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One-dimensional Solid-State Physics
Today most physicists work in the field of solid-state physics. One reason is that,
apart from chemistry, solid-state physics provides the fundamental basis of materials science and thus for most new branches of technology. This book is an introduction to a particular area within solid state physics, namely, one-dimensional solid
state physics. The main difference between solid-state physicists and other scientists
is their familiarity with reciprocal space (Figure 3-1). This difference is similar to
that between scientists and nonscientists, where familiarity with exponentials is the
boundary. Every educated person knows the definition of 107 or 10–24, but only scientists use powers and logarithms as readily as journalists use a notepad. The explanation of symbols alone does not help, what counts is familiarity.
In the first two chapters, reciprocal space was mentioned a few times. This chapter is an attempt to make you more familiar with the conventions of reciprocal
space, as they apply to the concepts of dimensionality. For an in-depth treatment of
the topic, please see textbooks on solid state physics (for example [1–3]).
Figure 3-1
Solid-state physicists differ from other people by their familiarity with reciprocal space.
One-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll
Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-30749-4
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3 One-dimensional Solid-State Physics
3.1
Crystal Lattice and Translation Symmetry
The three states of matter – solid, liquid, and gas – can be grouped in several different ways according to their physical properties. Solids and liquids are, when
grouped together, called condensed matter’ because they are indeed dense and difficult to compress: the atoms are already close together, with very little room for irregularities in interatomic distances. A gas, on the other hand, is highly compressible.
Liquids and gases can be classified as fluids – they take the shape of their container.
There are no, or only very small, shear forces in fluids. Solids do not flow, because
strong forces keep the distance between the atoms fixed and also lock the relative
angular positions of the atoms.
We classify two types of solids: crystalline solids and amorphous solids. In crystalline solids the molecules or atoms are arranged in a regular way; in amorphous solids there is a large amount of disorder within some inherent limitations. The existence of shear forces in the solid implies order. The angular positions of the atoms
cannot be completely random because of steric reasons and chemical bonding: for
example, a silicon atom is found in the center of a tetrahedron with an angle of
109.5 between the bonds to its nearest neighbors. Therefore, there is short-range
order even in amorphous solids. In crystalline solids the order is long-range, and it
can be assumed that the ground state of all matter is crystalline and that all amorphous solids will crystallize sooner or later, although this process might require geological times of millions of years in some instances.
Of course, the degree to which something is amorphous or crystalline can be
important for some purposes. How defective must a crystalline solid become before
it is amorphous? What is glassy’ behavior? To better understand the degree of organization, physicists frequently resort to statistical measures as the correlation function. This is simply the probability of finding an atom–atom separation within the
solid. Some base all discussion of condensed matter on this function, as the most
basic sense of order. Here, it is important to understand only that a disordered solid
may also reflect symmetries and that there are well-defined ways to describe it.
3.1.1
Classifying the Lattice
What does long-range’ order mean? In single crystals the order covers the entire crystal, which is from a few micrometers up to many centimeters depending on its size.
Most solids, however, are polycrystalline. They are composed of microcrystalline
domains or grains, which stick together to form the bulk solid. The diameter of a
grain can be from several fractions of a micrometer up to several millimeters.
The long-range order in a crystal leads to the crystal lattice. In a crystal lattice the
regular arrangement of the atoms (or molecules) is periodic; the same pattern is
repeated over and over again (e.g., sodium lattice, Figure 3-2; sodium chloride lattice, Figure 3-3 – three repeating units in each direction are shown). To classify the
many different lattice types their symmetry properties are used. (Many textbooks on
3.1 Crystal Lattice and Translation Symmetry
Figure 3-2
Crystal lattice of sodium
metal.
crystallography exist; however, Ashcroft and Mermin [1] has long been a standard
for understanding crystal symmetries at the elementary level.) The lattices in Figures 3-2 and 3-3 are cubic. A cubic elementary cell, which is a commonly found
repeating unit, can be constructed as indicated in bold in the upper-right corner. For
the sake of clarity, only the atoms at the surface of the crystal are shown and the
atoms inside the crystal are omitted. (Examples of textbooks on crystal structure and
chemistry are [4] and [5].)
Our first example is a simple cubic system. That is, each site of the mental construct of the lattice (the mathematical arrangement of points in a regular square
array) is occupied by an atom to form a real lattice of the solid. The subtlety (and
formality) of forming the construct mentally, then filling it in with atoms is an
important one. After all, we need not imagine only one atom associated with a given
Figure 3-3
chloride.
Crystal lattice of sodium
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site. Consider associating more than one with each site of the square lattice. The
picture could get rather complex. The associated group of atoms is referred to as a
basis’.
For sodium chloride, a two-atom basis, one of Na and one of Cl, is associated with
each site of the square lattice. The crystal of sodium chloride is built of sodium and
chloride atoms according the chemical formula NaCl. This crystal does have a
unique set of symmetries that arranges neighbors equivalently. There are as many
sodium atoms as chlorine atoms; however, a particular sodium atom does not have
only one single chlorine atom as a partner, but six equivalent chlorine neighbors.
