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The decrease in the hall angle and the sensitivity of silicon magnetic sensors at high electric fields
225
Sensors and Actuators, 20 (1989) 22.5-232
The Decrease in the Hall Angle and the Sensitivity of Silicon Magnetic Sensors at High Electric Fields* P. J. A. MUNTER and S. KORDIC** Electronic Instrumentation Laboratory, Department of EIecrrical Engineering, De@ Unfvetsity of Technology, P.O. Box 5031, 2600 GA Derft fThe Nethetla~~ (Received November 4, 1988; in revised form April 20, 1989; accepted May 11, 1989)
Abstract
A theory of the decrease in the sensitivity and the Hall angle in silicon magnetic sensors is presented. The mean collision-free time decreases in the presence of a strong electric field, which results in a smaller Hall angle. The total deflection of the charge carriers in a vertical dual-collector bipolar ma~etotransistor is the sum of the contributions of the deflection in the collector-base depletion layer and the deflection in the ohmic collector region. The sensitivity of a mag~etotrausistor decreases due to the decrease in the Hall angle in the depletion layer and the increased power dissipation and temperature of the device. The sensitivity decreases by 25% when the collector-base voltage is increased by 10 V.
been neglected in favour of the deflection in the depletion layer.) By contrast, measurements on a magnetotransistor [4-71 indicate a Hall angle and sensitivity reduction for an increasing collector-base voltage Ucb, which probably takes place in the depletion layer of the ma~eto~nsistor where the electric field is the largest. The ma~eto~ansistor used in the measurements is depicted in Fig. 1. The buried layer ~derneath the bipolar npn transistor is spiit in two and the current, which mainly flows down into the bulk, can be deflected by the magnetic field to either one of the collector contacts. The current difference A& is [4,fl: ~~~=~~~~B~~~=2~tane,irl;
(1)
The Hall angle 0,+ can be defined as [ 1,2]: 1. Intr~uctioll
tan&=p,B=rszB
To date, the theory of carrier deflection by a magnetic field has been developed well for low values of the electric field only [ 1,2]. However, silicon magnetic-field-sensitive devices, such as ma~etotransistors and caper-domain devices, have significant portions of the device (depletion layers) that operate at high electric fields. One’s first intuitive reaction is to assume that the deflection of current increases with a rising electric field, because charge-carrier velocities in silicon increase as well. The larger velocity, it is thought, results in a higher Lorentz force being exerted on the charge carriers so that the defiection of the current and, therefore, the Hall angle should consequently increase. (In ref. 3 the deflection in the ohmic part of the collector has even
with L’ the effective deflection length, W, the effective emitter width, 8, the Hall angle, &# the Hall mobility, B the magnetic induction, r the Hall factor, a the common-base current gain and ZE the emitter current. The collector-base bias voltage U,,, however, does not explicitly appear in the expression for the output signal of the vertical magnetotransistor. The reduction in sensitivity must, therefore, come about through dependence of one of the variables in eqn. ( 1) on U,,. ~eas~ements have shown that the 01dependency on U,, is so slight that it cannot significantly influence the sensitivity of the magnistor. Also, in our magnetotransistor structure with the base contacts ‘on top’ of the emitter, the crowding of the current along the emitter-base junction cannot infiuence the sensitivity. The width W, of the emitter and the emitter current (constant current bias) are independent of U,,, which leaves the Hall mobility JJ~ = rp and the deflection length L’ as the possible candidates [4]. A simple qualitative explanation of the Hall angle decrease at high electric fields has been given
*Paper presented at EUROSENSORS ‘87, First European Confereace on Sensors and their Applications, Cambridge, U.K., September 22-24, 1987. **Present address: Philips Resew& Laboratories Eindltovea, Nederlandse Philips Bedtijven B.V., P.O. Box 80000, 5600 JA Eindhovea, The Netherlands.
@ Elsevier Sequoia/Printed
(2)
in The Netherlands
226
ze
p- substrate
Fig. 1. Cross-section of a doublecollector (one-dimensional) vertical magnetotransistor. L’ is the effective deflection length.
in ref. 6. If we for the moment assume that the mean collision-free time ‘t is independent of the applied electric field E, (which is the case for low field values), from eqn. (2) we can see that the Hall angle is independent of the electric field. This can also be seen from Fig. 2, which depicts the trajectory of an electron at two values of the electric field for a constant collision-free time r. In other words, if the mean collision-free time is independent of the electric field, an increase in the collector-base bias will not change the sensitivity of the device because the Hall angle remains constant. The mean collision-free time is, however, dependent on the carrier energy. From Fig. 3 and eqn. (2) we can see that the result of a reduced collision-free time 7 is the reduction of the Hall angle which, in turn, means that above a certain value of the electric field in the depletion region of the magnetotransistor, the sensitivity of the device to magnetic fields decreases. It is not sufficient to consider the drift mobility ,u instead of the Hall mobility, as the Hall factor r depends on the electric field as well [8]. However, ref. 8 cannot be used here because the galvanomagnetic coefficients are given only for certain restricted electric fields ranges, which do not include the intermediate electric fields. In order to calculate the total Hall angle in the device, the
-Y
F 2
Fig. 2. The trajectory of an electron for different values of the electric field. The mean collision-free time is kept constant (independent of the applied electric field).
Fig. 3. Above a certain value of the electric field, the mean collision-free time starts decreasing. The result is that the Hall angle also decreases.
expression for the Hall mobility we are looking for should be valid at the high electric fields in the depletion layer and the low electric fields in the ohmic layer, as well as in the intermediate electric field range. As will be shown later on, a more straightforward and concise approach is to start from single expressions for the Hall angle and the electron energy distribution function that are valid in the whole electric field range of interest. A derivation of the sensitivity and the Hall angle in the magnetotransistor will be given. First, the electric field strength and the distribution of electrons in relation to the electron temperature will be presented. Secondly, the Hall angle will be derived as a function of the electron temperature (the electron temperature is a direct measure for the electric field strength). An expression that combines the Hall angle in the ohmic layer and in the depletion layer will be used to find the total Hall angle. Finally, the sensitivity will be derived using the total Hall angle and the effects of the temperature rise of the device due to increased power dissipation. The results of the study have been presented elsewhere [4,9]. 2. Theory of the Hall Angle in Current Deflection Sensors Electrons in a magnetotransistor pass consecutively through the emitter, base and collector regions. Current deflection in the emitter and base regions is negligible because of the high impurity concentration in the emitter and in the thin base, which means that the ohmic collector region and the collector-base depletion layer determine the sensitivity of our magnetotransistor.
