THE ELECTRICAL PROPERTIES OF SEMI CONDUCTORS

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• Collection: General CIA Records
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• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 164
• Document Creation Date: December 23, 2016
• Document Release Date: April 1, 2014
• Sequence Number: 3
• Publication Date: October 1, 1959
• Content Type: REPORT

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TRflSLRTIOfl
THE ELECTRICAL PROPERTIES OF SEMICONDUCTORS
By S. S. Shalyt
October 1959
157 Pages
STAT
PREPARED BY
LIAISON OFFICE
TECHNICAL INFORMATION CENTER
MCLTD
WRICHT-PATTERSON AIR FORCE BASE. OHIO
STAT
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STAT
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Elektricheskiye Svoystva Poluprovodnikov
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Contents
Introduction
Chapter I. General Treatment of Conductivity in Solids. Concentration and
Mobility of Current Carriers.
Chapter II. The Sign of the Current-Carrier Charge: Electrons and Holes
Chapter III. Intrinsic and Impurity Conductivities of Semiconductors
Chapter IV. Temperature-Dependence of Semiconductor Conductivity
The Influence of Electromagnetic Radiation (Light) on the Conduct-
ivity of Semiconductors.
Chapter VI. The Influence of Strona Rlor+vir
VialAQ nn -Eh= rnnAnn+4,74+.,
Semiconductors.
Chapter VII. The Influence of Various Types of Corpuscular Radiation on the
Conductivity of aaMicdnductors.
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Chapter VIII, The Influence of Deformation of Semiconductors on their
Conductivity.
Chapter IX. On the Conductivity of Liquid, Amorphous, and Polycrystalline
Bodies.
Chapter X. The Electrical Properties of Boundary Layers of Semiconductors.
7 SaFETJT57073717E
Elecirieal:Properties of Semiconductors
1 1 Technichl Editor: D.' P. Freger.
.1.
Iff ouse for Scientific and Technical Propaganda in Leningrad (LDNTP), Nevskiy pr. 58.
. ;
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INTRODUCTION
One of the foremost among the scientific-technical problems which absorb the
attention of research scientists and working engineers at the present time is the
problem of semiconductors. This problem consists in study of the special physical
properties which appear in the class of substances known as semiconductors, as well
as development of technological applications for these materials.
Of the physical properties which distinguish seniconductors as a special class
of solids, their electrical properties are the most important. Indeedlit is the
utilization of the specific electrical properties of semiconductors which has re-
cently resulted in progress in technical fields such as the rectification of lo-
and. high-voltage high-voltage alternating currents, the amplification and conversion of electro-
magnetic signals (radio engineering), the monitoring, regulation and automation of
industrial processes, heat engineering, and electric-power engineering on both
large and small scales.
The class of semiconductors comprises chemically heterogeneous substances
characterized by definite electrical properties peculiar to them. In the majority
intae-s-t'b Lmnces-represent nothing new from the standpoint of chemi-
cal composition.
However, the electrical properties of these materials are of such great in-
terest and importance that the physicists, whose attention was attracted to them
1several decades ago, prosecuted rapid and thorough investigations of the field,
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assembled, a large body of new informationland were soon indicating prospects for
numerous technological applications of semiconductors.
It would be wrong to think, however, that the increased interest exhibited by
scientists and engineers in semiconductors at the present time is based solely upon
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the fact that their properties were not previously known. It would be possible
to cite a long series of historical examples which demonstrate that the former
neglect of semiconductor materials for use in technology was based on reasons other
than a lack of knowledge on the part of scientists of some of their important
properties. We need refer only to the thermoelectric properties of metallic and
nonmetallic conductors, which were investigated as much as a hundred years ago.
While the thermoelectric properties of metals have long since become the property
of measurement technique (thermocouples), the outstanding thermoelectric properties
of semiconductors have been utilized for measuring purposes and for conversion of
heat energy into electrical energy only in relatively recent times. Most curious
in this particular case is the fact that the materials used in contemporary thermo-
electric generators are just those which were investigated in the first half of the
last century. A gap of halt a century between the discovery date of an effect and
the date of its utilization in practice may also be noted in the history of semi-
conductor rectifier and photocell development.
Such a state of affairs may be 'explainedon the One hand by the fact that the
present period is an epoch of tumultuous scientific and technological progress, and,
on- the other as a result of the fact that only contemporary technology has presented
needs-which can best be satisfied by the use of semiconductors.
' 'It should-alsn bp-nriti.A.fh=+wcrerkb1c
eleLt IcuL properties or semi-
conductors have been discovered only recently, and that they are still far' from
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having been exhaustively exploited for technological purposes.
The majority of semiconductors belong neither to the class of substances rare-
ly encountered in nature nor to the category of products of new and complex
,
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chemical syntheses. The materials which interest us here are available in prac-
tically unlimited quantities, since the iajority. of solid minerals with which we
-
deal in nature are essential* semiconductors.' 'Oxides, sudea; tellurides and
selenides of -numerous metals, as' well as certain intermetallic compounds, possess
,
typical semiconductor properiiei,
Do any of the chemical elements, in the pure state, possess semiconductor
properties? No positive answer to this question was available until comparatively
recently. This state of affairs has been considerably clarified as a result of a
,
tremendous volume of research, both in the direction of obtaining 'chemically pure
elements, and that of the study of their properties. It has been found, for ex-
ample, that silicon and tellurium, both of which were formerly considered to be
metals, are typical semiconductors in the pure state. As it developed, such in-
sulators as sulphur, phosphorus and iodine must be included in the category of
semiconductor substances. Figure lpresents that part ofMendeleyeris periodic system
in which elements with semiconducting properties are concentrated. It may be seen
from their arrangement in the table that they form a compact group of elements
which occupies an intermediate position between the metals and the nonconductors:
located to the left of and below the group of semiconductor elements are the highly
1 conductive metals, for which the weak bond between valence electrons and 'atoms
characteristic; and to the right of and above this group are those elements
in the solid state, behave as insulators characterized by considerable affinity to
2
electrons, i.e.la tendency to acquire the latter in the course of chemical reactions
with metals.
.imil.r 0,1.0"vmant may be nImervPd within the iRolated armn_of_elements
Itself: the elements located in the upper right side--C (diamond), PI S and I--are
'essentially nonconductors, and fall into this group because of their high photo-
!
electric conductivity, which is a characteristic of all semiconductors; the elements
In the lower left corner-- Sn and Sb--behave as metals in their stable modifications,
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Iand semiconductor properties are exhibit0, only by specially prepared films of
these elements. This location of the semiconductors in the periodic table is not 1
I accidental. It will be shown below that as is true for all solids, the electrical,
properties of semiconductors are determined in the last analysis by the structure
of the outermost electron shells of the atoms, while the periodic completion of
electron shells is the basic factor which determines the chemical properties of the
elements, and, consequently, their location in the periodic table. In a large
circle to the right of the symbol for each semiconductor element (Fig. 1) appears
Fig. 1. Elements with semiconductor properties as they are arranged in
the periodic table.
a value basically characteristic of the element's electric properties; its physical
significance will be explained below. (Width of forbidden band; seepages 42- 43).
The regularity of the variation of this number with respect to both the columns and
lines of the table indicates a close association between the electrical character-
istics of semiconductors and their location in the Mendeleyev system. These elec-
:tricalcharacteristics are .compared in the same table to valuesvhichpresent, in
?
nhp.mTg-hry_ a rough measlurp O. -1" --F ++ +4?. a--ranon ofl_thp valprIpp. salon-Ernnm ..h.n crone.
acting atom, and; are called electrongativities of the elements(small circles on
,
:the left). , There is no doubt as to the marked,correlation between these values.
? ..-?, :
,
Ebiiever, the fact that the same element may have completely different elec-
, ? ?
itrical properties in -different crystalline modifications (for instance, carbon in
.Ithe form of'diamond. or graphite), indicates that these properties are determined ,
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not only by the element's position in the table- of the elements, but by the nature
of the chemical bonds between the atoms of the solid as well.
Let us characterize briefly the 12 elementary semiconductors which have been
isolated in Fig. 1. The most typical semiconductorsl'whose electrical properties
have been thoroughly investigated, are germanium, silicon and tellurium. Germanium
is the material most important to contemporary semiconductor. technology, and has
therefore been, investigated with the utmost thoroughness--certainly with greater
care than any other element; its disadvantage, from a practical point of view, is.!
its high cost, which results from the fact that germanium occurs in nature only in
scattered small deposits and from the difficulty encountered in extracting it.
Silicon is one of the most widely distributed terrestrial elements (28% of the
earth's crust); certain of its physical characteristics, which are of great signif-
icance in technological applications, are superior to those of germanium, but since
the technique of industrial-scale preparation of highly pure silicon?which mani-
fests its remarkable properties only in this state--is still in the familiarizatin
stage, we can speak at present only of its tremendous possibilities. Tellurium
a semiconductor which has found.no application in its pure state--is the basis for
the preparation of a number of semiconductor alloys which, in particular, are used
in the fabrication of thermal generators' and coolers.
Antimony and arsenic exhibit semiconductor properties only when prepared in
the form of films by sublimation in a vacuum onto a cool base; at temperatures ex-
ceeding 00 C for antimony and 3000 C for arsenic, the semiconductor modifications
of these elements become unstable, and the specimens are converted to the normal
metallic state.
Tin lases its ordinary metallic properties and becomes a semiconductor upon the
polymorphous conversion of the white modification to the gray modification (below
Still insufficiently investigated are the electrical properties of pure boron.
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However, there exists no doubt as to its classification among semidonductors.
At roam temperature, gray selenium, Ired phosphorus, sulfur and iodine are
;
essentially nonconductors; their semiconductor properties appear in the, form of a
' great increase in cenductivity when exposed to light (photoconductivity). Sulfur,
for example, which is one of the best insulators at roam temperature undergoes a
rill4o-'o1d increase in conductivity under the action of light.
Carbon, finally, is a semiconductor which, depending on the spatial arrangement
of the atoms in its crystal lattice, forms solid. bodies with either insulating,
semiconductor or metallic properties. In the form of diamond,carbon is an insu-
lator sensitive to electromagnetic radiation; in the form of graphite, it is either
a semiconductor or a metal, depending on. the orientation of the current with respect
to the axes of the graphite crystal.
While pointing out that there is nothing new in principle in semiconductor
chemistry, we should nevertheless make the reservation that the successful applica-
tion of semiconductors in contemporary technology involves the solution of diffi-
cult problems in physics and chemistry. The manufacture of perfect rectifiers,
ammlifiers and. photocells requires that the material be purified so thoroughly that
not even the most skillful chemist would have undertaken it ten years ago. Today
the technique of obtaining such high purity?for example, in germanium--has become
generally accessible. The problem of utilization of silicon,which, in this partic-
ular field of semiconductor application, has considerable advantages over germanium,
. 'remains essentially the unsolved problem of its purification. In another important
'ffield of semiconductor technology, the goal of which consists in manufacturing
- " -
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economically profitable thermal gef-Jeidtdes, the i)roblem Of-the purity of materia-Is
?
is replaced. by another, no less difficult problem of manufacturing heat-resistant
semiconductor materials with specified electrical and thermal properties.
At the Present time, numerous groups:ofchemists, physicists, metallurgists, .
:radio and :electrical engineers and other specialists are at work in most countries of
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the world on the problems of mastering semiconductor techniques and in thorough
?.
investigation of the physical properties of semiconductors. The Study of semicon-
ductors is of scientific value in that it enables us to establish most completely
the general physical laws which govern, to varying degrees, lAie:behavior of the
:
different solids--the metals, semiconductors, and insulators. It is not an exag-
geration to state in conclusion that the scientific, technological and economic
'im-
portance of the semiconductor problem, and the attention which it invites, place
this problem on the same level with that of the utilization Of nuclear energy.
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CHAPTER I
GENERAL TREATMENT OF CONDUCTIVITY IN sums.
CONCENTRATION AND MOBILITY OF CURRENT CARRIERS
The term "semiconductors" itself implies that the general properties which
unify this chemically heterogeneous group of substances are their electrical prop-
erties -- or, to be more accurate, the laws which govern the passage of electric
currents through them.
The conception according to which the electric current generated in a solid by
an external electric field constitutes a directed migration of charged particles,
or current carriers, which is superimposed as a "drift" upon the chaotic motion of
the particles which prevails in the absence of an external field, has long been in
existence. These charged particles are either electrons or the positive or nega-
tive ions of the substance. Simultaneous participation in the electric current of
all these types of carriers is possible and has already been Observed. In the
technological application of semiconductors which is of interest to us, the fact
that either electrons or iOns.may participate separately in the electric current
?
Ls important- ,In the first instance, we speak of electronic conductivity, and in
the second.ofionic conductivity. The ionic, or, as it is often called, electro-
. ,
lytic 'conductivity of solids is a result of their gradual chemical decomposition
'during passage of an.pLi...1ctric Current through them. This fact compels us to dis-
,?- ,
_
'-i.egard.sUbstances.in:WhiCh'condactivity is, due to any appreciable extent to the
: Migratfon,:of-.ions,,since the. utilization Of such materials would not satisfy the
.'bdia-specifioations Set fOrth for semiconductor equipment -- that is, operational
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stability and long service life. The experience of several years indicates that if
?
_
electric current is generated by the motiOn of elect/6= only,'no'dhemical effect
of the current will appear. Hence at the present time,...when we speak'of semicon-
-
ductors, we refer to, solid substances inA whiah current-ia generated scilely by the
motion of electrons, despite the fact that formerly, when semiconductors had been
inVes.tigated only very poorly, this term applied to all solid substances with small
.11
.?
but detectable conductivities, irrespectiVe of the nature of tbe'Current carriers.
?
It is known that since the electric current is subject to Ohm's law, the'spe-
'cific electrical resistance p, or the uniauely-related value referred to as the
1 1 , -1 -1N
specific conductivity T (a_
p(ohm am) ?'-2:1
a quantitative measure of the conductivitSr of a solid.
may be taken as
The specific conductivities of good conductor-metals vary within a range of
two orders of magnitude between 104 and 106 ohm-1 cm-1. In the other extreme case
-- that of the very poor conductors, or insulators -- this value is usually smaller
-1
than 10-10 ohm cm-1 . The specific conductivities of the bodies referred to as
-1
semiconductors occupy the entire extensive range of values from 103 to 10-10 ohm
cm-1, thus varying through a vast region Which encompasses 13 orders of magnitude.
It is not the quantitative distinction between semiconductors and metals, how-'
?
ever, whia=leads us to single them out as a special group of substances and per- '
mits their utilization in various specific technical fields. A, basic and character-
istic property of a semiconductor is that the value of its specific conductivity ,
may vary greatly under the influence of various factors, such as temperature, light,
pressure, etc., .and, what is of great importance, that the conductivity of semicon-
ductors may be controlled by adding very smell ventities of impurities to them.
It should also be pointed out that the numerous and varied applications of semicon-
ductors in contemporary technology are due not only to the above characteristic
effects of impurities, temperature, illumination, etc., on conductivity, but also
:to a number of other specific physical properties, inherent to one degree or another
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in the different groups of semiconductors. These properties include the ideal
rectifying and amplifying properties of semiconductor junctions, high thermo-and
photoelectromotive forces, the high electronic emission of semiconductor cathodes,
outstanding luminescent and catalytic properties, a remarkable combination of high
electric resistance and ferromagnetism, and many other properties upon which the
various uses of semiconductors in technology are based.
Proceeding from absolutely general concepts of the mechanism of the electric
current, it may be indicated that the specific conductivity of a solid is deter-
, mined by the expression
0:= enu,
where e = 1.6.10-19coulomb is the electronic charge, n is the concentration of cur-
rent carriers (i.e., their number per unit volume), and u is a quantity which is
termed the mobility of the current carriers.
Let us clarify the physical significance of this expression. It has already
been noted above that electric current is created by the directed-motion of free
electrons, which exist in a solid irrespective of the presence of an external field,
and move in a randan manner similar to that of molecules of a gas. However, while
tne trajectories of the gas molecules are determined by the collisions that occur
between them, the trajectoy4es of the electrons, based on a number of physical
considerations which we shall consider later, are determined not by collisions,
,even with the quiescent atoms of the substance, but by collisions with thermal
vibrations of.atoms and with lattice defects, a more detailed discussion of which
-
will also follow below.
, We assume that the plectrons move freely in the intervals between collisions,
and, consequently, rectilinearai and uniformly. Let 1 denote the length of free
?1
path :4!Lnd the corresponding time of free path. These two quantities stand in
f
-
the obvious..Trelationship 1= v , where v represents the velocity of motion which,1
In the genera case, differs with different electrons. The 'chaotic motion of
y
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electrons in the absence of an external electric field does not create an electric
current; when a field is applied, the electrons experience an acceleration in :the
4
- ,?- - -
direction opposed to that of the field under the influence of the'-:electriCal
?-?!. '
The secondary mean directed velocity ?which -the electrons' atiaiii-d'ier
the free'
?
path will be jut the velocity which determines the magnitude of the-electric cur-
rent. Figure 2 illustrates the considerations outlined here: the Solid:arrows
,
?
? . ; . ?
pointing in different directions represent the velocities of the -random
the electrons in the absence of a field; the solid arrows pointing from right "to
?
left represent the mean secondary directed velocities attained by electrons over
the free path when an external electrical field of intensity E is applied; the
dotted arrows represent the resultant velocities. In this simplified diagram, all
_
---random velocities are represented as being equal, while the secondary directed
...
velocities attained by such electrons will not be equal due to the disparity between
the lengths of free path for random motion. Such is the pattern of the motion of
electrons in metals.