These again are neighbors to other sodium atoms. In other words, there is no
uniquely defined NaCl unit in the crystal. This coincidence does not always happen
when there is a basis set. (More precisely one should speak of sodium and of chlorine ions, rather than atoms, but this is not important in this context.)
The distinction between solids with complex basis sets and solids that are elemental (or nearly so) is frequently used to further classify solids. Atomic crystals and molecular crystals can be distinguished along exactly these lines. Most inorganic materials form atomic crystals; examples include silicon, copper, and sodium chloride. In
contrast, most organic compounds form molecular crystals. In a molecular crystal
most properties of the solid are determined by the properties of the molecule. Crystallization introduces only small changes. For example, benzene crystal contains
well-defined C6H6 units. The interactions between the carbon atoms within such a
unit (molecule) are much stronger than those between carbon atoms belonging to
two different molecules. Benzene molecules exist as benzene vapor, as liquid benzene, as mixtures of benzene with other solvents, and also as incorporated into a
molecular crystal. (For a monograph on organic crystals, see [6].)
Fundamentally, molecular crystals are described with the same symmetry language
as atomic lattices. Thus, we can speak of all crystal structures in the same way.
3.1.2
Using a Coordinate System
The elementary cell is perhaps best explained with the example of the two-dimensional rectangular lattice in Figure 3-4. The elementary cell is presented as hatched
rectangle. The lattice is built up by moving this cell parallel to the crystallographic
a axis and the crystallographic b axis. The length a and the width b of the elementary
cell are called lattice parameters, and the crosses in the figure mark the lattice points.
In primitive’ crystal structures the lattice points are the sites at which the atoms or
molecules are located. (Nonprimitive structures have additional atoms.) All lattice
points are identical and the lattice is assumed to extend infinitely in all directions of
space (i.e., the crystal is composed of an (infinitely) large number of atoms or molecules). Any lattice point can be chosen as the origin of a coordinate system.
The origin in the figure is marked by a circled cross. The arrows along the crystallographic a and b axis, with lengths a and b respectively, are called the unit lattice
vectors a and b. They are said to span the elementary cell. Any lattice point can be
reached by a linear combination of the unit vectors:
3.1 Crystal Lattice and Translation Symmetry
Figure 3-4 Two-dimensional rectangular lattice.
¼ ha þ kb
T
(1)
is a lattice vector. The integers h and k are the indices of the lattice point to which
T
points. These indices are used to characterize directions and planes in
the vector T
crystals. Because all lattice points are equivalent, the physics at each lattice site is
the same, or, in mathematical terms, the physical laws are invariant to the addition
of a lattice vector to the special coordinate. This is called the translational invariance
of the crystal lattice and of the laws governing the physics within a crystal.
We have, in fact, already seen a two-dimensional lattice in Chapter 2. The graphene layer is a hexagonal two-dimensional lattice. The translation vectors we gave
were termed (an, am). In fact, the simplest way to describe this lattice is as a triangular lattice with a two-atom basis. How would one know this? That is, how do we
know when we have the simplest description for a lattice? The easiest way is to pretend to stand on a given lattice point facing, say, north. As you glance to your left
and to your right, the view from any lattice point should be the same. This is not
true for the hexagonal system in Chapter 2. Try it – adjacent sites will have the
bonds oriented differently relative to a given direction.
Once the most simple lattice structure has been determined, it is important to
realize how many possibilities exist for distinctively different lattices. When in a rectangular lattice a = b, the rectangular lattice is a square lattice. The three-dimensional analog of the square lattice is, of course, the cubic lattice. In lattices of lower
symmetry the dimensions of the elementary cell might differ, a „ b „ c; also the
angles between the axes do not need to be 90. In fact, there are 18 possible variations of this most simple lattice. When the symmetries of rotations, translations,
etc. are considered, a finite enumeration of all possible lattice symmetries can be
made (again, see Ashcroft and Mermin [1]).
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3.1.3
The One-dimensional Lattice
The simplest lattice is a one-dimensional crystal lattice, which is merely a row, or
line, of equidistant points (Figure 3-5). In a strictly one-dimensional lattice there is
no need to consider the angular positions of the atoms, but many real’ one-dimensional substances have the atoms arranged in a zigzag line, as indicated in Figure 3-6. Such a zigzag chain can melt’ by generating defects as illustrated in Figure 3-7, where the strict alternation is disturbed. Another way of looking at these
defects is to allow for rotations around a bond between two atoms.
Figure 3-5
One-dimensional lattice.
Figure 3-6
Zigzag chain.
Figure 3-7
Zigzag chain with defects.
As we saw in Chapters 1 and 2, the conducting polymer system is generally constructed of alternating double and single carbon–carbon bonds. To see an example
of how the above applies to a real polymer, consider polyacetylene. The structure is
given in Figure 1-12 but is reproduced here in its schematic form (Figure 3-8):
3.1 Crystal Lattice and Translation Symmetry
Figure 3-8 The schematic structure of polyacetylene along with
its lattice. Notice that the lattice points do not fall on every carbon; rather, there is a two-carbon (along with some hydrogens)
basis.
