This can be done if contacts are placed at the ends of rectangular strips (Fig. 3). From field
10
GILBERT DE MEY
FIG.3. Hall generator with contacts placed at the ends of strips.
calculations presented in the next sections, it is shown that disturbances due to the magnetic field (i.e., AJ # 0) only occur near the contacts. These areas are shaded in Fig. 3, and an integration path is drawn which only goes through the shaded areas at D and B. In Hall measurements, however, these contacts are used to measure potentials, and hence the total current through them is zero. Therefore J and, afortiori, AJ are very small in the neighborhoods of D and B. The conclusion is that the correction (27) will be small for the geometry of Fig. 3 where the contacts cannot be considered “point shaped.” Finite contacts with dimensions comparable to those of the Hall plate offer several advantages. The resistance between two contacts is low. Hence the noise induced in the measuring circuitry will be reduced. Small contacts require precise mask positioning and hence cause technological problems which can be avoided by using bigger contacts. The only disadvantage is the introduction of an error ApH; however, in the following section we prove that ApH can be calculated from the potential distribution in a Hall plate. It is then rather convenient to call ApH a geometry correction instead of an error.
V. CONFORMAL MAPPING TECHNIQUES A. Basic Ideas
Conformal mapping is a very useful method for solving the Laplace equation, provided that a geometry can be found for which the problem is solved by inspection. For a Hall-plate problem such a geometry can be found (Fig. 4). Indeed, for the parallelogram of Fig. 4, field lines are perpendicular to the contacts AB and A’B‘. The current lines are parallel to the free sides AA’ and BE’. The homogeneous E field obviously satisfies the Laplace equation (6). One can easily verify that the boundary condition (18) is
POTENTIAL CALCULATIONS IN HALL PLATES
11
A
FIG.4. Homogeneous field and current lines in a Hall plate having thz shape of a parallelogram.
exactly fulfilled if the angles at A, B, A', and B' have the values (n/2) - OH or ( 4 2 ) + as shown in Fig. 4. By mapping a given Hall-plate geometry on the parallelogram of Fig. 4, the potential problem can be solved for a given value of PHB = tan 8,. Wick (1954) was the first to use conformal mapping techniques for the study of the Hall effect. By using the Schwarz-Christoffel transformation formula, the mapping function could be determined. For quite regular Hall plates, however, calculations turned out to be very complicated, even for a simple rectangular shape with two contacts (Lippman and Kuhrt, 1958a,b). A considerable amount of work has been done by Haeusler (1966,1968,1971) and Haeusler and Lippmann (1968), who calculated the Hall voltage for a rectangular Hall plate provided with four finite contacts. The calculations are so complicated that this method cannot be considered a general rule, i.e., applicable to all shapes of Hall plates. We present a semianalytical version of the technique in which the Schwarz-Christoffel transformation formula is still used, but the evaluation of various integrals and constants is done numerically (De Mey, 1973a,b); we shall give an example of this later.
B. Approximate Analysis of the Cross-Shaped Geometry A very useful geometry for Hall-mobility measurements is the crossshaped sample (De Mey, 1973b). It combines broad contacts with a low geometry correction (27), which is now calculated. The cross-shaped form (Fig. 5b) can be mapped onto a circle by the following equations (Haeusler and Lippmann, 1968): y = tan2 26
h/l
=
{2[(k/h) - l]}-'
(28)
12
GILBERT DE MEY
b c d e f g H I J K L M N 0 A B C D D , E F G
FIG.5. Conformal mapping of a cross-shaped geometry (b) onto a parallelogram (a).
where F is the hypergeometric function. In the case that c sz 0, i.e., the mapped contacts on the circle are small, one can introduce several approximations in Eqs. (28)-(30) so that one obtains finally (De Mey, 1973b)
2c = 2.58e-"hi'
(31)
In his famous article, Van Der Pauw (1958) has given the following formula for the geometry correction in the case of a Hall sample with circular geometry and one finite contact: ACIHICIH
=
2ch2
(32)
13
POTENTIAL CALCULATIONS IN HALL PLATES
A
2010\
P
-
5-
\s-
2 2I=
*0.50
0.5
I.5
I
h/l
2
FIG.6. Geometry correction as a function of h / / for a cross-shaped sample.
Note that Eq. (32) is only valid if 6 is sufficiently small. For four contacts, one can simply multiply Eq. (32) by 4. For the cross-shaped sample, this yields Ap(H/p(H = 4(2€/a2) = 1.0453e-"h/'
(33)
This relationship is shown in Fig. 6. Note that for h = 1, the correction ApH/pH becomes 4.5%, whereas for h = 21 a negligible value of 0.19% is found. A correction of 5% means that experimental pH values have to be increased by 5% in order to obtain the exact value. C . Exact Analysis of the Cross-Shaped Sample
In order to calculate the potential distribution and hence the Hall voltage in a cross-shaped geometry, the Hall plate has to be mapped onto the parallelogram of Fig. 5a (De Mey, 1973b). For this simple geometry, the field distribution is homogeneous. The current density is found to be IJI = I/(IANIcos8(H)
(34)
I
where I is the current supplied through the contacts AN and HG. I AN denotes the length of the contact A N . For unit values of I and p, the Hall voltage is nothing other than the vertical distance between the Hall contacts CDE and JKL (Fig. 5a), or VH = IEI(IACI - ILNI)COS8H
X
(IACI - I L N I ) / ( A N )
(35)
14
GILBERT DE MEY
According to Van Der Pauw's theory, V , should be equal to pHB= tan OH, and by comparing it with the value given by Eq. (35), the geometry correction &/pH is found. The conformal mapping of a circle into a semiinfinite plane is well known and is described by the relation x = tan(O/2)
(36)
This half-plane is mapped onto the cross-shaped geometry according t o the Schwarz-Christoffel transformation formula:
From Eq. (37), the values of h and I can be found:
I
=
h=
I-+; (u2
job
-dxx 2 ) 1 ' 2
((c2
dx [(b - x)(x - a)]'"
>"'
(b2 - x2)(f2 - x 2 ) - x2)(e2 - x2)(g2 - x2)
x)(b2 - x 2 ) ( f 2 - x') (x + u)(c2 - x2)(e2- x2)(g2- x 2 )
(
(b
-
These are Gaussian-type integrals and can be easily evaluated numerically (Abramowitz and Stegun, 1965). The conformal mapping of the half-plane onto the parallelogram is performed by dwldz = A ( z - u ) - ~ ~ / ~ (C )z- ( P ~ / ~ ( Z - d ' ) ( z - e)-v2/n ( z - g)-(Pl/n x (z
+
+ C ) - ~ ~ / ~+( Zd " ) ( z + e)-'+'lln(z + g)-'+'2/n
U)-+'~/~(Z
(39)
The unknown constants d' and d" in Eq. (39) are determined by saying that the point E should coincide with C and J with L. From Eq. (39) one can determine the distances I ACI - I L N I and I A N I needed to evaluate the Hall voltage V,. These calculations also lead to Gaussian-type integrals, but further details about the numerical procedure are omitted here. Figure 6 shows several numerically calculated values of ApH/pH. We observe good agreement with the approximate equation (31). These results also prove that conformal mapping can be adequately applied to rather complicated geometries if one starts immediately with numerical integrations once the Schwarz-Christoffel transformation formula has been written. D . Properties oJ the Cross-Shaped Hall Plute
The cross-shaped sample has contacts whose lengths are comparable to the dimensions of the semiconductor layer. This means that the resistance
POTENTIAL CALCULATIONS IN HALL PLATES
15
between two contacts is low, which offers many advantages when low Hall voltages are to be measured. Technological problems caused by small contacts no longer occur. Moreover, the geometry correction ApH/pH is moderate. Even for h = 1, less than 5% is found. This also means that the position of the contacts is not very critical. High-precision mask positioning before evaporation of the contacts is no longer required.
VI. RELAXATION METHODS The relaxation method is the only purely numerical method that has been used to calculate the potential distribution in a Hall-plate medium (Newsome, 1963; Grutzmann, 1966; Chwang et al., 1974; Mimizuka, 1971, 1978, 1979; Mimizuka and Ito, 1972). If one is only concerned with the potential distribution for given voltages at the contacts, the relaxation method gives good results; however, if the current through a contact has to be calculated, the method is far from accurate. Impedance calculations carried out by Mimizuka (1978) show relative errors of 10%. On the other hand, the relaxation method allows us to include nonlinear effects such as the temperature effects inside the Hall plate, a problem which cannot be treated with other techniques (Mimizuka and Ito, 1972; Mimizuka, 1979). We now use the relaxation method to calculate the geometry correction ApH/pH for a cross-shaped plate. Because this involves knowledge of both the Hall voltage and the supply current, we expect moderate accuracy. This is mainly owing to the tangential derivative V 4 u, in the boundary condition ( 1 8), which is difficult to represent numerically. In order to represent the Laplace equation numerically, the cross-shaped form has to be divided into a mesh (Fig. 7). The Laplace equation is then approximated by the well-known five-points formula : 44i.j - 4i+1 . j - A-
1.j
-
4i.j+1
-
4i,j- 1 = 0
(40)
At a contact, the known potential values have to be inserted into Eq. (40). The boundary condition (18) can be written as (Fig. 8) 1 4i,j - 4i- 1 . j = pHBT(4i,j+ 1 - $i,j-
1
-
(41)
At a free boundary, the Neumann boundary conditionV4 u, = 0 is written as (42) 4i.j = 6(4i,j+1 + 4i.j-1 + W i - 1 . j ) which corresponds to putting 4’ = 4i-l . j (Fig. 8). This condition (42) gives
good results in numerical calculations. Note that Eq. (41) cannot be reduced to Eq. (42) when p H B tends to zero. If pHB = 0, Eq. (41) reduces to 4i.j=
16
GILBERT DE MEY
0
0
0
0
0
0
0
0
0
0
0
FIG.7. Grid pattern for the finite-difference approximation in a cross-shaped Hall plate.