In semiconductors, the situation 1s much the same as in gases: the velocities
of the electrons involved in conduction have all values from zero to infinity, and
are distributed according to the Maxwell law. The case which occurs in metals is
characterized by nearly absolute temperature-independence of the electron veloci-
ties and is therefore termed the degenerate case. Just as in the case of gas mole-
cules, the electron velocities in semiconductors increase with rising temperature,
and there are few instances in which the electron gas in semiconductors approaches
the degenerate state in which electron velocities depend only slightly on the
temperature of the substance. (The conditions in which degeneration of electron
gas occurs will be discussed below on page 58.) Eence, the classical Maxwell-
Boltzmann statistics can nearly always be applied to semiconductors. The more
tgeneral Fermi-Dirac quantum statistics, which includes the classical statistics as
la particular and, mathematically speaking, more simple case applies for the
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eq
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degenerate electron gas.
-CC
Let US turn to the diagram in Fig. 2,
and, applying the known laws of elementary
physics, calculate the average supplementary
velocity . The acceleration of an elec-
tron a tmlirc.r, and the average supplementary
velocity of one electron over the length of
, eE
free path AV L=342 !ff-1:, where in is the
?o
electronic maso and T is the free time
for this electron.
Fig. 2. Diagram illustrating crea-
tion of electric current in a solid. Taking the statistical distribution
of time intervals T into account in the
normal manner, the average additional velocity of the entire ensemble of electrons
in the solid will be
( 1)
where iE is the mean free time of the electrons.
The electric-current density is determined by the quantity of charge which the
electrons carry across a unit cross section of the conductor per unit time:
T?
mo -
By definition, the specific conductivity C is given by the ratio of the cur-
rent density d to the field intensity E:
_
e t- , , ?
. , _
The value u is referred to as the mobility o? the current carriers,
ICI ? ...
r;-?
?
2
?
= e n u.
?
(2)
? . ?
, .
?
since it determines the average sumplementary velocity
/Iv
electrons in a field of
, -.: ?
unit_intensity (for example, 1 volt/Cm
attained by the
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If we express the charge in coulombs (e = 1.6.10-19 ), n andu-in
?
1 -1
cm2/ volt .sec, then a will be obtained in ohm',-1 era
Expression (2) is 'universal, since i applies both for the mechanism of elec-
tron conduction and. the mechanism of ionic conduction (in the latter instance the
numerical.va2.ues of the quantities which characterize migrating charges will, of
course, be different). Nor does this expression, obtained, with the aid of simple
classical considerations, change its form when we subject the phenomena of,con-
ductivity to rigorous investigation in terms of quantum mechanics; in the quadtum
theory, however, certain of the values in Expression (2) assume interpretations
significantly different from those of the classical theory. This applies, in the
first place, to the free time -c , which is defined in both the classical and the
1
quantum theories as the ratio t =.---. In the classical theory of conductivity,
the free path 1 was determined by interatomic distances, since it was assumed that
electrons are scattered upon collision with the atoms of the body which obstruct
the directed motion of the electrons even when at rest; as to the velocity ir, it
is assumed in the classical theory that this quantity is determined by the simple
t.
condition of uniform distribution of thermal energy among the electrons and the
vibrating atoms in the lattice. The quantum theory has essentially changed our
concepts regarding both the free path of electrons in solids, and the conditions
by which their velocities are determined. It has become evident that the length of
free path in an ideally ordered crystal lattice is equal to. infinity (and that the .
electrical resistance equals Zero), so that the quiescent atoms of a strictly -
periodic lattice do not present obstacles to the electrons which participate in the.
electric current. In accordance with the .quantum theory, electrical resistance in
a crystal occurs only when the ideal ordered state of the atoms (or ions) is dis-
turbed by thermal agitation of the atoms or ions) themselves or lattice defects.
Lattice defects include lattice points replaced by fw.eign atoms, lattice vacancies,
atoms (or ions) implanted in interstices, shear between crystal layers, cracks,
? t3
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grain boundaries, etc. Thus in perfect mOnocrystals the free path of an electron
can reach tens, hundreds and -- at low temperatures -- even tens of thousands of
interatomic distances. The velocity of the random motion of free electrons in
solids is determined in the general case from specific quantum conditions which,
by the way, lead to the same classic results in the case of the degenerate semi-
conductors under consideration.
In the above elementary derivation of the formula for the conductivity of a
solid, an essential simplification has been made: we have assumed that an electron
moves freely along its free path. In actual fact, we know very well that at the
'distances which separate free electrons from atoms in a solid, very strong electric
fields are active, and that-their forces exceed by far those of the external fields
in which we usimlly.place the solid. These fieldi must necessarily exert great in-
fluence on the motion of the electrons which, for this reasons may not be considered
as a free motion even in rough approximation. The quantum-theory shows that the
effect of strong atomic fields maybe taken into account by formally ascribing to
the electron a certain mass different from the mass of the freely moving electron. ,
'It develops that such an operation enables us to study electron motion in solids
without taking the effect of strong internal fields into account. The mass which
we ascribe to an electron of a solid is called the effective mass m. This value
is determined. by the structure of the medium in which the motion of the electron
occurs. In different bodies it may be either larger or smaller than the mass of
the free electron mo 9.7_0-28 g. We shall repeatedly be confronted by this
-
problen in the material -which follows.
It will be seen from Formula (2) that the direct. proportionality between the
current and the field knowh as Ohm's law will occur only in the case when the MO-
! , .-... --...."
. .?, .- ?
? , ,
-
bility u and the, caacentration of current carriers n. dO. not- depend. on the field. E;,
?
- . ..
1 . ? .,.. ....,..... -_, - ., ..
, -
.in the, contrary case, the specific conductivity er 'will be" a function of the field
, ?
law vill,not apply. In metals, Ohm's lav remains valid, under all
'
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!
iconditions; only in the presence of tremendous current densities which are practi-
cally never used can infinitesimal deviations be Observed; the reality of-these,
:... I 1 .-'. ? - ' - ?:. ' .-.:. -??? : .- - : ..-
.1
'however, is still a'matter of dispute. Deviations from Ohm's law maybe Ohierved
in semiconductors under practically realizable conditions; and even find applica-
tions in technology. The range of electric-field strengths In whichOhm's law-is
-
satisfied is termed the weak-field range; the deviations from Ohm's law will be re-'
lated to the influence exerted by strong electric fields on mobility and'concentra-'
tion. Due to its importance for semiconductors, this phenomenon will be discussed
i
'in detail in a separate chapter: ?
Thus, according to (2), the conductivity of a solid is determined by the con-
centration n and the-mobi:ity u of current carriers. Let us now establish the
reason for the lower conductivities of semiconductors as compared with metals: is
it due to the fact that concentration in semiconductors is lower, or does it result:
from lower mobility of their current carriers? To solve this problem experimentally,
it is necessary to determine, in addition to the readily measured value 6 , either
n or u by an independent method, and then each of these quantities separately witht ,
the aid of Formula (2).
Fig. 3. Scheme of experimental Ball-
effect measurement.
There exist various independent
methods for determination of the concen-
tration and mobility of electrons on the
basis of measurements of the various gal-
vanomagnetic and thermomagnetic effects
-- i.e., of physical phenomena which arise
In semiconductors in a magnetic field
when free electrons acquire a directed
motion in this field under the influence
of potential or temperature differences.
We shall show below why the Hall effect, among eight of these effects, is the most ,
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suitable for this purpose.
? Since we shall refer repeatedly to the Hall effect as the experimental basis
for numerous important conclusions regarding the properties of semiconductors, let
113 first recal_ the experimental procedure used for measurement of this effect.
It is convenient to use a test conductor having the shape of an elongated
plate of rectangular cross section. If we place this conductor in a magnetic field
H and pass through it a current I, as indicated in Fig. 3, the potential difference
0
AV appears acrcRs the conductor under the influence of the magnetic field; it
-H
disappears with the elimination of the magnetic field. The occurrence of this po-
tential difference is called the H1] effect. This effect is the result of the de-
flecting action of the magnetic field on the current carriers moving along the
plate. Theory and experiment indicate that the magnitude of Hall's potential dif-
ference in not to strong magnetic fields is determined by the expression
TH
AV R_
H -
where is the current density j.
a b
The sign of this potential difference -- i.e., the direction of the Hall elec-
tric field -- is uniquely related to the sign of :he current carriers, and may be
determined according to the left-hand rule, which is known to us from high-school
physics. With directions as noted in Fig. 3, the current will be deflected down-
-4
ward; consequently, electrons moving against the external field E must be deflected
downward, thereby imparting a negative charge to the lower edge of the plate. Should
positive charges moving along the electric field E participate in the electric cur-
rent; the lower edge of the plate would become positively charged, and the direction
-of the Hall pOtential difference would be reversed. Thus the qualitative investiga- ?
,. ,
., - .- --,?
_ . .
tioli:Of:tlii6 effect enables us to verify the sign of the current carriers. The
,, - ,
-,..;....,,,,-;;,? - . ?
. .. .,
ue7..of'ihe concentration of current carriers n which is of interest to us can be
. ,
determined, from the value of the Hall coefficient R computed from experimental data
- - - - - ? --- -a - Y
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since it is possible to show by simple deductions based on the eleMentary laws of
electro-dynamics that in the case of a single species of current carrier
1 ?
cm/coulomb.
-" -
ng
Indeed, it is possible to determine the Hall field from the condition of
equality of the force evH exerted upon the electron by the magnetic field to the
force of the magnetic field 3EH. Taking into account that the current density
j = evn, and = R we obtain RE =
R -en
Upon absolutely rigorous derivation;taking the statistical distribution of
electron velocities into account, this expression takes the form
A
na
amicouloMh,
(3)
where A is some numerical coefficient deliendent upon the degree of degeneration of
the electron gas and on the mechanism of current-carrier dispersion. The advantage
offered by the Hall effect in the choice of a method to determine n wadi." is due to
the fact that the numerical value of the coefficientAl which depends in a complex man-
ner on the above-mentioned factors, varies in the relatively narrow range from 1
to 2. In the other methods, we are.also'confronted by numerical coefficients which
also depend in a complex manner on the same factors; in these instances, however,
the intervals of variation of the respective coefficients are much larger. Since !
ithe exact determination of the degree of 'degenerationof the electrons and the
mechanism of their dispersion constitutes a separate and difficult problem, we
shall incur the smallest error from a none too reliable solution if we use the
Hall effect to determine n and u. The fact that the Hall effect is nearly inde-
pendent of the anisotropy of the crystal when measured for a weak magnetic field
is a further advantage of this method.
Experiments designed for measurement of the Hall effect in metals and the I
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isemiconductors have indicated that the 1 ver conductivity of semiconductors may be
explained by smaller concentrations of current carriers. Concerning the mobility
llithis quantity can be both larger and smaller in semiconductors than in metals.
2
Whereas the mobility in metals will normally amount to hundreds of cm at
1 v.sec
room temperature, this quantity varies in semiconductors from unit numbers to tens,
cm?
of thousands of (InSb - 8o,0oo; Inks - 30,000).
v.sec
It is therefore of interest to note that high mobility of current carriers
can be observed in solids despite small conductivity values. If, for instance, a
diamond -- normally a good. insulator -- Is subjected to electromagnetic or corpus-
cular radiation, free electrons may be created in it and the passage of a weak cur-
rent through it may also be observed; in this case, it develops that the nobility
2
cm
of the electrons participating in the current attains a value of 1000
v.sec
i.e., exceeds that of the electrons in typical metals. The same can be observed
1
in semiconductors at low temperatures, wilen the mobility attains hundreds of
2
cm
v.sec
thousands of
with very low c cnductivity. We shall return to the problem
of the mobility of current carriers and Of its physical significance after having
familiarized ourselves with a number of important aspects of the electronic theory
of crystals.
Experiments in the measurement of the Hall effect in semiconductors have shown
that the strong influence of temperature, light and impurities on their conductiv-
ity is basically the result of a change in the concentration of current carriers,
although it should be noted that appreciable though much more weakly manifested
changes in mobility occur upon a change in temperature and upon the introduction
of impurities. Similar experiments made with metals indizate that their current-
carrier concentrations are practically constant while the effects of temperature
and impurities are asserted in the length of free path only, or, which is the same
thing, in the mobility value (which, however, is not always used in describing the
electric properties of metals).
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I.
The very sharp drop in the concentration of current carriers which is often
observed in semiconductors upon a decline in temperature suggests that the conduct-1
ing state of such materials is a state of excitation, and that-semiconductors mast
become nonconductors at absolute zero due to the total "condensation" of the
elec-
tron gas, which begins -to "evaporate" as the temperature rises above absolute zero,'
and to a greater extent as the temperature of the body increases. The strong in-
fluence of light also becomes understandable from this point of view: electromagnetic
quanta (photons) which are absorbed by the solid provide the current carriers with
the energy necessary for their "evaporation," or, better said, their activation.
This conclusion, combined with what we have said above concerning the nature
of the current carriers, enables us to establish points of resemblance and dissim-
ilarity between the three classes of solid conductors: metals, semiconductors and
electrolytes. Considered current carriers, semiconductors are similar to metals
and not to electrolytes; from the viewpoint of the energy aspect of the conductive
state itself, semiconductors are similar not to metals but to electrolytes, in which
the state of conductivity is also one of excitation: The concentration of current
carriers is'practically independent of the effects of temperature and light; in
semiconductors these factors exert decisive influence on concentration, although
it is necessary to make the reservation that this conclusion is not universally
valid. In every semiconductor there exists a temperature range in whidh,the.con-
centration remains constant, and the influence of light, even when absorbed by the,
semiconductor, is not necessarily asserted in its conductivity.
The concentration of current carriers in some semiconductors remains constant
over a very wide range extending to the lowest temperatures, as in the case of
metals. Such substances are not normal metals, since, first, their concentration
of current carriers is relatively low and, second, it increases rapidly with rising
temperature as soon as the latter reaches a certain value which is quite definite
-for each of these substances (see page 87 ). Such materials, which possess metallic
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properties at low and medium temperatures! and the properties of semiconductors at
elevated' and. high temperatures are called semimetals. Semimetals are used in tech-,
nology at the present time because their electrical and thermal properties still
come closest to satisfying the requirements made of materials for thermogenerator
Similarly, even our conclusion concerning the strong effect of light on the
construction.
conductivity of semiconductors is not universal. The influence of light varies
from a degree barely perceptible even with the most precise measurement to a tre-
mendous effect in which the conductivity under illumination is increased thousands
,and even millions of times as compared to the dark conductivity. This phenomenon,
which has been naMed photoconductivity, is highly selective: this means that it is
not observed. under the influence of light of any wave length, but only under that
of electromagnetic radiation at wave lengths confined to a relatively narrow range
which is frequently situated outside the visible band of the spectrum.
Continuing our reservations in regard to the universal prevalence of wide
variation in the concentration of current' carriers in semiconductors, it should be
pointed out that even the presence of Impurities does not always exert any great
.influence. In certain, semiconductors, the introduction of trace (.0 0.01%) quan-
tities of impurities increases conductivity thousands or even millions of times;
the same procedure in other semiconductors interferes with their conductivity to a
(nearly imperceptible degree. Trace impurities of one chemical nature may prOve
highly effective with a given semiconductor, while other chemical impurities intro-
duced. in Much larger quantities have but a weak effect on conductivity; here it
sometimes happens that conductivity decreases, instead of increasing, as a result
of the introduction of impurities;
20
1
1
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??.
CHAPTER II
?
THE SIGN OF THE CURRENT-CARRIER CHARGE'
IN SEMICONDUCTORS: EIECTRONS AND HOLES
itvaspointedout above in explaining the the natureofthe effect that the di-
rection of the transverse Hall electric field is uniquely related to the sign of the
current-carrier charge; it was noted further that the mechanism of conductivity in,
the semiconductor materials which interest us here is electronic. It might be
assumed that these two conditions should make it possible to draw a definite con-
clusion as to the direction of the Hall electric field which should be obtained in
investigations of such materials. However, it was observed long ago--and long re-
mained an enigma--that the "correct" direction of the Hall electric field occurs asi
often as the "incorrect" direction--i.e.,. the reverse direction. The correct direc-
tions, which correspond to the negative sign of the electronic charge, came to be
referred to as cases of negative or n-conductivity, and the "incorrect" cases are
said to be characterized. by positive or p-conductivity. Experiments have indicated
7
and theory has confirmed that the sign of the conductivity observed in the Hall
effect also appears in certain other gaivanomagnetic and thermomagnetic effects,
as well as in the thermoelectric effect. The latter provides the simplest method
.by which to determine the sign of conductivity, since all that is necessary for
this purpose is to determine the direction of the thermoelectromotive force which
arises upon nonuniform heating of the conductor; this method does not require ex-
ternal electric-or magnetic-field sourcel.
Experiment indicates further that the sign of conductivity in a given
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semiconductor may change either upon variation of temperature or the introduction
of one or another impurity. Thus, for instance, for tellurium of average purity
the sign of conductivity is always positive at the temperature of liquid air,
irrespective of the nature of the impurities: as the temperature is elevated to
, room temperature, the sign of conductivity becomes negative, and with further
elevation of temperature to several hundred degrees above room temperature, it be-
comes positive again. In some semiconductors, including germanium, silicon, sul-
fide of lead, telluride of lead and many others, the sign of conductivity changes
in dependence upon the chemical nature of the impurities which they contain. In
others, the sign of conductivity is independent of the chemical nature of the im-
purities; thus, for example, NV deal constantly with cuprous oxide, copper sulfide,
selenium, manganese [sesqui-] oxide and Other similar semiconductors whose con-
ductivity is always positive; on. the other hand, the oxides of zinc, aluminum-and
titanium and the sulfides of silver, cadmium and mercury are examples of semicon-
ductors whose conductivity is always negative.