For the one-dimensional system, however, this is not the end of the story. As mentioned before, rotations about a bond are allowed in many systems. In fact, these
can occur in an ordered way, leading to another crystal structure for the same material. Again, for an example let’s turn to polyacetylene. In fact, the structure could
occur in several ways (Figure 3-9).
As noted in Chapter 2, the one-dimensional nature of such systems as polyacetylene means that the properties of the material, especially the electronic transport
characteristics, are extremely sensitive to defects (or mistakes) within the lattice.
Consider only an occasional bond rotation within the lattice, as seen in Figure 3-7.
It makes sense then that our expectation of the electronic properties of such a sys-
Figure 3-9 The difference between cis and trans polyacetylene is
the rotation of a few bonds. This gives different crystal structures in one dimension.
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tem of randomly distributed defects is that they are dominated by the electronics of
the individual segments as though they are isolated. In this way the prediction of
phenomena in one dimension can easily be as complex as that for a three-dimensional system. We will return to this point in later chapters.
3.2
Reciprocal Lattice, Reciprocal Space
The crystal lattice has two functions. On the one hand it is an idealized model of the
crystal. Any novel chemical compound is first characterized by determining its crystallographic structure: the symmetry properties of the lattice and the lattice parameters. The second function of the crystal lattice is to furnish a convenient coordinate
system for the description of physical phenomena within the crystal. In simple
terms – we prefer to give the location of a coffee shop as 5 miles form exit 12 on
highway A81, rather than to specify its geographic longitude and latitude. Once the
crystal lattice is accepted as a convenient geometrical tool specifically designed to
simplify the description of physical phenomena within a given crystal, it is easy to
accept other geometrical (mathematical) tools, which are derived from the first one:
the reciprocal lattice, or more generally, the reciprocal space.
3.2.1
Describing Objects by Momentum and Energy
The concept of the reciprocal lattice was developed to describe X-ray diffraction patterns of crystals. Reciprocal space is used to describe wave-like phenomena in a crystal. It was already mentioned Section 1.2 in explaining open Fermi surfaces and in
Section 2.2 in the context of the Kohn anomaly.
Reciprocal space is not used to describe the positions of objects, but to characterize the nature of mobile objects independent of their actual position. For example,
consider cars traveling along a highway. To characterize the cars we would determine their velocity v and their weight (mass) m. To determine the velocity a measure
of length would be necessary, with the units being miles, kilometers, or unity vectors
of the crystal lattice. A physicist would prefer to use momentum and kinetic energy
rather than velocity and mass, because with this choice some of the physical equations become more decent – the transition from the particle picture’ to the wave
picture’ is especially eased. The wave quantity related to the momentum of a particle
is the wave number, which is the reciprocal of the wavelength, and the wave quantity
related to the energy is the frequency, something expressed in oscillations per unit
time, i.e., a quantity with the dimension of reciprocal time. So some solid-state phenomena are simplified if described in reciprocal space, but the price of this simplification is that we need to get accustomed to reciprocal space. For most solid-state
scientists this intellectual investment is worthwhile.
3.2 Reciprocal Lattice, Reciprocal Space
3.2.2
Constructing the Reciprocal Lattice
There is a simple algebraic procedure to derive the reciprocal lattice from the crystal
lattice. The crystal lattice is defined by the dimensions of the elementary cell, or, in
other terms, by the three unit vectors a, b, and c , which span the elementary cell.
The reciprocal lattice is also defined by three unit vectors, now a, b , and c . The
direction of a is perpendicular to the plane defined by b and c . In a cubic lattice, its
length is just 2p times the reciprocal of the lattice parameter a, hence the name
reciprocal lattice’. For a general lattice type the length of a is obtained by dividing
the area between b and c by the volume of the elementary cell, which in vector algebra is expressed as
a ¼ 2p ðb c Þ = ða b c Þ
(2)
or 2p bc/abc for a rectangular lattice. For example, if in any given lattice a is 3 ,
in the reciprocal lattice the length of a* is 2p times one third of a reciprocal Angstrom. The factor 2p was chosen to simplify the equations. Simplification in one
particular field of physics often means a complication in another field. The factor 2p
is often incorporated into other quantities. Examples are the angular frequency x =
2pm for a wave, used instead of the linear frequency m, or Planck’s constant existing
in two versions, h and h = h/2p, etc.
In practice, the reciprocal lattice and reciprocal space are mainly used for graphic
representations. Crystallographers, for example, are interested in predicting the
positions of X-ray reflections. They will draw the reciprocal lattice or rather a particular plane of it, choose a certain scale for it (e.g., cm) to represent one reciprocal
Angstrom, and then proceed according to the description above (a* perpendicular to
the plane of b and c , and so on). They will mark the wave vectors of the incoming
and presumed outgoing X-ray beams (the direction to the detector). The difference
between these two vectors is the momentum transfer’ associated with the scattering
event. A sphere with radius equal to the momentum transfer is the Ewald sphere. If
the Ewald sphere touches one of the lattice points on the reciprocal lattice, it will
show as a peak in the X-ray intensity (a reflection) at the chosen detector position. If
not, the sample (and with it the reciprocal lattice) will have to be rotated until a reciprocal lattice point meets the sphere and the reflection condition is fulfilled.
