TAG | THEORY
2.1 Introduction
This chapter will cover the physics behind the operation of semiconductor devices and show how
these principles are applied in several different types of semiconductor devices. Subsequent chapters
will deal primarily with the practical aspects of these devices in circuits and omit theory as much
as possible.
2.2 Quantum physics
“I think it is safe to say that no one understands quantum mechanics.”
Physicist Richard P. Feynman
To say that the invention of semiconductor devices was a revolution would not be an exaggeration.
Not only was this an impressive technological accomplishment, but it paved the way for develop-
ments that would indelibly alter modern society. Semiconductor devices made possible miniaturized
electronics, including computers, certain types of medical diagnostic and treatment equipment, and
popular telecommunication devices, to name a few applications of this technology.
But behind this revolution in technology stands an even greater revolution in general science: the
¯eld of quantum physics. Without this leap in understanding the natural world, the development of
semiconductor devices (and more advanced electronic devices still under development) would never
have been possible. Quantum physics is an incredibly complicated realm of science, and this chapter
is by no means a complete discussion of it, but rather a brief overview. When scientists of Feynman’s
caliber say that “no one understands [it],” you can be sure it is a complex subject. Without a basic
understanding of quantum physics, or at least an understanding of the scienti¯c discoveries that led to
its formulation, though, it is impossible to understand how and why semiconductor electronic devices
function. Most introductory electronics textbooks I’ve read attempt to explain semiconductors in
terms of “classical” physics, resulting in more confusion than comprehension.
Tiny particles of matter called protons and neutrons make up the center of the atom, while
electrons orbit around not unlike planets around a star. The nucleus carries a positive electrical
charge, owing to the presence of protons (the neutrons have no electrical charge whatsoever), while
the atom’s balancing negative charge resides in the orbiting electrons. The negative electrons tend
to be attracted to the positive protons just as planets are gravitationally attracted toward whatever
object(s) they orbit, yet the orbits are stable due to the electrons’ motion. We owe this popular
model of the atom to the work of Ernest Rutherford, who around the year 1911 experimentally
determined that atoms’ positive charges were concentrated in a tiny, dense core rather than being
spread evenly about the diameter as was proposed by an earlier researcher, J.J. Thompson.
While Rutherford’s atomic model accounted for experimental data better than Thompson’s, it
still wasn’t perfect. Further attempts at de¯ning atomic structure were undertaken, and these eff orts
helped pave the way for the bizarre discoveries of quantum physics. Today our understanding of
the atom is quite a bit more complex. However, despite the revolution of quantum physics and the
impact it had on our understanding of atomic structure, Rutherford’s solar-system picture of the
atom embedded itself in the popular conscience to such a degree that it persists in some areas of
study even when inappropriate.
Consider this short description of electrons in an atom, taken from a popular electronics textbook:
Orbiting negative electrons are therefore attracted toward the positive nucleus, which
leads us to the question of why the electrons do not °y into the atom’s nucleus. The
answer is that the orbiting electrons remain in their stable orbit due to two equal but
opposite forces. The centrifugal outward force exerted on the electrons due to the orbit
counteracts the attractive inward force (centripetal) trying to pull the electrons toward
the nucleus due to the unlike charges.
In keeping with the Rutherford model, this author casts the electrons as solid chunks of matter
engaged in circular orbits, their inward attraction to the oppositely charged nucleus balanced by their
motion. The reference to “centrifugal force” is technically incorrect (even for orbiting planets), but
2.2. QUANTUM PHYSICS 17
is easily forgiven due to its popular acceptance: in reality, there is no such thing as a force pushing
any orbiting body away from its center of orbit. It only seems that way because a body’s inertia
tends to keep it traveling in a straight line, and since an orbit is a constant deviation (acceleration)
from straight-line travel, there is constant inertial opposition to whatever force is attracting the
body toward the orbit center (centripetal), be it gravity, electrostatic attraction, or even the tension
of a mechanical link.