FIG. 8. Boundary conditions for the finitedifference approximation.
THEORETICALVALUE= L 51 90
: s : : : : : : : : : : : 10
M f=NI
15
+
FIG.9. Geometry correction as a function of M :p H B= 0.1 ;h/l = 1.
POTENTIAL CALCULATIONS IN HALL PLATES
17
h/I
N
FIG. 10. Geometry correction versus h/l calculated with finite difference approximation: and 13 ( A ) .
= 7 ( 0 ) .10 ( W),
4i- a boundary condition leading to a reduced accuracy and hence never used. For Hall plates, only Eq. (41) can be applied. This fact can explain the poor results. Figure 9 represents the numerically calculated geometry correction ApH/pH for h = 1 as a function of M (= N).Note that the results show errors of more than SO%, and the convergence is extremely poor when the mesh number M increases. Figure 10 represents the geometry correction A/.iH/pH as a function of h / l . The analytic approximation (33) has also been drawn. Only for low h / l values were acceptable results obtained. For higher h / l values the calculations are meaningless. We conclude that the relaxation method is not suitable for calculating the geometry correction ApH/pH of a Hall plate. Several reasons can be advanced. It turns out that contact current cannot be calculated with sufficient accuracy because it involves numerical differentiations. Also, the numerical representation (41) of the boundary condition (18) is not optimal from a numerical point of view. It should be noted, however, that the high relative errors found here are made on a geometry correction, i.e., the relative difference between the numerical and theoretical Hall voltage, two numbers that are normally close together. The accuracy on the geometry correction will hence be much lower than the accuracy obtained on the Hall voltage.
18
GILBERT DE MEY
VII. THEBOUNDARY-ELEMENT METHODFOR POTENTIAL CALCULATIONS IN HALLPLATES A . Introduction
Several years ago, the boundary-element method (BEM) was a rather unknown technique. Its popularity has since grown considerably, and the method now competes with the finite-element method (FEM). In recent years, some textbooks on the BEM have been published (Jaswon and Symm, 1978; Brebbia, 1978a,b). The BEM replaces the given Laplace equation and the boundary conditions by a single integral equation. This equation involves only the boundary of the given geometry. For a numerical solution the boundary has to be divided into a number of elements. This explains the term boundaryelement method. Because only the boundary is involved, the complexity of the problem is reduced. The method requires less storage allocation and computation time if it is solved numerically. The BEM can also be programmed on small-size computers (e.g., 60K memory) in contrast to the FEM, which requires a large memory. The integral equation technique was very well known to people working in the area of electromagnetic fields. Presumably, the reason is that many problems in electromagnetic theory can be directly formulated in terms of integral equations (Edwards and Van Bladel, 1961; Mei and Van Bladel, 1963a,b). Integral equations have also been used in other fields such as thermal diffusion problems (Shaw, 1974; De Mey, 1976a, 1977a), driftdiffusion problems (De Mey, 1976b, 1977b), elastic problems (Brown and Jaswon, 1971; Symm and Pitfield, 1974),calculation of eigenvalues (De Mey, 1976c, 1977c), semiconductor-component analysis using abrupt depletionlayer approximation (De Visschere and De Mey, 1977; De Mey et al., 1977), and potential distribution in Hall plates (De Mey, 1973c, 1974a, 1976d, 1977d). B. Integral Equation f o r the Potential Distribution in a Hall Plate
We consider the rectangular Hall generator shown in Fig. 11. The potential satisfies the Laplace equation (6) and the boundary conditions
4=
V,
at
A'B',
4 = 0 at A B
V4.u" = p H BV ~ . U , at AA' and BB'
(43)
In order to construct an integral equation, the Green's function G(r I r') of the Laplace equation is used: G(r1r') = (27c)-' lnlr - r'I
(44)
POTENTIAL CALCULATIONS IN HALL PLATES
19
FIG. I I . Rectangular Hall plate.
One can easily verify that G(r I r’) satisfies the Laplace equation provided that a delta function is put in the right-hand member: V:G(rlr’) = d(r - r’)
(45)
The solution of our problem 4(r) is now written as
where p(r) is called the source function and is only defined along the boundary C of the Hall-plate medium. Owing to Eq. (49, the proposed solution (46) automatically satisfies the Laplace equation for every point r inside C. In order to determine the unknown source function p(r), one has to impose the boundary conditions (43) on the proposed solution (46). This gives rise to
- r’(dC’=
&fcp(r’)lnlr
V,,
0,
rEA’B’ rEAB
(47)
where r E AA’ and r E BB’. Equations (47) and (48) constitute an integral equation in the unknown source function p. Once this integral equation has been solved, the potential 4 (and hence all quantities which can be deduced from 4) can be found at an arbitrary point r by evaluating the integral (46). The first term - p ( r ) / 2 ~occurring in Eq. (48) is a so-called fundamental discontinuity. It is caused by the discontinuity of the normal component V$*un at the boundary. The gradient V+ can be calculated at an arbitrary point r by V4(r) = (271.)-’ $C
p(r’)[(r - r’)/(lr - r’l’)]
dc‘
(49)
20
GILBERT DE MEY
C FIG. 12. Coordinate system (t, q ) to explain the fundamental discontinuities around ro.
If r lies on the boundary, the integrand of Eq. (49) becomes infinite because r can coincide with r'. For the normal component the integrand should be treated carefully by taking r close to the boundary at the interior side (Fig. 12). The boundary C can then be approximated by a straight line and p(r) can be considered a constant p. Introducing a (5, q ) Cartesian coordinate system, one has
5
2
By taking the limit 5 + 0 (i.e., r tends to the boundary C), the term -p(r)/2 remains. This problem does not occur for the tangential component (r - r')-u, because this integrand is an odd function of q. There are no fundamental discontinuities for tangential components.
C. Numerical Solution of the Integral Equations
In order to solve the integral equations (47) and (48) numerically, the boundary C has to be divided into n elements ACi. In each of them, the unknown source function p is replaced by an unknown constant p i . Denoting ri as the center point of the element ACi, the expression (46)for the potential can be rewritten as n
(#l(r)=
1 p.G(r(ri)IACjl
j= 1
J
(51)
where IACjI denotes the length of the element ACj. In a similar fashion, the integral equations (47) and (48) can be discretized. However, if r coincides with one of the ris, one obtains G(ri 1 ri) In 0 = co ! In order to avoid this infinity, the self-potential, i.e., the potential at ri due to the source pi itself on
-
POTENTIAL CALCULATIONS IN HALL PLATES
21
Aci, has to be calculated otherwise. Replacing pi by a uniform constant function along Aci, the divergence no longer occurs and the self-potential turns out to be
The integral equations are then replaced by PiIAciIClnOlAciI) 1 + C" -pjIn(ri
j=l27L
11 -
rj(IAcjl =
j#i
= 0,
V,,,
0,
riE A'B' ricAB
(53)
riE AA', BB'
Equations (53) and (54)constitute a linear algebraic set with n equations and n unknowns pi, which can be easily solved numerically by the Gauss pivotal elimination method, for example. The potential can then be calculated at an arbitrary point r by using Eq. (51). D. Application t o a Rectangular Hall Plate
The rectangular Hall plate shown in Fig. 11 has also been treated by conformal mapping techniques (Lippmann and Kuhrt, 1958a,b; De Mey, 1973a). The Hall voltage V , induced between the opposite point contacts P and Q has been calculated accurately. For a square-shaped Hall plate, one has V , = 0.522654 if V , = 1 and 0, = 45" or c(HB = 1. By comparing this value with the numerical results, the accuracy of the BEM can be checked. Figure 13 represents the relative error on the Hall voltage V , as a function of n, where n is the number of unknowns on each side (hence 4n unknowns have to be determined for this problem). One observes that good accuracies are obtained. Precision of better than 1% can be attained with a moderate number of unknowns. The linear behavior of the results shown in Fig. 13 is remarkable. When solving a problem with the BEM the relative accuracy always varies according to a l/n law. In contrast to most other numerical methods, this l / n law does not change when other algorithms for representing the source function are used. It should also be noted that a theoretical explanation for this phenomenon has not yet been published.
22
GILBERT DE MEY
I
I
3
I
1
1
5
1
7
1
1
1
1
1
1
9 11 13 151719
*
n
FIG.13. Relative error on the Hall voltage versus n for a rectangular plate: p H B= I ; V, = 1.
E. Zeroth-Order Approximation
It is a general experience in numerical analysis that higher order approximations yield more accurate results for the same number of unknowns. For example, we can cite that the fact that Simpson's rule for the numerical evaluation of an integral will be more accurate than the trapezoidal rule. In the preceding section a rather crude approximation for the source function p was introduced. Instead of replacing p by a series of constants pi concentrated at the points ri, one can see p as a piecewise constant function. In each element Aci, p equals a constant pi.The expression (51) for the potential now has to be written as
The integral equations (53) and (54) now read
1" p j g1I A c J l n l r -i r ' I d c ' =
j= 1
V,,
0
riEA'B' riEAB
where ri E AA', BB'. This method has also been applied to the square-shaped Hall generator. Figure 14 represents the relative error of the numerically calculated Hall voltage VH. The results obtained in the preceding section
POTENTIAL CALCULATIONS IN HALL PLATES
23
43-
E 4
2-
3 \
c
1-
0.5-
3
-4
5
7
n
9 11 13151719
FIG. 14. Comparison between the direct method and a first-order approximation.
have been redrawn in Fig. 14. One observes a slightly lower error due to the zeroth-order approximation. It is also remarkable that the same l/n law appears again. Similar results have also been found with higher order approximations of the source function (Stevens and De Mey, 1978). These results are in full agreement with the statements made at the end of the foregoing section. A l/n law always occurs, and the relative errors d o not decrease remarkably when higher order approximations are used for the source function. The integrals appearing in Eqs. (55)-(57) can easily be evaluated by complex integrations. By drawing an ( x , y ) coordinate system with the x axis directed tangentially and the origin at ri (Fig. 15), the integral appearing
t’
r z x Cx*yCy
*
Zzxejy
X
FIG. 15. Complex plane used to evaluate some integrals.