3.2.3
Application to One Dimension
What does the reciprocal lattice of a one-dimensional crystal look like? Look again at
the crystal lattice in Figure 3-5. We anticipate that the reciprocal lattice will also be a
linear arrangement of lattice points, the a axis pointing in the same direction as the
a axis, and the length of the unit vector a being 2p times the reciprocal of the
length a.
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Tetragonal lattice, the limit b fi ¥ and c fi ¥ is the
infinite anisotropy’ approach to one-dimensionality and demonstrates that the reciprocal lattice of a linear chain is also a linear
chain.
Figure 3-10
However, the previously described procedure to design a reciprocal lattice works
only if we use all three dimensions: we absolutely need all unit vectors a, b, and c , to
find a unit vector of the reciprocal lattice. With some mediation over Figure 3-10 we
try the anisotropy approach of Section 1.2 to find the reciprocal lattice in one dimension. We start with an anisotropic tetragonal lattice, having three axes at right
angles, but two lattice parameters much larger than the third (b >> a and c >> a),
and increase b and c to infinity. The direction of a is perpendicular to the plane
defined by b and c . Because the crystal lattice is already tetragonal, a is perpendicular to that plane and hence a is pointing in the same direction as a. What then is
the length of a ? 2p times the area (b·c) divided by the volume (a·b·c) gives 2p/a, a
finite value for any value of b and c , valid also for the infinite b and c . What about
the lengths of b and c ? b* = 2p(a·c)/(a·b·c) = 2p/b and similarly c = 2p/c. Both
approach zero as b and c take infinite values. So the infinitely anisotropic’ onedimensional tetragonal lattice indeed consists of a linear arrangement of points as
the reciprocal lattice.
In Section 2.1 we mentioned that experiments are usually not carried out on isolated chains, but on bundles’. Bundles are arrays of parallel chains. Of course, a
three-dimensional crystal is a bundle, but the term bundle is more general. It also
covers arrays of chains with random origin and random distances between the
chains.
In other words, there is no order in the b and c directions and elementary cells
with many different b and c vectors are conceivable. In this case we still get
a* = 2p(b·c)(a·b·c) = 2p/a, but b* and c* are undefined. They are not zero as with
the infinitely anisotropic lattice when b fi ¥ and c fi ¥. In a disordered bundle b*
and c* can take any value. Consequently, the reciprocal lattice of a disordered bundle
is not a linear arrangement of equidistant points, but an arrangement of equidistant
planes (Figure 3-11). A construction like this has a very practical background. It
3.3 Electrons and Phonons in a Crystal, Dispersion Relations
Reciprocal lattice of a disordered bundle (order
in one dimension only) consisting of equidistant planes.
Figure 3-11
actually describes the intensity of X-rays (or neutrons) diffracted by real samples.
When the X-rays are detected by the blackening of a photographic film, the planes
create lines on the film.
Figure 3-12 shows the diffraction pattern of KCP (see Section 2.2). The Peierls
distortion (see Chapter 4) leads to reorganization of the platinum atoms within the
chains. At room temperature the chains act independently of each other and, as far
as the Peierls distortion is concerned, they form a disordered bundle. The X-ray evidence for a disordered bundle is continuous streaks, as shown on the left side of the
diffraction diagram. At low temperatures the chains interact, a coherent threedimensional order is established, and the streaks disintegrate into spots [7].
X-ray diffraction pattern of KCP [7]. The streak
indicates disordered bundle behavior of the Peierls distortion in
the platinum chains.
Figure 3-12
3.3
Electrons and Phonons in a Crystal, Dispersion Relations
We can already see how the constructs of the reciprocal lattice might relate to the
wave-like phenomena in a crystal. After all, an electron will have specific wavelengths within the crystal. If the crystal has a characteristic lattice spacing that is
some rational fraction of one of those wavelengths, it is natural to expect unusual’
behavior at that electron energy.
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3.3.1
Crystal Vibrations and Phonons
The regular arrangement of atoms or molecules in a crystal is caused by forces that
keep the atoms in their positions. The forces are, of course, the chemical bonding
forces. In a mechanical model they are represented as elastic springs (Figure 3-13).
When an atom is deflected from its position of equilibrium and then left alone, the
spring pulls the atom back. Due to inertia it overshoots its initial position; the
springs now push it in the opposite direction. The process repeats, and the atom
oscillates.
The strain in the spring causes adjacent atoms to oscillate as well: a wave will
develop. These waves are nothing other than the well-known sound waves. The
Greek word for sound is phonos and the particles corresponding to the sound waves
(according to the quantum mechanical wave–particle dualism) are called phonons.
The human ear perceives sound up to some 10–100 kHz. The phonon frequencies,
however, lie in the GHz range; thus most of the phonons in solids are in the ultrasonic region.