The Electric Field Strength The thickness of the depletion layer id and the maximum value of the electric field E_ in the
227
collector region as a function of the bias voltage UC, are given by
r, = 1 E
=
2(ubi
.xm
+
ucb)
Li
li2
(3)
f(t) dt = 5 exp( -t/r)
(4)
with t the relaxation time. If we assume that the average velocity just after the collision is zero, which is the case for a randomizing collision process, we can calculate the average velocity for small magnetic fields (w$* Q 1):
Ubi is the built-in voltage. With a base impurity ~on~ntration NA of about 10” cmw3 and an epilayer concentration ND of about 6 x lOI cmP3, the depletion layer will extend mainly into the epilayer. For a reverse bias UC,of 10 V, & is about 1.5 pm and E_ becomes 1.4 x lO’Vm-‘. This value of the electric field is well above lo5 V m-’ at which the decrease. of the mobility, and hence the decrease of the mean collision-free time, sets in [IO]. In comparison, the electric field strength in the ohmic part of the expitaxial layer is much smaller, i.e., Eti = 2.2 x lo4 V m-r. The elevated electric field in the depletion layer will influence the energy distribution of the electrons. Given an electron concentration that is high enough to dominate the energy relaxation of the hot electrons, their energy dist~bution is Maxwellian [ 11, 121:
f@J = G ew( - Elb, T,>
(5)
with &, Bolzmann’s constant and T, the electron temperature, which is defined as a measure for the average kinetic energy of the electrons. An approximate formula for the electron temperature, which is reasonably accurate for the warm region and the onset of the hot region of the mobility characteristic, is given by [ 13, 141: T,=
T,+aE,
The probabili~ that an electron will collide between times t and I + dt is given by [ 151: dt
(10)
m
(%> =
s
v~(~)~(~) dt = - -f+Ex c
0
(11) ccl
(vy>=
f
v,,(r)f(t) dt = - f
w,r’E, C
0
with
where m, is the conductivity effective mass of electrons [ 161. These expressions are a function of energy, because t is energy dependent. Using the fact that the number of electrons is electric field dependent too [ 12, 16], the cm~nt-density components become
(13)
(6)
with a = 1.27 x 10v4 Km V-‘. The Hall angle in magnetotransistors and other sensors based on current deflection is defined as ‘(see Figs. 1 and 2): tan 0, = J,/J,
(7)
J=
63)
-en
where (v) is the average carrier velocity, J, and Jy the current densities parallel and perpendicular to the electric field, respectively. Starting from the equations of electron motion in crossed electric and magnetic fields, the x and the y components of the electron velocity can be calculated:
dJ, and dJ,, are current-density components consisting of electrons with energy E. Integrating over all energies with the Maxwe~an dist~bution as a weighing function, eqn. (5), yields J,=E,
(14) J,, = E, $
ocn
co
C, E312exp( - E/kbT,)
n =
s
dE
(15)
0
E v,(t) = - f sin o,t
Z
EX(cos C&t YJf) = F 1) z
(9)
with n the total number of electrons in the eonduction band and C, a constant. The energy averages of t(E) and r*(E) are given by
228 cc (r(E))
= -$
T(E)E3”
exp( -E/k,,T,)
dE
6
I 0
c
m
(t’(E))
=
$
exp( -E/k,T,)
THEY’D
dE
(16)
4-
s 0
n’ = n/C,
2-
The Hall angle, eqn. (7), can now be written tan
0
= H
B
e e*(E))
as (17)
Imc (49)
0
The quotient of the average relaxation times determines the electric field dependence of the Hall angle. The Average Relaxation Time The collision-free time T as a function of electron energy is given by the following expression [17]:
Fig. 4. The functions q(T,/O) ture.
2
g(E) =
and g(T,/@)
at room tcmpcra-
- l)E’/* + w2a,
x [E + kb@)‘/* + exp(O/T,)(E
wza, =2r
(20) At room temperature g(E) can be approximated by the following function
0
g(E) = -0.19E
(T) = 3 [exp(O/T,) - l] n’ oc E312 exp( -E/k, T,) dE x (exp(O/T,) - l)E”‘+ w,a,[(E + kbO)‘/* + exp(@/T,)(E - k,,O)“‘] (19) and
+ 0.266k,,O
kbO < E < 1.2k,,O
g(E) = 0.02E + O.O136k,O
1.2k,@
The average relaxation 4): (r)
= lo-*(k,T,)*
[ 4,181 (Fig.