24
sAc,
GILBERT DE MEY
in Eq. (55) can be written as In ) r - r‘ I dC’ = Re
s
+ IAcJ1/2
- lAcJl/2
In(z - z’)dz’
(58)
Indeed, the real part of ln(z - z’) is nothing other than lnlr - r’I and dc’ = dz’ along the x axis (Fig. 15). The ( x , y ) plane is considered to be a complex plane, and the points z and z‘ correspond to r and r’. The complex integral appearing in Eq. (58) is easily evaluated analytically:
s
+ lAcj112
-IAcjI/Z
In(z - z’)dz’ = [-(z - z’)ln(z - z’) + (z - z ’ ) ] ~ : ~ ! l ~ ~ l \ ~
If one uses a computer with complex-number facilities, all integrals appearing in Eqs. (55)-(57) can be easily calculated as the real part of an analytic function. F. Numerical Calculation of the Current through a Contact
Using the constitutive relation (14) between the current density J and the electric field E, the total current through a contact AB (Fig. 16) is given by I = C J - u n d C= 0
E.u,dC
(59)
JAB
because the tangential component of E is zero at a metallic contact. From Eq. (46) the electric field in an arbitrary point r is found to be E(r) = -V$
=
-
r
p(r’) VG(r1r’)dC’ 9 C
The current I is then given by I = -Os~dCun.~~p(r’)VG(rIr.)dC.
(61)
If one intends to calculate Eq. (61) numerically, the electric field has to be found at a number of points on the contact AB, and then the summation along AB should be carried out. This procedure will require a lot of computation time, and the accuracy will be moderate because two numerical
FIG. 16. Use of the angle a to calculate current through AB
POTENTIAL CALCULATIONS IN HALL PLATES
25
approximations will be carried out. It is more convenient to interchange the sequence of the integrations in Eq. (61), which gives rise to I = -a~~p(r’)dC~~~’dCu..VG(rlr.)
(62)
The last integral appearing in Eq. (62) is nothing other than the flux of VG through AB. As G(r I r’) depends only upon the distance I r - r‘ 1, the field VG will show a radial symmetry around r’. Hence the flux of VG through AB is given by the angle a normalized to 2n (Fig. 16). The integral (62) reduces to I = -(a/2n)
h
dC’p(r’)a(r’,A, B )
where a(r‘,A, B) denotes the algebraic value of the angle under which AB is seen from the point r’. For the numerical calculations the evaluation of an angle poses no particular problems, and a single summation is sufficient. If r’ coincides with a point on AB, then a = n and the flux is then 4.This is another view of the fundamental discontinuities discussed in Section VII,B. The numerical calculation of contact currents is very important for the study of Hall generators because these devices are normally operated under a constant current through two opposite contacts generating a Hall voltage at the other contacts. G. Applicution to the Cross-Shaped Geometry
The BEM is now applied to the calculation of the geometry correction of the cross-shaped sample. The boundary conditions are shown in Fig. 17.
v0.+IH
q
8V@.
V@
.rn=,+,so@ ..;
--
qj=h/2
IH
IH
--
@ =-vH/2
v@..‘,=p,,[email protected]
.
~#.u;l=~,[email protected];
26
GILBERT DE MEY
FIG. 18. Hall voltage ( VH), supply current (I), and geometry correction (ApH/pH)as a function o f m : h / / = I ; p H B= 0.1.
Normally, the source current I is given and the Hall current is zero. It is difficult to impose the total current through a contact as a boundary condition because this often leads to an ill-conditioned algebraic set. Therefore the potentials at the contacts are given. The current-supplying contacts have potentials 4 = + 1 and - 1. The supply current I is then found from the potential distribution, as outlined in the preceeding section. The calculations are carried out twice, once with V , = 0 and once with V , = pHB.Each time the Hall current I , is also calculated. As V , varies linearly with I,, the correct Hall voltage is then found by putting I, = 0. Figure 18 shows some numerical results obtained for a geometry with h/l = 1. The Hall voltage V,, the supply current I, and the geometry correction ApH/pH have been drawn as a function of m where m is the total number of unknowns used during the numerical procedure. The geometry correction is far from accurate, as can be seen by comparing the numerical value of 12% with the exact value 4.5%. From this consideration one may not conclude that the BEM method is useless. One calculates the Hall voltage numerically and compares it with the theoretical value (25) according to Van Der Pauw's theory. In order to obtain the geometry correction these two values, which are normally close to each other, have to be substracted. This will increase the relative error so that the numerically calculated geometry correction cannot be used. Nevertheless, the potential and field calculations in the Hall plate are still accurate. If one wants to calculate ApH/pH with a 10% accuracy (for example), the Hall voltage and the supply current have to be known with much higher precision. In Section VII,H, a method is presented for finding
POTENTIAL CALCULATIONS IN HALL PLATES
27
t
- 0.86
-
5
21
60 m
120
FIG. 19. Difference between calculated and exact geometry correction versus m. Note that double logarithmic scale is used.
ApH/pH directly, without the intermediate calculation of the Hall voltage
VH or the supply current I .
The same calculations have also been carried out with other h/l values and other numerical approximations for the source function. The results and hence the conclusions remain identical. If the difference between the calculated and exact geometry correction is plotted on a double logarithmic scale as a function of m, one obtains linear behavior with a slope -0.86 (Fig. 19). This agrees with the results shown in Figs. 13 and 14. It also proves that the numerical values converge, although very slowly, to the exact value. H . Direct Calculation o j the Geometry Correction
The geometry correction can be calculated directly, and, as is proved later, this can be easily performed if one introduces the stream potential II/ defined by Eq. (16). The technique is outlined first for a Hall plate with two point-shaped Hall contacts, and a formula is then derived for an arbitrary geometry. Let us consider the rectangular Hall plate with two point-shaped Hall contacts P and Q (Fig. 20). A current 1 is supplied. Figure 20 indicates the boundary conditions if the stream potential t,b is to be used. From Eqs. (15) and (16) one finds the electric field
E = PP,BVt,b
+ P(U,
x V*)
(64)
28
GILBERT DE MEY
FIG.20. Boundary conditions for the stream potential on a rectangular plate.
where uz is the normal unit vector perpendicular to the Hall plate. In order to obtain the Hall voltage VH, the electric field (64) has to be integrated between the Hall contacts P and Q. One obtains VH
= j:E-dr
= pp.sj;V+-dr
= PPHB[$(P) -
(Q)]- P
IQ P
= pPHB(l/d) - P
J;
v$ ’(u:
+ psypC, x v+).dr v$.(U;
X
dr)
dr)
(65)
By Eq. (65),the Hall voltage VHcan be found from the stream potential $. It is remarkable that the first term in the right-hand member of Eq. (65)is nothing other than the theoretical Hall voltage (25) given by Van Der Pauw’s theory. Hence, the remaining term in Eq. (65)is the difference between the theoretical and the actual Hall voltage, or the so-called geometry correction. We remark
FIG.21. Stream potential used on an arbitrary shape.
POTENTIAL CALCULATIONS IN HALL PLATES
29
that the last integral in Eq. (65) is the flux of V+ through a line connecting P and Q. It has also been assumed that the flux V + * (u, x dr) is zero if PHB = 0, which is certainly the case for the symmetrical configuration of Fig. 20. The conclusion is that the geometry correction can be calculated directly as the flux of the V+ field. This can be calculated numerically by the method outlined in Section VII,F. We now consider an arbitrary Hall plate provided with four finite contacts (Fig. 21). The electric field is still given by Eq. (64). On the Hall contacts two points P and Q have been chosen. The integration of the electric field gives us the Hall voltage:
where A$(P) and A+(Q) denote the differences between the II/ values at P and Q, respectively, and the values at the ends of the contacts. For pointshaped contacts these differences are evidently zero and Eq. (66) reduces to Eq. (65).The first term in the right-hand member of Eq. (66) is the theoretical Hall voltage according to Van Der Pauw. The geometry correction is now given by a term containing the stream-potential differences A+(P) and A+(Q) and a second term involving the flux of V+ through PQ. If necessary, the ohmic potential drop occurring when pHB= 0 has to be subtracted from Eq. (66); however, most Hall-plate configurations show sufficient symmetry for this operation to be dropped.
I . Application to a Rectangular Hall Generator
The method outlined in Section VII,H is applied to the rectangular plate shown in Fig. 20. Because the Hall voltage has also been calculated with conformal mapping methods, the accuracy of the numerical values can be controlled. Figure 22 shows the relative error of the geometry correction for various values of PHB obtained by evaluating the last term of Eq. (65). In contrast to previous results, the accuracy is fairly high. It should be noted that a relative accuracy of 10% on the geometry correction means that the Hall voltage is known with a precision much better than 10%. We remark again that the
30
GILBERT DE MEY
20
-t
FIG.22. Relative error on the geometry correction as a function of IM for a square-shaped Hall plate: p H B= 0.1 ( 0 ) , 0 . 3 ( + ) , 0 . 5 ( A ) , a n d I ( 0 ) . IMisthenumberofunknownsperside.
error decreases in inverse proportion to the number of unknowns used in the numerical procedure. We may conclude that the accuracies are sufficient for practical purposes. J . Application to a Cross-Shaped Geometry Equation (66) is now used to analyze the cross-shaped geometry. Two points P and Q have been placed at the centers of the Hall contacts (Fig. 23).