To characterize a wave, two basic quantities are necessary, the frequency m and the
wavelength k. (An additional quantity is the amplitude. In the particle picture this
corresponds to the number of phonons.) Frequency and wavelength are related by
the velocity t.
m= t /k
(3)
The wave velocity depends on the properties of the medium in which the wave
propagates, for example, on the force constant of the spring in Figure 3-13. For
small values of frequency and amplitude, the force constant is independent of these
quantities – otherwise we would not call it a constant. However, this is not true for
very high frequencies. Here, waves with different frequencies move with different
velocities and a wave package disperses. Similarly, in optics a prism disperses a polychromatic light beam because the refractive index of the prism material (and hence
the light velocity) depends on the light frequency. Because of possible frequency–velocity dependence, Eq. (3) is called dispersion relation.
For a complete description of a wave, the wavelength is not sufficient. The direction of the wave propagation is also an important factor. Both pieces of information,
the direction of propagation and the reciprocal wavelength, are combined in the
wave vector k, which points in the direction of propagation and has the length of the
reciprocal wavelength multiplied by 2p:
|k| = 2p / k
(4)
(It counts the number of waves within a unit length, multiplied by 2p.) Using the
wave vector, Eq. (3) becomes
m = tk / 2p
(5)
3.3 Electrons and Phonons in a Crystal, Dispersion Relations
From quantum mechanics and the wave–particle duality we know that a wave
with frequency m and wave vector k corresponds to a particle with energy E = hm and
momentum p = hk/2p, so the dispersion relation can be written in yet another
form:
E = tp
(6)
For a free particle is p = mt, where m is the mass, so that finally
mt2/2 = p2/2m
(7)
In a crystal, phonons (and electrons) are not free particles; they are bound to the
crystal and strongly influenced by the crystal lattice. Therefore hk/2p does not really
represent the momentum, it is just a quasi-momentum’ or a crystal momentum’.
The peculiarity of the crystal momentum is that its value cannot exceed the value of
a reciprocal lattice vector. If the momentum of a quasiparticle is increased, the
momentum value hs will be transferred to the crystal as a whole (s is a reciprocal
lattice vector, see above). The recoil energy associated with the momentum and
transferred to the crystal as a whole does not show up in the energy balance because
of E = p2/2M. M is the total mass of the crystal, which is almost infinite compared
to the mass of an electron (there are typically 1021 atoms in a crystal of 1 cm3 and an
atom is about 104 times as heavy as an electron).
Similarly, waves in the discrete lattice of a crystal also differ from waves in a continuum. Because of the discreteness of the lattice there is a minimum wavelength.
Waves with shorter wavelengths than the interatomic distance are not possible,
because there is nothing between two atoms that could oscillate. Minimum wavelength means maximum wave vector and, consequently, maximum momentum. A
fancy way of expressing this is to say that the momentum is only defined modulo
hs’, where s is a reciprocal lattice vector.
The dispersion relation for phonons in the linear model of Figure 3-13 can easily
be calculated (refer to textbooks on solid-state physics). The dispersion relation is
shown graphically in Figure 3-14, with the energy as ordinate and the wave vector k
as abscissa. The dispersion relation is plotted up to the aforementioned maximum
value for the wave vector k. The graph does not have to end at that point, but there is
no additional information for extending it any further, because the energy does not
change if integer multiples of a reciprocal lattice vector are added to the wave vector.
Figure 3-13
Model of a one-dimensional crystal. The lattice forces are represented as elastic springs.
Figure 3-14 shows a typical representation in reciprocal space. The wave vector is
used which has a reciprocal length (the wave number) and to which reciprocal lattice
vectors can be added. Reciprocal space can either be regarded simply as a convenient
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Figure 3-14
Dispersion relation for the model in Figure 3-13.
tool for discussing dispersion relations or as a generalization of the reciprocal lattice.
The reciprocal lattice consists of discrete lattice points, connected by reciprocal lattice vectors with the origin of the reciprocal lattice. Reciprocal space also includes
the space between the points of the reciprocal lattice. For most purposes, only the
space between the origin and the reciprocal lattice points nearest to the origin is
important. This part of reciprocal space is the so-called first Brillouin zone.
In Figure 3-14 the dispersion relation is plotted for only half of the first Brillouin
zone, from the zone center 0 to the edge of the right-hand zone at k = p/a. For symmetry reasons, the other half is just the mirror image.
To introduce the concept of Brillouin zones we look at Figure 3-15. It shows the
reciprocal lattice of a linear chain. The origin of the reciprocal lattice lies in the center of the figure. The interval between the origin and the first lattice points on both
sides is divided into halves; the region 1. BZ’ marks the first Brillouin zone. It is
surrounded by the second Brillouin zone and so forth. The concept can easily be
generalized to two- and three-dimensional space. For phonons, only the first Brillouin zone is relevant. In discussing the wave vector and the quasi momentum, we
noted that this quantity was only defined as modulo hs. A wave vector in the second
Brillouin zone would correspond to a wavelength shorter than a lattice period.
Figure 3-15
Brillouin zones.
3.3 Electrons and Phonons in a Crystal, Dispersion Relations
3.3.2
Phonons are Electrons are Different
Usually phonons are associated with waves (sound waves’); however, from quantum
mechanics we know that waves also have particle aspects. The reverse is true for
electrons: they are looked upon as particles although they also have wave character.
The wave character of electrons is particularly important for electrons in a crystal
lattice.
There are many electrons in a crystal. Most of them circulate around atomic
nuclei in inner shells. These electrons are not important in the present context.