time becomes
f$ [exp(O/T,)
- Ilp(T,/O)
,$TJ@)=[7+(-26+0.6g)exp(-s) +(21-O.O4:)exp(-1.2;)]
(22)
The (r *) function can be approximated in a similar manner to that shown above (Fig. 4): (‘*)
be solved analytically,
E
g(E) = 0.07E
I
cannot
- k,O)‘/*]
(18)
0 is the characteristic temperature of the phonons (630 K), To is a reference temperature that can be chosen to fit the magnitude of 5, while w, relates the strength of coupling of the electrons to acoustic phonons. The E ‘I* term in this expression is due to the acoustic intravalley scattering, while the second term is due to the f-type intervalley scattering, and it can be divided into two parts. The first part (E + k,,O) ‘I* is due to the absorption of energy by the electron from the lattice. The second part (E - k,,O)“* describes the transfer of energy from the electrons to phonons. This transfer can only occur if the electron energy E is higher than kbO. If this is not the case, the (E - k,O) II2 term is zero. If we substitute eqn. (18) in (z), eqn. (16), we obtain
This equation
4
~312
(exp(O/T,)
(E+kbO)“2+exp(O/T,)(E-k,~)1’2 exp(O/T,) - 1
a = (kb To)3’2 1 w,kb TI
3
we are forced to approximate the integrand without the exponent g(E) with a simpler expression
al
+w
2 cle
1 ElJ2 -=7
1
(kt,T,)* a: = 1o-3 ~(k&)1/2 n’ [exp(@/T,)
q(TJ@)=[6.2+(-45.2-0.2g)exp(-
-
112dTe/@) E)
+(39.05-l.O9$exp(-l.l;)]
The expression eqn. (17):
(23) for tan 0” can now be calculated,
229
(24) 6.2 + ( - 45.2 - 0.2 -=r -26+0.6
7+ [
(
The quotient of the two functions q~~~/~~ and p(TJO) is given in Fig. 5, From this Fikure it can be seen that the Hall angle decreases with rising electron temperature, and, therefore, with a rising electric field (see eqn. (6)).
We can represent the ~ll~tor-~~e~t difference AI, caused by the total deflection in the depiction layer by
(25) 0
Now that the electric field dependence of the Hall angle has been derived, the Hall angle needs to be determined as a function of the position in the magnetotransistor. The electric field in the depletion layer is not a constant, but it is a function of the position within this layer. As a consequence, the Hall angle is also a function of the position in the depletion layer. The minimum value of f?, is at the collector-base junction, where the electric field is the strongest (see Fig. 6). &$ will gradually increase further away from the junction as the electric field decreases, and it will reach a maximum in the ohmic part of the epitaxial collector region.
tan eHd = J_ tan OHdx ld 1 --id
(26)
with e,, the equivalent Hall angle in the depletion layer. The contribution of the ohmic region deflection Al, is
where L’ is the total deflection distance. With the total collector-current difference AZ, (see Fig. 7) given by AI, = AId i- AIn
(281 we obtain after substitution of eqns. (25) and (27)
Fig. 5. q(T,/@)/p(T,/O) as a fitnction of T# at room temperature. The Hall mobility electric field dependence is proportional to this ratio.
Fig. 6. A sketch of the trajectory of an electron passing through the depletion layer ( -/, G x d 0) and the ohmic co%?ctor region ( - .&’C x < -&). 8,, is the equivalent Hall angle in the depletion layer, wbib @,, is the Hall angle in the Ohmic coll&or region.
230
tan 13,~ we have to resort to an approximation as eqn. (24) cannot be integrated analytically in eqn. (26). A reasonable approximation of tan &_, is
x
I depletion
layer
tan8,=A,T;ii2
T, < 0
tan 8, = 25.9A, T;’
O
tan 0” = 0.7754, T;‘12
T,>20
A, = 0.0484 a,w, kk,2 [exp(O/T,)
(30)
- l]
After integration of eqn. (26) over the possible values for the electric field, this yields for the Hall angle in the depletion layer
‘=-e
+ 0.77[( T, + aE,)
Fig. 7. Contribution of the ohmic and the depletion the total deflection of the collector current.
into eqn. (28) and introduction angle 8,,
layers to
I” - (T, + aE,,) 1’2] (31) > with the integral boundaries E,, and E,, corresponding to the interval boundaries for the electron temperature T, as chosen in eqn. (30):
of the total Hall I?,,=(@--T,)/a=3OO/aK E,, = (20 - Tl)/a = 960/a K
tan 8,, = tan 8,o - 2
[tan 0,,
- tan
4-d
The total Hall angle can now be calculated using the above results for the average Hall angles in the depletion and the ohmic layers. If we assume No = 6 x 1015 cmP3 in the epitaxial collector region and L’ = 6 pm (values used in the experimental devices), we obtain for the total Hall angle tan 0,,( UC, = 0.6 V) = 46.94’
A’=2Y
I
I
I
I
I
I
2
4
6
8
IO
12
(32)
3. Comparison of the Theory to the Experiments
tan f?,,(V,,
I
for T, = 20
(29)
This equation shows the effect of the depletion layer and the depletion layer width on the total Hall angle. We find that the total Hall angle will decrease for an increasing depletion layer width Id. The total Hall angle will also decrease when the Hall angle in the depletion layer, &, decreases due to the increased electric fields in this layer. At this point we can combine all previous results to calculate the total Hall angle. The Hall angle in the Ohmic layer can be obtained by setting T, = Tl in eqn. (24). For the calculation of
0
for T,=O
I
u,, IV1
Fig. 8. Calculated Hall angle ratio as a function of the collector-base bias at room temperature for A’,, = 6 x lOIs crrm3 and L' = 6 pm.
= 10.6 V) = 41.1A’
(33)
A,
This indicates a drop in the sensitivity of 12.4% when UC, is increased from 0.6 V to 10.6 V. The constant factor A’ contains the reference temperature To, which can be chosen to fit the measurements [ 131. The decrease in the Hall angle in the depletion layer with respect to the Hall angle in the ohmic layer is given as a function of UC, in Fig. 8. From experimental results we find that an increase in UC, from 0.6 to 10.6 V reduces the sensitivity of the magnetotransistor by 20.8%, 26.0% and 28.5% for emitter currents of 10 mA,
231
2OmA
Fig. 9. Calculated Wall angle ratio as a Functionof the collector-base bias for No= 6 x 11Pcm-~ and L’ =iipm. 1, is used as a .parameter.