The real problem with this explanation, however, is the idea of electrons traveling in circular
orbits in the ¯rst place. It is a veri¯able fact that accelerating electric charges emit electromagnetic
radiation, and this fact was known even in Rutherford’s time. Since orbiting motion is a form of
acceleration (the orbiting object in constant acceleration away from normal, straight-line motion),
electrons in an orbiting state should be throwing off radiation like mud from a spinning tire. Electrons
accelerated around circular paths in particle accelerators called synchrotrons are known to do this,
and the result is called synchrotron radiation. If electrons were losing energy in this way, their orbits
would eventually decay, resulting in collisions with the positively charged nucleus. However, this
doesn’t ordinarily happen within atoms. Indeed, electron “orbits” are remarkably stable over a wide
range of conditions.
Furthermore, experiments with “excited” atoms demonstrated that electromagnetic energy emit-
ted by an atom occurs only at certain, de¯nite frequencies. Atoms that are “excited” by outside
in°uences such as light are known to absorb that energy and return it as electromagnetic waves of
very speci¯c frequencies, like a tuning fork that rings at a ¯xed pitch no matter how it is struck.
When the light emitted by an excited atom is divided into its constituent frequencies (colors) by a
prism, distinct lines of color appear in the spectrum, the pattern of spectral lines being unique to
that element. So regular is this phenomenon that it is commonly used to identify atomic elements,
and even measure the proportions of each element in a compound or chemical mixture. According
to Rutherford’s solar-system atomic model (regarding electrons as chunks of matter free to orbit at
any radius) and the laws of classical physics, excited atoms should be able to return energy over
a virtually limitless range of frequencies rather than a select few. In other words, if Rutherford’s
model were correct, there would be no “tuning fork” eff ect, and the light spectrum emitted by any
atom would appear as a continuous band of colors rather than as a few distinct lines.
A pioneering researcher by the name of Niels Bohr attempted to improve upon Rutherford’s
model after studying in Rutherford’s laboratory for several months in 1912. Trying to harmonize
the ¯ndings of other physicists (most notably, Max Planck and Albert Einstein), Bohr suggested
that each electron possessed a certain, speci¯c amount of energy, and that their orbits were likewise
quantized such that they could only occupy certain places around the nucleus, somewhat like marbles
¯xed in circular tracks around the nucleus rather than the free-ranging satellites they were formerly
imagined to be. In deference to the laws of electromagnetics and accelerating charges, Bohr referred
to these “orbits” as stationary states so as to escape the implication that they were in motion.
While Bohr’s ambitious attempt at re-framing the structure of the atom in terms that agreed
closer to experimental results was a milestone in physics, it was by no means complete. His math-
ematical analyses produced better predictions of experimental events than analyses belonging to
previous models, but there were still some unanswered questions as to why electrons would behave
in such strange ways. The assertion that electrons existed in stationary, quantized states around
the nucleus certainly accounted for experimental data better than Rutherford’s model, but he had
no idea what would force electrons to manifest those particular states. The answer to that question
had to come from another physicist, Louis de Broglie, about a decade later.
De Broglie proposed that electrons, like photons (particles of light) manifested both particle-
like and wave-like properties. Building on this proposal, he suggested that an analysis of orbiting
electrons from a wave perspective rather than a particle perspective might make more sense of
their quantized nature. Indeed, this was the case, and another breakthrough in understanding was
reached.
The atom according to de Broglie consisted of electrons existing in the form of standing waves,
a phenomenon well known to physicists in a variety of forms. Like the plucked string of a musical
instrument vibrating at a resonant frequency, with “nodes” and “antinodes” at stable positions along
its length, de Broglie envisioned electrons around atoms standing as waves bent around a circle:
antinode
node
“Orbiting” electron as a standing wave around
the nucleus. Three cycles per “orbit” shown.
antinode
antinode
antinode antinode
antinode
node
node
node
node
node
Electrons could only exist in certain, de¯nite “orbits” around the nucleus because those were the
only distances where the wave ends would match. In any other radius, the wave would destructively
interfere with itself and thus cease to exist.