Here we are interested only in electrons that do not belong to a particular atom but
which are delocalized all over the crystal: the conduction electrons of metals as well
as, in some respect, the valence electrons of doped or of photoexcited (occasionally
also thermally excited) semiconductors. These electrons are described as Bloch
waves (the theory of Bloch waves is not discussed in detail here). The Bloch theorem
states that, in a periodic lattice, electrons behave (can move) pretty much like phonons. They have a k vector and an energy value, and a dispersion relation connects
the two. The shape of the dispersion relation, however, is different. For phonons the
dispersion relation initially is linear at the origin of the reciprocal space, resembling
a classical wave with constant velocity: v = tk/2p. The electron dispersion relation
initially takes a quadratic course E = h2k2/2m, as expected for a classical particle
with kinetic energy of E = p2/2m. In this equation m is usually not identical with the
free electron mass. It is called the effective mass and is designated by m*. In general,
the effective mass is different for k pointing in different directions. To keep the formula E = h2k2/2m*, the effective mass is even allowed to change with k. These tricks
with m* allow us to still treat the electrons in a crystal almost as if they were free
particles.
Another difference between electrons and phonons is that they obey different statistics. Phonons are bosons, and for bosons the Pauli exclusion principle does not
exist. It is possible to have several phonons in the same quantum state, i.e., more
than one phonon with the same k-vector in one crystal. Electrons, however, are fermions, which follow Fermi statistics and have to observe the Pauli exclusion principle. They cannot populate the same quantum state in duplicate. For each k-value
there are only two electrons possible in a crystal, one with spin-up and one with
spin-down. At absolute zero, all phonons are in the lowest energy state at k = 0, and
the atoms just vibrate in their zero-point motion but the electrons fill the k-states of
the Bloch waves up to very high energies. The energy of the highest occupied state
is the Fermi energy EF, or the Fermi level. The corresponding wave vector is the Fermi
wave vector kF and its reciprocal is the Fermi wavelength kF (divided by 2p).
In most metals the Fermi energy is very high, several eV (electron volts), much
higher than the thermal energy kT (k: Boltzmann constant; T: temperature), which
is only about 30 meV at room temperature. Only the electrons close to the Fermi
energy can be thermally excited into an unoccupied k-state. The same is true for
acceleration of electrons by an applied electric field, which also involves a change of
the k-state, and for many other responses of the electron system to outside influ-
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ences. Consequently almost all properties of an electron system are determined by
the electrons at the Fermi energy, and that is why Fermi energy, Fermi wave vector,
and Fermi wavelength are so important.
The dispersion relation for the electrons is different in different directions, but in
each direction there is a well-defined Fermi vector. The tips of these vectors define a
surface in reciprocal space. This surface is called the Fermi surface, which reflects
h2kF2/
the symmetry of a crystal. For an isotropic solid it is spherical. Because EF = 2m*, the symmetry of the Fermi surface is related to the symmetry of the effective
mass.
To construct the Fermi surface of a one-dimensional metal we again use the anisotropy approximation introduced in Section 1.2. A small effective mass means
that it is easy to move the electrons. If, in a bundle of chains, the electrons move
easily along the chains but much less easily perpendicular to the chains, then the
effective mass tensor and the Fermi surface look like a lens (a flat ellipsoid) with a
short axis in the chain direction and two long axes perpendicular to the chains; the
Fermi surface distorts into two planes, as indicated in Figure 3-16.
In Section 1.2 (Figure 1-6) open’ Fermi surfaces were mentioned. The Fermi surface in Figure 3-16 definitely is open, because it consists of two parallel planes. Even
if the Fermi surface is slightly curved it can still remain open. This effect can be
understood with the help of the Brillouin zones. If the Fermi surface is a very flat
lens, its diameter might be larger than the dimension of the first Brillouin zone.
The Fermi surface would close only in the second or even a higher Brillouin zone.
However, then we would be allowed to subtract the reciprocal lattice vectors from kF,
thus yielding an open Fermi surface, which is a criterion for one-dimensionality.
Figure 3-16
1D Fermi surface.
3.3.3
Nearly Free Electron Model, Energy Bands, Energy Gap, Density of States
It has been mentioned that in contrast to the phonon dispersion relation, which is
linear for small wave vectors, the dispersion relation of electrons is quadratic,
E = h2k2/2m. The quadratic behavior is characteristic of classical particles, where
E = mv2/2. The free electron model treats the electrons in a crystal as classical particles. This model is surprisingly successful in explaining the electronic transport properties of a solid, such as electrical conductivity and thermal conductivity. We emphasized the word surprisingly, because it is not easy to understand that electrons can
3.3 Electrons and Phonons in a Crystal, Dispersion Relations
freely move between oppositely charged metal ions. However, the Bloch theory justifies the free electron model. As often happens in science, the theoretical justification
came years after the successful application.
The classical free electron model fails to explain specific heat and magnetic susceptibility. The Pauli exclusion principle has to be taken into account, i.e., Fermi statistics have to be applied to the electrons, leading to the free electron Fermi gas’. To
include the specific properties of a given solid and to explain, e.g., why silicon is a
semiconductor or sodium is a metal, we have to go one step further and allow for
some influence of the crystal lattice on the electron gas. This led to the nearly free
electron gas’ model, which differs from the free electron Fermi gas in two ways:
.