2 a temperature dependence of -0.7%/K of the ma~etotransistor ~nsiti~ty has been measured. With a magnistor thermoresistance of 150\K W-i f4,9] the calculated results, including power dissipation, are presented in Fig. 9. For Z, of 10 mA, 15 mA and 20 mA the calculated decrease in the sensitivity is 21.1%, 25.6% and 31.6% respectively, which agrees much better with the measurements. We know that the IIall angle at 1 T in the ohmic part of the collector region is about 0.14, so that constant a, in eqns. ( 18) and (24) becomes 3.95 x 10ez3 s J-‘j2. Finally, if we write the sensitivity of the magnetotransistor as the collectorcurrent difference AZ, per tesla, see eqn. ( 1): S = AIJB = A,p.,(T,
15 mA and 20 mA respectively. The first discrepancy with the calculated results is the magnitude of the s~nsiti~ty decrease. According to the measurements, the decrease in the sensitivity is also dependent on the value of the emitter current of the magn~totransistor. Both of these discrepancies can be accounted for by the influence of the increased temperat~e of the device due to power dissi~tion. The power dissipated in the magnetotransistor is equivalent to the Z,U, product, where U, is the collector-e~tter bias voltage of the device (the base current has been neglected). The dissipated power will result in a temperature increase AT of the device with respect to the surroundings. In ref.
U,,)I,
(34) as a geometry facWith A, =0.182Vsm-2T-’ tor, we can calculate the sensitivity of the magnetotransistor as a function of the emitter current and collector-base voltage. The calculated sensitivity curves are compared to the measurements in Fig. 10. 4. Conclusioas It has been shown that the decrease in the sensitivity of a magnetotransistor is caused by two effects. (1) A significant portion of the device consists of a collector-base depletion layer in which high electric fields are present. High electric fields in the depletion layer reduce the mean collision-free time, and the Hall angle with it. To calculate this effect, the Hall angle has been determined as a function of the mean collision-free time and the electron temperature. The electric field in the depletion layer is not a constant but is a function of the position, which makes the Hall angle in the depletion layer position de~ndent, too. The equivalent Hall angle in the depletion layer has been determined, as well as the total equivalent Hall angle of the device. (2) A decrease in the Hall angle at high electric fields is not enough to describe fully the measurements. The temperature of the device rises with UC, and Z, due to power dissipation, which is accompanied by a further drop in the sensitivity of the magnetotransistor. The combined results of the above two effects agree reasonably well with the measurements.
References Fig. 10. Comparison of the cahlated surements.
sensitivity to the mea-
1 S. Kordic, Integrated silicon magnetic-field sensors, Sensors and Actuators,
10 (1986)
347-378,
232 2 V. Zieren, Integrated silicon multicollector magnetotransistors, Ph.D. Thesis, Delft University of Technology, 1983. 3 M. Ishida, H. Fujiwara, T. Nakamura, M. Ashiki, Y. Yasuda, A. Yoshida, T. Ohsakama and Y. Kawase. Silicon magnetic vector sensors for integration, Proc. 4th Sensor Svmv., Javan, 1984. DD. 79-83. 4 S: Kordic, d&t reduction and 3-D field sensing with magnetotransitors, Ph.D. Thesis, Delft University of Technology, 1987. 5 S. Kordic, V. Zieren and S. Middlehoek, A magnetic-fieldsensitive multicollector transistor with low o&t, Int. Electron Devices Meet., Washington, DC, U.S.A., 1983, IEDM Tech. Digest, pp. 631-634. 6 S. Kordic and P.C.M. van der Jagt, Theory and practice of the electronic implementation of the sensitivity-variation offset-reduction method, Sensors and Acfuators, 8 (1985) 197-217. 7 S. Kordic and P. J. A. Munter, Three-dimensional magnetic-field sensors, IEEE Trans. Electron Devices, ED-35 (1988) 771-779. 8 Z. S. Kachlishvili, Galvanomagnetic and recombination effects in semiconductors in a strong electric field, fhys. Status Solidi (a), 33 (1976) 15-51. 9 P. J. A. Munter and S. Kordic, The decrease in the Hall angle and the sensitivity at high electric fields in silicon magnetic sensors, Tech. Digest, Eurosensors, ‘87, First European Conf. on Sensors and their Application, Cambridge, U.K., 1987, pp. 200-201. 10 D. L. Scharfetter and H. K. Gummel, Large-signal analysis of a silicon Read diode oscillator, IEEE Trans. Electron Devices ED-16 (1969) 64-77. 11 B. R. Nag, Electron Transport in Compound Semiconductors, Springer, Berlin, 1980, pp. 312-332. 12 R. A. Smith, Wave Mechanics of Crystalline Soliak. Chanman and Hall, London, 2nd edn., 1969, pp. 355-357. _ 13 P. K. Basu and B. R. Nag, Room temperature conductivity anisotropy and population redistribution in n-type silicon at high electric fields, Phvs. Rev. B, 1 (1970) 627-631. 14 C. UT Duh and J. L. Mall, Temperature dependence of hot electron drift velocity in silicon at high electric field, SofidState Electron., II (1968) 917-930.
15 H. Weiss, Structure and Application of Galvanomagnetic Devices, Pergamon, Oxford, 1969, pp. 1-6. 16 R. A. Smith, Semiconductors, Cambridge University Press, Cambridge, 2nd edn., 1978, pp. 135-140. 17 D. Long, Scattering of conduction electrons by lattice vibrations in silicon, Phys. Rev., 1.20 (1960) 2024-2032. 18 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals. Series, and Products, Academic Press, New York, 4th edn., 1980, pp. 248-256.
Biographies Peter Munter was born in Doetinchem, The Netherlands, on March 17, 1964. He spent the first half of 1988 working at IC Sensors, Milpitas, CA before gaining an M.Sc. (cum laude) in electrical engineering from the Delft University of Technology, The Netherlands. He is currently working towards his Ph.D. at Delft; his thesis is to deal with magnetic-field sensors. Srdjan Kordic was born in Belgrade, Yugoslavia, on February 27, 1958. Following a leave of absence at the IBM Thomas J. Watson Research Center, Yorktown Heights, NY, where he worked on the simulation of latch-up in CMOS structures, he returned to The Netherlands to gain an M.Sc. at Delft University of Technology in 1983. He received his Ph.D. in 1987; his thesis dealt with offset-reduction and three-dimensional field sensing in magnetotransistors. At present he is working at the Philips Research Laboratories in Eindhoven, The Netherlands.