1
Since the possibility of participation of positively charged ions in the elec-
tric current is excluded by the experimentally established fact of the absence of
electrolytic decomposition in the substances of interest to us, the thought of
possible participation of positrons in the electric current occurs to us first when
we turn to consideration of positive conductivity in electron conductors. Yet this
hypothesis will be rejected immediately when we consider that positrons are not
stable in a medium containing electrons, ;and that the formation of positrons is
1
associated with the consumption of a tremendous energy (1 Mev) which cannot possi-
bly occur in a nonradioactive material; the average thermal energy of atoms in a
1
1
solid.atroomtemperatUre amounts to several hundredths of an ev, while the energy
1
increase in current carriers in an external electric field, does not even exceed a
.hundredth of this value. Thus there remains only to search for some other expla-
nation. 'ofthe fact that'electronic condudtivity may manifest itself in different
'
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_/-
?
ways--either in the form of a simple motion of negatively charged electrons which
is readily understood from elementary concepts, or in the form of some other, more
d I
complex motion of these particles which is equivalent to the-mOtion of -positive.
, ? .1 - .
charges which, as experiments show, havelapproximately the sane mass as the elec-
.
!
trons participating in negative conductivity. A valid and rigorous explanation ofi
' this paradoxical phenomenon has been found in the course Of the theoretical study of
1
electron behavior in the crystal lattice of a solid. In essence, this explanation
1
? is associated with the particular wave properties of electrons and with the period-
ic structure of the crystal lattice. The rigorous treatment of this problem is
carriea out with the aid of the complex mathematical apparatus of contemporary wave
mechanics, which does not 7,o-1nit exposition of the physical essence of this phenow.
enon on the basis of the graphi- concepts of classical physics. Nevertheless, and
accurate, though not rigorous interp.:,=!tation of the experimentally estdbliched
fact that two types of conductivity exist may be presented without reference to the
difficult mathematics involved in the quantm theory, using as a basis the general,
concepts of atomic structure which are known to us from general physics courses.
By generalizing these concepts, it is podsfble to construct a descriptive model of
a solid which demonstrates, firstly, the cause of the experimentally observed dif-
ferences in the electrical properties of metals, semiconductors and insulators,
and, secondly, explains the observed effects of temperature, light and impurities
on the conductivity of semiconductors which form the basis for their technological
applications.
Let us proceed from the familiar ass:umption that the electrons of free atoms
may not possess arbitrary energy values, but only quite definite or, so to speak,
discrete, quantized values which are separated from one another by wide forbidden
intervals. The energy levels of electrons in an isolated atom, as is known, may
be represented schematically in the form tof a vertical series of lines; their posi-
.
tions in this series correspond to the energies of a given state (see the right
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side of Fig.. 4).
forbidden energy
1
The intervals between the lines correspond to the intervals of
values. A second circumstance which we must take into account
in constructing the general scheme of the motion of electrons in a solid consists
in the fact that according to the Pauli principle,which is the basis of the quantum
theory of the atom, no more than two electrons can exist in one quantum state,
which is determined, by three quantum nuMbers. In the course of formation of a
solid through the union of individual atoms, the effect of the above restrictive
laws begins to apply to the entire microscopic volume of the solid which, in this
instance, should. be considered as a gigantic molecule consisting of a great number
of atoms. Quantum theory indicates that as identical atoms draw together and
atomic interaction begins to influence the energy states of electrons, levels which
were identical in the case of the free atoms are now displaced along the energy
1
scale by an amount which increases with increasing interaction between
1
with their approach to one another. There forms in the solid, from N identical
levels of the widely separated atoms, an lenergy band consisting of N closely
juxtaposed different levels. It will be recalled that 1 cm3 of a solid contains
approximately1022 - 1023 atoms. The entire tremendous number of electrons in the
1
section of the solid. with which we are experimenting form a collective and present
1
a single system which is subject to the same quantum laws as the limited electron
1
system of the individual atom. Yet instead of the restricted level of an individ-
ual atom,which contains no more than two electrons (if the level is not degenerate),
A
wide energy bands containing as many levels as there are atoms in the section under
1
consideration form in the solid. If an energy level in a free atom is g-fold de-
generate, the energy band formed from it may also be degenerate, and in this case
2g electrons may be distributed, in each level of the band; but instances are also
?
possible when, under the influence of the internal electric field of the crystal,
?
this,. energy band is ,decomposed into several bands (the maximum number of, which may,
not;.'of course, exceed g). It will be recalled that the integer g=. 1, 2
24
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which characterizes the multiplicity of degeneration of an atomic energy level, is I
the number of different states of the given atom which possess equal energy.
Moreover, it Is necessary to note that the order of band distribution in the 1
1
energy diagram of a solid may not correspond to the order of energy-level
,
butiOn in the free atom; in this case, electrons are redistributed according to the
?
Pauli principle in.the course of formation of the solid, .in such a way as to occupy
all lower levels of the energy spectrum. The left side of_Fig. 4 ,presents ache-
Fig. 4. Splitting of levels occuring on the rapprochement of
atoms, and the formation of the energy spectrum in a solid.
Distances between energy levels are not given to scale; ro
stands for interatomic distance in a crystal.
1
1
matically the genesis of energy bands in a solid as atoms draw together. The bands,'
or, better said, the intervals of allowed energy values are separated. from one an-
Ler by intervals of forbidden energy values; for the external, so-called valence
'electrons, the latter have a width of the same order as that of the alloyed energy
intervals (see Fig. 5). These intervals (bands or ranges) of allowed and forbiddeni
energy values are referred to as energy bands. As seen from Fig. 51 the width of
the allowed bands increases with increasing energy of state, while the width of the '
Iforbidden bands decreases in the same direction. As in the case of an individual 1
atom, in which a quantum level may be either filled or unfilled (or partially fille4
he energy bands in a solid may be filled Lto different degrees (in extreme intancesl,
they can be either completely filled or completely empty). And just as in an
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individual atom, where the transition of an electron from a lower quantum level
? 1
to ahigheronerequires an external energy source equal to the difference in energy
between these levels, it is necessary for the displacement of an electron from a
Dover to a higher energy band in a solid to consume an energy equal to the width
ofrthe forbidden band which lies between them. The same condition must also be
Observed in the case of electron displacement within the boundaries of a single
allowed. band. An energy equal to the difference between the respective levels is
lfberated in the reverse transitions.
The density of the levels in the
allowed bands is very large, since the
width of the energy bands in a solid does
not exceed several electron-volts, while
the number of levels in them is equal to
the number of atoms in the volume of the
solid. With 1022 atoms--a number corre-
sponding to a solid volume of approxi-
mately 1 cm3? the difference in energy
between the neighboring levels amounts to
10-22
ev. This condition permits us to
Fig. 5. Scheme of energy bands in
a solid. assume that the energy levels in a band
1
form a practically continuous spectrum, Tid that the displacement of electrons
within the boundaries of one band occurs quite readily. In. particular, the energy
twhich an electron acquires over its free path (10 to 10 ev) greatly exceeds the
energy which separates adjacent levels in a band. It should be noted here, mci-
-dentally, that the average thermal energy of a vibrating atom in a solid at room
temperature amounts to 0.05 ev, and that the thermal energy of the crystal lattice
,
maybe tranSferred to electrons, which corresponds in the energy,diagram tip the
? ? ? ?
transition of electrons to higher levels. It follows from these definitions that
,
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a
either an external electric field or thermal vibrations of the atoms in a crystal
lattice may bring about displacement of electrons within the limits of a given
k
_ -
band. Turning to the possibility of transitions between adjacent bands, which,
for valence electrons, are separated by forbidden zones-the width of which is of
the order of one to several electron:-'volts, we find that the mechanisms of elec.-
trical and thermal excitation are to be contrasted. An ordinary (not very strong):
electric field cannot impart to an election, over its mean free path, an energy
any greater than thousandths of an electron-volt, while the mechanism of thermal
excitation, due to the statistical distribution of the thermal energy of the en-
tire crystal among its individual atoms and the possible resulting fluctuations, ,
is subject to no upper limit for the energy which can be transferred to an elec-
tron. Thus the thermal mechanism of excitation can bring about both intraband and,
interband transitions. Statistical fluctuations in the free path of the electrons,
in which the energy imparted to an electron by the external electric field proves .
sufficient for an interband transition -clay a negligible part by comparison to the
thermal fluctuations indicated above. It should be noted here, however, that an
electric field can also bring about electron transitions from one band to another,
if the allowed bands overlap in the same manner as the 32 and 3p bands in Fig. 4.
It will be necessary for us to return repeatedly to these problems. The extent to
which light quanta (photons) acting umcn electrons can give rise to energy transi-
tions may be judged from the fact that the energy of one photon of visible light
entirely absorbed. by one electron is amtroximately equal to 2-3 ev. This ques-
tion, too, will be discussed in greater detail, let us return to the effect of the
external electric field created by an electric current.
Let us emphasize that for the present purpose we draw no distinction between
semiconductors, metals and insulators, and refer to the most general case of a
solid consisting of atoms (or ions) distributed in an ordered manner.
We already know that an electric current is a directed motion of charged
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'particles, produced by the application of an external electric field to the sub-
stance. Under the influence of the field) the charged particles (the motion of
which in the absence of a field is chaotic) are accelerated in the direction of the
field (electrons are accelerated from the cathode to the anode--i.e., against the
field) and experience an increase in energy over the region of their free path. On
collision with the scattering centers at the end of the free-path region, the elec-
trons transfer the energy which they have accumulated in the electric field to the
lattice in the form of heat. Such is the microscopic pattern of the heat effect of
an electric current. Should the solid represent an ideal crystal, the wave proper-
ties of electrons, according to quantum mechanics, would assure their unobstructed
motion within such a crystal. Only deviations from strict periodicity as a result
of thermal vibrations of atoms, the various lattice defects, and impurities cause
the electrons to move in a zig-zag fashion in a solid, abandoning their rectilinear
course as do the molecules of a gas. It follows from this physical scheme that the
existence of an electric current is a result of a process of acceleration of elec-
trons over their free path by the external field, to which the displacement of elec-
trons across adjacent levels of an allowed band corresponds in the energy-band dia-
gram; the reasonfor the inability of an electron to transfer into the next superior
band in an or?irRry moderate field(iT and the
concentration of similar impurities i not too small, the following formula
will apply:
,3/2
V. 2(2TuticT) e
? 113
(7)
2) If 2LE, tT, or if, as before, Ai > kl but the concentration of
Impurities is small enough,
?
=
(8)
-- i.e., in these cases, all of the similar impurity atoms to which these con-
ditions apply are ionized, or, as we say, depleted.
?
-- Yet another reservation must be made in connection with Formula (7); this re-
!
I lates to Formula (4) for the intrinsic conductivity. This arises from the de-
!
' generacy of the electron gas at higher concentrations, which we mentioned briefly .
on pp. 10-11. Formulas (4) and (7) relate to nondegenerate states. The critical
' concentration of current carriers n at which degeneration sets in can be deter-
mined from the formula
_ _
=4 3/2
se me 1.35 ? 101"12T)
?
Assuming m mb 9010 g, we obtain
pc = 3.6 ? 1015T3/2
At room temperature (T = 290? K) this gives
:Re 1019 cm3.
The concentration of current carriers in those semiconductors with which we
deal in practice seldom attains such litre values. However, if the concentration
approaches the Iritical concentration, and if the semicondgctor is a good con-
ductor being studied at low temperatures', at which the critical concentration
_diminishes, Formulas (4) and (7) may not:be used, and it will be necessary to
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perform the calculations by a more cumbersome approximate method, which, neverthe-
lesslenables us to obtain a very accuratle result.
It follows from the numerical calculations which can be carried out by means
of Formulas (4) and (7) that even if the concentration of impurities in the semi--
conductor is small in the usual chemical sense, but AE liE the bulk of
-o
current carriers will be supplied at normal temperature by the impurities and the 1
semiconductor will be of the extrinsic type. With a further purification by
modern techniques which, in some instances, enable us to reduce impurities to a
millionth part of a per cent, its conductivity will diminish greatly, and the
semiconductor will come to be an intrinsic one. By adding the appropriate impuri-
ties to such a material, it is possible to produce a semiconductor of any previous-
ly specified conductivity, and -- a fact of greatest importance in some cases --
to create in the pre-purified semiconductor regions whose conductivities are of
different signs. It becomes clear on comparison of Formulas (4) and (7) why the
injection of trace amounts of impurities into a semiconductor might at times alter
its conductivity by a factor of several million. Clearly, this will take place
in a semiconductor in which A >>.
E.
What is the physical reason for the ease with which electrons break away
from impurity atoms? After all, the first ionization potential of a free atom
of any element does not fall lower than 4 v, and the activation energy of the
same atoms in the crystal lattice is often less than 0.1 ev. The physical factor
which facilitates the breaking away of an electron is the polarizability of the
material medium into which the impurity atom is introduced. This property of the 1
physical medium is characterized by the dielectric constant e. Dielectric polariz?
ation of an atomic or ionic medium occurs as a result of the redistribution of
electric charges caused by the introduction of an impurity atom into this medium.
In the crystal, the polarization process results in a reduction in total energy,
while in the impurity atom it leads to the loosening of bonds between the electron,
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1
T
1
and the nucleus. Therefore the e1ectro4 orbits of the impurity atom grow larger,
1
and the ionization energy -- i.e., the mprk required for the removal of an elec-
tron from an atom -- diminishes significantly.
Enlargement of the electron orbits of the impurity atoms leads to the forma-
tion of the impurity conduction band described on page 54 , even at comparatively
small impurity concentrations.
Upon the abrupt enlargement of a valence-electron orbit, the impurity atom
may be regarded as a hydrogen-like atom, and the theory of the hydrogen atom may
be applied to the determination of the radius of the enlarged orbit as well as of
the ionization energy. According to this theory, the radius of the normal orbit
Eb
of the electron of a hydrogen atom in a dielectric medium is r 0.53.10-8 e cm,
while the ionization energy is given by,
== 13.53 a ev .
e 2 igo
For germanium, for which e = 16 and in = 0.3 m , we obtain r = 28.10-8cm and
?o
E1 = 0.016 ev. The latter value may be confirmed directly by experiment since
?
E. = AEE,and the impurity-activation energy A E can be measured by several
independent methods which we shall discuss below. In numerous instances, the
experimental value for germanium is close to 0.015 ev, which is in close agree-
ment with theory. Formula (11) is often used for the determination of the effec-
tive mass m of the current carriers, from experimentally known values for AEI
1
and. ?.
Polarization of a semiconductor medium also directly affects the excitation
1
ellei?gy of intrinsic conductivity by facilitating the break-away of an electron
1
from the basic atom of the lattice. In most instances, experimental data confirm
1
that the width of the forbidden band AK of a semiconductor diminishes with
13
increasing dielectric constant t. The product itE a-2 is constant for a number
of semiconductors, in agreement with Forirula (11).
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As a concrete example illustrating he above, let us consider germanium,
whose forbidden-bandwidth E = 0.75 ev. At room temperature, according to
-0
Formula (4), the number of its intrinsic current carriers, the electrons and.
holes, is n = 21013cm-3. Since the activation energy of many chemical impurities
in germanium does not exceed 0.015 ev, at room temperature, when the value of
2kT = 0.05 ev, all impurity atoms are ionized at the expense of the thermal energy
of the substance, and the concentration of impurity current carriers -- according
to Formula (8) -- amounts to ni =Er Thus at a concentration of impurity atoms
of only 0.001% -- i.e., with NI = 51017cm13, the concentration of impurity current
carriers ni = 51017 cm-3 exceeds the concentration of the intrinsic current car-
riers by a factor of 25,000! Another conclusion follows from this: namely, that
in order for the intrinsic conductivity of germanium to become apparent at all at
room temperature, it must be purified to such an extent that its concentration of
impurities does not exceed 1013cm-3, which amounts to one ten millionth of a per
cent! This condition may be expressed in other terms: no more than one foreign
atom may be present per one billion germanium atoms! The most astonishing fact
is that such a degree of germanium purification has been accomplished at the
present time and that, moreover, this method of purification has been adopted by
industry which produces germanium of an even higher degree of purity. At room
temperature, specimens of such germanium show intrinsic conductivity, and a spe-
cific resistance p = 50 to 60 ohm-am. The concentration of intrinsic current
carriers decreases rapidly in germgnium with falling temperature, in accordance
? '
with Formula (4), while the concentration of impurity current carriers ni remains
constant dawn to very law temperatures, as a result of the small value of
AE = 0.015 ev. Therefore, a specimen of pure germanium which exhibits intrinsit
-1
conductivity at room temperature becomes at low temperatures an impurity semicon-
ductor whose current carriers are of onyign. The temperature of transition froml
intrinsic to extrinsic conductivity depends on impurity concentration: the lower I
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this concentration, the lower the transition temperature (an illustration of this
rule is shown in Fig. 18). Naturally, the instances of purely extrinsic and purelr
intrinsic conductivity are nothing but ideal extreme cases, since every real pure
Isemiconductor always contains a certain amount of impurities, and the excitation
of impurity current carriers in an extrinsic semiconductor is always accompanied
by silmiltaneous excitation of its intrinsic current carriers. This fact is not
essential for numerous problems which arise in practice, and the influence of a
small fraction of intrinsic current carriers in an extrinsic semiconductor may be
neglected. In some instances, however, it may be of importance to take into ac-
count not only the majority of current carriers with the sign of charge dharacter-
istic of the impurity semiconductor, but also the small number of current carriers
with the opposite sign of charge which occurs in an extrinsic semiconductor by
' virtue of its intrinsic conductivity. In this connection it is of interest to
note that if the intrinsic conductivity is characterized by the condition ni
the following simple relationship is valid for a nondegenerate extrinsic semicon-
ductor in which a majority of current carriers of one sign is furnished by the im-
purities while its intrinsic atoms supply a complementary number of current.car-
riers of both signs: i
?p ni2. (9)
In. this formula n and p represent the total numbers of electrons and holes
actlipmy present in the given extrinsic semiconductor, and n is the con-
centration of current carriers of one sign which would occur in the given semi-
conductor if it were completely pure -- i.e., the concentration determined by
meansofFormula(1.).Sincen.depends upon temperature only for a given semi-
-I
conductorin. ? p = const at a given temperature. This means that one of these two
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Fig. 12. Diagram illustrating the appearance of current carriers
In a semiconductor in the course of heating. a - intrinsic con-
ductivity; b, c - extrinsic conductivity ( b - electron conductiv-
ity, c - hole-type conductivity).
concentrations increases and the other diminishes, not only in the relative but
also in the absolute sense, as the quantity of impurities is increased.