FIG.23. Cross-shaped Hall plate analyzed with the stream potential.
POTENTIAL CALCULATIONS IN HALL PLATES
31
Figure 24 shows the relative error on the numerically calculated geometry correction for h/l = 0.5. For practical purposes these accuracies are more than satisfactory. For high values of pHB(>0.5), the error curve seems to diverge. This does not mean that the method should fail. The reason is that the relative error was calculated taking Eq. (33) as the exact value. However, there is a slight nonlinearity between the Hall voltage and the applied magnetic field B, a secondary effect not included in Eq. (33). Figure 25 shows the geometry correction as a function of h/l calculated for PHB = 0.1. The theoretical relationship (33) has also been drawn. For h/l < 1.5, numerical results are close to the theoretical ones, especially for high values of ZM,which is the number of unknowns for one side (the total number of unknowns is then 12ZM). For h/l 2 1.5, the deviations become remarkable. Nevertheless, the numerical results still converge to the exact values. In order to obtain insight into the practical consequences, let us take h/l = 2. The exact geometry correction is 0.2%, whereas the numerical value obtained with ZM = 9 turns out to be 0.5%. It makes no difference in practice if an experimental value is corrected by 0.2% or OS%, because this correction is much smaller than the accuracy of the Hall-mobility measurements. Only I 20
-
1
+
-ae 10
$
-
20 10
5-
5-
2-
2-
9' 0 z Q
0
: Q b Q
1-
I-
0.5-
0.2-
0.5
+
-
0.2
-
FIG.24. Relative error on the geometry correction versus IM for a cross-shaped Hall plate:
(a) h/l = 0.5; p H B= 0.3 (+).
32
GILBERT DE MEY
!i
+8 z E
9
1-
0.5-
az
\
1
0.5
1
I
1
I h/l
I
1.5
I
I
2
*
FIG.25. Geometry correction as a function of h /lfor a cross-shaped Hall plate: p H B= 0.1 ; I M = 3 (A),5 (OL7 (+), 9 ( 0 ) .
=-
when the geometry correction is greater (e.g., 1%) can it be calculated with sufficient accuracy. The conclusion is that the method can be applied in actual cases. Only when the geometry correction is negligible can it not be calculated with high precision; but in these cases knowledge of it is of no importance. It should be noted that the cross-shaped geometry has four reentrant corners, which makes its use difficult for numerical treatments. Some equipotential lines have been drawn in Fig. 26. Owing to Eq. (16) these lines are also the current lines or streamlines. Only 50% of the Hall plate is shown because the remaining part can be found by symmetry. From Fig. 26 the hypothesis formulated in Section IV can be verified.
POTENTIAL CALCULATIONS I N HALL PLATES
33
FIG.26. Streamlines in a cross-shaped Hall plate.
K . Application to Some Other Geometries
We first consider a rectangular Hall generator provided with four finite contacts (Fig. 27). This type of Hall plate is often used experimentally because monocrystalline semiconductors are usually available in the form of a rectangular bar. It is not always possible to cut a monocrystalline semiconductor slice in the form of a cross; a rectangular Hall plate is then used to carry out Hall-mobility measurements. This geometry has been extensively studied theoretically by Haeusler, using conformal mapping techniques (Haeusler and Lippmann, 1968). Nevertheless, this shape is far from optimal. Even for small contacts, the geometry correction turns out to be 30-40%. A
34
GILBERT DE MEY
h
FIG.27. Rectangular Hall plate provided with four finite contacts.
small shift in the position of the contacts causes a nonnegligible change of the geometry correction and hence an error in the experiments. Figure 28 shows the numerically calculated value of A,uH/PHfor h/b = 2 as a function of a/b. The geometry correction increases rapidly with the contact length a and can attain very high values. For this geometry Haeusler found the following approximate formula valid for sufficiently small contacts (Haeusler and Lippmann, 1968):
Equation (67) is accurate up to 4% for h/b > 1.5 and a/b < 0.18. The first condition is met, but the second one is not fulfilled because the numerical BEM cannot be used if a contact is much smaller than the other sides of the Hall plate. Hence we expect that Eq. (67) will not coincide perfectly with the numerical results. For h/b = 2 and 8, small, Eq. (67) reduces to APH/PH
= 1 - (1 - e-")[l
- (2/a)(a/b)] = 0.0432 + 0.609(a/b) (68)
The linear relationship (68)has also been drawn in Fig. 28, and the agreement is fairly good. One observes a better fit when the number of unknowns per side IM increases. A second particular geometry is the octagon provided with four equal contacts (Fig. 29). Although this shape has not been used in experiments, it is the first geometry that has been studied theoretically by Wick using conformal mapping techniques (Wick, 1954). The reason is that the conformal mapping of the octagon into a circle leads to calculations which can be
POTENTIAL CALCULATIONS IN HALL PLATES
35
< 0.L
0.5
0.6
0.7
0.8
O/b
FIG. 28. Geometry correction for the rectangular Hall plate: h/b = 2; pHB = 0.1 ; IM = 3 (+I, 5 (A), 7 ( x ), 9 (Oh I 1 13 (01, 15 ( 0 ) .
(m),
performed analytically. Figure 30 represents the numerically calculated geometry correction as a function of l / h for various values of the number of unknowns. One observes again that the geometry correction is high. Wick did not calculate any geometry correction, but from several graphs in his article it was possible to deduce approximately ApH/pH. These results are
t1
FIG.29. Octagonal Hall plate.
36
GILBERT DE MEY
+
I
I
I
0.2
0.3
i/h
0.4
I
0.5
*
FIG.30. Geometry correction for the octagonal Hall plate: p H B= 0.1 ; IM = 3 (+), 5 ( A ) , (m),1 3 ( 0 ) , 1 5 ( 0 ) .
7(x).9(0). I1
shown in Fig. 30 and referred to as “theoretical.” One sees good agreement between these theoretical points and the numerical results. Chwang ef al. (1974) analyzed the octagon using finite difference techniques. From the results published in their article, the geometry correction could be derived for small values of l/h. These results are not very accurate because they differ a lot from the results obtained by Wick (conformal mapping) and the BEM. It is also possible to obtain an approximate formula for ApH/pH for the octogon. Van Der Pauw published geometry corrections for a circular shape provided with finite contacts. An octagon can be approximated by a circle with a diameter h and a contact length 1. The geometry correction is then (Van Der Pauw, 1958) AA+/PH
=
4(2/nZHI/h)= (8/n)’(1/h)
(69)
This relation is also drawn in Fig. 30, and it turns out to be a good approximation for l / h % 0.5. For other l/h values the replacement of an octagon by a circle seems to be less adequate.
37
POTENTIAL CALCULATIONS IN HALL PLATES
FIG.31. Unsymmetrical cross-shaped form.
A final interesting geometry is the unsymmetrical cross-shaped form. Thin-film semiconducting Hall plates are usually made by evaporating the materials through a metallic mask. For the contacts on the Hall plate another mask is required, and thus a shift between the mask positionings can never be avoided. This gives rise to an unsymmetrical cross-shaped form. A fortiori, when the contacts are made manually with a conducting ink, for example, a perfect symmetry cannot be guaranteed. We now investigate a cross-shaped form where one arm differs from the others (Fig. 31). The numerical results are shown in Fig. 32, representing the geometry correction ApH/pH as a function of u/l. The geometry correction increases when the arm length shortens, which can easily be understood. These results can also be found theoretically. In Section V,B, we made the assumption that each arm gives its own contribution to the geometry correction. This being the case, the following formula was found for four equal arm lengths:
Applying Eq. (70) to three arms with length h and one arm with length u, one obtains
*
PH = 4L nze x p ( i l n 2 =
For h / l
=
1.04,3[:exp(
- :)[:exp( -n)>
-n:)
+ aexp( -
+ aexp( - n 3 1
31
1, Eq. (71) reduces to ApH/pH = 0.03387
+ 0.2613e-"""
(72)
This relation has also been shown in Fig. 32, and the agreement with the numerical results is very good. This also proves the hypothesis that each
38
GILBERT DE MEY
13
+ 12
A
11
10-
-
s
9
+
8
L
z I
x
d
7
6 5
L
I
0.4
I
0.5
I
0.6
1
0.7
I
0.8
I
0.9
I
I
C
0 4
FIG.32. Geometry correction for the unsymmetrical cross-shaped Hall generator: p H B = IM = 3 (+), 5 (A), 7 ( U ) . 9 (0).
0.1; h / l = I ;
arm generates its own contribution to the geometry correction ApH/pH independent of the other ones.
VIII. IMPROVEMENT OF THE BOUNDARY-ELEMENT METHOD A . Introduction
In this section we present a modified BEM. By investigating the method it was found that the highest errors occurred at the corners of the geometry. Therefore, a method is given here which uses analytic approximations at the corners while the remaining part of the potential is calculated numerically. This technique can be seen as a combination of conformal mapping and the
POTENTIAL CALCULATIONS IN HALL PLATES
39
BEM. The conformal mappings are used to calculate analytic approximations at the corners of the Hall plate. The remaining part of the potential distribution is found by numerical solution of an integral equation. The main advantage of this method is that singularities occurring at a corner can be represented exactly by the analytic approximation (De Mey, 1980).