Sensors and Actuators, 20 (1989) 22.5-232
The Decrease in the Hall Angle and the Sensitivity of Silicon Magnetic Sensors at High Electric Fields* P. J. A. MUNTER and S. KORDIC** Electronic Instrumentation Laboratory, Department of EIecrrical Engineering, De@ Unfvetsity of Technology, P.O. Box 5031, 2600 GA Derft fThe Nethetla~~ (Received November 4, 1988; in revised form April 20, 1989; accepted May 11, 1989)
Abstract
A theory of the decrease in the sensitivity and the Hall angle in silicon magnetic sensors is presented. The mean collision-free time decreases in the presence of a strong electric field, which results in a smaller Hall angle. The total deflection of the charge carriers in a vertical dual-collector bipolar ma~etotransistor is the sum of the contributions of the deflection in the collector-base depletion layer and the deflection in the ohmic collector region. The sensitivity of a mag~etotrausistor decreases due to the decrease in the Hall angle in the depletion layer and the increased power dissipation and temperature of the device. The sensitivity decreases by 25% when the collector-base voltage is increased by 10 V.
been neglected in favour of the deflection in the depletion layer.) By contrast, measurements on a magnetotransistor [4-71 indicate a Hall angle and sensitivity reduction for an increasing collector-base voltage Ucb, which probably takes place in the depletion layer of the ma~eto~nsistor where the electric field is the largest. The ma~eto~ansistor used in the measurements is depicted in Fig. 1. The buried layer ~derneath the bipolar npn transistor is spiit in two and the current, which mainly flows down into the bulk, can be deflected by the magnetic field to either one of the collector contacts. The current difference A& is [4,fl: ~~~=~~~~B~~~=2~tane,irl;
(1)
The Hall angle 0,+ can be defined as [ 1,2]: 1. Intr~uctioll
tan&=p,B=rszB
To date, the theory of carrier deflection by a magnetic field has been developed well for low values of the electric field only [ 1,2]. However, silicon magnetic-field-sensitive devices, such as ma~etotransistors and caper-domain devices, have significant portions of the device (depletion layers) that operate at high electric fields. One’s first intuitive reaction is to assume that the deflection of current increases with a rising electric field, because charge-carrier velocities in silicon increase as well. The larger velocity, it is thought, results in a higher Lorentz force being exerted on the charge carriers so that the defiection of the current and, therefore, the Hall angle should consequently increase. (In ref. 3 the deflection in the ohmic part of the collector has even
with L’ the effective deflection length, W, the effective emitter width, 8, the Hall angle, &# the Hall mobility, B the magnetic induction, r the Hall factor, a the common-base current gain and ZE the emitter current. The collector-base bias voltage U,,, however, does not explicitly appear in the expression for the output signal of the vertical magnetotransistor. The reduction in sensitivity must, therefore, come about through dependence of one of the variables in eqn. ( 1) on U,,. ~eas~ements have shown that the 01dependency on U,, is so slight that it cannot significantly influence the sensitivity of the magnistor. Also, in our magnetotransistor structure with the base contacts ‘on top’ of the emitter, the crowding of the current along the emitter-base junction cannot infiuence the sensitivity. The width W, of the emitter and the emitter current (constant current bias) are independent of U,,, which leaves the Hall mobility JJ~ = rp and the deflection length L’ as the possible candidates [4]. A simple qualitative explanation of the Hall angle decrease at high electric fields has been given
*Paper presented at EUROSENSORS ‘87, First European Confereace on Sensors and their Applications, Cambridge, U.K., September 22-24, 1987. **Present address: Philips Resew& Laboratories Eindltovea, Nederlandse Philips Bedtijven B.V., P.O. Box 80000, 5600 JA Eindhovea, The Netherlands.
@ Elsevier Sequoia/Printed
(2)
in The Netherlands
226
ze
p- substrate
Fig. 1. Cross-section of a doublecollector (one-dimensional) vertical magnetotransistor. L’ is the effective deflection length.
in ref. 6. If we for the moment assume that the mean collision-free time ‘t is independent of the applied electric field E, (which is the case for low field values), from eqn. (2) we can see that the Hall angle is independent of the electric field. This can also be seen from Fig. 2, which depicts the trajectory of an electron at two values of the electric field for a constant collision-free time r. In other words, if the mean collision-free time is independent of the electric field, an increase in the collector-base bias will not change the sensitivity of the device because the Hall angle remains constant. The mean collision-free time is, however, dependent on the carrier energy. From Fig. 3 and eqn. (2) we can see that the result of a reduced collision-free time 7 is the reduction of the Hall angle which, in turn, means that above a certain value of the electric field in the depletion region of the magnetotransistor, the sensitivity of the device to magnetic fields decreases. It is not sufficient to consider the drift mobility ,u instead of the Hall mobility, as the Hall factor r depends on the electric field as well [8]. However, ref. 8 cannot be used here because the galvanomagnetic coefficients are given only for certain restricted electric fields ranges, which do not include the intermediate electric fields. In order to calculate the total Hall angle in the device, the
-Y
F 2
Fig. 2. The trajectory of an electron for different values of the electric field. The mean collision-free time is kept constant (independent of the applied electric field).
Fig. 3. Above a certain value of the electric field, the mean collision-free time starts decreasing. The result is that the Hall angle also decreases.
expression for the Hall mobility we are looking for should be valid at the high electric fields in the depletion layer and the low electric fields in the ohmic layer, as well as in the intermediate electric field range. As will be shown later on, a more straightforward and concise approach is to start from single expressions for the Hall angle and the electron energy distribution function that are valid in the whole electric field range of interest. A derivation of the sensitivity and the Hall angle in the magnetotransistor will be given. First, the electric field strength and the distribution of electrons in relation to the electron temperature will be presented. Secondly, the Hall angle will be derived as a function of the electron temperature (the electron temperature is a direct measure for the electric field strength). An expression that combines the Hall angle in the ohmic layer and in the depletion layer will be used to find the total Hall angle. Finally, the sensitivity will be derived using the total Hall angle and the effects of the temperature rise of the device due to increased power dissipation. The results of the study have been presented elsewhere [4,9]. 2. Theory of the Hall Angle in Current Deflection Sensors Electrons in a magnetotransistor pass consecutively through the emitter, base and collector regions. Current deflection in the emitter and base regions is negligible because of the high impurity concentration in the emitter and in the thin base, which means that the ohmic collector region and the collector-base depletion layer determine the sensitivity of our magnetotransistor.