Fig. 12 illustrates the physical pattern of the appearance of intrinsic and
extrinsic conductivity in a semiconductor in the course of heating as described
in the present paragraph. a) relates to the case of intrinsic conductivity,
b) and c) to cases of extrinsic conductivity -- the electron-and hole-type con-
_
! ductivities,.respectively.
For practical purposes it is often necessary to know the influence exerted
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?
by impurities not on concentration but directly on specific resistance. For this
purpose it is convenient to use the graphs of the type shown in Fig. 13 for
germanium and silicon. The difference in resistance of the n and p specimens can
be explained by the differing mobilities of electrons and holes (see table of the
principal semiconductors at the end of Chapter IX). This diagram shows graph-
ically the tremendous influence exercised by impurities on the conductivity of a
semiconductor. Germanium shows a millionfold range of variation of resistance,
while the same property for silicon ranges through a billionfold interval.
Fig. 13. Effect of impurities on electrical resistance of
germanium and silicon at room temperature.
Impurity atoms may be distributed in the crystal lattice in two ways: either
at lattice points, or by implantation in the interstices. fn the first case we
speak of substitutional and in the second of interstitial impurities. Let us
subject the effect caused by a substitution atom in the germanium (or silicon)
lattice to more detailed examination. For this purpose, it is essential to know
that germanium is a tetravalent element of group IV of the Mendeleyev periodic
system, and that it has a crystalline stilucture of the diamond type (see Fig. 14). 1
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In the structure of this lattice, each atom is surrounded by its four closest
neighbors, with which it interacts through covalent bonding forces. A dharacter-,
istic of these forces is their saturation -- i.e., the closed state of the bonds,
which prevents a given atom from interacting with any additional fifth atom.
To clarify the problem with which we are concerned here -- the effect caused
by a substitution atom in the germanium lattice -- let us present the actual three-
dimensional lattice as shown in Fig. 14 in the form of a plane network in which
each atom is surrounded by its four closest neighbors of the same chemical nature
(see Fig. 15). The outer electrons of these atoms, by which the valence bonds are
Fig. 14. Structure of diamond (presented here to illustrate
the formation by each atom of four bonds with its closest neigh-
bors).
formed, cannot break away from their atoms, migrate through the crystal, and thus
participate in the electric current without expending a considerable amount of
energy. But if a foreign atom of a different valence and chemical nature comes
to replace one of the atoms of the basic substance at any lattice point, the
system of the valence bonds in the vicinity of this substitution atom is disturbed4
and one of two things happens -- see Fig; 16. If the valence of this impurity atom
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3.
is higher than that of the atoms of the basic lattice -- i.e., if, for example,
an atom of phosphorus or arsenic, which has five valence electrons, finds its way
into a point of the germanium lattice, the fifth valence electron of the impurity
atom, which is superfluous in the bond system of the germanium lattice, can break
away from the impurity atom with relative ease (but with the expenditure of a
certain small energy), begin to migrate through the crystal, and, in the case of
application of an external electric field, this electron can participate in the
electric current. In the case considered here, an atom of group V of the periodic
table is a donor.
We had spoken previously of the event in which an electron breaks away from
an atom as an event of excitation of the crystal in which the electron jumps from
an impurity level to the conduction band (Fig. 12, b). The first explanation is
more graphic, while the second, associated with Fig. 12, is a more convenient and
shorter expression of the same physical process. The mechanism of NAC1 conductivity,
Fig. 15. Scheme of electronic bonds in the structure of diamond.
which was described in graphic terms on pages 29-33 , can also be described
briefly as the transfer of an electron under the influence of electromagnetic
radiation from the valence band 3pel into the conduction band 3sNa (Fig. 6).
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t?
Fig. 16. Distortion of tetravalent germanium lattice, caused
a) by an atom of trivalent boron, b) by an atom of penta-
valent phosphorus.
A different result will be Obtained in the case of replacement, of a germanium
atom by a trivalent boron, aluminum, or indium atom. In order to maintain the bond
system characteristic of a diamond-type lattice, such an impurity atom may capture'
an electron from an adjacent germanium atom. As in the preceeding case, a certain
energy must be consumed in order to accomplish this event. The value of this
energy' which we referred to above as the energy of excitation or impurity acti-
vation, may differ between the event of removal' of a surplus electron in the
previous case and the borrowing of an electron as in the present case; it depends
on the concentration of impurities, and, in most cases, declines as this concen-
tration increases. For example, the eneigy of activation of donors in silicon
amounts in many cases to 0.08 ev, and the activation energy of acceptors to 0.06.ey.
0 0
According to experimental data, the dependency of the energy of activation of im-
purities )1E upon their concentration Ni in n-type silicon takes the form
1
a 3
0
A gi (1D);
whereLiEo is the activation energy of donor atoms present in vanishingly small
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.nuMbers, which is equal to 0.08 ev, and or. 4.3-10-8. According to this empiri-
cal formula, when N= 6-1018cm -- i.e., when the concentration of impurities is
only 0.01%--AE1 0, and silicon becomes a semimetal whose electron concentration:
-
is independent of temperature in the impurity region of its conductivity. The
1
physical significance of Formula (10) becomes quite clear if one considers that
the activation energy of impurity atoms may depend upon their interaction, and
that the latter must be determined by the distance d. between them, which,
on the
basis of simple considerations, changes according to the law
?I
In germanium with small concentrations of group III and V impurities, the
1
; activation energies of the donors and acceptors are equal at 0.015 ev.
1 I
Let us return, however, to consideriltion of the case in which a substitutional
1
impurity atom of low valence borrows one electron from a tetravalent lattice
1
, (see Fig. 16). In this case there appears 'in the lattice an electron hole which
does not localize at any lattice point but migrates at random through the crystal.
If such a crystal is subjected to an electric field, the motion of this hole be-
comes directionallmanifesting itself in experiments as positive, hole-type con-
ductivity. Actually, only electrons move in this case, but their successive jumps
from one atom to another may be described formally as the motion of a single hole
1
in the direction opposite to that of the electrons. It may be said that the relayr
1
, like translocation of many electrons is in this case replaced by the equivalent
1
motion of one hole. The question as to why the motion of this hole is equivalent
1
to the motion of a single positive charge cannot be satisfactorily answered. by
graphic presentations, since this theoretical conclusion proceeds from the non-
objective properties of electrons (see pages 43-49). Thus the influence of
substitutional impurity atoms on the atomic lattice of a diamond is determined by
their,valence, and. .we can nearly always predict the sign of conductivity which
substitutional impurities of one type or
another will create in such a lattice.
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S.
If impurity atoms are implanted in the lattice interstices of semiconductors
belonging to group IV of the periodic table, the sign of the impurity conductivity
is determined, by. the dimensions of the i planted atoms and by their electronega-
tivity (see page 4). It is known from experimental data, for example, that con-
trary to the above simple rule of valency, the behavior by lithium (group I) in
1
the germanium lattice is that of a donor, while that of oxygen (group VI) is that
of an acceptor. In order to explain these factors it is assumed that lithium and
oxygen are implanted in the interstices of the germanium lattice. The implantation
of the aimensionally large lithium atom in the tight interstices of the germanium
lattice is possible by virtue of the loose bonds of its valence electron which,
in a medium with a high dielectric constant, breaks away from its atom with rela-
tive ease (see page 60). The small dimensions of the lithium on permit its
implantation in the tight interstices of the lattice, while the released electron
imparts electron-type conductivity to the germanium crystal. On the other hand,
the implantation in the interstices of an oxygen atom, which is of relatively smalL
dimensions and high electronegativity, is accompanied by capture of electrons of
? the basic lattice with the result that the latter acquires hole-type conductivity..
If a Ge or Si atom is transferred (for instance, by thermal agitation) into
an interstice, two impurity levels are created: The implanted atom acts as a
1
donor, and the vacant lattice point as an acceptor.
The effect of impurities on the conductivity of intermetallic semiconductors 1
may be judged from experimental data in regard to the most thorongbly investigated!
group of compounds of elements of groups III and V of the table of the elements
(InSb, GaSb, InAs, etc). The sign of conductivity is determined in these semi-
conductors by the same simple valency rule as in the case of the elementary semi-
conductors of group IV (Ge, Si), namely:, substitutional impurity atoms of Group
II (Mg, Zn,' etc), the valency of which is lower, create acceptor levels, while
impurity atoms of group IV (Te, Se), the. valency of which is higher, form donor 1
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'
levels. If we introduce impurity atoms of group IV into these intermetallic corn-
pounds, the result will depend on which atom of the compound is replaced by the
impurity atom. If, for example, a tetravalent impurity atom replaces a trivalent
In atom in the InAs lattice, it will act as a donor; if the same impurity atom
replaces a pentavalent As atom in the same lattice, its behavior is that of an
acceptor. Which of the two atoms is substituted in each particular instance
apparently depends on the relationship between the dimensions of the atoms. Thus,
for instance; on finding its way into the InSb lattice,a Pb impurity atom, which
is characterized by relatively large dimensionslreplaces an In atom therein and
acts as a donor; entering an AlSb lattice, the same impurity atom assumes the
position of an Sb atom and acts as an acceptor. Substitution of the components
1
of an intermetallic compound by other elements* of the same groups of the periodic
table (III and V) has no distinct effect on the conductivity of the given inter-
metnilic semiconductors. Very little is as yet known concerning the effects of
interstitial impurities on the lattices of these materials.
For ionic lattices (PbS, CdS and oxides) such simple and general rules cannot
yet be formulated; therefore the development of new semiconductors of this group
proceeds principally by trial and error. It is still not clear why a given im-
purity, when injected into these semiconductors, may prove utterly ineffective on
one occasion i.e., to have no .influence on the concentration of current carriers
-- and on another occasion confer conductivity of the sign opposite to that which
one might expect by analogy. This may result from the fact that in semiconductors
of the,ionic type, the bonds between the particles are maintained in actlinlity not
only by'the electrostatic attraction of milike ions but by a combination of
1
electrovalent and covalent bonds as well. General considerations justify the
0
conclusion that a metallic atom may be implanted in the tight interstices of the
lattice only in cases where its dimensions are greatly diminished by the loss of
outer electrons, -which, migrating througt the crystal, impart electron-type
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40.
1
conductivity to it. It also follows from general considerations thations of
many nonmetals cannot be implanted in interstices due to their large ionic radii.
If a certain number of negative lat4ce points are vacant in an ionic lattice
as maybe the case with an excess of the metal, it follows from the consideration'
of the electrical neutrality of the body taken as a whole that the same number of
positive ions must be neutralized by acquisition of electrons. This state of
electrical neutrality attained by acquisition of the electrons can migiate through.
the crystal, thus creating electron-type conductivity. By the same reasoning, an
excess of the nonmetal realized at the expense of vacant positive lattice points
must lead either to the neutralization of adjacent negative ions, or to further
ionization (if this is more favorable in terms of energy) of the metollic ions.
In either case, the vacant lattice point binds an electron of the lattice --
creates in the lattice an electron hole which, migrating through the crystal, im-
parts hole-type conductivity to it. In both of the above cases, the conducting
state of the ionic crystal is a state of excitation. The physical significance
of this statement consists in the fact that the breaking away of an electron in
the first case and the formation of a-hole in the crystal in the second require
the expenditure of activation energy. Therefore at the absolute zero of tempera- I
ture, in total darkness, and in the presence of a weak external electric field,
such an ionic crystal should be a perfe4 insulator.
From experimental data on ionic semiconductors we may conclude that the
following is usually the case with oxides and sulfides. If the semiconductor can
attain conductivity of either sign, as, for instance, PbS, a stoichiometric excess'
of sulphur or the admixture of oxygen imparts hole-type conductivity to this sub-
stance, and an excess in metal a correspOnding electron-type conductivity. In
semiconductors with impurity conductivities of one sign, an increase in the
number of holes is obtained in a hole-type semiconductor through an excess of
oxygen or sulfur; an increase in the nuMber of electrons in an electron-type
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semiconductor may be obtained by reducing the concentrations of these elements.
It is weilknom from experience, for example, that the introduction of hole-type
cuprous oxide (Cu20) into an atmosphere of elevated oxygen concentration results
in a rise in conductivity, while the same experiment with zinc oxide has the
.reverse effect.
For practical purposes it is important that the sign of the impurity con-
ductivity of a semiconductor change either as a function of the impurities which
, are added to it, or generally inttpendence on the treatment to which it is sub-
jected. As was noted above, by no means all semiconductors comply with this
1
condition. For instance, the conductivity of Cu20 is always of the hole type,
while that of ZnO is only of the electron type. Nobody has ever succeeded in
producing tellurium or selenium of negative impurity conductivity. The impurity
conductivity of many semiconductors changes sign depending on the chemical nature
of the impurities injected, but sometimes this depends on their thermal treatment
or the exposure to radiation by electrons, neutrons, etc., as well. These are
called amphoteric semiconductors. They include germanium, silicon, lead sulfide,
silicon carbide, lead telluride and many others.
Of great importance is the circumstance that not only may different fragments
of an amphoteric semiconductor possess conductivities of different signs, but
1 regions of different types of conductivity may also be formed at different places
in the same specimen of such a semicondubtor. For example, by taking a pure
: germanium or silicon lamina with electron-type conductivity and building up a
thin layer of boron or indium on one of its sides, we may, after appropriate
heating of this lamina, obtain two sections with different conductivity types.
That part of the lamina to which the atoms of boron or indium have found access
by diffusion will have hole-type conductivity, while the remainder will have
electron-type conductivity. For the practical use of such semiconductors as
rectifiers, amplifiers and phototubes, i is of great importance that the transition
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layer between these two parts, which is referred to as the electron-hole or p-n
junction, be of the order of 10 au* in width. The electronic processes Nbidh
1
unfold in this transition region are the physical basis of the technological
applications mentioned above, and form alseparate important division of semicon-
ductor physics. (This problem is discussed in Chapter X.)
It was pointed out above that simple lattice defects; such as cracks and
shpars, may act as impurities in regard to their effect on the concentration of
current carriers. Indeed, it has recently been demonstrated that, for example,
plastic shears in a germanium lattice act as acceptors which furnish hole-type
current carriers.
It is not possible in all cases to explain the conductivity experimentally
observed in solid substances on the basis of the band theory of the electronic
energy spectrum. Thus, for instance, the reason why such transition-metal oxides
as NiO, Co0? MnO and others do not possess metallic conductivity cannot be ex-
plained within the framework of this simple theory. After all, according to this
theory, the NiO crystal should include, in addition to the filled 2p0-- band, an
.++ 1
unfilled 3dNi band which contains only eight electrons for ten vacant places.
In the Co0 crystal, only 7 electrons occupy the 10 places of the 3dCo++ band, etc. 1
Why, then, are such crystals insulators and not metals at the stoichiometric compo-
sition? The explanation of this fact apparently involves taking into account the 1
interaction of atoms, which the band theory does not consider. Impurity conductiv-
ity of the p-type can be observed in these crystals after heating in an atmosphere
IMMO
of oxygen, when acceptor levels appear in the energy spectra of these crystals.
In concluding this section, which has been devoted to the explanation of the
nature of intrinsic and extrinsic conductivity in semiconductors, let um consider .
briefly one more specific variety of intrinsic conductivity which we encounter in '
the semiconductor group called ferrites. This semiconductor group has recently
*Translator's Note: illegible in original. 10-5?