.
The free electron mass m is replaced by the effective mass m*. The effective
mass can be different for electron motions in different crystallographic directions (see above, discussion on the Fermi surface for a one-dimensional
solid). Consequently, m* is a tensor, not a scalar (which, when concentrating
on one dimension, does not matter, of course). Moreover, m* depends on the
magnitude of the wave vector k, and although the dispersion relation is given
as E = h2k2/2m*, the functional dependence of E on k deviates from the parabolic shape. The dispersion relation can even have an inflection point at
which m* changes sign and becomes negative.
The dispersion relation of a nearly free electron gas is not continuous at the
edge of a Brillouin zone but has jumps. Because of these discontinuities the
allowed energy values are confined to energy bands separated by forbidden
gaps.
In Figure 3-17 the dispersion relation for free electrons and the nearly free electron Fermi gas are compared. In the first case we have a simple parabola, in the
latter the parabola is distorted and shows jumps where k passes from the first into
the second Brillouin zone. At the zone edge (k = p/a) the dispersion function has
two different energy values for the same k value. The energy values are separated by
Eg, the width of the forbidden energy gap.
The energy gap can be explained as an interference phenomenon, which is very
simple in a one-dimensional lattice. The electrons (or Bloch waves) are scattered by
the electrostatic potential of the positively charged metal ions at the lattice points.
The lattice points are separated by the lattice constant a. Waves back-scattered from
adjacent ions have a path difference of 2a, and hence waves with this wavelength
Figure 3-17
Electron dispersion relation for free Fermi gas and nearly free Fermi gas.
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3 One-dimensional Solid-State Physics
interfere constructively. Wavelength a implies a wave vector k = p/a. At this point
the discontinuities in the dispersion relations occur. Waves with wave vector k = p/a
are in geometrical resonance with the crystal lattice. They are standing waves, not
propagating waves. Two types of standing waves with k = p/a can be formed: one
type has nodes at the lattice points, the other has maxima at the lattice points. In
one case the electrons are between’ the positive ions; in the other they are at’ the
ions. Evidently the energy for the two cases is different so that there are two energy
values for k = 2p/a.
Certain electron energies are not allowed in a crystal for resonance reasons. The
resonance is a consequence of the periodic potential of the positive ions in the solid.
Here an analogous example might be helpful. A similar situation exists on certain
unpaved roads in the Sahara Desert. Because of the camels traveling along it, there
are periodic bumps in the road. French 2CV cars resonate at 30 2 km h–1. So the
driver is forced to either stay below 28 or above 32 km h–1. In between there is a
forbidden gap (Figure 3-18).
An important quantity for solids is the electronic density of states (DOS), which
is the number of electronic states per energy interval. Evidently, in the forbidden
gap the DOS is zero. Figure 3-19 shows the electronic dispersion relation between
k = 0 and k = p/2 (as explained above, one half of the first Brillouin zone), together
The structure of certain Sahara roads allows cars
to ride well only below or well above 30 km h–1. Between 28 and
32 km h–1 there is a forbidden’ gap.
Figure 3-18
3.3 Electrons and Phonons in a Crystal, Dispersion Relations
Figure 3-19
1D electron dispersion relation and DOS.
with the DOS. The number of allowed k values in a Brillouin zone is large, but
finite. It is discrete because the electronic states are discrete when the electrons are
confined in a box. A crystal can be understood as such a box in which the electrons
are confined. The allowed k values are equally spaced along the x axis (Figure 3-19).
To get the DOS the dispersion relation is projected onto the energy axis. At places
where the dispersion relation is flat, the DOS is high and vice versa. At horizontal
parts of the dispersion relation the DOS is infinite. The points of infinite density of
states are called van Hove singularities and play an important role in one-dimensional
solid-state physics.
Figure 3-19 exhibits van Hove singularities for k = 0 and for k = p/a, i.e., the bottom and the top of the energy band. The DOS in Figure 3-19 shows van Hove singularities for a one-dimensional solid. If the solid is three-dimensional, k-vectors point
to all directions of space and the number of k-values within a k interval increases
with k, thus compensating for the van Hove singularity at k = 0. This k count finally
leads to the characteristic dimensionality behavior of DOS as indicated in Figure 1-18 (Chapter 1): parabolic in shape in three dimensions, step function in two
dimensions, and square root singularity in one dimension.
Many properties of a solid are determined by the DOS at the Fermi energy, N(EF).
We recall that electrons are fermions and obey Fermi statistics, i.e., we can accommodate only one electron in a quantum state (labeled by k). In simple words: we
pour’ the electrons into the crystal, they fill up the dispersion relation and at places
where we run out of electrons there is the Fermi level, the highest occupied electronic state. If there are as many electrons as there are k values in the first Brillouin
zone, the Fermi level is at the top of the band. Adding the next electron causes it to
jump to the bottom of the next band. Strictly speaking, this is only true for absolute
zero, where the Fermi distribution is sharp and where there are no thermal excitations in the electron system. Otherwise the Fermi level is defined in a somewhat
more complicated way, and for a completely filled band the Fermi level is placed in
the center of the gap above the filled band.