De Broglie’s hypothesis gave both mathematical support and a convenient physical analogy
to account for the quantized states of electrons within an atom, but his atomic model was still
incomplete. Within a few years, though, physicists Werner Heisenberg and Erwin Schrodinger,
working independently of each other, built upon de Broglie’s concept of a matter-wave duality to
create more mathematically rigorous models of subatomic particles.
This theoretical advance from de Broglie’s primitive standing wave model to Heisenberg’s ma-
trix and Schrodinger’s diff erential equation models was given the name quantum mechanics, and it
introduced a rather shocking characteristic to the world of subatomic particles: the trait of prob-
ability, or uncertainty. According to the new quantum theory, it was impossible to determine the
exact position and exact momentum of a particle at the same time. Popular explanations of this
“uncertainty principle” usually cast it in terms of error caused by the process of measurement (i.e.
by attempting to precisely measure the position of an electron, you interfere with its momentum
and thus cannot know what it was before the position measurement was taken, and visa versa), but
the truth is actually much more mysterious than simple measurement interference. The startling
implication of quantum mechanics is that particles do not actually possess precise positions and
momenta, but rather balance the two quantities in a such way that their combined uncertainties
never diminish below a certain minimum value.
It is interesting to note that this form of “uncertainty” relationship exists in areas other than
quantum mechanics. As discussed in the “Mixed-Frequency AC Signals” chapter in volume II of
this book series, there is a mutually exclusive relationship between the certainty of a waveform’s
time-domain data and its frequency-domain data. In simple terms, the more precisely we know its
constituent frequency(ies), the less precisely we know its amplitude in time, and vice versa. To quote
myself:
A waveform of in¯nite duration (in¯nite number of cycles) can be analyzed with
absolute precision, but the less cycles available to the computer for analysis, the less
precise the analysis. . . The fewer times that a wave cycles, the less certain its frequency
is. Taking this concept to its logical extreme, a short pulse { a waveform that doesn’t
even complete a cycle { actually has no frequency, but rather acts as an in¯nite range of
frequencies. This principle is common to all wave-based phenomena, not just AC voltages
and currents.
In order to precisely determine the amplitude of a varying signal, we must sample it over a very
narrow span of time. However, doing this limits our view of the wave’s frequency. Conversely, to
determine a wave’s frequency with great precision, we must sample it over many, many cycles, which
means we lose view of its amplitude at any given moment. Thus, we cannot simultaneously know the
instantaneous amplitude and the overall frequency of any wave with unlimited precision. Stranger
yet, this uncertainty is much more than observer imprecision; it resides in the very nature of the
wave itself. It is not as though it would be possible, given the proper technology, to obtain precise
measurements of both instantaneous amplitude and frequency at once. Quite literally, a wave cannot
possess both a precise, instantaneous amplitude, and a precise frequency at the same time.
Likewise, the minimum uncertainty of a particle’s position and momentum expressed by Heisen-
berg and Schrodinger has nothing to do with limitation in measurement; rather it is an intrinsic
property of the particle’s matter-wave dual nature. Electrons, therefore, do not really exist in their
“orbits” as precisely de¯ned bits of matter, or even as precisely de¯ned waveshapes, but rather as
“clouds” { the technical term is wavefunction { of probability distribution, as if each electron were
“spread” or “smeared” over a range of positions and momenta.
This radical view of electrons as imprecise clouds at ¯rst seems to contradict the original principle
of quantized electron states: that electrons exist in discrete, de¯ned “orbits” around atomic nuclei.
It was, after all, this discovery that led to the formation of quantum theory to explain it. How
odd it seems that a theory developed to explain the discrete behavior of electrons ends up declaring
that electrons exist as “clouds” rather than as discrete pieces of matter. However, the quantized
behavior of electrons does not depend on electrons having de¯nite position and momentum values,
but rather on other properties called quantum numbers. In essence, quantum mechanics dispenses
with commonly held notions of absolute position and absolute momentum, and replaces them with
absolute notions of a sort having no analogue in common experience.