B. Integral Equation We consider a Hall plate as having the shape of an arbitrary polygon (Fig. 33). Along each side zi the following boundary condition holds: a4
+ BV4.U" + yV4.q
= fo
(73)
Forametalliccontactatpotential Vo,onehasa = 1,p = y = O,andf, = Vo. For a free boundary a = 0, /?= 1, y = -pHB, and fo = 0. All types of boundary conditions along a Hall-plate medium can be represented by Eq. (73). It makes no difference in the subsequent analysis whether one is interested in the electrostatic potential or the stream potential II/. We assume now that in the neighborhood of each corner an analytic approximation for the potential is known. At the ith corner one has
4(r)
AiPi(r)
(74)
where cpi(r) is a known function satisfying the Laplace equation; Ai is a still unknown proportionality constant. The constants Ai will be determined together with the solution of the integral equation. The potential inside the Hall plate is now written as
4(r) =
Aicpi(r) i
+ f C p(r')G(rlr') dC'
(75)
The function cpi(r) includes all the singularities at the corners. Note that cpi(r) is not only defined at the ith corner, but in the entire region. This means that cpi(r) will not necessarely be zero at the other corners but that the functions cpi(r) will be chosen in such a way that cpi(r) is a smooth potential
\
z,-i
FIG. 33. Boundary condition at one side of an arbitrary polygon: aiQ yIvQ.u,=jO.
+ fii V+.u. +
40
GILBERT DE MEY
distribution at the other corners. The basic idea is that 1 Aiqi(r)is not a first-order approximation but a particular solution involving all singularities, especially those occurring in the gradient field. The remaining part will then be a perfectly smooth surface and is represented by the last term in Eq. (75). Applying the boundary condition (73) to the proposed solution (75) gives rise to C A i [ a q i + PVqi.un + ~ v q i . ~ ] i
- P+[p(r)l
+ (jcP(rWNG(rIr7
+ P V G ( r l r ’ ) - u , + yVG(rIr’)*u,]dC’ = fb(r)
(76)
Equation (76) constitutes an integral equation in the unknown function p. We remark that the constants Ai are also unknown and have to be determined simultaneously with the numerical calculation of p. C. Culculation of the Functions qi(r)
The purpose of this section is to determine the q i , potential functions being good approximations at corners, and to include all singularities: especially those occurring in the gradient fields Vq,.The method is outlined for the corner A of the Hall plate shown in Fig. 20. The corner can be replaced by two infinite sides. The function qi is then a potential satisfying the boundary conditions indicated in Fig. 20 along two infinite sides. This can be easily done, as it is always possible to map one corner onto another at which the potential is reduced to a constant and homogeneous field. This conformal
FIG.34. Approximation of a corner by two circular arcs.
POTENTIAL CALCULATIONS IN HALL PLATES
41
mapping is done by the simple analytic function 2'. Some problems may arise, however. Consider again Fig. 20 for the particular case p H B = 0. In each corner one side has a constant potential and the other side requires that the normal gradient should be zero. The field is then homogeneous because the corners are rectangular. The potential pi then varies linearly with distance. The same situation is valid for all corners, however, which means that the four functions cpl, cpz, cp3, and cp4 will be linearly dependent. In this case it is not possible to determine the constants A , , A z , A 3 , and A, numerically. Linearly dependent functions should always be avoided. In order to eliminate this problem, a corner will not be approximated by two infinite sides but rather by two circular arcs, as shown in Fig. 34, representing a corner with angle cp. One must now solve the Laplace equation V Z q i= 0 with the condition cpi = 0 at one circular arc and Vcpi-u,+ p H BVqi u, = 0 at the other. In this case it is almost impossible to obtain linearly dependent solutions at two corners. Because the circles are tangent at the sides of the corners, qi is still a good approximation for the potential. The solution of this potential problem is carried out with conformal mapping techniques. The geometry bounded by two circular arcs can be mapped onto a semiinfinite plane using (Fig. 39,
-
2' =
A"z/(z -
Zo)]n'e
(77)
where A' is a complex constant. Let us consider now the particular case where c2 is a free boundary and c1 a metallic contact held at constant potential. We
FIG.
35. Conformal mapping on a corner with angle ( 4 2 ) - OH
42
GILBERT DE MEY
then have to transform into a corner with an angle 4 2 - OH ( w plane in Fig. 3 9 , because for this geometry the potential can be written immediately. The transformation from the z to the w plane is then w = A ’ ( 1 / 2 ) - ( @ H / 2[) z / ( z - zO)](n/2o)-(@H/9) = A [ z / ( z - ~ ~ ) ] ( n / 2 1 ) - ( @ H / O ) (78) For a point z located on c1 (Fig. 35), one has
- zo) = ( z / ( z - zo)l e-jqlz
z/(z
(79)
then, w = A 1 z/(z - z o )1 ( ~ / W - ( @ H / V ) exp( - j i n
+ jl-0 2 H)
(80)
A point z on c1 is mapped onto a point w situated on the real u axis. Hence Eq. (80) has to be real, from which A can be determined:
A = exp(jin - j + O H )
(81)
Because the function cpi is multiplied by a constant Ai in the proposed solution (75), the absolute value of A has no influence on our problem. From the transformation formula (78) one obtains dw/dz = w [ ( $ n - &)/~][-zO/z(z - ZO)]
(82)
Equation (82) is used later to calculate the transformation of the electric fields. In the w plane, the potential cp corresponds to a homogeneous field directed perpendicular to the u axis (Fig. 35) because the angle was taken to be $n - 0,. Hence pi can be written as Vi(U,O)
=
(83)
IJ
The potential function can be derived from the complex potential W : W = -jw
[cpi
(84)
= Re( W ) ]
with d W / d w = -j
The potential in the z plane can be easily determined because the complex potential W is invariant. The electric field components in the z plane are determined by dW - - - - E , + j E. dz
y
acpi ax
= - - j - = -acpi -= ay
dwdw dw dz
-j-dw
dz
(86)
Inserting Eq. (82) into Eq. (86) directly gives us the field components in the original z plane. It has been proved that the potential cpi in a corner can be determined by a simple conformal mapping.
POTENTIAL CALCULATIONS IN HALL PLATES
43
It should be noted that cpi has only been determined up to a proportionality constant. Because the functions cpi are preceded by an unknown constant Ai in Eqs. (74) and (75), this causes no problems. For other situations at the corners, e.g., two adjacent free boundaries at a corner, the procedure to determine cpi remains identical; however, some formulas may be appropriately changed. D. Application to a Rectangular Hall Generator
Figure 36 shows a rectangular Hall plate with the circles for the approximation of the potential in the corners. First, the influence of the radius R was investigated. Figure 37 indicates the relative error of the Hall voltage and the current I as a function of the radius R for various values of I M , which is the number of unknowns per side. One observes that the error is almost independent of R. As for the current, it seems to be an optimum around R = 1; however, this is owing to the fact that the numerical value of the current minus the exact value changes its sign when R x 1, so one cannot consider it to be optimal. Figure 38 shows the relative error of the Hall voltage and the supply current as a function of IM. The numerical results obtained without the analytic approximation C Aicpi have also been drawn in Fig. 38. The accuracy of the Hall voltage remains practically unchanged, but the accuracy on the current calculation increases by almost two decades. Because both quantities are needed for practical applications, an overall improvement has been achieved. The geometry correction has also been investigated. Figure 39 shows the relative error on the geometry correction as a function of IM. The same
FIG.36. Square-shaped Hall plate with approximate arcs at the corners.
5-
-z
2-
6
1-
2
a5-
E c
2 a20.1-
M5-l
-., t
a1
R
FIG. 37. Relative error as a function of radius R : p H B = 1 ; AVH/VH ( 0 .
(0.0. A).
A); A I / l
4
3
5
9
1M
15
29
FIG.38. Relative error on the Hall voltage and the supply current: R = 1 ; pHB = I , with ( + ) and without ( 0 )analytic approximation.
POTENTIAL CALCULATIONS IN HALL PLATES
45
t
i
5
rb
IM
.?a
50
FIG.39. Relative error on the geometry correction as a function of IM:p,J = 0.I (+, x ); R = I ; with analytic approximation (+, A, 0 , 0); 0.3 (A, A); 0.5 ( 0 ,0 ) ; I (W, 0); without analytic approximation ( x , A , 0, m).
results without the analytic approximation are also drawn in Fig. 39. Generally, the introduction of analytic approximations yields better results, especially for low values of p H B ,when the Hall voltage is low and difficult to detect. E. Application to a Cross-Shaped Form
The cross-shaped form has also been analyzed using the same method. In this case, there are 12 corners and hence 12 qifunctions have to be evaluated. The geometry correction was calculated using the method outlined in Section VII,H. The results are shown in Fig. 40.The data obtained in Section VII,J are also drawn in Fig. 40. One now observes, contrary to the results found in the preceeding section for a square-shaped form, that the introduction of analytic approximations at the corners yields no improvement in the numerical results. Noting the fact that the calculation of the cpi functions involves computation time, one must conclude that the method of Section VII,H should now be preferred. These results indicate that the present method is only useful for a limited number of corners. The cross-shaped sample has also been analyzed using an analytic function at a reduced number ( c12) of corners, but this did not lead to any improvement.