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acquired great technological importance by virtue of its highly favorable combi-
nation of those physical properties which are required for the cores of high-
frequency transformers -- namely, ferromagnetism and high ohmic resistance. Two
circumstances are significant for the mechanism of conductivity in this group of
! ionic semiconductors: 1) the position of metallic ions in the crystal lattice,
and 2) the variable valency of these ions. Ferrites are oxi,32s of which magne-
tite, Fe304, is the chief representative. Upon partial replacement of the iron
atoms of this compound by divalent atoms (Ni, Co, Zn, Cd, etc.), we obtain mate-
rials with a wide variety of magnetic and electrical properties. These substances
crystallize in a close-packed cubic lattice of the inverse spinel type -- see
Fig. 17. The unit cell of this structure contains 32 oxygen ions, 16 iron ions
. which are implanted between these in octahedral positions, and the 8 remaining
ions of the metal in tetrahedral positions. Electron exchanges may occur between
' the doubly-charged and. triply-charged ions, which are distributed statistically
among the octahedral vacancies of the cell. The transition of an electron from a
doubly-charged to a triply-charged ion of the same substance actpaliy amounts to
a simple exchange of places between these ions. Since these ions occupy similar
positions in the crystal lattice, the energy of the initial and final states of the
crystal cell remains unchanged even though these states are separated by an energy
' barrier. Thus the motion of electrons between closely situated identical doubly-
and triply-charged ions assumes an entirely disordered nature in the absence of a
: field, and is accomplished by transition ,of electrons anross an energy barrier;
therefore its intensity -- i.e., the frequency of electrontransitions -- increases
with rising temperature: On the application of an external electric field, these
disordered electron transitions acquire a preferential directional tendency, with
the result that an electric current arises in the substance, and, in a given
electric field, increases rapidly with rising temperature according to the exponen-
,
tiAl law, characteristic of semiconductors, with an activation energy of '?'0.1 ev
_
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Fig. 17. Unit inverse-spinel cell of Fe30.
in the case of Fe304. The above exposition-el,ould facilitate understanding of
the difference between the mechanism of conductivity in the ordinary semicon-
ductors of the germanium, silicon, lead sulfide and other types, which were dis-
cussed previously, and the oxide semiconductors with their metallic ions of
variable valency. In the former, an increase in temperature causes an increase
in the number of current carriers in both the intrinsic and extrinsic regions of
conductivity; in the latter, temperature affects not the number of current car-
riers but their effective mobility outside the conduction band. Thus impurities
also affect the electric properties of these two semiconductor groups in completely
different ways. We already know that the introduction of infinitesimal amounts
of impurities into pure specimens of germanium and lead sulfide produces a sharp
increase in their conductivity. In semiconductors with variable valency, the in-
jection of impurities disturbs the syitem of electron transitions which we have
described and causes a decline in conductivity. It is known, for example, that
on the addition of nickel or zinc impurities to the good conductor magnetite
-I /
kFe304), the conductivity of the stoichiometric composition declines, This may
be explained as a result of inhibition of electron transitions by le+ or Zia++
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I. V
ions distributed statistically among the octahedral vacant spaces-in
the crystal cell. The transition of an electron from Ni++ to Fe+++, which is
possible in principle, involves a considerable change in the energy of the crystalf
and therefore its occurence is of small probability.
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CHAPTER IV
DEPENDENCE OF SEMICONDUCTOR CONDUCTIVITY ON TEMPERATURE
In order to analyze the effect of tenperature on the electric conductivity
of semiconductors, one should proceed from the expression (5), the more general
form of which is
a ..enini + 22+ en3u3 +
This sum is arithmetical for the reason that if the signs of the charges e
of the current carriers differ, the sign of the mobility of a current carrier will
change simultaneously with that of its charge, so that the products of these
values, which enter the expression for specific conductivity (12), have the same
sign for all current carriers. The physical significance of this statement is
obvious: in an electric field, current carriers with different charge signs move
in opposite directions, and the total current equals the arithmetic sum of the
individual currents. Although we generally deal with only one type of current
carrier in the region of extrinsic conductivity, and in the region of intrinsic
conductivity with two such types which differ with rcespect to both sign of charge
and degree of mobility, the participation of an even larger number of current-
carrier species in the electric current is still possible. Thus, for example, it
(12)
is now known that two species of hole-type current carriers of differing mobility
exist in germanium; moreover, the participation in the electric current of impuri-
ty-band current carriers whose mobility differs from that of the valence-band holes
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becomes apparent in the extrinsic-conductivity region of germanium at low tempera-
tures. Two types of hole have been observed in the valence band and two types of '
!
electron in the conduction band of tellurium.
The theoretical determination of the specific conductivity of a semiconductor
is such a complex problem that its solution has never been fully achieved for any
specific semiconductor. True, there exist theoretical papers in which the width
of the forbidden band is determined for diamond, silicon, germanium and certain
other semiconductorsIbut the results of these calculations are still not suf-
ficiently in keeping with experimental data. Our knowledge of the other values
by which conductivity is determined -- such as the mobility and the effective mass
of current carriers -- has thus far been obtained from experimental data. The
experimental method for determining.cancentration and mobility by means of the
Hall effect was mentioned at the beginningoon page.15. Eight independent methods
for the determination of the effective mass in exist at the present time; however,
the results obtained from investigations of one and the same substance by dif-
ferent methods do not, for the most part, coincide -- sometimes not even in order
1
of magnitude. This can be explained both by the imperfection of the theory accord-
ing to which the relationship of the effective mass to the experimental data is
determined and by the complex nature of this value. The values for the effective
masses of current carriers in semiconductors investigated thus far range from
several units to hundredths of the true mass of the electron. The most complete
and authentic information on the effective mass which, in a crystal, constitutes
an anisotropic value -- i.e., may, as a consequence, have different values for
different directions ire' the crystal -- is furnished by the diamagnetic-resonance
method. We cannot dwell here on the explanation of this method since it is of no
direct concern to the present problem of the dependency of conductivity upon ?
temperature.
The experimental determination of tile width of the forbidden band. of a crystal
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1
and the activation energy of impurities lis carried out on the basis of the temper
tu_re curve of either the Hall coefficient or the electric conduct1vity:2:
-
and directly concerns the subject of thi paragraph.'.*MO-dei:n theory satisiactbril,y
>.
explains the effect of such physical factors as temperature, impurities, pressure,
1
dielectric constant, elasticity coefficient, the crystal-lattice parameters, the
nature of binding forces acting in the lattice, lattice defects and the like on -
1
the concentration and mobility of current carriers. Of greatest interest-in -7:-
1
practice are the effects of impurities and temperature. The effect of impurities
1
has been discussed above. Let us turn now to the effect of temperature.
The effect of temperature on the conductivity of a semiconductor is a conse-
quence of changes in the concentration and mobility of current carriers with vari-
ation of the temperature of the substance. In the first place, therefore, let us
clarify the influence which temperature exerts on each of these factors individua*
1
ly. We shall do this with reference to the simple, yet practically most important!
cases of the existence of a single type of current carrier in the extrinsic-
conductivity region of the semiconductor, and of two types in its region of in-
trinsic conductivity -- for which last case, as we know, nn ..a2. The dependency'
of concentration on temperature has already been discussed for these two eases,.
and is sumnwrized by Formulas (4) and (7). The general result Obtained for the
change in current-carrier concentration with temperature is conveniently repre-
sented. in the form of a graph'whose axes are calibrated for values of log n and
1 1
Fig. 18 shows the typical form of this graph. The choice of semilogarith-
mic coordinates is convenient because the entire curve of the variation la con-
centration may be presented in the form pf a broken line consisting of three
rectilinear segments. The slopes of the segments ab and cd determinepaccording
1
to Formulas (7) and (4), the activation energiesilEI and ZiE . Such graphs are
-o
plotted in experimental determinations of these energies. More accurate current-
'
i carrier activation-energy values are obtiained when the experimental results
1
are
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Fig. 18. Graph showing typical dependency of current-carrier
concentration on temperature in a semiconductor. .
4
plotted, on a graph with log (RT*) laid Off on the axis of ordinates if AEI is
to be determined or log (RT*) if AE, isto be determined and 1 2 plotted against
kT
I ? ?
the axis of abscissas; see Formulas (3)/ (4), (6) and (7). In the low-temperatur7
region, on segment ab, we deal with extrinsic conductivity only, since the number
of intrinsic current carriers is still very small due to the low temperature, and
the slope of the line is determined in this segment by the impurity-activation ;
energy AE With rising.temperature, segment ab) the number of Impurity carriers
will increase until the electron reserves of the impurity atoms are depleted. In
the, segment bc, impurity reserves are already exhausted, while the intrinsic
conductivity has not yet become apparent; hence this temperature range, in which
. -
_
the concentration does not vary, is called the region of depletion. Finslly, the
. ?
temperature in segment cd has become so high that the number of intrinsic current
?carriers* begins to increase rapidly, so that we enter the region of intrinsic
conductivltj which is lisracterized by the activation
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?
'
-
energy 11E0. Experiment indicates that the slope of the segment ab depends on
the impurity concentration NI. The physical cause of this phenomenon has already
been discussed above in connection with Formula (10), which determines the de-
.
1
pendency of ?L1E on N in the particularl case of silicon. Fig. 18 illustrates
-I -I . ?
the rule set forth on page 61 according to which the temperature of the transition
from extrinsic to intrinsic conductivity. shifts toward higher temperatures with
increasing impurity concentration::-
?
I
The slope of the line does not depend on impurity concentration in the region,
of intrinsic conductivity and is a constant characteristic of the electric prope+-
ties of a given semiconductor. At some higher impurity concentration (,---1%),
when LIE the entire curve may be represented in the form of the two segments'
a"c" and c"d". In the extrinsic-conductivity region eel the current-carrier
concentration of the semiconductor is independent of temperature from the lowest
temperatures (a") to the temperature of onset of the intrinsic conductivity (2").1
This behavior of semiconductors -- i.e., their assumption of properties character-
istic of metals (constancy of concentration) in the region of extrinsic conductiv-
ity -- induces us to segregate them into a special group of Semimetals. It should
be clear from the above that any semiconductor may become a semimetal at a suf-
ficiently high concentration of dissolve& impurities, and conversely that a semi-
metal should become a typical semiconductor after thorough purification, with the
concentration of its current carriers strongly dependent on temperature. The
current-carrier concentration in semimetals is usually so high that the electron
gas in these materials enters the region of degeneracy (see page 58).
Before discussing the mechanism by which temperature affects the nobility of
current carriers in semiconductors, let us first explain the factors which deter-
mine its absolute value. After all, it may seem strange at first glance thai the
mobility of current carriers in certain semiconductors whose conductivities are,
1
as a rule, insignificant by comparison to those of metals proves to be hundreds of
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. I
times greater than those of typical metallls (see page 18). The fact, also mentioned
in the same place, that certain insulators show mobilities higher than those of
metals may seen even more surprising. What is the reason for this? Why can
electrons sometimes move more "freely" in poorly conducting crystals than in the
highly conductive metals? First of all, let us remember that according to one of
the most basic conclusions of the electronic theory.of crystals, as discussed on
page 41, it is not the question of Why the free paths of an electron within a
crystal may exceed the interatomic distances by tens and even hundreds of times
'which we must subject to scrutiny, but rather the question of what, in a real
crystal, limits the infinitely large mobility which an electron should theoretical-
ly have in an ideal crystal (at absolute zero).
According to Formula (2), which determines the mobility of the current car-
riers
e 1
3111111
= m T ?
DI V
(29 1
1
a high mobility may be due to a small effective mass (m) of the current carrier
and a large value of the free time or, more precisely, of the relaxation time 1C.
The numerical value of the effective mass is, as we know, determined by the
conditions of motion of the electron over the length of its free path in the in-
ternal field of the crystal. In tne theory of conductivity, as in classical
meehanics, this value essentially determines the directional acceleration which
the electron acquires in an external electric field in the intervals between col-
lisions; it is this directed acceleration which creates a current in a solid
(see page 11). The general conditions of the mOtion of electrons, which determine
-
their effective masses, are identical in metallic and nonmetallic crystals, so
?
that we cannot draw distinctions between these two classes of solid substances in
discussing the problem of the effective mass of current carriers. In both metals
and. semiconductors-the effective mass of the current carriers may be either larger
,
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or smaller than the mass of a free tleet on.;'"
The relaxation time Tin. Formula ( determines the interval of time in. the
Connie of which the decay of the originally established current occurs following !
cutoff of the external field. The numerical value of this quantity is deternined
by collision processes or, as one would say, the scattering processes to which
electrons are subject when moving within .a crystal. It is obvious that the more
intensive and more frequent these procesSes are, the smaller will be the time of
?
relaxation and, consequently, the mdbiIity.aslwell.
Let us dwell in somewhat greater detail on the physical patternof the elec-
tron-scattering process in a crystal and clarify the significant differences be-
tween these processes in metals and semiconductors Which affect relaxation time
and, consequently, mobility.
The conditions of scattering of electrons in a crystal are determined by the
itwave properties. It will be recalled that these wave properties of electrons have
I i
been observed directly in certain electron-diffractionexperiments (see Shpol'ski,
I
Atomic Physics). Essentially, the wave izoperties of electrons form the founda-
tion on which the entire modern theory of conductivity in solid substances is
i
based. Consideration of these properties has made it possible to demonstrate the
I .
physical significance of such important concepts as the length of free path of a
i
current carrier -- a concept which found no satisfactory interpretation in the
'
i
old prequantum theories. !:
_ I
1
A
In modern quantum theory, it is not the classical Newtonian equation which
describes the motion of an electron within a crystal, but Schroedinger's wave
equation (see Shpol'skil Atomic Physics). In the simplest case, in which the
interaction between an electron and the atoms (or ions) of a crystal lattice is ac-1
1
counted for by introduction of the effe4ive mass into the Schroedinger equation
(see page 45), the wave which describes the behavior of such a quasi-free particle1
is a plane wave. As is known from the elementary physics course, a plane wave is
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I ,
characterized, by two parameters: the vibrational frequency* and. wavelength X. On
1
comparison of the wave presentation of the motion of the electron with the corpus-
cular presentation, in which the basic parameters of the motion are the values of
mass, velocity, and energy, we should assume that the frequency v is determined by
the energy: y 2117-1 and. the wavelength X by the momentum of the electron: x lo
m v
At room temperature, the average random velocity of the free electrons in a I
nondegenerate semiconductor (and, if they are presentrin an insulator) is 107
cm/sec. According to the formula given above, the wavelength will be 7.10-7am.
In a metal, where the discrete, quantized character of the energy spectrum is
' essentially in evidence, the velocity of the free electrons is determined not by
i the temperature of the substance, but by their concentration (this circumstance is
1
also referred to as degeneracy). At the high values assumed by the free-electron
221 ,
concentration in typical metals (..--e-10 cm-3), the velocity of their motion
attains values which exceed the average velocity of electrons in semiconductors
1
by several thousand percent. Thus the wavelength of the electrons participating
1
In. the conductivity of a metal proves to be one order less (,--.5.10-8 an) than in
1
a semiconductor.
Let us note here that the very fact of the higher electron velocities in
metals must result, according to Formula (2), in decreased mobility (other con-
ditions being equal) by comparison with semiconductors..
Let us now ascertain the manner in which the established difference in wave-
lengths affects the process of electron scattering. Proceeding from the wave pre-
sentation of the motion of the electron, we should assume that the process of
electron-wave scattering is basically determined, Just as with any other waves,
by those nonuniformities in the disposition of atoms which extend to distances of
the order of one-quarter of an electron-wavelength. Nonuniformities of smaller
dimensions do not produce any appreciable wave scattering.
,
As an example which explains this conclusion of wave theory, let us recall
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the explanation for the experimental fact of the transparency of glasses and. -
-, .. .._.
.. ? ,
liquids. The characteristic nonuniformi1 ies of Such substanceS:; which are caused-
by the random distribution of atoms in amorphous media, are of atomic dimensions .
-8 N 1 ? , -- - ., ..-'
. ,
(......10 am), while the wavelength of light is a thousand times longer (.10-5am)t
- i-
... ,.
This dimensional ratio being the case, the propagation of light waves in.a ,

ordered amorphous medium occurs in the same manner as in an ordered transparent
crystal -- i.e., with no appreciable scattering. In turbid medial in which the
dimensions of the spatial nonuniformities attain values comparable to the wave-
length of light, a strong scattering effect is to be observed. These nonuniformi-j
ties of turbid media may be caused eithei by pollution of the medium by foreign 1
particles of appropriate dimensions, or by fluctuations in molecular density, suchl
as occur in gases as a result of chaotic thermal motion of molecules. This motion4
as we know, provides the explanation for 'the blue 'color of the sky (the short-vave.1
length rays of the solar spectrum experience the most intensive scattering in the
atmosphere). Spatial nonuniformities of atomic dimensions may cause appreciable
scattering in the instance when the wavelength of the electromagnetic oscillation
is of the same order of magnitude. As we know, such a condition occurs in the
propagation of X-rays in glasses or liquilds.
The scattering of electron waves in 'a crystals just as that of light in a turbid
medium, is effected by nonuniformities of two different types: disturbances of the
homogeneous structure of the crystal due to impurities, as well as by lattice de-
fects and by fluctuating nonuniformities which arise from the thermal vibration of
the atoms in the lattice. The above-established difference in the electron wave-
lengths enables us to understand the natiire of the difference between the con-
ditions for electron scattering in metal s and in semiconductors (as in insulators),
and why, in certain cases, the free path of an electron may be longer in the lattexl
than in the former. As we see, this consists in the fact that due to the long
wavelength of the electrons in semieondudtor and insulator crystals, nonuniformitiels
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of an atomic order do not give rise to appreciable electron scattering, while in
1 1
a metallic crystal, in which the wavelength of the electron is shorter by one
order, such nonuniformities cause appreciable scattering, due to which the mobili-
ty of the electrons is reduced.
It does not follow from the above, of course, that semiconductors in general
are distinguished from metals by the higher mobility Of their current carriers.
From the table presented at the end of Chapter IX it may be seen that high mobili-
ties, exceeding 10,000 am2fv.sec, are obsekved in only a few semiconductors (which
are prepared, moreover, in a-very pure form). In many instances, the mobility of
the current carriers in semiconductors proves so small that the free paths com-
pUted from these data do not amount to more than fractions of the interatomic
1
distances in the lattice. Serious difficulties arise in theoretical interpreta-
1
idon of the results of measurement in these instances, since by its very physical
. significance in the theory of electric cimductivity, the free path may not be
1 1
smaller than the distance between neighboring atoms. This means, essentially, that
1
4 the conductivity of semiconductors with low current-carrier mobilities cannot be
investigated, within the scope of the existing theory and. that one of the problems
to be met in further development of this theory is that of the clarification of
this question.