Even though electrons are known to exist in ethereal, “cloud-like” forms of distributed probabil-
ity rather than as discrete chunks of matter, those “clouds” possess other characteristics that are
discrete. Any electron in an atom can be described in terms of four numerical measures (the previ-
ously mentioned quantum numbers), called the Principal, Angular Momentum, Magnetic, and
Spin numbers. The following is a synopsis of each of these numbers’ meanings:
Principal Quantum Number: Symbolized by the letter n, this number describes the shell
that an electron resides in. An electron “shell” is a region of space around an atom’s nucleus that
electrons are allowed to exist in, corresponding to the stable “standing wave” patterns of de Broglie
and Bohr. Electrons may “leap” from shell to shell, but cannot exist between the shell regions.
The principle quantum number can be any positive integer (a whole number, greater than or
equal to 1). In other words, there is no such thing as a principle quantum number for an electron
of 1/2 or -3. These integer values were not arrived at arbitrarily, but rather through experimental
evidence of light spectra: the diff ering frequencies (colors) of light emitted by excited hydrogen
atoms follow a sequence mathematically dependent on speci¯c, integer values.
2.2. QUANTUM PHYSICS 21
Each shell has the capacity to hold multiple electrons. An analogy for electron shells is the
concentric rows of seats of an amphitheater. Just as a person seated in an amphitheater must choose
a row to sit in (for there is no place to sit in the space between rows), electrons must “choose” a
particular shell to “sit” in. Like amphitheater rows, the outermost shells are able to hold more
electrons than the inner shells. Also, electrons tend to seek the lowest available shell, like people in
an amphitheater trying to ¯nd the closest seat to the center stage. The higher the shell number,
the greater the energy of the electrons in it.
The maximum number of electrons that any shell can hold is described by the equation 2n2,
where “n” is the principle quantum number. Thus, the ¯rst shell (n=1) can hold 2 electrons; the
second shell (n=2) 8 electrons, and the third shell (n=3) 18 electrons.
Electron shells in an atom are sometimes designated by letter rather than by number. The ¯rst
shell (n=1) is labeled K, the second shell (n=2) L, the third shell (n=3) M, the fourth shell (n=4)
N, the ¯fth shell (n=5) O, the sixth shell (n=6) P, and the seventh shell (n=7) Q.
Angular Momentum Quantum Number: Within each shell, there are subshells. One might
be inclined to think of subshells as simple subdivisions of shells, like lanes dividing a road, but the
truth is much stranger than this. Subshells are regions of space where electron “clouds” are allowed to
exist, and different subshells actually have different shapes. The ¯rst subshell is shaped like a sphere,
which makes sense to most people, visualizing a cloud of electrons surrounding the atomic nucleus
in three dimensions. The second subshell, however, resembles a dumbbell, comprised of two “lobes”
joined together at a single point near the atom’s center. The third subshell typically resembles a
set of four “lobes” clustered around the atom’s nucleus. These subshell shapes are reminiscent of
graphical depictions of radio antenna signal strength, with bulbous lobe-shaped regions extending
from the antenna in various directions.
Valid angular momentum quantum numbers are positive integers like principal quantum numbers,
but also include zero. These quantum numbers for electrons are symbolized by the letter l. The
number of subshells in a shell is equal to the shell’s principal quantum number. Thus, the ¯rst shell
(n=1) has one subshell, numbered 0; the second shell (n=2) has two subshells, numbered 0 and 1;
the third shell (n=3) has three subshells, numbered 0, 1, and 2.
An older convention for subshell description used letters rather than numbers. In this notational
system, the ¯rst subshell (l=0) was designated s, the second subshell (l=1) designated p, the third
subshell (l=2) designated d, and the fourth subshell (l=3) designated f. The letters come from the
words sharp, principal (not to be confused with the principal quantum number, n), diff use, and
fundamental. You will still see this notational convention in many periodic tables, used to designate
the electron con¯guration of the atoms’ outermost, or valence, shells.
