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Determination of the electric field in a Hall generator under influence of an alternating magnetic field
Solid-State
Electronics,
1974, Vol.
17. pp. 917-979.
Pergamon
Press.
Printed
in Great
Britain
DETERMINATION OF THE ELECTRIC FIELD IN A HALL GENERATOR UNDER INFLUENCE OF AN ALTERNATING MAGNETIC FIELD Laboratory
G. of Electronics,
DE MEY Ghent State University,
Belgium
(Received 17 October 1973; in revised fotm 2 February
1974)
Abstract-A method is described to calculate the electric field in a Hall generator under alternating magnetic field. The problem is divided into two separate problems, the describes the Hall effect while the second gives the behaviour of the eddy currents. problems can be reduced to an integral equation, which is easily solved on a digital method is illustrated for a rectangular Hall generator.
influence of an first of which Each of these computer. The
The calculation of the potential distribution in Hall generators is very important for Hall mobility measurements [ 1,2] and magnetic field strength measurements[3]. The field caused by a constant magnetic field in a semicbnducting sample has been the subject of several articles [4-81. For special applications the magnetic field varies with time[9]. The calculation of the potential becomes then more involved. This problem has been treated analytically for simple geometries by De Sabata[lO]. We present here a numerical method, based on an integral equation technique, which enables us to solve the problem for arbitrary geometries. We consider a plane n-type semiconductor sample (Fig. 1) placed in a time dependent homogeneous magnetic field B, directed perpendicular to the plane of the sample. The fundamental equations for the electric field B and the current density J are:
vx&_aB v.J=
at
0
cx=P+~LH(JxI3)
(1)
Fig. 1. Semiconducting
magne-
(2)
potential function. (B = Bii,):
(3) and the
where u and pH denote the conductivity Hall mobility of the semiconductor. The magnetic field B(t) is a known quantity. We assume that the current density _i gives a negligible contribution to the magnetic field. We can then use a scalar potential 4 defined by:
J = u(Vc$ x ii,).
sample in time dependent tic field f?.
From
(l),
(2) and
(4) we get
At the free boundaries cz and cq (Fig. l), the current density vector J should be tangent. Due to (4), this yields the following boundary condition for the potential 4 :
(4) C$= V, 4 = Vz
It should be mentioned that the potential C$ is not the electrostatic potential, because due to (1) the electric field e cannot be derived from a scalar
where 977
V, and
on on
Vz are constants.
cq cz
(6) (7) The current
sup-
GILBERT DE MEY
978
plied at a metallic boundary c, or ci is given by the potential difference of the two adjacent free boundaries: I = (+( v, - V,).
C
(8)
At the metallic boundaries c, and ci, the electric field J? is directed normally. In connection with (3) and (4) this gives the following boundary condition for C#J: V$.~~,+~~BVC$J.~,=O
on c, and c,.
(9)
It should be emphasised that the potential 4 as defined by (4) may be a varying function along a metallic boundary. The potential 4 is now completely determined by the equation (5) and the boundary conditions (6) (7) and (9). We try a solution of the form: 4 =(#J,+&
V?$, = 0
(11)
4, = V,
on
cr
(12)
f$, = V?
on
cI
(13)
(15)
& = 0
on cz and cq
(16)
zz
at
.1 I=1 Q&=
v,=-0,.