The above differences in the physical conditions in which electron scattering
occurs in metals and in semiconductors are manifested not only in the numerical
1
values for mobility, but also in the dependency of mobility on temperature, with
1
whichweare-conbernedialthischapter..leaso essential in this dependency is the
1
fact that in semiconductors, as opposed to metals, electron-velocity is a function
of temperature (as in the classical ideal gas), which has a direct effect on con-
ductivity, as seen from. Formula (2)-.
The manner in. which temperature affects the length of free path is variable,
? .depending OU-the meehAnism of scatteringl
1
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-
We should consider the following fundamental mechanisms of scattering
semiconductors:
,
(I) by thermal vibrations of the atoms or ions of which the crystal lattice '
'is composed,'
(
by the impurities which may be present in a semiconductor, in either the
ionized or neutral state,
( 3 )
" ?
?
by all types of lattice defects, such as vacancies, distortions caused-
by atom implantations, shears, cracks, crystal-grain boundaries., etc.; modern;
semiconductor theory neglects reciprocal' scattering by the electrons themselves,
(due to the small concentration).
1
In atomic lattices (Ge, Si, Te and Others) we deal primarily with -two- ---
mechanisms o.. ,.urrent-carrier scattering whichproduce two essentially different
temperature dependencies of mobility -- one with scattering due to the thermal
1
vibrations of the lattice, and another with scattering due to ionized impurities.
Theory indicates that for scattering due' to themnal vibrations of the lattice, the
1
free path 1 has the same value for all current carriers irrespective of velocity,
and is inversely proportional to the absolute temperature of the substance -- i.e.
1
1... This last result can be understood from the following simple reasoning.
T
It follows from the general classic considerations that the scattering of current
1
carriers should be directly proportional to the cross section of the volume occu-
pied by the vibrating atom; on the basis of simple geometrical considerations,
1
this cross section may in turn be considered directly proportional to the squared
1
amplitude of vibration of the atom, while the latter, which determines the energy
of the lattice, is known to increase as a linear function of temperature. Since,
1 according to Formula (2).
? and v
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(13)
This law is not always confirmed experimentally. Along with instances in which
the dependency takes the form (13), instances of still more marked dependency of
mobility upon temperature (up to u'~?T-3) are also encountered. At law tempera-
- !
tures the effectiveness of. thermal scattering according to (13) diminishes, and
another mechanism of scattering becomes predominant in atomic lattices -- the
mechanism of scattering on ionized impurities. In essence, this mechanism of
scattering is identical to the well-known mechanism of alpha-particle scattering,
which was first investigated by Rutherford in his classical experiments on the
structure of atoms. Characteristic of this mechanism is the decrease in the
effectiveness of scattering of a moving charge as its velocity v increases, with
the result that the length of free path of the current carrier increases with
velocity according to the law 1 ' In this case, for the mobility of a cur-
rent carrier u c, the theory gives the expression:
-
3/2
c (14)
?
NI
If both the thermal and ionic mechanisms participate in the scattering of
current carriers, the resulting mobility may be presented approximately in the form
- 1
_ ONO ?
? ?
. -
,
and. jtaifi temperature curve of mobility by ,
1- aT73/2
bT3/2
u
- ?
_
At. low temperatures the first term pred.ominates, while the second term comes
to ;.i.tr..1cr temperatures.
(15
88
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? -1?? ,r?
.. ? a?"..
The approximate graphic form of the relationship (15) is presented in 71g. 4.
I
-4 c
,19. The position of the maximum of thiE curve depends on the con6entraiion'..bf
?Y
?,;.?, ? ti
? , - - ? t. ? ? `II.
. . ? . ? 4
? ,-impurities;-ai the latter-increases, the maximum Movia'-in the direction Ofjiighert--
? . -? - ?
'temperatures. The scattering due to neutral impurities does not depend on.temper -
ture and its importance is,Usually:seco
- , ? ?
? ?????
Turning to the question of the mobility of Current carriers in*idnic crystals -
(oxides, sulfides, etc.), the following general statement shoulolbe made at the
. . .
? ' ^
, .. ?1 , V . . ,..
. . .
outset. The interaction between the current carriers and the vibrating charged
particles in the ionic lattice (i.e., the ions) is, on :the whole, much stronger..
than the interaction between the current! carriers and. the neutral atoms of the
i
atomic lattice. Therefore, the Scattering of current carriers due to thermal
!
vibrations in ionic crystals occurs wIt1 far greater intensity, and the mobility
I
proves to be lower in most cases in this semiconductor group. The qualitative
I
curve of the temperature-depen&zuce of mobility is the same in this case as in
--
I
. atomic crystals: mobility diminishes with rising temperature, but the specific
form of the theoretical formula differs for high and low temperatures, the
I
. 1
. boundary between these two regions being determined by the maximal frequency rm
t !
-
of the so-called longitudinal optical vibrations of the ions, which is a_character-
hY
istic parameter for any given ionic crystal. At low temperatures,
and at high temperatures u ...-...., ...L Experimental data pertaining to this problem
1
are still meagre and, on the whole, do not agree satisfactorily with the theoreti
Fig. 19. Typical curve of temp?erature-dependency of current-
.
,carrier mobility in a semicondnctor.
89
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. . ,
`-?
The approximate graphic form of the relationship (15) is presented in Jig.
.--. -
,
. ?
19. The position of :the maY-fmnai of this curve depends on the Coit.6entrationt.bf
, .....; ,z,!:,-0.--,.;---. - ? -:.--i ..,,.. z..- .1,-..,
4'44,1;4
-impurities; as the latter 'increases, the Maximum moves in the direction'of`hi ,/
. _ _ ..
_ .? ,, . , ? ? S.1-::47.?
'-
temperatures. The scattering due to neutral impurities does not depend. On.tempera=- -
-s . . ..
,
1 1
?
ture and its importance is.Usually:seco
1
-
Turning to the question of the mobility of Current carriers in ionic crystals
,
(oxides, sulfides, etc.), the following general' statement should he made at the
outset. The interaction between the current carriers and. the vibrating charged.
particles in the ionic lattice (i.e., the ions) is, on the whole, muCh stronger" ?
than tie interaction between the current carriers and the neutral atoms of the
atomic lattice. Therefore, the scattering of current carriers due to thermal
vibrations in ionic crystals occurs wit li far greater intensity, and the mobility
proves to be lower in most cases in this semiconductor group. The Qualitative
I
curve of the temperature-dependence of mobility is the same in this case as in
-
, atomic crystals: mobility diminishes wiith rising temperature, but the specific
form of the theoretical formula differs :for high and low temperatures, the
i
. boundary between these two regions being determined by the maximal frequency Fla
of the so-called longitudinal optical vibrations of the ions, which is %character-
'? -Ef I
istic parameter for any given ionic crystal. At law temperatures, u?-?-?1 / i
1 1
and at high temperatures u...-.....- -1.... Experimental data pertaining to this problem
la
are still meagre and, on the whole, do not agree satisfactorily withthe theoreti-.
1
1
-
.Fig. 19. Typical curve of temperature-dependency of current-
carrier mobility in,a semicond4ctor. _
89
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7
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cal formulas. This may possibly be related to the general observation on ionic
? - semiconductors put forth on page 70.
Now, having explained, the separate effects of temperature on the concentra-
tion and mobility of current carriers, let us describe the general course of the
1
variation of conductivity with temperature. Since in atomic lattices (and, at
higher temperatures, in ionic lattices) mobility may be a comparatively low power
ftnction of temperature, while concentration may vary in accordance with a very
, high exponential law, the overall temperature curve of conductivity will have
approximately the same appearance as the curvefor concentration shown in Fig. 18,
with the exception that in the depletion region, in which concentration is con-
stant, the temperature curve of conductivity is determined, by the variation of
1
temperature with mobility. If the chief factor in the depletion region is the
thermal mechanism of scattering, in which' mobility aiminishes with rising tempera-
ture, the overall conductivity curve will have the form abed. indicated in Fig.
20; if mobility is determined in the region of depletion by scattering due to
Ionized. impurities, in which case mobility increases with rising temperature,
1
the general form of the curve will approximate
1 web
At high temperatures, when the current-carrier concentration becomes so
large that the electron gas enters the region of degeneracy (see page 61), c in-
creases with temperature at a slower rate than indicated by the exponential law
1
presented in. Fig. 20 for the intrinsic-conductivity segment cd in the non-
-.
1
degenerate case. Fig. 20 illustrates the frequently-applied experimental method
_ -
by .means of which the width A1E of the forbidden band of a crystal and the acti-
.
-vaticmcenergy of the impurities tiE are determined. Figs. 18, 19 and.' 20 do not
- ?I
- - -
_
?apply.to any _specific semiconductor; they are plotted on the basis of the general
_ , _ .
;,-.-_t? e - f... : ,, ? :.,, - , - , ,-,,, - .:, :7--:. '''::, 7 7:
._ _,.,.. .
:.,.theoretical_laws set forth above for the influence of ieMperettire on the conceu7, ..,.
....?
iii:..r?.0,:-.?,-.-_-,:::-..?,, -,,,,.:- ,- ... - .--,,..........? .7,4 ,..:_ ?,,,-- .-!:. . .. ...,: ......
tratitiii mobility and. conductivity of an idealized semiconductor."?', , ,....
? -- : -..?.,.., ,,,g4?-jalke,4 . - ' ,-.,..
?? - ; __.,.... .?
rr.,??? a. I: ..; - ...; ...,?.? ,?.1.?:'.....,..... , ..` ? .s'..,? i; ?,-, A!?? d # + 0.r.: .. ? &"ti), :f.r.,,-..
??????),' z
_
4tr.9,
;>,..t.? ?
? ." -`, r" vo 1 with the
1 1
result that the total electromagnetic energy incident on the body is absorbed in a.
thin layer close to the surface of the body, and the increase in the number of cur-
rent carriers occurs only in the thin surface layer. Such an increase in number ot
1
current carriers may exert a slight influence on the total conductivity of a mas-
sive body both because the rate of recombination of electrons and. holes is higher
I
at the Surfacethan in the volume, and. because of the intensified recombination in
\:' ..-?
the volume due to diffusion of secondary current carriers into an. impurity semi-
:
, ?- - ? -
conductor-
'
i
- - ? .., , 1
knegative photoeffect which is accompanied by a decrease, not an increase, in
--_
..f
-tne-conductiyity of -a body exposed to radiation is sometimes observed and is ex-
paainedas a, consequence of this intensified. recombination.. Capture of current
dirriers of the opposite sign by the impurity centers is believed to be the cause
98
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oEft,?
of the diffusion of current carriers with only one sign deep into the body:, There-
,
?
phenomena of capturing("stickineof current carriers have not been ex-epriMemial4
;fr %., V; _
demonstrated in all cases, but are still' often referred. to because they:fuppckrt
..? - -
.. .- ,-
? ,
..
. . ? _ . ..
concepts of the complex mechanisms of the phOtoConducti?ve effect which aid in?ex-
'
--'
?
plaining the Complex rtlationships observed in experiments in the study of photo-
conductivity. The future development of this branch of 'semiconductor Physics will
show to what eicient these concepts are true. Thus far, theexperimental fact is
that the frequency range of photoconductivity is comparatively very narrow. The
situation which prevails in thisrespect is shown schematically in Fig. 28. In
this figure the wavelength X =c / I where c is the velocity of light,
is plotted against the x-axis rather than the frequency i. The absorption co-
efficient 1 and the photoelectric current 2 are plotted against the raxis. It
1
is seen from the curves in Fig. 28 that the photoconductivity of nonmetallic
crystals is observed to be within a comparatively narrow Spectral region at the
long-wave edge of the continuous-absorption band of the crystal. The curve which
corresponds to the photoeffect in the impurity centers is not shown on the sche-
matic graph in Fig; 28; it mould have formed one or several smaller peaks in the
long-wave band of the spectrum.
Since the experimental curves havextended and unclear extremities and. are not
as distinct as the schematic curves in Fig. 28, a wavelength situated on the
right-hand descending atm of the photoconductivity curve and corresponding to the
half-value of photosensitivity ( A1 n Fig. 28) is taken as the threshold
2 _ph
wavelength which determines the activation energy. The activation energy iLao
determined from optical data may be compared with that determined from the tempera
ture curve of conductivity, A.E0 20).
There are theoretical considerations according to which the above two methods I
of measuring the forbidden-bandwidth may, in some cases, produce numerically dif-
ferent results. We shrll discuss these considerations at somewhat greater length, 1
I
?
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for they involve important problems of semiconductor physics.
The first consideration proceeds from the specific properties of the ionic
lattice.. In -hider case theory leads to the conclusion thatAi-22 7. Al .% and
explains this discrepancy on the basis of polarization of the ionic lattice. Let
us take into account the fact that an electron is displaced into space upon exci-
tation. A displacement Of the surrounding electrons and nuclei into new equilibrium
positions occurs as a response, with the result that the energy of the crystal
diminishes: In the atomic lattice, this
polarization process is effected solely
by displacement of elections, which is
very rapid. The polarization process in
the ionic lattice is due to both the rapid
displacement of light electrons and. the
comparatively slaw displacement of the
heavy ions. The energy of optional eXci-
Fig. 28. Illustrative spectral
distribution curves for optical tation in an ionic crystal is determined
absorption 1 and. photoconductivity
2 in crystals. by the difference in energy between the
!two states of the crystal: the nonexcited
initial state and. that after polarization -- i.e., after displacement of the ions
into new equilibrium positions. Therefore the energy A Oh of optical excita-
t -o
tion is greater than the energy Al E of the thermal excitation. by. the quantity
receives as a result of the displacement of ions.
- ,
in atomic crystals, since the polarization of the medium, which
is effected. by the displacement of electrons only, keeps up with the process of
I - 7
of energy which the lattice
A. e=gAL
-
excitation.
The experimental fact which we are now discussing may become more compre-
hensible if explained in the following manner. The phenomenon of the electron
excitation in an atomic crystal is distinguished from that in an ionized crystal
3:00
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I
by the absence of the energy of polarization displacement of ions in the energy
? .
balance of the atomic crystall'which is present in the energy balance of the ionic
crystal; this energy is converted'into t ermal energy of the ,crystal in both opti-
cal and thermal excitation mechanism's. Therefore if the excitaticiii In an ionic
crystal is caused by a source of thermal energy, the final excitation,.energy ex-
._ ,
_
pended by the source is reduced bY the amount of energy returned to the source by
. ?
ionic polarization. In the case of the ? optical excitation Mechanism; the electro-
magnetic quantum must furnish the electron undergoing excitation:with-the-ie'qUire-B-.
, 1
...1??
quantity of' energy in full.
. ?
The difference in the numerical values of A EPh and. A Et for the sub-
stance may be linked to yet another circumstance. The fact is that optical elec-
tron transitions in a solid (as in individual atoms) obey specific laws of' selec-
tion which, in some cases, do not permit transition from the valence band to the
conduction band. by the shortest energy route; in such cases, the only possible
transitions are those occurring between levels at some distance from the energy-
band boundaries. This circumstance, which applies equally to both ionic and
atomic crystals, may also give rise to an inequality of the type A EPh> Z
-o -o?
No such restrictive rules apply for the thermal mechanism of excitation, and the
thermal transfer of electrons can proceed along the shortest energy route between
bands.
a third and last cause of' possible discrepancies between the experi-
Finally,
mental values
tation in the
ph
for A E and
-o
E may arise from the excitonic mechanism of exci-
-o
photocond.uctive effect, which will be discussed on page 108. We
indicate here only that the discrepancy in experimental data on the width of the
1
forbidden band which results from this is, expressed. by the converse inequality
STAT
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NI
It should be noted, however, that existing experimental data from determina-
tions of the forbidden-bandwidth in atomic and ionic crystals by optical and
electrical methods reveal noticeable divergency between A6E and A E
?o
only in rare cases.
The next problem to be dealt with is the establishment of the dependency of
1
the equilibrium concentration of the current carriers excibed by electromagnetic
radiation on the intensity of the latter, Theory and experiment indicate that the
. result depends on whether the photoconductivity constitutes a small addition to
the conductivity due to thermal excitation (i.e., to the dark conductivity) or is
the primary generator of current carriers in the crystal. The equilibrium concentra-
1 -
tion of current carriers results from two competitive processes -- excitation,
1
which depends on the radiation intensity ,L and the reverse process of recombina-
tion, which depends on the total equilibrium concentration. When the dark con-
ductivity of the semiconductor is small and can be disregarded, the equilibrium
1
concentration n is determined by a formula of the type:
-ph
n -
- alit2
1
1
where a is a coefficient of proportionality which depends on the photoelectric
activity of the light and the rate of recombination of current carriers.
1
In the other case, in which the number of current carriers created by the
photoconductive effect is small as compered to the number of dark current carriersi
- ?
-
-
the additional concentration at equilibrium is determined by the expression
'"V ?
A" ? ?
:
?q
,
.1?_
.Experiments indicate that, the? mobilities of, both the photocarriers and the
-c..,;.-_,--:-_-:?,-...",?:??,.....:.?;._ ?-2.? ,':-..u.:::':T'.-7c.;:::';',-,'f?'-',;:---- ''''; '."( '-:' - --, ._-? - .-._-?f-- = . ..: ,?-7-,. -7,' ?-,):? .---.. ..--..,, -
dark carriers are equal.. From this ,it f llowethat the photoconductivity as
4. 4 . ...
determined: by the productph phph
is proportional either to the square root
. ---
of the-4radiatiorCintensity when the photoconductive effect is the principal factor
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-
-d
in the excitation of current carriers, or to the radiation intensity_itaelc:when_
, 1
photoconductivity constitutes a small adaition to the dark cohductivitj: "It should-r,..,
? .... ____ - --- -
be noted, however, that these laws are h t universal, for dt
r some,SemiconucOrith
. , - . . , - . _ _
. .