46
GlLBERT DE MEY
A
1
a5
I
1
1
1
I
1.5
I
1)
2
h/ I
FIG.40. Geometry correction versus h / l for a cross-shaped sample; pHB = 0.1 ; with analytic approximation, IM = 3 (A), 9 ( 0 )without ; analytic approximation, IM = 3 (A),9 ).(
F . Application to Some Other Geometries
Because the number of corners seems to influence the numerical accuracy, two geometries with eight corners are now investigated. The rectangular and octagonal Hall plates treated in Section VII,K are now analyzed. It should be noted that the rectangular Hall plate provided with four finite contacts should be viewed as having eight corners. Because the electric field at both ends of each Hall contact shows a singular behavior, the introduction of analytic functions may be useful. Figure 41 represents the geometry correction as a function of a/b. Some results of Figure 28 have been redrawn. One sees that more accurate results are now found.
60
-
2
-A
M-
*5 z
Q
LO-
30
I
I
0.5
0.4
I
I
0.7
d.6
0.6
o/b
FIG.41. Geometry correction for a rectangular Hall plate: hlb = 2; pHB= 0.1 ; without analyticapproximation, IM = 3 ( A ) , 15 ( 0 ) withanalyticapproximation, ; IM = 3 ( A), 15 (0).
0.2
0.6
0.3
as
I/h
FIG. 42. Geometry correction for an octagonal Hall plate: pHB= 0.1, without analytic approximation, IM = 3 ( A ) , 15 ( 0 ) ;with analytic approximation, IM = 3 ( A ) , 15 ( 0 ) .
48
GILBERT DE MEY
Similar data for the octagonal shape are shown in Fig. 42. The same conclusion still holds because one observes that the numerical results with analytic approximations are closer to the theoretical curve. As a general conclusion of this section, one can state that the introduction of analytic approximations that include the singularities at the corners is only useful if the number of corners is limited to eight.
1X. CONCLUSION In this article two objectives concerning Hall-effect devices have been outlined. First, a review was given concerning several methods for calculating the potential distribution in a Hall plate. Second, an attempt was made to answer the question: What can be done with such a potential distribution? It turns out that the Hall voltage depends upon the geometry of the plate, an effect which can only be calculated if the potential problem is solved. Because many applications of Hall effects involve measurement of the Hall voltage, it is important to know all the influences on this voltage. When several methods for potential calculations were tested, not only did we draw streamlines or equipotential lines, but we emphasized the evaluation of the geometry correction to the Hall voltage. The first objective of this article was to check several methods for potential calculations in Hall plates. The potential satisfies the Laplace equation, but the boundary conditions are rather unusual, so that common techniques such as separation of variables or eigenfunction expansion cannot be applied. Three methods were found to give acceptable results: conformal mapping, finite difference approximation, and the BEM, the last two being purely numerical. Conformal mapping was the first method used to investigate the Hall effect. Some geometries can be analyzed with a completely analytical treatment; however, the calculations are very lengthly and complicated. Conformal mapping has also been applied successfully in a semianalytical approach : after the Schwarz-Christoffel transformation formula is written down further steps are performed numerically. In this way the high accuracy associated with conformal mapping techniques can be achieved with only moderate computational effort. The finite difference method is the most obvious numerical method, but complications arise owing to the special boundary conditions. This method can be used for a potential distribution, but if current and impedances are calculated the finite difference approach gives unacceptable results. In the BEM the special boundary condition presents no particular difficulties. Potential calculations can be done accurately. With some modifications, even small geometry corrections may be
POTENTIAL CALCULATIONS IN HALL PLATES
49
calculated. Finally, we can say that either theconformal mapping or the BEM can be used. Conformal mapping yields high accuracies, but for every new geometry, calculations have to be done again. The BEM can be applied to all geometries but its accuracy is lower. One of the two methods will be appropriate for each individual case. In the last section a combination of these two methods was presented. The potential was still calculated by the BEM, but at the corners analytical approximations obtained with conformal mapping were introduced. I t turns out that this combined method only constitutes an improvement for geometries with a limited number of corners. The second objective of this article was to show the influence of the geometry on the Hall voltage, i.e., the geometry correction. Van Der Pauw’s theory gives a formula for the Hall voltage, but owing to geometic effects (i.e.,finitecontacts) a lower value will be measured in practice. Thiscorrection has been calculated for several geometries, and the conclusion is that the cross-shaped form requires the smallest correction. This may be useful because some geometric parameters are not always known with sufficient accuracy. The cross-shaped form is also fitted with contacts comparable to the dimensions of the Hall plate, so that the resistance between two contacts and hence the noise in the measuring circuitry will be minimal. APPENDIX 1. THETHREE-DIMENSIONAL HALLEFFECT The Hall generators studied in this article are essentially two dimensional. Van Der Pauw’s theory, as outlined in Section 111, cannot be extended to a three-dimensional semiconducting volume having four contacts on its surface and placed in a magnetic field. Even when all contacts are point shaped, a general formula for the Hall voltage cannot be given. For every geometry one has to solve the potential problem from which the Hall voltage is found. Neglecting terms of order &B2, it turns out that the potential still satisfies the Laplace equation (De Mey, 1974b). At a metallic contact the potential is given, but on a free surface one has where u, is the unit vector perpendicular to the semiconductor surface and u, is the unit vector in the direction of the magnetic field. Note that u, and u, need to be perpendicular, as in the two-dimensional case. Because u, and u, can include all possible angles, the right-hand side of Eq. (87) turns out to be a complicated boundary condition. A perturbation method is therefore used to solve the problem. Treating PHB as a small quantity, the potential 4 can be
50
GILBERT DE MEY
written as a zeroth-order approximation 4oand a first-order perturbation
4
=
40
+ P(HB4I
The equations and boundary conditions for do and
v24, = 0 v40 'U, = 0 4o = applied potential v24, = 0 V 4 , 'u, = (V40 x uz)*u, 41
=o
:
(88)
4, are then (89) (90)
on a free surface on a metallic contact
(91) (92)
on a free surface
(93)
on a metallic contact
(94)
These equations are to be solved for a cylindrical semiconducting bar, as shown in Fig. 43. If a potential difference V is applied, the zeroth-order potential is easily found: 40
=
(95)
(V/a)x
The zeroth-order current I, through the contacts is given by I0
(96)
= (V/p)(nR2/a)
For the first-order perturbation, an eigenfunction expansion is used. In a circular section, the Dirichlet eigenfunctions are
where N,, is the normalization constant, Jn is the Bessel function of the first kind of order n and xnpis the pth root of the transcendental equation The first-order potential
41 is then written
where
,
the other coefficients being zero. The normalization constant N, is given by
3
FIG.43. Cylindrical Hall medium placed in a magnetic field.
as
0
1
1
1
1
1
5
FIG.44. Hall voltage for a cylindrical semiconductor.
.-
52
GILBERT DE MEY
From Eqs. (96), (99), and (lOO), the Hall voltage V , between the points P and Q can be found for a given current I , : VH = -4/&BpI,(nR)-'c ( x t p - l)-'[l - cosh(x,,/R)(a/2)]-'
(102)
P
Figure 44 shows the Hall voltage normalized to pHBpIo/R as a function of u/R. If u/R + co, i.e., the semiconducting bar becomes infinitely long, the normalized Hall voltage equals 0.6324. This means that the cylinder generates the same Hall voltage as a two-dimensional Hall plate with the same width 2R and a thickness R/0.6324.
APPENDIX 2. ON THE EXISTENCE OF SOLUTIONS OF INTEGRAL EQUATIONS If the Laplace equation for a given geometry with suitable boundary conditions is replaced by an equivalent integral equation, generally a Fredholm equation of the first kind will be obtained. On the other hand, the existence theorems for integral equations deal only with equations of the second kind (Hochstadt, 1973).For these equations it is possible to construct an iteration procedure, which can be used to prove the existence theorems. With regard to Fredholm integral equations of the first kind, the literature is quite contradictory. Some authors claim that the existence has been proved (Symm, 1963), whereas other authors allege the reverse (Tottenham, 1978). A particular geometry is now treated in order to investigate the existence of a solution. The conclusion cannot be extended to arbitrary geometries, but it will provide us with a deeper insight into the problem. Also, the contradictions occurring in the literature will be better understood. As integral equations for potential problems are mainly used as numerical approximations, the treatment in this appendix is also done from a numerical point of view. According to Fredholm theory (Courant and Hilbert, 1968) an integral equation can be regarded as the limit of a set of algebraic equations whose number of unknowns goes to infinity. We shall proceed now in a similar way. In order to investigate the integral equation for the circular geometry shown in Fig. 45, the boundary is divided into n equal parts Acj. The integral equation fcp(r')G(rlrc)dC'= V(r),
is then replaced by the finite algebraic set
reC
(103)
POTENTIAL CALCULATIONS IN HALL PLATES
53
I'
FIG.45. Circular geometry used to study the existence of a solution of the integral equation.
This set will have a unique solution if the determinant of the matrix [A,,] is not zero: (105) det[A,,] # 0 Similarly, the integral equation will have a unique solution if the Fredholm determinant does not equal zero. In our problem, the easiest way to check if Eq. (105) is fulfilled is to calculate all the eigenvalues of [A,,]. When all the eigenvalues are other than zero, the determinant will also be nonzero and vice versa. For the particular geometry of Fig. 45, one can easily verify that =
A(l-J)mOdn
(106)
if the boundary is divided into equal parts AC,. Equation (106) means that the matrix [A,] is circulant. It can be proved that the eigenvalues are then given by 11
i.,
=
'm
=
C
k= 1 n- I
1A l . k + l
k= 0
(107)
A,,
e - 2 n jmkln
,
m = l , 2 ,..., n - 1
(108)
If the matrix dimension increases, the matrix still remains circulant. In the limit for n + 00, the expressions (107)and (108)are then replaced by integrals.