Magnetic Quantum Number: The magnetic quantum number for an electron classi¯es which
orientation its subshell shape is pointed. For each subshell in each shell, there are multiple directions
in which the “lobes” can point, and these different orientations are called orbitals. For the ¯rst
subshell (s; l=0), which resembles a sphere, there is no “direction” it can “point,” so there is only
one orbital. For the second (p; l=1) subshell in each shell, which resembles a dumbbell, there are
three different directions they can be oriented (think of three dumbbells intersecting in the middle,
each oriented along a different axis in a three-axis coordinate system).
Valid numerical values for this quantum number consist of integers ranging from -l to l, and
are symbolized as ml in atomic physics and lz in nuclear physics. To calculate the number of
orbitals in any given subshell, double the subshell number and add 1 (2l + 1). For example, the ¯rst
subshell (l=0) in any shell contains a single orbital, numbered 0; the second subshell (l=1) in any
shell contains three orbitals, numbered -1, 0, and 1; the third subshell (l=2) contains ¯ve orbitals,
numbered -2, -1, 0, 1, and 2; and so on.
Like principal quantum numbers, the magnetic quantum number arose directly from experimental
evidence: the division of spectral lines as a result of exposing an ionized gas to a magnetic ¯eld,
hence the name “magnetic” quantum number.
Spin Quantum Number: Like the magnetic quantum number, this property of atomic elec-
trons was discovered through experimentation. Close observation of spectral lines revealed that each
line was actually a pair of very closely-spaced lines, and this so-called ¯ne structure was hypothesized
to be the result of each electron “spinning” on an axis like a planet. Electrons with different “spins”
would give off slightly different frequencies of light when excited, and so the quantum number of
“spin” came to be named as such. The concept of a spinning electron is now obsolete, being better
suited to the (incorrect) view of electrons as discrete chunks of matter rather than as the “clouds”
they really are, but the name remains.
Spin quantum numbers are symbolized as ms in atomic physics and sz in nuclear physics. For
each orbital in each subshell in each shell, there can be two electrons, one with a spin of +1/2 and
the other with a spin of -1/2.
The physicist Wolfgang Pauli developed a principle explaining the ordering of electrons in an
atom according to these quantum numbers. His principle, called the Pauli exclusion principle, states
that no two electrons in the same atom may occupy the exact same quantum states. That is, each
electron in an atom has a unique set of quantum numbers. This limits the number of electrons that
may occupy any given orbital, subshell, and shell.
Shown here is the electron arrangement for a hydrogen atom:
Hydrogen
Atomic number (Z) = 1
(one proton in nucleus)
K shell
(n = 1)
subshell
(l)
orbital
(ml)
spin
(ms)
0 0 1/2 One electron
Spectroscopic notation: 1s1
With one proton in the nucleus, it takes one electron to electrostatically balance the atom (the
proton’s positive electric charge exactly balanced by the electron’s negative electric charge). This
one electron resides in the lowest shell (n=1), the ¯rst subshell (l=0), in the only orbital (spatial
orientation) of that subshell (ml=0), with a spin value of 1/2. A very common method of describing
this organization is by listing the electrons according to their shells and subshells in a convention
called spectroscopic notation. In this notation, the shell number is shown as an integer, the subshell
as a letter (s,p,d,f), and the total number of electrons in the subshell (all orbitals, all spins) as a
superscript. Thus, hydrogen, with its lone electron residing in the base level, would be described as
1s1.
Proceeding to the next atom type (in order of atomic number), we have the element helium:
K shell
(n = 1)
subshell
(l)
orbital
(ml)
spin
(ms)
0 0 1/2
Spectroscopic notation:
Helium
Atomic number (Z) = 2
(two protons in nucleus)
0 0 -1/2 electron
electron
1s2
A helium atom has two protons in the nucleus, and this necessitates two electrons to balance the
double-positive electric charge. Since two electrons { one with spin=1/2 and the other with spin=-
1/2 { will ¯t into one orbital, the electron con¯guration of helium requires no additional subshells
or shells to hold the second electron.