. .
- ......, - .
dependence of photoconductivity on radiation intensity is expreased:eitherlyy:a..,-
, . . . _
'... _
different exponent or is of more complex%dharacter in general. ;As wai*ted--.,
?
earlier, some semiconductors even show .a decrease rather than.an increase in con-
.
ductivity when exposed to radiation,in a certain temperature range (negative photo-
-
effect). . ., - - . -,,
. , , -
Experimental studies indicate that not only the nobility of the photocurrent
carriers but also the sign of their charre is the same as that of the dark car-
riers. This is natural in the case of electromagnetic excitation of current car-
riers from impurity levels. However, it is also found that when electrons are
excited from the valence band into the cOnductivity band -- in which case, as we
know, two current carriers are formed (an electron and a hole) -- the charge car-
rier which participates in dark conductivity usually possesses mobility in the
crystal lattice. This experimental fact is not yet accounted for; it is unclear
why the carrier with the opposite sign becomes trapped, as it were, in the lattice
very shortly after formation and does not participate in the current. The fact
that such carriers actvPlly arise in the photoconductive effect is confirmed by
our experiments with secondary radiation: additional carriers with the opposite
sign appear appear when a semiconductor previoulsly excited by photons which transfer elec
trons from ihe valence band into the coniduction band is subsequently exposed to
radiation by photons whose wavelength is such that they are unable to effect the
above transfer ( hv .c 4IE
-o ). These additional current carriers do not appear
when such a crystal is exposed to radiation by the latter photon type only. Bence
we conclude that the secondary exposure to radiation does not create new current
carriers but merely liberates those charges which have been trapped in the various
secondary "capture" levels formed by impurities .and. crystal defects in the first
c.+Imap nf PlmitatiOn.
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bc,
_
??????
The dependence of the intensity of the photoelectric current on the strength
of the external electric field acting on
the photoconductive substance (the so-
called current-voltage characteristic) has its own characteristic properties. The
primary photocurrent created. by those charges which are directly activated by
light conform to Ohm's law only in a low4field region with an upper boundary nu-
merically different for different photocInductive substances. As the field in-
creases beyond this boundary, the primary photoelectric current increases at a
rate slaver than that prescribed. by a linear relationship, with a tendency to
1
saturation in strong fields which is illustrated by the theoretical curve in
Fig. 29.
Physically, this phenomenon stems from the mechanism of generation of the
primary photoelectric current itself. Charge carriers activated* light migrate
at random in the crystal in the absence of a field during the period 1:0 of their
free existence, and then 'recombine or are captured. by impurities or defects of the
crystal. If a weak external field E is applied to the body exposed to radiation,
1
the photocurrent carriers drift in the field during the period from the moment of
excitation to the moment of their recombination, undergoing displacement toward
the electrodes through a distance d, whili is determined by the product d = urco,
where u is the mobility. The quantity d is called. the drift path of the current
.
carriers andIo their lifetime (see page 38). The strength of the photocurrent, '
1?
oh
which is proportional to the number ,of primary photocarriers liberated by
?o t
o
light in unit time and to their mean drift d ( lph 12.1ph d) in the field,
. . .
will increase linearly -With the field only until the drift path attains
-
? ?
-
the ? magnitude Of theinterelectrode distance D. The increase of the pri-
_
mary photodurrent with increasing field Strength slackens as the drift of the
?
mdi-
'vichi-1 carriers begins to experience restriction by; the distance to the
?
:eleetrOde,"and is 'limited by a certain value I
?
equal to :the toial-charge of all
--max
current:.:carriers liberated.by the light er unit time over the entire cryitial 'and
?
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transported to the electrodes by the fieLd: I = en .
-max ?1111
,
Fig. 29. Current-voltage characteristic of primary photocurrent.
This picture is observed in pure form only in nonconductor crystals. Thus the
saturation point of photocurrent was attLined in a 15,000-v/cm field in the diamonb.
specimens investigated and in a 2,000-v/cm field in AgBr. The current-voltage
characteristics of many typical semiconductors exposed to light differ considerabli
from the theoretical curve shown in, Fig. 29. Two circumstances are responsible I
this departure: first, the fact that the above limiting condition (Ii ):
concerns only the primary photoelectric Component of the total current and. not itsi
?
dark component (which considerably exceeds the former in many semiconductors) and,'
second, the fact that various seconaRry phenomena which distort the above mechanism
of formation of.the photocurrent appreciably are superimposed on the primary phot4
current in semiconductors.
? The fact that photoconductivity is observed both in semiconductors and'insu- I
lators confirms the correctness of the consistent concept of electroconductivity 1
for all solids which has been established by the band. theory. The experimentally 1
observed differences in the manifestations of photoconductivity in semiconductors I
and insulators arises basically from the fact that, in the former, the photocurreni
often constitutes only an inconsiderable
addition to the dark current. It is an
empirically established fact that photoconductivity does not occur in all pure
105 STAT
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0
semiconductors and insulators, but only in those with dielectric constants e
greater than 4. The photosensitivity may be imparted to substances which fail to
meet this condition by adding impurities' to them. In addition to impurities, the
condition of a solid's surface (e.g., its roughness) also influences its photo-
sensitivity, especially in cases in which the absorption of light occurs in a
thin surface layer. This is explained 1br the fact that the condition of the sur-
face of a body strongly affects the rate of surface recombination -- i.e., the
1
rate of the process which returns the current carriers to their nonconductive state.
Also of interest is the question of the number of current carriers which
correspond to I absorbed quantum (photon); 'this value is called the quantum yield
of the -photoconductive effect. In the majority of cases the quantum yield within
the photoelectrically active band of .the electromagnetic spectrum is equal to
unity -- i.e., each photon engenders one pair of current carriers in the excitation
of the basic lattice. This is shown clearly, for example, in the case of germanium,
1
where each photon with a wavelength from 1.0 to 1.8 forms a single pair consisting
1
of one electron mad one hole. Sometimes? however, comparatively small amounts of
1
radiation produce intensive photoconductivity in experiments with certain sub-
stances, and. the quantum yield is found to exceed unity. According to the general
concept of the mechanism of the photoconductive effect which was developed earlier,
. each primary excitation event in the intrinsic lattice can create only one pair of
current carriers. To account for experimental facts involving a quantumyteld
which exceeds unity, therefore, various secondary processes are assumed to arise
from the imperfection of the crystal after excitation of primary current carriers.
As an example, the very strong photoconductivity effects observed in fine-grained
crystalline specimens of same semiconductors are explained by the presence of in-
sulating, oxidized intercrystalline layers whose conductivity changes sharply on
the appearance of the primary photoelectrons. Experiments indicate that the In-
fluence of light on such a semiconductor manifests itself in a sharp decline in
-
106
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1
the resistance of the specimen, although the conductivity of the material of the
semiconductor itself changes inconsiderably inside the crystals. Experiment indi-
cates that secondary processes do actu1ly occur in the photoconductive effect;
.- ? , .
The secondary processes are distinguished from the primary by the noticeable 'but.
-
sometimes quite small time lag in their development. Therefore an'ultra-short
pulse irradiation technique which prevents the development of the secondary proc-
esses is used in experimental separation:of these processes.-
?
The quantum yield is found in some cases to exceed 1 due tO the fact that the.
,
photoelectric activity of the light extends into the frequency range in which'the
energy of a single photon hv is several times greater than the excitation energy of
The primary current carrier in this case proves capable
of exciting additional secondary current carriers by virtue of the excess energy
(h14--LiE ) which it acquires on absorption of a photon.
It was ftndicated in the discussion of Fig. 28 that in the majority of cases
only a comparatively narrow range of the electromagnetic spectrum at the longwave
edge of the continuous-absorption band of a substance is photoelectrically active.
This means that photons which ao not belong to this narrow range do not excite
current carriers even when absorbed by a solid and possessed of an energy sufficient
to accomplish an excitation event 0.4Y > AE:0). This photoelectrically inactive
absorption of light has suggested to physicists the idea of the existence of a
certain mechanism of?excitation which produces no current carriers. However, it
was recently proposed that this mechanism of excitation also participates in photo
electrically active vcitation -- i.e., that the formation of current carriers unde6
the influence of light proceeds in two stages. During the first stage a photon
transfers an electron not directly into the conduction band but into an excited level
the current carriers dE .
-o
-
located just below the bottom of the conduction band. An electron in this state
does not completely escape_the-attractionl of its atom, but remains related to the
hole which it had left behind in the latter. Such an excited state of an atom is
107
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T
1
called an exciton and maybe transmitted from one atom to another -- i.e., migrate
1
at random within the atom. It must be understood here that in transmission from
one atom to another, the excited state returns the first atom to its normal unex-
cited. state. The application of an external electrical field has no influence on
the random movement of excitons in the crystal lattice and therefore does not cre-
ate a current, since the exciton as a whole coAstitutes a neutral combination of an
electron and a hole and the bond cannot be broken by a low external field. Accord
jug to the above concept, an exciton can "break" into two current carriers (an
electron and. a hole) or recombine -- i.e., return the atom to its normal state --
only upon collision with impurity centers. The former (i.e., the second stage of
the photoconductive effect) occurs at the expense of the thermal energy which it is
1
necessary to impart to the exciton to effect the break -- or, in other words, the i
1
energy which is necessary to transfer the electron from the exciton level to the
conduction. band. The latter (i.e., the recombination of the exciton) is accompanied
either by emission of an electromagnetic quantum or -- which is more frequently
1 the case -- by transfer of the exciton' energy to the lattice in the form of heat..
1 1
An electron and a hole are formed after the second stage of the photoeffect, but
0
1
one of these current carriers may prove ineffective due to its low mobility or
capture by impurity centers. This, briefly, is the physical essence of the modern:
theory of excitons. 1
1
The Above enables us to understand one of the causes of the phenomenon which
was discussed. on page 101. It may be seen that the explanation reduces to the
:
-
fact thatwhen the photoconductive effect -is due to the excitonic,mechanism of exci-
?
tation,-%the threshold.excitation frequency vo may be below pEo/h and, consequently,
. .
ph
the forbidden-band. (AB ) determined by the optical method as AE
. ?o . 1
may be found. smA11 er thayi A Er . It should. be noted that the excitonio concept
7-0
_
bAs received experimental confirmation in the absorption lines recently observed
iwthe:long-giavelength edge of the fundamental absorption bands of some '
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-
?.,
,
semiconductors (i.e., at the right end. of the absorption curve;,iig,_218).
. . -
? '
? .
How does variation in the temperature of a body influence photoconductivity
, A
? I
. ,??
The relative importance of photoconductivity mast, naturally, increase 7with
? -
decreasing temperature, since the dark conductivity which serves as the background
i
against which the photoconductivity mPnifests itself declines with decreasing
temperature. Theory and experience indiCate that the absolute magnitude of-phOto
? I ? ?
conductivity increases with decreasing temperature. This is explained*by-the_fact.
that as the concentration of dark current carriers' - - the electrons and
-
decreases, the probability of recombination of current carriers is also reduced.
-
Variation in temperature also influences the red photoconductivity threshold'
( Xl, in Fig. 28); here, the algebraic sign of the shift is found to be
dif-
ferent for different semiconductors -- i.e.,?some substances show a displacement
of the threshold to the right with decreasing temperature, while in others it
shifts to the left. This experimental result is explained as due to constriction
of the forbidden band in some semiconductors and its expansion in others with in-
creasing temperature (see page 37).
Characteristic of the photoelectric effect of electromagnetic radiation is the
fact that the change in the electrical properties of semiconductors exposed. to
radiation is of tempordry nature (with the exception of the case where ).--quanta
induce artificial radioactivity). This Means that the pre-irradiation value of the
dark conductivity is recovered more or less rapidly in the majority of cases after
irradiation ceases; some semiconductors require microseconds for this process,
while in others it requires minutes and even hours. This circumstance is of ex-
treme importance for a whole series af technical applications of the phenomenon of
photoconductivity, since high specifications are sometimes imposed on devices of
this class (photoresistors) with respect to their response inertia.
Th /
e most suitable materials for pho oresistors are selenium, germanium, and
I
silicon, as well as sulfides and seledides of cadmium, lead and bismuth. Thin
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l.ayers of PbSI PbSe and. PbTe are photose itive in the far infrared band. of the .
I
spectrum extending to 4 )1. . This photos4nsitivity boundary moves in. the direction
,
of longer wavelengths with decreasing temperature.
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??
?
_,CHA.P1 ,ER VI 1
TEE INFLUENCE OF STRONG ELECTRIC FIELDS ON ATIE CONDUCTIVITY OF SEMICONDUCT6RS
It follows from the physical considerations by which we substantiated Ohm's_
. , . ?
law see Formulas [(1) and (2)1 that this law must apply only if the current car-
rier mobility and concentration are independent of the electric-field strength.
Actually, the specific conductivity S is not a function of the field strength E
in this case, and the current is proportional to the field in agreement with Ohm's'
law. 1
Let us discuss the physical significance of these two premises. Let us begin'
with mobility. According to (2),u . e7/m , so that the dependence of the mobil7
ity u on the field E may manifest itself only through the free timeT . The valuei
1
v itself begins to depend on the field when the increase in the velocityvl
over the free path]. becomes comparable to the velocity -- i.e., NrhenAv 1 as I
determined by Formula (1), is not negligibly small as compared to v. The quantity'
1C is both an explicit (denominator) and implicit function of v, since the nuneratoIl
the length of free path 1, also depends on v in some important scattering mechanis4s.
Consequently, the physical premise of the field-independence of mobility is re-
duced to one of negligibly small influence of the field on the velocity. On the other
hand, however, it follows from the considerations on which Ohm's law is based_thati
1
the influence of the electric field consists precisely in a change in velocity, ind
it night appear that no explanation of the occurrence of the current could be foul.
if this change were completely disregarded. How to account for this? The fact isi
that the velocity of materiaLparticle is a vector quantity, and the e1STATC___j
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field acts on this quantity in two ways:i it changes both the absolute magnitude
of the velocity and. its direction in spaCe. In the low-field range in which Ohm's
law is valid, the change in the magnitude of electron velocity over the free path
is small by comparison with the absolute magnitude of the initial velocity of the
electron in its free path, and may be neglected in this case. The other effect of
the field -- the change of the direction of electron velocity suffices to ex-
plain the occurrence of the current and substantiate Ohm's law.
In. the absence of an external field, the directed average velocity found by
geometrical summation of the velocities of the individual electrons is equal to
zero (see Fig. 2). Application of an external field produces a certain directed
resultant velocityAv which, though small by comparison with the absolute average
velocity of the electrons, appears distinctly enough against the background of the
zero directional velocity which prevails in the absence of the field.
'It can be imagined that with increasing field strength, the change in the
absolute value of the velocity over the free path of an electron becomes, at suf-
ficiently high fields, comparable to the magnitude of the initial free-path
velocity.
An increase in. the average random energy of electrons represents, essentially
i an increase in their average thermal energy measured in degrees of the temperature
scale. This means that in a strong enough field the steady-state temperature of
I the electron gas. maynoticeably exceed. that of the lattice. As a result, the
itransfer of energy by current carriers to the lattice increases, and a stationary
condition is established in which the total energy received from the external field
l.'s transmitted by the current carriers to the lattice. Since the thermal velocity
- - ?-
, ? ,
of current carriers increases in this case in proportion to the square root of the
. ?
field strength, the mobility. of semiconductors with the length of free path lade-
,-
perident of velocity must vary according tp the law 21'I/VE [see Formula (1):1.
? -
For)thesefields, which we refer to as strong fields, Ohm's law no longer holds,
. ?
_
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F
_ ? ?
since a noticeable increase in the velocitrover the length of the free peth.giies
rise to an appreciable dependence of bot
? the free time T and. the mobility U.
the field. strength E; this.results .in dependence of the
. .f
a departure. from Ohm's law. It follows from the relationship r = 1. (1)1v? that
. .
and, consequently, the mobility u, may either decrease or increasewith'increasing
field strength E. Thus, "C and, consequently, u must decrease with increasing field
strength (causing a marked increase in v when the free path i does not depend on -
1
the velocity vl as is the case in the thermal mechanism of scattering in atomic '
I
crystals. Butt and, consequently, u must increase with increasing field strength
when the free path l.-.v4, as is the case in the scattering of current carriers by
ionized impurities.
Thus a departure from Ohm's law due
to dependence of nobility on the field
must occur when the external field causes a perceptible increase in the velocity
of electrons. The minimum field strengt1 which causes the departure from Ohm's
? lay to become perceptible is called the critical field E. What are the substances
in which this departure should occur earliest? Sinceliv= luE 1 a noticeable in-
crease in velocity with increasing field:will manifest itself earlier in substances
with greater mobility than in those with lower mobility. The mobility of electrons
in germanium at room temperature, for example, is u, = 3900 cMiv.sec, so that even
a field with an intensity of 103 vicm should produce in it a noticeable increase
in velocity over the free path. The relative &II:Inge in velocity in this case
amounts to
114 V
uE1-
3900
? 103
3.101..
STAT
113
-
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Fig. 30. Experimental curves illustrating departure from Ohm's
law due to change in mobility of current carriers in germanium
in strong fields.
