Hall effect

Hi, We are discussing physics. remember that!!! I would like you to write me 4 pages talking about those Three articles I have attached. if you think you need other sources please do. what I want is a good paper with a very very great explanation. I do not want to see a collection of an academics words. Try to make your writing as the level as the articles. resistivity, conductivity, hall voltage, directions, and hall effect will be involved here a lot. 1- The subject is related to physics, which is Hall Effect. 2- 4 pages regarding to the three articles. 3- use American journal of physics citation format. ( I will not accept any other citation format). 4- no plagiarism please please please please please. Good luck and show me a good quality, with good explanation.
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1414
November
PROCEEDINGS OF THE I.R.E.
Hall Effect*
OLOF LINDBERGt
Summery–The Hall effect is one of the rich sources of informa
tion about the conduction properties of semiconductors. The mobility
and carrier concentration can be obtained from the Hall constant in
conjunction with the resistivity; this cannot be done with the resistivity alone. The mobility is pertinent to the understanding of
transistors since such things a s high-frequency cut-off and the intrinsic current gain of the transistor are related to thie property of
germanium. The Hall effect and associated thermomagnetic and
galvanomagnetic (Ettingshausen, Nernst, Righi-Leduc, and Ohmic)
effects are discussed The elimination of the effect of associated
phenomena from the Hall measurement can be achieved in several
ways. Some of the methods which are used today in the study of
germanium are discussed, and typical apparatus is described.
INTRODUCTION
HE TRANSISTOR makes use of the special conduction properties of semiconductors to gain the
T advantages it has over the vacuum tube. To
understand the advantages and limitations of the
transistor one must understand these special conduction properties. For example, the high-frequency cut-off
of the type A transistor can be predicted from the injected carrier’s transit time which is related to the
mobility. The intrinsic current gain in the type A
transistor is related to the ratio of the mobility of electrons to the mobility of holes. In dealing with current
flow in semiconductors, one must take account of the
fact that other carriers than electrons may be present.
The nature of these carriers (whether they are electrons
or holes), the number of carriers per unit volume, and
the ease with which they respond to an applied electric field (mobility) are all important quantities to
know. The Hall effect provides a direct measurement
of the carrier type and concentration and, in conjunction with the resistivity, yields the mobility. The
density of carriers in semiconductors is determined in
part by the density of foreign atoms in the material
(for example, aluminum in germanium). Thus the Hall
measurement is used to determine the density of impurity atoms. This measure of impurity density in
high-purity samples is several orders of magnitude more
sensitive than the best chemical procedures. Because of
this, the measurement of the Hall constant is one of the
basic procedures in experimental studies of semiconductors.
The Hall effect occurs when a substance carrying a
current is subjected to a magnetic field perpendicular
to the direction of the current, If the current is flowing
in the x-direction and the magnetic field is applied in
the s-direction, a potential gradient will appear across
the sample in the y-direction. This transverse potential
gradient is found to be proportional to the product of
the current density in the sample and the applied mag* Decimal classification R282.12. Original manuscript received
by the Institute, June 30, 1952.
t Westinghouse
Research Laboratories, Eaet Pittsburgh, Pa.
netic field ; the constant of proportionality is called the
“Hall constant.” Mathematically this can be expressed
as follows:
Grad VII = – RiH.
(1)
Grad VH– (the transverse potential gradient) = – EH,
where EII is the Hall field.
i-the current density.
H-the applied magnetic field.
R-the Hall constant.
If the sample is a rectangular solid of width a and thickness b and if the distribution of current is assumed
uniform, (1) can be rewritten in terms of the total current I, the Hall voltage VH, and the dimensions of the
sample.
– RIB
Grad V”H = k – a
ab
– RIH
VII =-,
b
(2)
The Hall effect can be explained on the basis of the
particle nature of conduction. The current consists of
streams of charged particles drifting under the influence of the electric field. When not under the influence’
of the magnetic field, the current flows longitudinally
in the sample. On application of the magnetic field, the
current carriers experience a force e/c(?xH) and are
swept to the edges of the sample. The charge continues
to build up on the edges of the sample until the field
due to the nonuniform charge distribution exerts a
force equal to the deflecting force of the magnetic field.
L
Fig. l-Rotation of equipotentials by Hall effect, The Hall effect
rotates the equipotentials so that th ey are no longer normal to the
current flow.