The Electric Field Strength The thickness of the depletion layer id and the maximum value of the electric field E_ in the
227
collector region as a function of the bias voltage UC, are given by
r, = 1 E
=
2(ubi
.xm
+
ucb)
Li
li2
(3)
f(t) dt = 5 exp( -t/r)
(4)
with t the relaxation time. If we assume that the average velocity just after the collision is zero, which is the case for a randomizing collision process, we can calculate the average velocity for small magnetic fields (w$* Q 1):
Ubi is the built-in voltage. With a base impurity ~on~ntration NA of about 10” cmw3 and an epilayer concentration ND of about 6 x lOI cmP3, the depletion layer will extend mainly into the epilayer. For a reverse bias UC,of 10 V, & is about 1.5 pm and E_ becomes 1.4 x lO’Vm-‘. This value of the electric field is well above lo5 V m-’ at which the decrease. of the mobility, and hence the decrease of the mean collision-free time, sets in [IO]. In comparison, the electric field strength in the ohmic part of the expitaxial layer is much smaller, i.e., Eti = 2.2 x lo4 V m-r. The elevated electric field in the depletion layer will influence the energy distribution of the electrons. Given an electron concentration that is high enough to dominate the energy relaxation of the hot electrons, their energy dist~bution is Maxwellian [ 11, 121:
f@J = G ew( - Elb, T,>
(5)
with &, Bolzmann’s constant and T, the electron temperature, which is defined as a measure for the average kinetic energy of the electrons. An approximate formula for the electron temperature, which is reasonably accurate for the warm region and the onset of the hot region of the mobility characteristic, is given by [ 13, 141: T,=
T,+aE,
The probabili~ that an electron will collide between times t and I + dt is given by [ 151: dt
(10)
m
(%> =
s
v~(~)~(~) dt = - -f+Ex c
0
(11) ccl
(vy>=
f
v,,(r)f(t) dt = - f
w,r’E, C
0
with
where m, is the conductivity effective mass of electrons [ 161. These expressions are a function of energy, because t is energy dependent. Using the fact that the number of electrons is electric field dependent too [ 12, 16], the cm~nt-density components become
(13)
(6)
with a = 1.27 x 10v4 Km V-‘. The Hall angle in magnetotransistors and other sensors based on current deflection is defined as ‘(see Figs. 1 and 2): tan 0, = J,/J,
(7)
J=
63)
-en
where (v) is the average carrier velocity, J, and Jy the current densities parallel and perpendicular to the electric field, respectively. Starting from the equations of electron motion in crossed electric and magnetic fields, the x and the y components of the electron velocity can be calculated:
dJ, and dJ,, are current-density components consisting of electrons with energy E. Integrating over all energies with the Maxwe~an dist~bution as a weighing function, eqn. (5), yields J,=E,
(14) J,, = E, $
ocn
co
C, E312exp( - E/kbT,)
n =
s
dE
(15)
0
E v,(t) = - f sin o,t
Z
EX(cos C&t YJf) = F 1) z
(9)
with n the total number of electrons in the eonduction band and C, a constant. The energy averages of t(E) and r*(E) are given by
228 cc (r(E))
= -$
T(E)E3”
exp( -E/k,,T,)
dE
6
I 0
c
m
(t’(E))
=
$
exp( -E/k,T,)
THEY’D
dE
(16)
4-
s 0
n’ = n/C,
2-
The Hall angle, eqn. (7), can now be written tan
0
= H
B
e e*(E))
as (17)
Imc (49)
0
The quotient of the average relaxation times determines the electric field dependence of the Hall angle. The Average Relaxation Time The collision-free time T as a function of electron energy is given by the following expression [17]:
Fig. 4. The functions q(T,/O) ture.
2
g(E) =
and g(T,/@)
at room tcmpcra-
- l)E’/* + w2a,
x [E + kb@)‘/* + exp(O/T,)(E
wza, =2r
(20) At room temperature g(E) can be approximated by the following function
0
g(E) = -0.19E
(T) = 3 [exp(O/T,) - l] n’ oc E312 exp( -E/k, T,) dE x (exp(O/T,) - l)E”‘+ w,a,[(E + kbO)‘/* + exp(@/T,)(E - k,,O)“‘] (19) and
+ 0.266k,,O
kbO < E < 1.2k,,O
g(E) = 0.02E + O.O136k,O
1.2k,@
The average relaxation 4): (r)
= lo-*(k,T,)*
[ 4,181 (Fig.