Experiment has actually shown that a perceptible departure from Ohm's law
may be observed in electronic germanium at as low .a field as 900 vicm at room
temperature (see Fig. 30). The critical field of hole-type germanium at room
temperature is somewhat stronger -- 1400 vicm; this is in accord with the fact
that the mobility of holes in germanium Cu ..1900 aj/v.sec) is lower than. that
of electrons. The experimental results for silicon may be considered a confirma-
1
tion of this rule for the dependency of critical field upon mobility. Thus for
electronic silicon Ec 2500 viam Cu . 1200 cm2iv.sec); for hole-type silicon
-
E' 7500 viam-Cu 500 am2ivssec). It:follows further from this rule that the
,
critical field should decrease if mobility increases with decreasing temperature.
.Indeedl. Fig. 30 shows-that the critical field in electron-type germanium falls to
- -
,
120 vicm. at T =770IC. In all of the above cases, the departures from Ohm's law
_
manif' _est themselves in transitions from the direct proportional i ty with
E 20 vjam. Howeverlithere exists another body of opinion as 1
STAT
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????
i i
to the treatment of these last experimen al facts (Fig. 31) which maintains thatf t e
observed. departure from Ohm's law is due Ito violation of its second premise ?
1
,
i.e., to nondependence of the current carrier concentration on the field. Let us
now turn to this problem.
Fig. 32. Curve illustrating the
influence of strong fields on con- rier concentrations of many semiconductors
ductivity- with mechanism of second-
ary increase in current carrier con- show a noticeable increase with increasing
centration due to the electric field.
so that the current rises at a much
faster rate than that required by Ohm's law.
Theory predicts that when the in-
tensity of the field in a pare semicon-
ductor exceeds 104 to 105 viam additional
current carriers produced by the field
should appear in it. Experiments have
=twiny shown that in the range of fields
with E = 104 to 105 viam and sometimes
MC 3
even with E 10 vicm, the current-car-
The experimental data may be divided into two groups. The dependence of the
conductivity on the field for those in the first group is characterized by Fool's*
empirical law cr mcscp, ? E'e ) (see Fig. 32), where cro = -o/2i voc_is a co-
efficient -which depends upon temperature and usually decreases as the latter in-
creases, and E'e is the critical field, Which depends on the nature of the semi-
,
conductor, temperature and the impurity concentration; the dependency of current
.on. field for the second group may be represented in the form of the Frenkel's theo-
retically-derived law 45::ao 131/2
'-What then, is the physical essence of this phenomenon -- i.e., the mechanism
of the influence of the electrical field which is responsible for the increase in
. . _
-
'
? " -
Translator's ,note: Name transliterated (Pul'). Spelling not
116--
S TAT
I
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,
current carrier concentration? It devellps that this phenomenon involves not one,
but as many as four different mechanisms, 'which characteristically become operaiiv
_ . . ?
,
not simultaneously but at different critical fields. We shall -concern Ourselvei--
only with the two meelw-Inisms which produce stationary states without':Cauiing de-
composition of the substance, since the other two (electrostatid4opliation- and
- 7
Stark's effect), according to theoretical estimates, first become effect1ve,inL the
_ -
region of7 vicm? when breakdown due to the-first-twomechanisms-las -
already occurred.
-
With increasing field strength, the mechanism of Frenkers thermoelectronicl
ionization is the first to become effective. The action of this mechanism -will
1
become clear when we consider that the eiternal electric *field acts on the electroli
1
with a force a and changes its energy state in the atom. As a result, the energy'
expenditure necessary to excite an election into the conductive state is reduced
e e
by a quantity 2e ? , while the probability of thermal excitation
' Ve7 17 -47-
increases correspondingly by a factor expk2e ? which results directly in
1
an increase in current-carrier concentration in accordance with Frenkers law,
above. A more exact theory of this phenomenon gives a dependency of the type of
Pool's law for conductivity in comparatively weak fields and of the type of Frenke1's
1
law for stronger fields. -Thus the action of the thermoelectronic ionization
mechanism reduces to facilitation of the escape of the electron from the atom; the
separation itself is effected by thermall 1
excitation.
1 1
To understand the physical essence Of the second mechanism, whichyfollowing I
1
the mechanism of thermoelectronic ionization, becomes effective in stronger fields),
it should be taken into consideration that a free electron undergoing acceleration,
on its free path, may, in a strong enough field, acquire an energy sufficient to
excite either a bound electron, an impurly atom, or an atom of the basic lattice. 1
In accordance with the physical essence of the phenomenon, thii mechanism of exci-1
tation of secondary current carriers is claled impact ionization. The mechanism
-
117
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7
of impact ionization may bring about an Increase in current-carrier concentration
only under the condition that the ionizing electron, after exciting a bound elec-
tronfitself remains excited -- i.e., in the conductive state. Clearly, this can
only occur when the kinetic energy of the ionizing electron is so great that it
retains (after effecting the excitation) an energy sufficient to keep it in the
conduction band -- i.e., when the ionizing electron is displaced only from a
higher to a lower level within the limits of the conduction band in the process
of impact ionization. This, in turn, requires that the conductivity-zone width
6xceed the forbidden-bandwidth when electrons in the valence band are excited.
Similar conditions may also be formulated for the impact ionization which may be
produced in the valence zone by a hole. Since the activation energy of impurities
is usually less than the forbidden-bandwidth, the excitation of current carriers
in an increasing electric field occurs first from impurity levels, following which
the impact ionization of basic atoms of the lattice becomes effective.
The above concept of the mechanism of impact ionization may give rise to the
misgiving that it cannot achieve a stationary state and mist always lead to break-
down. Indeed, does not an avalanche-like process of formation of steadily in-
creasing numbers of current carriers -- a process which should lead to breakdown -
occur at a given high field strength when conditions are such that the ionizing
electrons, after exciting secondary electrons, themselves remain in the conduction
band? This would. be the case if the process of excitation of current carriers were
not concurrent with another opposed and stabilizing process -- that of recothina-
llon,of current carriers. The formation of secondary current carriers by impact
,ionization is accompanied by either additional free impurity levels or additional-
, _ ? .- ?
iholeasin the valence band; the excited electrons recombine with these with a
? - - ?
probabilitywhich-increases with their nuMber. As a re
atil"1-;t7ers is' established. asa result Of impact
betifeenUationocesses-, some
e ? - -
_
. -
, -
SUlt of the competition
increased concentration
ionization, and increases
, ? .?-???..?
??
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with the field but remains unchanged in r given strop field.
When the field is strong enough (usually at E ?. 105 io 106 vim) the proc-
ess of impact ionization assumes an avalanche-like Character and leads, of course,
- to 'a breakdown.::
Since great quantities of energy which may considerably or even dangerously"
heat it are developed in the test specimen in strong fields, experimental studies
of the effects of strong electrical fields on the conductivity of
. -
are often carried out using the transient-Pulse method..
semiconductors-
,.
Let us note in conclusion that the conductivity increase in semiconductors
in strong electric fields is a phenomenon utilized in technical devices for the
1
' protection of electric transmission lines from overvoltage.
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CHAP ER VII
THE INFLUENCE OF VARIOUS TYPES OF CORPUSCULAR RADIATION ON THE CONLUCTIVITY OF
SEMICONDUCTORS
There are four reasons for the change in conductivity of semiconductors under
bombardment by streams of high-speed electrons, neutrons, deuterons and -particles.
1) Corpuscular radiation excites (ionizes) atoms, with the result that the
concentration of current carriers increases. This excitation is, in general, of
the same ephemeral character as in the cage of the photoconductive effect, and
vanishes rapidly after withdrawal of the radiation.
2) Bombarding particles cause displacements of ordered atoms (or ions), with
the result that stable and. unstable defects of two types appear in the lattice:
vacancies and interstitially implanted ions. Like chemical impurities, these de-
fects alter, the energy spectrum of electrons, causing a change in the equilibrium
poncentration of current carriers which arises from thermal excitation. The work
required.to remove an atom from a germanium lattice point amounts to about 25 to
30 ev, so that the displacements of atoms from the lattice points during boMbard-
ment of.this semiconductor by heavy high-speed particles are effected not only by
the.. bombarding particles themselves but also by those primarily displaced atoms
, . i ..,.. , , - .. _-_-...? - ?
- t ,
which haVe received energy sufficient for secondary displacement of other atoms in
-the-lattiCe. However, the number of stable secondary current carriers dislodged
- ,
. . . . , ?,"-..:.
...-
,t-One:'bombarding, particle ,does not exceed 4 or 5, although the total number of
_ _ , ,-: -.:.- - ..---,'?-,-,4 ,-,
- , 4 - - - . . , ..- - _ ., _..? ? ,
atoms displaced may amount to several hundred. This is explained by the' fact that
defects of the laitice.are for the most part quickly corrected after radiation
120_
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ceases. In many cases) even complete restoration of the original state of the
specimen maybe effected by means of appropriate annealing.
- 3) Under corpuscular radiation, as
well as when exposed to the 7,-rays of
the electromagnetic spectrum, radioactive transformations in the body exposedto
I _ ,
radiation may occur, resulting in the appearance of stable atoms or ions of elements
I
which were absent in the original substance. This development is similar-in its,A- --:---
- . .:. .
results to the process of injecting chemical impuritiesothich is- aften-usedto im-t.1._ ,
part definite, specified electrical properties to a semiconductor.- Since the'in-
jection of chemical impurities is a very importantprocedure in semiconductor '
I
technology, it is likely that techniques by which impurity centers are created by
various types of corpuscplAr radiation, which have their own peculiar advantages,
will compete successfully with the chemical method and find Important technical
application.
4) Defects of the lattice and impurities in the form of foreign elements
created by the action of corpuscular radiation reduce the current-carrier mobility!
(and, consequently, the conductivity) insofar as mobility is determined. by scat-
tering at impurity centers.
After thus generally characterizing the effects of all kinds of corpuscular
radiations, let us discuss in somewhat greater detail those properties which are
characteristic of certain particular forms of corpuscular radiation. Further dis-
cussion of this problem-will be Chiefly and necessarily concerned withgermanium
and silicon, since up to the present time these semiconductors had been studied. in
the majority of experiments made on the action of corpuscular radiation.
Electrons. It has already been indicated that a work of 25 ev per atom is
required to create stable effects resulting from displacement of atoms from the
lattice points of germanium. Calculation indicates that an electron may transmit
such an energy to an atom in an elastic Collision only if its initial energy ex-
Nev. _
ceeds 0.55 If the. energy of the boibarding electrons is lsss than this
_
121 STAT
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1
quantity, their ?effects on the semiconductor are reduced.
as in the case of the
photoconductive effect, to. a temporary change in the current-carrier concentration'
which vanishes when radiation ceases. Eicperimeuts with the exposure of germanium
to radiation by medium-energy electrons (
R and -- which is most important -- the conductivity of
?sc-
the semiconductor contact layer, in which the electron concentration
152
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Yr.
_
is greater in this case, is higher than that in its volume. The layer thus formed
constitutes an antibarrier layer and cannot serve to rectify alternating currents.
Similarly, it can be shown that formation of a barrier layer in an impurity acceptor
semiconductor is :possible only withA2C.Esc. When this inequality is reversed an
1 !
tantibarrier layer is formed in the conta't layer of a hole-type semiconductor.
3. Semiconductor-to-Semiconductor Contact
Junctions between two semiconductors may be realized in three variants:
a) junctions of two impuritSr semiconductors with current carriers of the same sign;
b) junctions of two impurity semiconductors ivith current carriers of different
signs; and c) a contact between two intrinsic semiconductors with two current-car-
rier species -- which will not be discussed here since it is the most complicated
in theory and of the least interest in practice.
a) The processes at the junction between two semiconductors with like signs
of conductivity are approximately the same as those at the junction of a semicon-
1
ductor with a metal. A, barrier layer with a sharply reduced current-carrier con-
centration is formed in one semiconductor and an antibarrier layer with a current-
carrier concentration higher than that in the volume is formed in the other. The
result depends both on the relation between the work functions of the contacting
bodies and on the sign of their conductivities. Thus, for example, if electronic
1
semiconductors form the contact, the contact layer of the semiconductor with the
lower work function loses free charges and. becomes a barrier layer; the contact
layer of the other semiconductor, which has the higher work function, becomes an
-I
antibarrier layer. The piaure is reverged at junctions between hole-type 1semi-
conductors: the barrier layer is formed 4..n. the conductor with the higher work
? 1
function'ind the antibarrier layer in that with the lower.
b) The jUnction'between two impuritirsemiconductors with nnlike signs of
conductivity is of the greatest practica_ interest. Its energy diagram is present
?
153
STAT
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I.
11.
'
in. Fig. 40. The contact field at the bIndary of the electron-type (21) and. the
hole-type (2) semiconductors is due to t
e loss of mobile charges by both boundary
levels. This also effects leveling of e ectrochemical potentials in the two e semi-
conductors. An excess positive. charge appears at the boundary of the n-r,egion in
the form of fixed ionized donors and an equal negative charge is created in the
boundary of the 2-region by acceptors fixed in the lattice; the entire contact
boundary layer becomes a layer of high electrical resistance and, fcr this reasoup
is called a barrier layer. The thickness of the electron-hole, or, as it is often
called, 2-n transition, is of the order Of 10-4 to 10-5 am in the most important
cases encountered in practice. The height (y.c) of the potential barrier concen-
trated in this layer attains values of several tenths of a volt. Since the cur-
rent carriers in a nondegenerate semiconductor acquire such energies only at a--
temperature of several thousand degrees, neither electrons from the n-type semi-
conductor nor holes from the 2.-type semiconductor can diffuse into the contact
layer in appreciable numbers at normal temperatures,,so that the latter becomes a
barrier layer for mobile charges. One can readily imagine that holes which dif-
fuse from left to right in the direction of the boundary of separation are acted
upon in the contact layer by the repulsive forces of the positively charged
donors, which are present in excess at the boundary of the n-type semiconductor.
Similarly, electrons which diffuse from right to left in their random thermal
motion are also acted upon in the conta4 layer by a force which repels them from
the uncompensated negatively-charged acceptors of the boundary layer of the 2.-type
semiconductor into the body of the n-typie semiconductor. The bend ok curves
limiting the forbidden band in the 2711 transition shown in Fig. 40 indicates the
increase in energy required for penetration of a mobile charge into one or another
point of the contact layer.
154
STAT
'
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a
?
?
ea.
Fig. 4o. Schematic diagram of formation of contact potential
difference V and. barrier layer d at electron-hole transition;
?c
-1-t- ionized donors, -- ionized acceptors.
If the concentrations of donors and acceptors are equal in the contacting
semiconductors, the contact field is symmetrical with respect to the boundary of
separation; if the concentrations of impurities are unequal, the contact field
penetrates deeper into the semiconductor with the lower carrier concentration.
Here we are considering the simple case In which the impurities are fully ionized
at the expense of the thermal energy of the solid.
The technical application of the phenomenon described here is based in one
case (photocells) on the very fact of the formation of a contact field in which
the separation of the electron-hole pairs created by light occurs; in another im-
portant case (rectifiers), the technical .application. of the contact field is based
on the strong dependence of the thicknesa of the contact layer in which this field
is concentrated, the energy level of the potential barrier (]..T.c) which forms, and,
consequently, the resistance of this layer on the direction of the external volt-
age applied. '
Of eat importance for the technical utilization of the contact phenomena is
the possibilitv of forming a diction between two semiconductors with wilike
155
STAT
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?
conductivity signs not by means of mechalical or welded jOints but by creating an =
electron-hole transition within the unmu ilated body of a monocrystalline semicon-
ductor . As experience shows, only by this method of forming 2.-n junctions taii.high,
, _ ? ,:.' -.0,--,:?. -
performance of semiconductor equipment utilizing the contact phenomena be ensured.:-
?-
_ ? e,zi
Special numbers of this series are devoted to more detailed consideration ottheie:,?,i
--- ? ' ' ' -
-s:, ..?_,:,_- ..,-h!,
?? - 4 y .E
problems.. ..
-
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/04/01 : CIA-RDP81-01043R004200060003-2
Literature
1. A. V. Joffe, Polu.Dro.rodniki I ikh primeneniye (Semiconductors and Their
Application), AN SSSR (Acad. Sci. USSR), Moscow-Leningrad, 1956.
2. A. V. Joffe, Fizika poluprovodnikov,(Physics of Semiconductors), AN SSSR,
1957.
3. G. Bush, Elektronnaya provodimostr nemetallov (Electronic Conductivity of
Nonmetals), UFN (Frog. Phys. Sci.), 47, 2, 258, 1952.
. I
F. Satz, Fizika m&allov (Physics of Metals), Moscow-Leningrad, 1947.
5. J. N. Frenkel, VVedeniye v teoriyu metallov (Introduction to the Theory of
MetalL), Moscow-LeninF.rad, 1950).
6. F. Seitz, Sovremennaya teoriya tverdogo tele (Modern Theory of Solids),
Moscow-Leningrad, 1949.
7. Foluprovoanikovyye materialy (Semiconductive materials). Collection of
Translations, Moscow-Leningrad, 1954.
8. W. Shockley, Teoriya electronuykh poluprovodnikov (Theory of Electronic
Semiconductors), IL (Fo.7e1gn Literature PUbliShing House), 1953.
9. Nauchnaya.literatura po poluprovodnikam (Scientific Literature Pertaining
to Semiconductors), bibliography aseembled under the direction of V. P. Zhuze.
. AN SSSR, 1955.
. 10. A. V. Shpol'skiy, Atomnaya fizika (Atomic Physics), Moscow-Leningrad, 1949.
.11. A. S. Kbmpaneyets, Teoreticheskaya fizika (Theoretical Physics), Moscow,
1955.. IV, Elements of Statistical Physics.
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