However, an atom requiring three or more electrons will require additional subshells to hold all
electrons, since only two electrons will ¯t into the lowest shell (n=1). Consider the next atom in the
sequence of increasing atomic numbers, lithium:
K shell
(n = 1)
subshell
(l)
orbital
(ml)
spin
(ms)
0 0 1/2
Spectroscopic notation:
0 0 -1/2 electron
electron
Lithium
Atomic number (Z) = 3
L shell
(n = 2)
0 0 1/2 electron
1s22s1
An atom of lithium only uses a fraction of the L shell’s (n=2) capacity. This shell actually
has a total capacity of eight electrons (maximum shell capacity = 2n2 electrons). If we examine
the organization of the atom with a completely ¯lled L shell, we will see how all combinations of
subshells, orbitals, and spins are occupied by electrons:
Often, when the spectroscopic notation is given for an atom, any shells that are completely ¯lled
are omitted, and only the un¯lled, or the highest-level ¯lled shell, is denoted. For example, the
element neon (shown in the previous illustration), which has two completely ¯lled shells, may be
spectroscopically described simply as 2p6 rather than 1s22s22p6. Lithium, with its K shell completely
¯lled and a solitary electron in the L shell, may be described simply as 2s1 rather than 1s22s1.
The omission of completely ¯lled, lower-level shells is not just a notational convenience. It
also illustrates a basic principle of chemistry: that the chemical behavior of an element is primarily
determined by its un¯lled shells. Both hydrogen and lithium have a single electron in their outermost
shells (1s1 and 2s1, respectively), and this gives the two elements some similar properties. Both are
highly reactive, and reactive in much the same way (bonding to similar elements in similar modes).
It matters little that lithium has a completely ¯lled K shell underneath its almost-vacant L shell:
the un¯lled L shell is the shell that determines its chemical behavior.
Elements having completely ¯lled outer shells are classi¯ed as noble, and are distinguished by
their almost complete non-reactivity with other elements. These elements used to be classi¯ed as
inert, when it was thought that they were completely unreactive, but it is now known that they may
form compounds with other elements under certain conditions.
Given the fact that elements with identical electron con¯gurations in their outermost shell(s)
exhibit similar chemical properties, it makes sense to organize the different elements in a table
accordingly. Such a table is known as a periodic table of the elements, and modern tables follow this
general form:
Dmitri Mendeleev, a Russian chemist, was the ¯rst to develop a periodic table of the elements.
Although Mendeleev organized his table according to atomic mass rather than atomic number, and
so produced a table that was not quite as useful as modern periodic tables, his development stands
as an excellent example of scienti¯c proof. Seeing the patterns of periodicity (similar chemical
properties according to atomic mass), Mendeleev hypothesized that all elements would ¯t into this
ordered scheme. When he discovered “empty” spots in the table, he followed the logic of the existing
order and hypothesized the existence of heretofore undiscovered elements. The subsequent discovery
of those elements granted scienti¯c legitimacy to Mendeleev’s hypothesis, further discoveries leading
to the form of the periodic table we use today.
This is how science should work: hypotheses followed to their logical conclusions, and accepted,
modi¯ed, or rejected as determined by the agreement of experimental data to those conclusions.
Any fool can formulate a hypothesis after-the-fact to explain existing experimental data, and many
do. What sets a scienti¯c hypothesis apart from post hoc speculation is the prediction of future
experimental data yet uncollected, and the possibility of disproof as a result of that data. To boldly
follow a hypothesis to its logical conclusion(s) and dare to predict the results of future experiments
is not a dogmatic leap of faith, but rather a public test of that hypothesis, open to challenge
from anyone able to produce contradictory data. In other words, scienti¯c hypotheses are always
“risky” in the sense that they claim to predict the results of experiments not yet conducted, and
are therefore susceptible to disproof if the experiments do not turn out as predicted. Thus, if a
hypothesis successfully predicts the results of repeated experiments, there is little probability of its
falsehood.