Fig. 1 shows that the equipotentials are no longer
normal to the current flow, but have been rotated
through an angle 0, called the “Hall angle.” Examination of the vector diagram of the fields in Fig. 2 (opposite
page) shows that the Hall angle 6 isdetermined by the
following relation:
lf52
Lindberg: Hall Effect
EH
tan 0 = x- E 0
I
3* 1
R=–,
for small angles.
(3)
RiHu
Now En = RiH and E, = 2, thus-? =-= RHa,
u
i
where Q is the conductivity.
Fig. 2-Vector diagram for the Hall effect. The Hall angle 8 is the
angle of rotation of equipotentials. EH is the Hall field if the carriers are e1ectrons.
It is possible to calculate an expression for the Hall
constant on the basis of the particle theory. In the
derivation we assume that only electrons are present;
but if only holes are present, the procedure is analogous.
From the discussion above, the condition for the steady
state is that the deflecting force of the magnetic field
on a current carrier just equals the force exerted by the
transverse electric field due to the charge build-up at
the edges of the sample. This condition can be met
mathematically by setting the y-component of the
electric-field force equal in magnitude but opposite in
sign to the force experienced by a charge moving in the
magnetic field.
V.H.
eElr = e c ’
(4)
i.H*
EH – – , the current density i, = ncv=
ncc
(5)
where n the carrier concentration; i, and II, can be
replaced by i and H since i, = i, = II, = E?,, = 0.
iH
EH = – = RiH
net
1
R =-*
net
1415
(6)
The Hall constant derived by this method is valid
only insofar as the particle picture of conduction is
valid; that is, it applies to simple metals and impure
semiconductors, To obtain an expression of more general applicability the mechanism of conduction must be
examined more closely on the basis of a Boltzmann
distribution of velocity of carriers. The result of such
an investigation gives a relation which is generally used
in semiconductor work when only one type of carrier is
involved.’
1 W. Shockley, “Electrons and Holes in Semiconductors,” D. Van
Nostrand Co., Inc., New York, N. Y., p. 277; 1950.
8 net
From the measurement of the Hall constant, the carrier concentration can be determined. In an impuritytype semiconductor the carrier concentration is determined by the density of the dominant impurity. For
example, in germanium at room temperature the intrinsic carrier concentration is approximately 5 X 1013
carriers per cubic centimetcr, and impurity densities
ten times as great are commonly found: thus it introduces a small error to attribute the entire carrier concentration to impurity atoms. From (6), the carrier concentration is proportional to l/R. This gives a quantitative
measure of the impurity density. The value of this
procedure can be seen from a calculation of the per cent
impurity concentration for an impurity density of
4 X 1016. There are approximately 4 X 1022 atoms of germanium per cubic centimeter; thus concentrations of
one part in 10’ are commonly met. To use chemical
analysis to determine such impurity concentrations is a
hopeless task. The Hall voltage, on the other hand, becomes larger for lower impurity concentration, and
therefore more easily measured.
The carrier concentration does not give the complete
picture of the conduction properties. The conductivity
u is related to the carrier concentration, the charge of
the electron and the ease with which the carriers move
in an electric field (mobility). The latter is defined as
the steady-state average velocity of the conducting
particle (cm/sec) in unit electric field (1 volt/cm). In
germanium the mobility will vary from 1,000 to 3,600
cm/sec per volt/cm. In a semiconductor with both
negative and positive (holes) carriers the conductivity
must be a function of the concentration and mobility of
both holes and electrons.
u = nlelbh + PlelcIp
hh
n-electron concentration (no/cc).
p-hole concentration (no/cc).
I.c,,-electron mobility (up to 3,600 cm*/volt-see in
germanium).
pp-hole mobility (up to 1,700 cm*/volt-set in germanium).
In the range where the concentrations of holes and electrons are about the same this expression for conductivity
must be used. At the operating temperature of a semiconducting device, the concentration of one type carrier
is much greater than the concentration of the other so
that the conductivity will be dependent on only one
term or the other in (8).
For p-type sample : u = p e N,,.
For n-type sample: u = 7t IeI P,,.
Carrier concentration can be determined from the
conductivity if the mobility is known or a value is assumed. The Hall effect gives a method for determining
the mobility and the carrier concentration for both
n-type and p-type semiconductors without assumptions.
November
PROCEEDINGS OF THE I.R.E.
1416
For an n-type sample the Hall constant is negative
since e is negative.
– .7r
3
1
R=–.
8
(9)
4elc
For a P-type sample the Hall constant is positive because the charge of a hole is positive.
31 1
R I—*
8 pet
(10)
The Hall-angle formula differs for holes and electrons
since mobility is different for holes and electrons.