time becomes
f$ [exp(O/T,)
- Ilp(T,/O)
,$TJ@)=[7+(-26+0.6g)exp(-s) +(21-O.O4:)exp(-1.2;)]
(22)
The (r *) function can be approximated in a similar manner to that shown above (Fig. 4): (‘*)
be solved analytically,
E
g(E) = 0.07E
I
cannot
- k,O)‘/*]
(18)
0 is the characteristic temperature of the phonons (630 K), To is a reference temperature that can be chosen to fit the magnitude of 5, while w, relates the strength of coupling of the electrons to acoustic phonons. The E ‘I* term in this expression is due to the acoustic intravalley scattering, while the second term is due to the f-type intervalley scattering, and it can be divided into two parts. The first part (E + k,,O) ‘I* is due to the absorption of energy by the electron from the lattice. The second part (E - k,,O)“* describes the transfer of energy from the electrons to phonons. This transfer can only occur if the electron energy E is higher than kbO. If this is not the case, the (E - k,O) II2 term is zero. If we substitute eqn. (18) in (z), eqn. (16), we obtain
This equation
4
~312
(exp(O/T,)
(E+kbO)“2+exp(O/T,)(E-k,~)1’2 exp(O/T,) - 1
a = (kb To)3’2 1 w,kb TI
3
we are forced to approximate the integrand without the exponent g(E) with a simpler expression
al
+w
2 cle
1 ElJ2 -=7
1
(kt,T,)* a: = 1o-3 ~(k&)1/2 n’ [exp(@/T,)
q(TJ@)=[6.2+(-45.2-0.2g)exp(-
-
112dTe/@) E)
+(39.05-l.O9$exp(-l.l;)]
The expression eqn. (17):
(23) for tan 0” can now be calculated,
229
(24) 6.2 + ( - 45.2 - 0.2 -=r -26+0.6
7+ [
(
The quotient of the two functions q~~~/~~ and p(TJO) is given in Fig. 5, From this Fikure it can be seen that the Hall angle decreases with rising electron temperature, and, therefore, with a rising electric field (see eqn. (6)).
We can represent the ~ll~tor-~~e~t difference AI, caused by the total deflection in the depiction layer by
(25) 0
Now that the electric field dependence of the Hall angle has been derived, the Hall angle needs to be determined as a function of the position in the magnetotransistor. The electric field in the depletion layer is not a constant, but it is a function of the position within this layer. As a consequence, the Hall angle is also a function of the position in the depletion layer. The minimum value of f?, is at the collector-base junction, where the electric field is the strongest (see Fig. 6). &$ will gradually increase further away from the junction as the electric field decreases, and it will reach a maximum in the ohmic part of the epitaxial collector region.
tan eHd = J_ tan OHdx ld 1 --id
(26)
with e,, the equivalent Hall angle in the depletion layer. The contribution of the ohmic region deflection Al, is
where L’ is the total deflection distance. With the total collector-current difference AZ, (see Fig. 7) given by AI, = AId i- AIn
(281 we obtain after substitution of eqns. (25) and (27)
Fig. 5. q(T,/@)/p(T,/O) as a fitnction of T# at room temperature. The Hall mobility electric field dependence is proportional to this ratio.
Fig. 6. A sketch of the trajectory of an electron passing through the depletion layer ( -/, G x d 0) and the ohmic co%?ctor region ( - .&’C x < -&). 8,, is the equivalent Hall angle in the depletion layer, wbib @,, is the Hall angle in the Ohmic coll&or region.
230
tan 13,~ we have to resort to an approximation as eqn. (24) cannot be integrated analytically in eqn. (26). A reasonable approximation of tan &_, is
x
I depletion
layer
tan8,=A,T;ii2
T, < 0
tan 8, = 25.9A, T;’
O
tan 0” = 0.7754, T;‘12
T,>20
A, = 0.0484 a,w, kk,2 [exp(O/T,)
(30)
- l]
After integration of eqn. (26) over the possible values for the electric field, this yields for the Hall angle in the depletion layer
‘=-e
+ 0.77[( T, + aE,)
Fig. 7. Contribution of the ohmic and the depletion the total deflection of the collector current.
into eqn. (28) and introduction angle 8,,
layers to
I” - (T, + aE,,) 1’2] (31) > with the integral boundaries E,, and E,, corresponding to the interval boundaries for the electron temperature T, as chosen in eqn. (30):
of the total Hall I?,,=(@--T,)/a=3OO/aK E,, = (20 - Tl)/a = 960/a K
tan 8,, = tan 8,o - 2
[tan 0,,
- tan
4-d
The total Hall angle can now be calculated using the above results for the average Hall angles in the depletion and the ohmic layers. If we assume No = 6 x 1015 cmP3 in the epitaxial collector region and L’ = 6 pm (values used in the experimental devices), we obtain for the total Hall angle tan 0,,( UC, = 0.6 V) = 46.94’
A’=2Y
I
I
I
I
I
I
2
4
6
8
IO
12
(32)
3. Comparison of the Theory to the Experiments
tan f?,,(V,,
I
for T, = 20
(29)
This equation shows the effect of the depletion layer and the depletion layer width on the total Hall angle. We find that the total Hall angle will decrease for an increasing depletion layer width Id. The total Hall angle will also decrease when the Hall angle in the depletion layer, &, decreases due to the increased electric fields in this layer. At this point we can combine all previous results to calculate the total Hall angle. The Hall angle in the Ohmic layer can be obtained by setting T, = Tl in eqn. (24). For the calculation of
0
for T,=O
I
u,, IV1
Fig. 8. Calculated Hall angle ratio as a function of the collector-base bias at room temperature for A’,, = 6 x lOIs crrm3 and L' = 6 pm.
= 10.6 V) = 41.1A’
(33)
A,
This indicates a drop in the sensitivity of 12.4% when UC, is increased from 0.6 V to 10.6 V. The constant factor A’ contains the reference temperature To, which can be chosen to fit the measurements [ 131. The decrease in the Hall angle in the depletion layer with respect to the Hall angle in the ohmic layer is given as a function of UC, in Fig. 8. From experimental results we find that an increase in UC, from 0.6 to 10.6 V reduces the sensitivity of the magnetotransistor by 20.8%, 26.0% and 28.5% for emitter currents of 10 mA,
231
2OmA
Fig. 9. Calculated Wall angle ratio as a Functionof the collector-base bias for No= 6 x 11Pcm-~ and L’ =iipm. 1, is used as a .parameter.