0
0.1
0,2
0.3
0.6
o,e
0.9
B
(h)
Fig. 2. (a) Current lines showing the pure Hall effect in a rectangular sample. (b) Eddy currents caused by alternating magnetic field in rectangular sample. The given values for #J?correspond to a sample of unit dimensions (1 x 1m) and alslat = 8.1 V/m’. V~$,.ti,+p~BV&.ii,=O
onc,andc,
(17)
4, describes the Hall effect caused by a time dependent magnetic field, whereas C#J? describes the behaviour of the eddy currents in the Hall generator. For the moment we have only to solve the equations for 4, and & for a given value of pLHB and a unit value of aBlat, because & depends linearly on aBlat. The resulting potential 4 is then given by:
0.4
j
~~
oar
on c, and c1 (14)
2E
VZ&
0,2
(10)
where
VI$, . U, + P~BVC#J,. R, = 0
x ‘=
-0,s
For the rectangular sample presented on Fig. 2. d, and & have been calculated numerically by transforming (1 l)-(14) and (15)-(17) into equivalent integral equations. This technique is described in the literature[l l-131. Figures 2(a) and (b) represent equipotential curves for D = I. pHB = 1 and aBlat = 8.1. As shown by (4), these curves are current lines. Both figures should be superimposed to give the true current distribution. REFERENCES 1. L. J. van
(a)
der Pauw, Phil. Res. Rep. 13, 1 (1958). 2. G. de Mey, Arch. Elektron. Uehertragungstechnik, 27. 309 (1973). 3. J. Haeusler and H. Lippmann, Solid-St. Electron., 11, 173 (1968). 4. R. Bonnefille, G. Chevalier and Q. Guichard, Solid-St. Electron. 11, 141 (1968).
Electric field in a Hall generator 5. 6. 7. 8. 9.
G. Chevalier, Rev. gCn. I’ilectricitt 12, 1178 (1968). J. L. Becquevort, Solid-St. Electron. 11, 147 (1968). G. de Mey Solid-St. Electron. 16, 955 (1973). R. F. Wick, J. appl. Phys. 25, 741 (1954). A. M. Hermann and J. S. Ham, Reo. Sci. Inst. 36, 1553 (1965).
979
10. I. de Sabata, Reu. E. 6, 355 (1971). 11. J. van Bladel, Electromagnetic fields, p. 92. McGraw Hill, New York (1964). 12. T. W. Edward and J. van Bladel, Appl. Sci. Res. 9, 151 (1961). 13. G. de Mey, Electron. Letts 9, 264 (1973).
Electronics,
1974, Vol.
17. pp. 917-979.
Pergamon
Press.
Printed
in Great
Britain
DETERMINATION OF THE ELECTRIC FIELD IN A HALL GENERATOR UNDER INFLUENCE OF AN ALTERNATING MAGNETIC FIELD Laboratory
G. of Electronics,
DE MEY Ghent State University,
Belgium
(Received 17 October 1973; in revised fotm 2 February
1974)
Abstract-A method is described to calculate the electric field in a Hall generator under alternating magnetic field. The problem is divided into two separate problems, the describes the Hall effect while the second gives the behaviour of the eddy currents. problems can be reduced to an integral equation, which is easily solved on a digital method is illustrated for a rectangular Hall generator.
influence of an first of which Each of these computer. The
The calculation of the potential distribution in Hall generators is very important for Hall mobility measurements [ 1,2] and magnetic field strength measurements[3]. The field caused by a constant magnetic field in a semicbnducting sample has been the subject of several articles [4-81. For special applications the magnetic field varies with time[9]. The calculation of the potential becomes then more involved. This problem has been treated analytically for simple geometries by De Sabata[lO]. We present here a numerical method, based on an integral equation technique, which enables us to solve the problem for arbitrary geometries. We consider a plane n-type semiconductor sample (Fig. 1) placed in a time dependent homogeneous magnetic field B, directed perpendicular to the plane of the sample. The fundamental equations for the electric field B and the current density J are:
vx&_aB v.J=
at
0
cx=P+~LH(JxI3)
(1)
Fig. 1. Semiconducting
magne-
(2)
potential function. (B = Bii,):
(3) and the
where u and pH denote the conductivity Hall mobility of the semiconductor. The magnetic field B(t) is a known quantity. We assume that the current density _i gives a negligible contribution to the magnetic field. We can then use a scalar potential 4 defined by:
J = u(Vc$ x ii,).
sample in time dependent tic field f?.