H/h
lt-type: e = c
W
HP%!
p-type: e = c
(12)
effect, the Nernst cffcct, and the Righi-Leduc effect.
These effects will give rise to a temperature gradient or
a potential gradient when either an electric current or a
thermal current is subjected to a magnetic field perpendicular to the direction of current flow. A tcmperature gradient as well as a potential gradient will cause
an error in the Hall-effect mcasurcmcnt. In Fig. 4, junction A is at temperature T2 and junction B is at temperature T1, which is less than 1’2. Since the probe
material is in general not the same as the sample material, the probes and the sample form a thermocouple
and produce a voltage dependent in sign and magnitude
on the materials of the probes and the sample.
Solving (9) and (10) for the carrier concentrations and
(11) and (12) for the mobilities, the following expressions are obtained:
31 -1
3* 1
P =-8 Ret
f$3-8 Ret
‘!I f
In measurements of the Hall voltage, certain associated effects give rise to potentials which must bc
corrected in order to avoid error in the measured value.
The largest of these effects is the potential which appears because of the experimental difficulty in aligning
the measuring probes on the same equipotential plane.
,
1
Fig. 4–Ettingshauscn effect. The Ettinghousen effect c a u s e s the edge
at probe A to be at temperature 1’1 grcatcr than 7’t the tcmperature of the ot her edge.
In the Ettinghousen
effect a p e r m a n t l y maintained temperature grndicnt will appear if an electric
current is subjected to a mngnctic field perpindicular
to its direction of flow. The temperaturc gradient is
found to be proportional to the product of the current
density and the magnetic field.2
?T
– = PIN
a
PIN
A
I
ix–.
ab
(13)
?T = _
I
b
tB
Fig. 3-Source of fR drop error. It is difficult to align the probes A
and B so that no voltage will be measured in the absence of a magnetic field.
If Fig. 3, A and B are probes for measuring the Hall
potential. With no field applied, the equipotentials are
ideally planes perpendicular to the lines of current flow
If probes ;Q and B are not exactly on the same equipotential, a potential will be measured between them,
giving a constant error to the Hall voltage measured.
This voltage can easily be of the order of magnitude of
the Hall voltage itself. This effect is sometimes referred
to as the ‘IR drop.” The IR drop is dependent only on
the current and the conductivity of the sample, not
being affected by a reversal of magnetic field.
In addition to the IR drop, there are three thermomagnetic or galvanomagnetic effects, the Ettingshausen
Ai”- the cliffcrcnce of tcmpcraturc bctwccn the edges
of the sample.
i-current density.
I-total current.
II-magnetic field perpendicular to the tlircction of
current flow.
P-Ettingshauscn coefficient. ’
u-width of sample.
b-thickness of sample.
As in the case of the Hall effect, uniform current distribution and a rectangular solid sample have been
assumed.
The Nernst effect and the Righi-Lcduc effect are
similar to the Hall effect and the Ettingshauscn effect
except that they arc produced by a thermal current
and a perpendicular magnetic field rather than an elec* P. W. BridEman, “The Thermodynamics of Electrical Phenomena in Metals,” Macmillan Co., New York, N. Y., pp. 135-138;
1934.
.
.
1

I.
1
’
, !952
Lindberg: Hall Efecl
1417
The Righi-Leduc effect produces a transverse temperature gradient when a longitudinal thermal current
flows in a magnetic field. As in the Ettingshausen effect
a transverse thermal current will flow from the cold
edge to the hot edge while the temperature gradient is
being built-up. The thcrmomotive force set up by this
transverse thermal current (Righi-Leduc effect) is in
opposition to the longitudinal thermal current, and is
&-Nernst transverse potential gradient.
therefore able to extract energy from the thermal curw-thermal current density.
rent and maintain itself.
I‘–thermal conductivity of the sample.
As has been explained above, the IX drop, the
@-Ncrnst coefficient.
Ettingshausen effect, the Ncrnst effect, and the RighiThe Righi-Leduc effect produces a temperature
Leduc effect all result in potentials at the probes
gradient in the y-direction when a thermal current flows measuring the Hall potential. Consequently, to measure
in the x-direction, and a magnetic field is applied in the Hall constant, these sources of error must be eliminated
z-direction.*
or minimized. The most common method of making Hall
WU
measurements utilizes dc magnetic fields and dc curAT
(15) rents. The experimental setup is illustrated in Fig. 4.
a = 57
The probes A and B for measuring the Hall voltage are
AT-difference in the temperature between the placed in contact with the sample as shown. The curedges of the sample.