54
GILBERT DE MEY
The eigenvalues of the integral equation (103) are then found to be r
i o= (2n)-'fClnlr - r,IdC
i,, = (2n)-
'
In I r - r, Ie-JmedC
where r, can be chosen arbitrarily on the boundary. FOPa circular geometry, the integrals (109) and (1 10) can be easily evaluated: j . , =
(R/n)
i,, = (R/n)
1; [i
ln(2R sin +6)d6 = R In R
(111)
ln(2R sin +6)cos m6 d6 = - R/2m
(1 12)
If R = 1, the eigenvalue I, turns out to be zero. The integral equation will then have no solution at all because the Fredholm determinant is zero. This fact has been investigated by Jaswon and Symm (1978) and Symm (1964). By using an appropriate scaling factor, this problem can be easily eliminated. It has been verified experimentally that eigenvalues of the matrix [ A i j ] (with dimension n) show behavior similar to the first n values l o ,R,, . . .,i n given by Eqs. (111) and (112). Especially, it was found that the smallest eigenvalue is inversely proportional to the matrix dimension n. This implies that I , , - , tends to zero when n + 00. One therefore concludes that the Fredholm determinant of the integral equation (103) will always be zero, meaning that Eq. (103) will have no solution at all. O n the other hand, as long as n remains finite, all the eigenvalues are other than zero and the algebraic set has a unique solution. It can thus be stated that the integral equation only has a solution if one considers it to be the limit solution of an algebraic set. If the solution of the set converges for n + co,one can define it as the solution of the integral equation. Many properties of Fredholm integral equations of the first kind have not been studied or explained because this matter turns out to be a difficult area of functional analysis. This explains why some numerical results, such as the l/n law for the relative error versus the number of unknowns, have not been declared yet. A theoretical background will indicate to us in which direction research should be continued in order to improve the BEM.
APPENDIX 3. GREEN'S THEOREM The method outlined in Section VIII,B is rather unusual for constructing an integral equation for the BEM. Normally, Green's theorem is used,
55
POTENTIAL CALCULATIONS IN HALL PLATES
leading to an integral equation where the unknown source function p has a physical meaning. If the potential is given on a part of the boundary, the unknown function turns out to be the normal component of the gradient and vice versa. However, this can only be done if the so-called natural boundary conditions are given, i.e., the boundary conditions only involve 4 or V 4 sun. The boundary condition (18) is not a natural one. An attempt has been made to integrate the tangential component along the boundary in order to obtain the potential 4 rather than V 4 mu,. The procedure is outlined later. Consider the Hall plate shown in Fig. 11. The potential satisfies
v24(r)= 0
(113)
and the Green's function satisfies V2G(r(r')= 6(r - r ' )
(1 14)
Multiplying Eq. (1 13) by G and Eq. (1 14) by 4 and substracting the resulting equations from each other, one obtains after applying Green's theorem fc [4(r)VG(rlr')*un- G(rlr')V4-un]dC = 4(r')
(1 15)
Equation (1 15) holds for each point r' inside C and can be transformed into an integral equation if r' is placed on the boundary. When 4 is given (e.g., on a metallic contact), then V 4 * u, is the unknown function. However, along the side BB', Eq. (18) is valid and V4.u" is not given, so 4 cannot be treated directly as the unknown function. Some intermediate steps are required:
The last integral in Eq. (1 16) can be transformed by partial integration: pHB
jBB'
G V$
*
U, dC
jBB'4
= pHB[G(r1r')4(r)]:r:~,- pHB
VG -u, dC
(1 17)
Equation (1 16) now reads jBB'[4VG-un- GV4.un]dC
IBB'
= pHBG(rB,lr')vo-I-
[4VG*U, - p~BvG'll,]dC
(118)
The tangential component V 4 -u, has been eliminated, and only 4 appears as an unknown function in the intergrand. Hence it is possible now to consider 4
56
GILBERT DE MEY
the unknown function along BB’. The same procedure can be carried out along AA’, so that Eq. (1 15) can be rewritten as n
n
- pHBG(rB lr’)Vo +
[ V o V G . u , - G V d * u , ] dC s.,A.
d[VG.U, - p H B V G ’ U , ] dC + JAA,
+ pHBG(rA,lr’)Vo,
CE
c
(119)
I f r’ E C , Eq. (1 19) is an integral equation that can be solved numerically; however, in Eq. (119) terms of the form VoG(rB,lr’) occur. Since G is a logarithmic function, these terms will be singular at the corner points A ’ and B’. Generally, it can be stated that singularities will occur at the end points of every contact staying at a nonzero potential. These difficulties d o not happen with the technique presented in Section VII,B, because the Green’s function then always appears under the integral sign. The integral equation (119) with r’E C has been solved numerically. In the term G(rA,lr’), r‘ was put equal to ri, which is the center point of the ith boundary element. Singularities were hence eliminated. Figure 46 shows the relative error on the Hall voltage. These results are compared with those
t
\
5
APPROXIMATION
I
1
3
1
1
5
I
I
9
7 n
1
I1
I
I
13 15
I
I
n 19
+
FIG.46. Relative error on the Hall voltage calculated by using Green’s theorem.
POTENTIAL CALCULATIONS IN HALL PLATES
57
obtained in Section V11. One observes that the error is very high, making the method unusable. For p H B = 0, accurate results were found because the singular terms in Eq. (1 19) vanish. These results explain why Green’s theorem is not used to construct an integral equation for the field problem in Hall plates. The methods outlined in Section VII should be preferred in this case. APPENDIX 4. THEHALL-EFFECT PHOTOVOLTAIC CELL The Hall effect can also be used to convert solar energy to electric power. The basic principle of all photovoltaic cells is the generation of electrons and holes in semiconductors owing to the absorbed light. If electrons and holes are accelerated in opposite directions a net current will be delivered to an attached load. In junction solar cells the separation of electrons and holes is done by the junction field. The same effect can be achieved with the Hall phenomenon because electrons and holes are deflected in different directions by the Lorentz force. A possible configuration of a Hall-effect photovoltaic cell is shown in Fig. 47. The incident light generates electron-hole pairs in the layer. Owing to the exponential decay of the light intensity, a concentration gradient for the charge carriers is built up. Hence electrons and holes diffuse in the y direction. Owing to the Lorentz force, charge carriers will be deviated along the x axis in such a way as to give a net current through the load resistor R,. A theoretical analysis has been published (De Mey, 1979).Assuming that one type of charge carrier has a much higher mobility than the other (which
gs
58
GILBERT DE MEY
is the case for InSb, InAs, etc.), the maximum attainable efficiency was found to be where p is the mobility, B the magnetic induction, E , the band gap of the semiconductor, L the diffusion length, and u the light absorption coefficient. For E , = 1 eV, and giving GILits optimum value, Eq. (120) reduces to = 0.00625(puB)2
(121)
For p B = 1, the efficiency turns out to be 0.6%, a low value compared with that of junction solar cells, which have efficiencies better than 10%. Note that p B = 1 is a rather high value because B = 1 Wb/m2 is difficult to attain, and there are only two semiconductors having p > 1 : InSb ( p = 7) and InAs ( p = 3). Because Eq. (120)was developed for monochromatic light, the efficiencywill be reduced at least by 0.44 in order to take the solar spectrum into account, and because both InSb and InAs have small band gaps, these conductors are not matched to the solar spectrum, which results in a much lower spectrum factor (< 0.44). The conclusion is that the Hall-effect solar cell is not suitable for energy production because of its low conversion efficiency.
APPENDIX 5. CONTRIBUTION OF THE HALL-PLATE CURRENT TO THE MAGNETIC FIELD A current must be supplied through two of the contacts of a Hall plate, which gives rise to a Hall voltage (in combination with an externally applied magnetic field). However, the current in the Hall place also generates its own magnetic field, which can also influence the Hall effect. The mathematical analysis of this secondary influence is rather complicated because the magnetic field caused by a current distribution in a plate turns out to be a three-dimensional problem. However, a crude approximate analysis indicates that the contribution of the currents in the Hall plate to the applied magnetic field is negligible. The situation changes if an alternating magnetic field is applied. Eddy currents are then generated in the Hall plate, creating an additional magnetic field and influencing the original Hall voltage. It is even possible to get a Hall voltage for a zero supply current, the eddy currents only being responsible for the Hall effect. If the frequency of the alternating magnetic field is high, the reaction of the eddy currents on the magnetic field can be important. A theoretical analysis carried out for a circular Hall plate in an ac magnetic field indicates that the parameter Q = 2n f o p o d / R should be considered (f,
POTENTIAL CALCULATIONS IN HALL PLATES
59
frequency; 0 , conductivity; po , permeability; d, thickness; and R, radius) (De Mey, 1976d). If R < 0.1, the contribution of the eddy currents to the magnetic field is negligible. Eddy currents are calculated directly from the applied field, and the Hall voltage is found by integrating the electric field between the Hall contacts. For high values of R, the calculation of the eddy-current pattern constitutes a complicated field problem. For practical purposes (e.g., magnetic field measurements in electrical machines), one is interested in working under the condition R < 0.1. The quantity d/R can then be replaced by the quotient of the thickness and a typical dimension of the Hall plate if the shape is not circular.
APPENDIX 6. LITERATURE Most books on the Hall effect describe the physical aspects of Hall mobility. Putley’s well-known book (1960) describes the galvanomagnetic properties of a large number of semiconductors. The book also provides many references. An excellent work has also been published by Wieder (1979), which describes both the physical nature and the measuring techniques related to the study of galvanomagnetic properties.
ACKNOWLEDGMENTS I wish to thank Professors M. Vanwormhoudt and H. Pauwels for their continuous interest in this work. I am also grateful to my collegues B. Jacobs, K.Stevens, and S. De Wolf, who collaborated on several topics treated in this article. I want also to thank Ms.H. Baele-Riems for careful typing of the manuscript and Mr. J. Bekaert for drawing the figures.
