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 carriers 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 VH = 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 12. 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 11 grcatcr than 7t 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
RIII
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 ...
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