Quantum mechanics, ¯rst as a hypothesis and later as a theory, has proven to be extremely
successful in predicting experimental results, hence the high degree of scienti¯c con¯dence placed in
it. Many scientists have reason to believe that it is an incomplete theory, though, as its predictions
hold true more so at very small physical scales than at macroscopic dimensions, but nevertheless it
is a tremendously useful theory in explaining and predicting the interactions of particles and atoms.
As you have already seen in this chapter, quantum physics is essential in describing and pre-
dicting many different phenomena. In the next section, we will see its signi¯cance in the electrical
conductivity of solid substances, including semiconductors. Simply put, nothing in chemistry or
solid-state physics makes sense within the popular theoretical framework of electrons existing as
discrete chunks of matter, whirling around atomic nuclei like miniature satellites. It is only when
electrons are viewed as “wavefunctions” existing in de¯nite, discrete states that the regular and
periodic behavior of matter can be explained.
² REVIEW:
² Electrons in atoms exist in “clouds” of distributed probability, not as discrete chunks of matter
orbiting the nucleus like tiny satellites, as common illustrations of atoms show.
² Individual electrons around an atomic nucleus seek unique “states,” described by four quan-
tum numbers: the Principal Quantum Number, otherwise known as the shell ; the Angular
Momentum Quantum Number, otherwise known as the subshell ; the Magnetic Quantum Num-
ber, describing the orbital (subshell orientation); and the Spin Quantum Number, or simply
spin. These states are quantized, meaning that there are no “in-between” conditions for an
electron other than those states that ¯t into the quantum numbering scheme.
² The Principal Quantum Number (n) describes the basic level or shell that an electron resides in.
The larger this number, the greater radius the electron cloud has from the atom’s nucleus, and
the greater than electron’s energy. Principal quantum numbers are whole numbers (positive
integers).
² The Angular Momentum Quantum Number (l ) describes the shape of the electron cloud within
a particular shell or level, and is often known as the “subshell.” There are as many subshells
(electron cloud shapes) in any given shell as that shell’s principal quantum number. Angular
momentum quantum numbers are positive integers beginning at zero and terminating at one
less than the principal quantum number (n-1).
² The Magnetic Quantum Number (ml) describes which orientation a subshell (electron cloud
shape) has. There are as many different orientations for each subshell as the subshell number
(l ) plus 1, and each unique orientation is called an orbital. These numbers are integers ranging
from the negative value of the subshell number (l ) through 0 to the positive value of the
subshell number.
² The Spin Quantum Number (ms) describes another property of an electron, and can be a value
of +1/2 or -1/2.
² Pauli’s Exclusion Principle says that no two electrons in an atom may share the exact same
set of quantum numbers. Therefore, there is room for two electrons in each orbital (spin=1/2
and spin=-1/2), 2l+1 orbitals in every subshell, and n subshells in every shell, and no more.
² Spectroscopic notation is a convention for denoting the electron con¯guration of an atom.
Shells are shown as whole numbers, followed by subshell letters (s,p,d,f), with superscripted
numbers totaling the number of electrons residing in each respective subshell.
² An atom’s chemical behavior is solely determined by the electrons in the un¯lled shells. Low-
level shells that are completely ¯lled have little or no eff ect on the chemical bonding charac-
teristics of elements.
² Elements with completely ¯lled electron shells are almost entirely unreactive, and are called
noble (formerly known as inert).
2.3 Band theory of solids
Quantum physics describes the states of electrons in an atom according to the four-fold scheme
of quantum numbers. The quantum number system describes the allowable states electrons may
assume in an atom. To use the analogy of an amphitheater, quantum numbers describe how many
rows and seats there are. Individual electrons may be described by the combination of quantum
numbers they possess, like a spectator in an amphitheater assigned to a particular row and seat.