REFERENCES Abramowitz, M., and Stegun, 1. (1965). “Handbook of Mathematical Functions,” pp. 888-890. Dover, New York. Bonfig, K. W., and Karamalikis, A. (1972a). Grundlagen des Halleffektes. Teil 1. Arch. Tech. Mess. 2, 115-118. Bonfig, K. W., and Karamalikis, A. (1972b). Grundlagen des Hall-effektes. Teil 11. Arch. Tech. 3, 137-140. Brebbia, C. (1978a). “The Boundary Element Method for Engineers.” Pentech Press, London. Brebbia, C., ed. (1978b). “Recent Advances in Boundary Element Methods.” Pentech Press, London. Brown, I. C., and Jaswon, E. (1971). “The Clamped Elliptic Plate under a Concentrated Transverse Load,” Res. Memo. City University, London. Bulman, W. E. (1966). Applications of the Hall effect. Solid-State Electron. 9,361-372. Chwang, R., Smith, B., and Crowell, C. (1974). Contact size effects on the Van Der Pauw method for resistivity and Hall coefficient measurements. Solid-Stare Electron. 17, 12171227.
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GILBERT D E MEY
Courant, R.,and Hilbert, D. (1968). “Methoden der Mathematischen Physik I,” pp. 121-124. Springer-Verlag, Berlin and New York. Davidson, R . S., and Gourlay, R. D. (1966). Applying the Hall effect to angular transducers. Solid-State Electron. 9,47 1-484. De Mey, G. (1973a). Field calculations in Hall samples. Solid-Slate Electron. 16,955-957. De Mey, G. (1973b). Influence of sample geometry on Hall mobility measurements. Arch. Elektron. Uebertragungsrech. 27, 309-3 13. De Mey, G .(1973~).Integral equation for the potential distribution in a Hall generator. Electron. Lelt. 9, 264-266. De Mey, G . (1974a). Determination of the electric field in a Hall generator under influence of an alternating magnetic field. Solid-State Electron. 17,977-979. De Mey, G. (1974b). An expansion method for calculation of low frequency Hall etTect and magnetoresistance. Radio Electron. Eng. 44,321-325. De Mey, G. (1975). Carrier concentration in a Hall generator under influence of a varying magnetic field. Phys. Status Solidi A 29, 175- 180. De Mey, G. (1976a). An integral equation approach to A.C. diffusion. In:. J. Heat Mass Transjer 19,702-704. De Mey, G . (1976b). An integral equation method for the numerical calculation of ion drift and diffusion in evaporated dielectrics. Computing 17, 169- 176. De Mey, G. (1976~).Calculation of eigenvalues of the Helmholtz equation by an integral equation. In:. J. Numer. Methods Eng. 10, 59-66. De Mey, G. (1976d). Eddy currents and Hall effect in a circular disc. Arch. Elektron. Uebertragungstech. 30, 312-315. De Mey, G . (1977a). A comment on an integral equation method for diffusion. In!. J. Heat Mass Transjer 20, I8 1 - 182. De Mey, G . (1977b). Numerical solution of a drift-diffusion problem with special boundary conditions by integral equations. Comput. Phys. Commun. 13,81-88. De Mey, G. (1977~).A simplified integral equation method for the calculation of the eigenvalues of Helmholtz equation. In:. J. Numer. Methods Eng. 11, 1340-1342. De Mey, G. (1977d). Hall effect in a nonhomogeneous magnetic field. Solid-State Electron. 20, 139-142. De Mey, G . (1979). Theoretical analysis of the Hall effect photovoltaic cell. IEE Trans. Solid Slate Electron Decices 3,69-71. De Mey, G . (1980). Improved boundary element method for solving the Laplace’ equation in two dimensions. Proc. In/. Semin. Recent Ado. Bound. Elem. Methods, I980 pp. 90- 100. De Mey, G., Jacobs, B., and Fransen, F. (1977). Influence of junction roughness on solar cell characteristics. Electron. Lert. 13, 657-658. De Visschere. P., and De Mey, G. (1977). Integral equation approach to the abrupt depletion approximation in semiconductor components. Eleclron. Lelt. 13, 104- 106. Edwards, T. W., and Van Bladel, J. (1961). Electrostatic dipole moment of a dielectric cube. Appl. Sci. Res. 9, 151-155. Ghosh. S. (1961). Variation of field effect mobility and Hall effect mobility with the thickness of deposited films of tellurium. J. Phys. Chem. Solids 19,61-65. Gray, R. M. (1971). “Toeplitz and Circulant Matrices: A Review,” Tech. Rep. No. 6502-1, pp. 16- 19. Information Systems Laboratory, Stanford University, Stanford, California. Grutzmann, S. (1966). The application of the relaxation method to the calculation of the potential distribution of Hall plates. Solid-State Electron. 9,401-416. Haeusler, J. (1966). Exakte Losungen von Potentialproblemen beim Halleffekt durch konforme Abbildung. Solid-Stare Eleclron. 9,417-441.
POTENTlAL CALCULATlONS 1N HALL PLATES
61
Haeusler, J. (1968). Zum Halleffekt Reaktanzkonverter mit vier Elektroden. Arch. Elektr. Uebertragung 22,258-259. Haeusler, J. ( I 971). Randpotentiale von Hallgeneratoren. Arch. Elekrrotech. (Berlin) 54, 77-81, Haeusler, J., and Lippmann, H. (1968). Hall-generatoren mit kleinem Linearisierungsfehler. Solid-Stare Electron. 11, 173- 182. Hochstadt, H. (1973). “Integral Equations,” Chapters 2 and 6. Wiley, New York. Jaswon, M. A., and Symm, G. T. (1978). “Integral Equation Methods in Potential Theory and Electrostatics.” Academic Press, New York. Kobus, A., and Quichaud, G. (1970). Etude d’un moteur a courant continu sans collecteur a commutation par un generateur a effet Hall en anneau. RGE, Rev. Gen. Electr. 79,235-242. Lippmann, H.,and Kuhrt, F.( l958a). Der Geometrieeinflussaufden transversalen magnetischen Widerstandseffekt bei rechteckformigen Halbleiterplatten. 2.Naturforsch. 13,462-474. Lippmann, H., and Kuhrt, F. (1958b). Der Geometrieeinfluss auf den Hall-effekt bei rechteckformigen Halbleiterplatten. Z. Naturjorsch. 13,474-483. Madelung, 0. (1970). “Grundlagen der Halbleiterphysik,” Chapters 37-39. Springer-Verlag, Berlin and New York. Many, A., Goldstein, Y., and Grover, N. B. (1965). “Semiconductor Surfaces,” p. 138. NorthHolland Publ., Amsterdam. Mei, K.,and Van Bladel, J. (1963a). Low frequency scattering by rectangular cylinders. IEEE Trans. Antennas Propagation AP-I 1,52-56. Mei, K., and Van Bladel, J. (1963b). Scattering by perfectly conducting rectangular cylinders. IEEE Trans. Antennas Propag. AP-11, 185-192. Mimizuka, T. (1971). Improvement of relaxation method for Hall plates. Solid-Stare Electron. 14, 107-110. Mimizuka, T. (1978). The accuracy of the relaxation solution for the potential problem of a Hall plate with finite Hall electrodes. Solid-State Electron. 21, 1195-1 197. Mimizuka, T. (1979). Temperature and potential distribution determination method for Hall plates considering the effect of temperature dependent conductivity and Hall coefficient. Solid-State Electron. 22, 157- 161. Mimizuka, T., and Ito, S. (1972). Determination of the temperature distribution of Hall plates by a relaxation method. Solid-State Electron. IS, 1197-1208. Nalecz, W., and Warsza, Z. L. (1966). Hall effect transducers for measurement of mechanical displacements. Solid-State Electron. 9,485-495. Newsome, J. P. (1963). Determination of the electrical characteristics of Hall plates. Proc. Inst. Electr. Eng. 110, 653-659. Putley, E. H. (1960). “The Hall Effect and Related Phenomena.” Butterworth, London. Shaw, R. (1974). An integral equation approach to diffusion. In:. J . Hear Mass Transjer 17, 693-699. Smith, A,, Janak, J., and Adler, R. (1967). “Electronic Conduction in Solids.” Chapters 7-9. McGraw-Hill, New York. Stevens, K., and De Mey, G . (1978). Higher order approximations for integral equations in potential theory. In:. J. Elecrron. 45,443-446. Symm, G. (1963). Integral equation methods in potential theory. Proc. R. Soc. London 275, 33-46. Symm, G. (1964). “Integral Equations Methods in Elasticity and Potential Theory,” Res. Rep. Natl. Phys. Lab., Mathematics Division, Teddington, U.K. Symm, G., and Pitfield, R. A. (1974). “Solution of Laplace Equation in Two Dimensions,” NPL Rep. NAC44. Natl. Phys. Lab., Teddington, U.K.
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Thompson, D. A., Romankiw, L. T., and Mayadas, A. F. (1975). Thin film magnetoresistors in memory, storage and related applications. IEEE Trans. Magn. MAG-11,1039-1050. Tottenham, H. (1978). "Finite Element Type Solutions of Boundary Integral Equations." Winter School on Integral Equation Methods, City University, London. Van Der Pauw, L. J. (1958). A method of measuring specific resistivity and Hall effect of discs of arbitrary shape. Philips Res. Rep. 13, 1-9. Wick, R. F. (1954). Solution of the field problem of the Germanium gyrator. J . Appl. Phys. 25,741-756.
Wieder, H. H. (1979). "Laboratory Notes on Electrical and Galvanomagnetic Measurements." Elsevier. Amsterdam.