From
(l),
(2) and
(4) we get
At the free boundaries cz and cq (Fig. l), the current density vector J should be tangent. Due to (4), this yields the following boundary condition for the potential 4 :
(4) C$= V, 4 = Vz
It should be mentioned that the potential C$ is not the electrostatic potential, because due to (1) the electric field e cannot be derived from a scalar
where 977
V, and
on on
Vz are constants.
cq cz
(6) (7) The current
sup-
GILBERT DE MEY
978
plied at a metallic boundary c, or ci is given by the potential difference of the two adjacent free boundaries: I = (+( v, - V,).
C
(8)
At the metallic boundaries c, and ci, the electric field J? is directed normally. In connection with (3) and (4) this gives the following boundary condition for C#J: V$.~~,+~~BVC$J.~,=O
on c, and c,.
(9)
It should be emphasised that the potential 4 as defined by (4) may be a varying function along a metallic boundary. The potential 4 is now completely determined by the equation (5) and the boundary conditions (6) (7) and (9). We try a solution of the form: 4 =(#J,+&
V?$, = 0
(11)
4, = V,
on
cr
(12)
f$, = V?
on
cI
(13)
(15)
& = 0
on cz and cq
(16)
zz
at
.1 I=1 Q&=
v,=-0,.
0
0.1
0,2
0.3
0.6
o,e
0.9
B
(h)
Fig. 2. (a) Current lines showing the pure Hall effect in a rectangular sample. (b) Eddy currents caused by alternating magnetic field in rectangular sample. The given values for #J?correspond to a sample of unit dimensions (1 x 1m) and alslat = 8.1 V/m’. V~$,.ti,+p~BV&.ii,=O
onc,andc,
(17)
4, describes the Hall effect caused by a time dependent magnetic field, whereas C#J? describes the behaviour of the eddy currents in the Hall generator. For the moment we have only to solve the equations for 4, and & for a given value of pLHB and a unit value of aBlat, because & depends linearly on aBlat. The resulting potential 4 is then given by:
0.4
j
~~
oar
on c, and c1 (14)
2E
VZ&
0,2
(10)
where
VI$, . U, + P~BVC#J,. R, = 0
x ‘=
-0,s
For the rectangular sample presented on Fig. 2. d, and & have been calculated numerically by transforming (1 l)-(14) and (15)-(17) into equivalent integral equations. This technique is described in the literature[l l-131. Figures 2(a) and (b) represent equipotential curves for D = I. pHB = 1 and aBlat = 8.1. As shown by (4), these curves are current lines. Both figures should be superimposed to give the true current distribution. REFERENCES 1. L. J. van
(a)
der Pauw, Phil. Res. Rep. 13, 1 (1958). 2. G. de Mey, Arch. Elektron. Uehertragungstechnik, 27. 309 (1973). 3. J. Haeusler and H. Lippmann, Solid-St. Electron., 11, 173 (1968). 4. R. Bonnefille, G. Chevalier and Q. Guichard, Solid-St. Electron. 11, 141 (1968).
Electric field in a Hall generator 5. 6. 7. 8. 9.
G. Chevalier, Rev. gCn. I’ilectricitt 12, 1178 (1968). J. L. Becquevort, Solid-St. Electron. 11, 147 (1968). G. de Mey Solid-St. Electron. 16, 955 (1973). R. F. Wick, J. appl. Phys. 25, 741 (1954). A. M. Hermann and J. S. Ham, Reo. Sci. Inst. 36, 1553 (1965).
979
10. I. de Sabata, Reu. E. 6, 355 (1971). 11. J. van Bladel, Electromagnetic fields, p. 92. McGraw Hill, New York (1964). 12. T. W. Edward and J. van Bladel, Appl. Sci. Res. 9, 151 (1961). 13. G. de Mey, Electron. Letts 9, 264 (1973).