2 a temperature dependence of -0.7%/K of the ma~etotransistor ~nsiti~ty has been measured. With a magnistor thermoresistance of 150\K W-i f4,9] the calculated results, including power dissipation, are presented in Fig. 9. For Z, of 10 mA, 15 mA and 20 mA the calculated decrease in the sensitivity is 21.1%, 25.6% and 31.6% respectively, which agrees much better with the measurements. We know that the IIall angle at 1 T in the ohmic part of the collector region is about 0.14, so that constant a, in eqns. ( 18) and (24) becomes 3.95 x 10ez3 s J-‘j2. Finally, if we write the sensitivity of the magnetotransistor as the collectorcurrent difference AZ, per tesla, see eqn. ( 1): S = AIJB = A,p.,(T,
15 mA and 20 mA respectively. The first discrepancy with the calculated results is the magnitude of the s~nsiti~ty decrease. According to the measurements, the decrease in the sensitivity is also dependent on the value of the emitter current of the magn~totransistor. Both of these discrepancies can be accounted for by the influence of the increased temperat~e of the device due to power dissi~tion. The power dissipated in the magnetotransistor is equivalent to the Z,U, product, where U, is the collector-e~tter bias voltage of the device (the base current has been neglected). The dissipated power will result in a temperature increase AT of the device with respect to the surroundings. In ref.
U,,)I,
(34) as a geometry facWith A, =0.182Vsm-2T-’ tor, we can calculate the sensitivity of the magnetotransistor as a function of the emitter current and collector-base voltage. The calculated sensitivity curves are compared to the measurements in Fig. 10. 4. Conclusioas It has been shown that the decrease in the sensitivity of a magnetotransistor is caused by two effects. (1) A significant portion of the device consists of a collector-base depletion layer in which high electric fields are present. High electric fields in the depletion layer reduce the mean collision-free time, and the Hall angle with it. To calculate this effect, the Hall angle has been determined as a function of the mean collision-free time and the electron temperature. The electric field in the depletion layer is not a constant but is a function of the position, which makes the Hall angle in the depletion layer position de~ndent, too. The equivalent Hall angle in the depletion layer has been determined, as well as the total equivalent Hall angle of the device. (2) A decrease in the Hall angle at high electric fields is not enough to describe fully the measurements. The temperature of the device rises with UC, and Z, due to power dissipation, which is accompanied by a further drop in the sensitivity of the magnetotransistor. The combined results of the above two effects agree reasonably well with the measurements.
References Fig. 10. Comparison of the cahlated surements.
sensitivity to the mea-
1 S. Kordic, Integrated silicon magnetic-field sensors, Sensors and Actuators,
10 (1986)
347-378,
232 2 V. Zieren, Integrated silicon multicollector magnetotransistors, Ph.D. Thesis, Delft University of Technology, 1983. 3 M. Ishida, H. Fujiwara, T. Nakamura, M. Ashiki, Y. Yasuda, A. Yoshida, T. Ohsakama and Y. Kawase. Silicon magnetic vector sensors for integration, Proc. 4th Sensor Svmv., Javan, 1984. DD. 79-83. 4 S: Kordic, d&t reduction and 3-D field sensing with magnetotransitors, Ph.D. Thesis, Delft University of Technology, 1987. 5 S. Kordic, V. Zieren and S. Middlehoek, A magnetic-fieldsensitive multicollector transistor with low o&t, Int. Electron Devices Meet., Washington, DC, U.S.A., 1983, IEDM Tech. Digest, pp. 631-634. 6 S. Kordic and P.C.M. van der Jagt, Theory and practice of the electronic implementation of the sensitivity-variation offset-reduction method, Sensors and Acfuators, 8 (1985) 197-217. 7 S. Kordic and P. J. A. Munter, Three-dimensional magnetic-field sensors, IEEE Trans. Electron Devices, ED-35 (1988) 771-779. 8 Z. S. Kachlishvili, Galvanomagnetic and recombination effects in semiconductors in a strong electric field, fhys. Status Solidi (a), 33 (1976) 15-51. 9 P. J. A. Munter and S. Kordic, The decrease in the Hall angle and the sensitivity at high electric fields in silicon magnetic sensors, Tech. Digest, Eurosensors, ‘87, First European Conf. on Sensors and their Application, Cambridge, U.K., 1987, pp. 200-201. 10 D. L. Scharfetter and H. K. Gummel, Large-signal analysis of a silicon Read diode oscillator, IEEE Trans. Electron Devices ED-16 (1969) 64-77. 11 B. R. Nag, Electron Transport in Compound Semiconductors, Springer, Berlin, 1980, pp. 312-332. 12 R. A. Smith, Wave Mechanics of Crystalline Soliak. Chanman and Hall, London, 2nd edn., 1969, pp. 355-357. _ 13 P. K. Basu and B. R. Nag, Room temperature conductivity anisotropy and population redistribution in n-type silicon at high electric fields, Phvs. Rev. B, 1 (1970) 627-631. 14 C. UT Duh and J. L. Mall, Temperature dependence of hot electron drift velocity in silicon at high electric field, SofidState Electron., II (1968) 917-930.
15 H. Weiss, Structure and Application of Galvanomagnetic Devices, Pergamon, Oxford, 1969, pp. 1-6. 16 R. A. Smith, Semiconductors, Cambridge University Press, Cambridge, 2nd edn., 1978, pp. 135-140. 17 D. Long, Scattering of conduction electrons by lattice vibrations in silicon, Phys. Rev., 1.20 (1960) 2024-2032. 18 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals. Series, and Products, Academic Press, New York, 4th edn., 1980, pp. 248-256.
Biographies Peter Munter was born in Doetinchem, The Netherlands, on March 17, 1964. He spent the first half of 1988 working at IC Sensors, Milpitas, CA before gaining an M.Sc. (cum laude) in electrical engineering from the Delft University of Technology, The Netherlands. He is currently working towards his Ph.D. at Delft; his thesis is to deal with magnetic-field sensors. Srdjan Kordic was born in Belgrade, Yugoslavia, on February 27, 1958. Following a leave of absence at the IBM Thomas J. Watson Research Center, Yorktown Heights, NY, where he worked on the simulation of latch-up in CMOS structures, he returned to The Netherlands to gain an M.Sc. at Delft University of Technology in 1983. He received his Ph.D. in 1987; his thesis dealt with offset-reduction and three-dimensional field sensing in magnetotransistors. At present he is working at the Philips Research Laboratories in Eindhoven, The Netherlands.