rent leads are soldered to the ends of the sample. A dc
S-Righi-Lcduc coefficient
magnetic field is applied in the e-direction and a curU, w, M, I<--as previously defined. rent flows in the x-direction. If we consider the probes Associated with a thermal current, there exists an to be perfectly aligned on an equipotential surface at no electron current. When two ends of a sample are at dif- magnetic field and we also assume that there is no ferent temperatures, the high-velocity electrons at the longitudinal thermal current flowing, then the potential hot end will diffuse toward the cold end; but for each mcasurcd at /i and H will be the sum of the Hall high-velocity electron that diffuses away, a low-velocity potential and the thermocouple potential due to the Ettingshauscn temperature gradient. electron must take its place so that a uniform charge distribution will bc maintained. v, = Vm + vsrA satisfactory description of the mechanism of the where Vg, the thermocouple potential due to the Ettingshausen effect on the basis of the classical theory Ettingshausen effect, and VU, the Hall potential, are of conduction by particles has not been given. However, both dependent on the current and the magnetic field. following a method given by Bridgman,3 some justifica- The only apparent way to separate the two in a dc tion of the Ettingshauscn effect can bc made on a measurement is to make the probes out of the same thermodynamic basis. Where an electric current flows material as the sample, and this is usually difficult if longitudinally in a material and is subjected to a mag- not impossible in the case of semiconductors. In making nctic field perpendicular to the current flow, a transverse dc Hall measurements in germanium, I/g is usually temperature gradient builds up because of a “thcrmo- considered to be of the order of 5 per cent, and is motive force.” This temperature gradient is maintained neglected. as long as the current and field are maintained. When From the following considerations it is apparent that the temperature gradient builds up, a transverse thermal a longitudinal thermal current will flow. When a longicurrent must flow from the cold edge to the hot edge. tudinal electric current flows, there will be a Peltier The energy required for the thermomotive force to drive effect at the junctions where the current leads arc the thermal current against the temperature gradient soldered to the sample. One junction will become heated must be supplied by the electric current in the sample. and the other coo1cd, and a temperature difference in Because of the Ncrnst effect, the transverse thermal the x-direction will be maintained so long as the current current produces a longitudinal potential; the electrical Rows in the junctions. Such a temperature gradient can current flowing against this potential provides the also occur due to nonuniform temperature of the sample energy to build up and maintain the Ettingshausen or its surroundings. In general, the combination of these temperature gradient. effects can add up to an important longitudinal temperature difference, and consequently, a longitudinal a In this discussion the assumption has been made that the thermal current and the electric current are independent of each otter. In thermal current will probably exist. Because of the reality a part of ‘the total thermal current arises from the electric field presence of such a thermal current, the Ncrnst effect present and similarly a part of the total electric current arises from the temperature gradient present. This interdependence is compliand the Righi-Leduc effect should also be considered. cated and it is beyond the scope of this article to discuss this effect. The potential E measured at the probes A and B will be The electric current as used here is current as would be measured by a meter and the thermal current is that determined by the temperature gradient and the thermal conductivity. tric current and a perpendicular magnetic field.8 In the Nernst effect, a potential gradient appears in the ydirection if a thermal current flows in the x-direction and a magnetic field is applied in the z-direction.* ’ A I V’~=potential due to Nernst effect. VRL = thermocouple potential due to the Righi-Leduc temperature gradient. If the sample current is reversed and the potential at the probes, A and B, is remeasured, then the potential will be E, = - VH - VE + VN + VRL. E - E , = 2(Vlf + E - El VIf + VE I-* 2 VN + RIN - VII = -, b R V,l = - I’ sin cd.. sin cd b RII’I’ RWI’ VII = - - - cos 2wf. 2b 26 Ve) (16) To take account of all first-order associated effects, the IR drop must also be considered. If the probes, A and B, were not properly aligned on equipotentials at no magnetic field (as is always the case when the experiment is performed), then the voltage measured at the probes A and B would be E = VII + Vs + These objections are overcome by making the measurements with an ac field and an ac sample current. Let the field and the sample current be the same frequency. The Hall potential from (2) is Substitute the instantaneous value of 1 and II in (2); Since the Hall and the Ettingshausen effects are directly related to the current and the Nernst and the RighiLeduc are independent of current, only the former change sign when the current is reversed. Subtract El from E; I November PROCEEDINGS OF THE I.R.E. 1418 VRL + VrR. VIR is the potential due to IR drop. I’,s is a function of the current but not the field. VM and Vg are dependent on both ... Purchase answer to see full attachment

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