Like spectators in an amphitheater moving between seats and/or rows, electrons may change
their statuses, given the presence of available spaces for them to ¯t, and available energy. Since
shell level is closely related to the amount of energy that an electron possesses, “leaps” between shell
(and even subshell) levels requires transfers of energy. If an electron is to move into a higher-order
shell, it requires that additional energy be given to the electron from an external source. Using
the amphitheater analogy, it takes an increase in energy for a person to move into a higher row of
seats, because that person must climb to a greater height against the force of gravity. Conversely,
an electron “leaping” into a lower shell gives up some of its energy, like a person jumping down into
a lower row of seats, the expended energy manifesting as heat and sound released upon impact.
Not all “leaps” are equal. Leaps between different shells requires a substantial exchange of energy,
while leaps between subshells or between orbitals require lesser exchanges.
When atoms combine to form substances, the outermost shells, subshells, and orbitals merge,
providing a greater number of available energy levels for electrons to assume. When large numbers
of atoms exist in close proximity to each other, these available energy levels form a nearly continuous
band wherein electrons may transition.
It is the width of these bands and their proximity to existing electrons that determines how
mobile those electrons will be when exposed to an electric ¯eld. In metallic substances, empty
bands overlap with bands containing electrons, meaning that electrons may move to what would
normally be (in the case of a single atom) a higher-level state with little or no additional energy
imparted. Thus, the outer electrons are said to be “free,” and ready to move at the beckoning of an
electric ¯eld.
Band overlap will not occur in all substances, no matter how many atoms are in close proximity
to each other. In some substances, a substantial gap remains between the highest band containing
electrons (the so-called valence band) and the next band, which is empty (the so-called conduction
band). As a result, valence electrons are “bound” to their constituent atoms and cannot become
mobile within the substance without a signi¯cant amount of imparted energy. These substances are
electrical insulators:
Multitudes of atoms
in close proximity
Electron band separation in insulating substances
Conduction band
Valence band
Significant leap required
for an electron to enter
the conduction band and
travel through the material
“Energy gap”
Materials that fall within the category of semiconductors have a narrow gap between the valence
and conduction bands. Thus, the amount of energy required to motivate a valence electron into the
conduction band where it becomes mobile is quite modest:
Multitudes of atoms
in close proximity
Conduction band
Valence band
for an electron to enter
the conduction band and
travel through the material
“Energy gap”
Electron band separation in
semiconducting substances
Small leap required
At low temperatures, there is little thermal energy available to push valence electrons across this
gap, and the semiconducting material acts as an insulator. At higher temperatures, though, the
ambient thermal energy becomes su±cient to force electrons across the gap, and the material will
conduct electricity.
It is di±cult to predict the conductive properties of a substance by examining the electron
con¯gurations of its constituent atoms. While it is true that the best metallic conductors of electricity
(silver, copper, and gold) all have outer s subshells with a single electron, the relationship between
conductivity and valence electron count is not necessarily consistent:
Element
Specific resistance
Silver (Ag)
(r) at 20o Celsius
9.546 W×cmil/ft 4d105s1
Electron
configuration
Copper (Cu) 10.09 W×cmil/ft 3d104s1
Gold (Au) 13.32 W×cmil/ft 5d106s1
Aluminum (Al) 15.94 W×cmil/ft 3p1
Tungsten (W) 31.76 W×cmil/ft 5d46s2
Molybdenum (Mo) 32.12 W×cmil/ft 4d55s1
Zinc (Zn) 35.49 W×cmil/ft 3d104s2
Nickel (Ni) 41.69 W×cmil/ft 3d84s2
Iron (Fe) 57.81 W×cmil/ft 3d64s2
Platinum (Pt) 63.16 W×cmil/ft 5d96s1
Likewise, the electron band con¯gurations produced by compounds of different elements de¯es
easy association with the electron con¯gurations of its constituent elements.