A solid with only completely filled and completely empty energy bands is an insulator. Electrons can only be excited across the gap’ and no small energy excitations
are possible. For electrical conductivity, however, such small excitations are neces-
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3 One-dimensional Solid-State Physics
sary. The electric field has to change the k values (and hence the energy) of the electrons to transport electrons form one side of the solid to the other. To have a metal
in which an electric field can accelerate electrons so that a electric current becomes
possible, partially filled bands are needed.
Metals have partially filled bands, semiconductors and insulators only completely
filled and completely empty bands. The difference between semiconductors and
insulators is just the size of the gap that separates the occupied and the unoccupied
bands. In semiconductors the gap is small enough to allow for thermal excitations
across the gap. Similarly, electrons can be lifted from a filled into an empty band by
photoexcitation. Furthermore, the band filling can be influenced by doping, i.e., by
replacing some of the atoms in the crystal with atoms having more or less electrons
than the host atoms (impurities). When, for example, silicon with four electrons is
replaced by arsenic with five, electrons are added to the crystal, or, using gallium
with three electrons, electrons are withdrawn. In addition to the substitutional type
of doping, interstitial doping can also be applied, e.g., Li in Si, in which the introduced atom is located between the regular lattice sites. Most dopants in polymers
are interstitial, e.g., iodine or sodium in polyacetylene. In chemistry, usually withdrawing of electrons is called oxidation, and adding is reduction. Thus doping is a
redox process that causes the Fermi level to shift (the Fermi level is comparable to
the chemical potential).
The dispersion relation in Figure 3-19 is symmetrical with respect to the bottom
and the top of the energy band. It initially proceeds parabolically at either side. This
type of symmetry is the so-called electron hole symmetry. Holes indicate missing
electrons. If a band is nearly filled it is more convenient to look at those (few) states
that are unoccupied than to look at all the many occupied states. Holes move’ in a
semiconductor pretty much in the same way as empty seats move’ in a concert hall
from the expensive front rows to the cheap back rows: people move forwards, the
empty seats backwards. Holes are positively charged quasiparticles, which are
responsible for the conductivity in p-doped (acceptor doped’, oxidized’) semiconductors.
In n-doped (donor-doped’, reduced’) semiconductors the electric current is carried by excess electrons. At the point where a p- and an n-doped zone of a semiconductor meet, a p–n junction results. A p–n junction has rectifying properties (for
textbooks on semiconductor physics, see [8–10]). More complicated arrangements of
p- and n-conducting zones lead to all kinds of marvelous semiconductor devices,
from laser diodes to megabyte chips, from resistance with negative temperature
coefficient to fractional quantum Hall effect.
Figure 3-17 shows the approach to the electron dispersion relation from the
nearly free particle. The opposite approach is the molecular orbital approximation,
which for many organic solids (and also for many inorganic solids) is more appropriate. It is based on the assumption that the electrons are localized at atoms or molecules. If these molecules approach close enough their orbitals interact, resulting in
energy splitting. In molecular pairs this is known as Davidov splitting. In a manyparticle system like a crystal the splitting leads to an energy band. The band is as
wide as the overlap of the wave functions (several meV to several eV) and the disper-
3.4 A Simple One-dimensional System
sion relation (band structure) is obtained by passing s-shaped curves through the
end points (parabolic approach as in Figure 3-19). Often, such a rough estimate is
sufficient for an approximate orientation. To calculate the actual band structure,
extended expertise in quantum chemistry and fairly large computer power are
essential.
3.4
A Simple One-dimensional System
Now that we have been reminded of a few of the tenants of solid-state physics and
how they apply to one-dimensional systems, let’s return to an object that we introduced in Chapter 2: the carbon nanotube. Recall that we built the nanotube in our
imagination by mapping it onto a graphene sheet and then rolling the sheet to form
a seamless cylinder. In fact as we shall see, there is something very special about
doing this. It allows there to be an unusual degree of decoupling between the carbon
lattice and the mobile charge. That is not to say that no coupling exists, it is just that
it is easy to approximate the charge as being free carriers. This may be the only onedimensional carbon system for which this is true. In other systems the charge and
the lattice are intimately locked in step, forming a much more complicated object.
Thus, the carbon nanotube allows us to understand the simplest system first.
Perhaps most astonishingly, a little knowledge of the electronic structure of
graphite and of the geometry of the tube leads one to fascinating conclusions with
regards to the electronic nature of these objects. Recall that the extra, small’, dimension of circumference was an issue earlier. In fact, it is quite easy to understand why
it should be. A half integral number of electron waves, for wave functions of electrons on the tube’s surface, must fit around the circumference. The part of the wave
function heading down the tube axis, however, looks like a plane wave. Now one
asks: in what states do the electrons exist in the graphene before rolling it (because
it is these electrons that are available for constructing the tube)? The Fermi surface
of the graphene looks like six points on the edge of a hexagon (Figure 3-20).
Notice that the hexagon is tilted with respect to the axial direction of the tube.
The amount of tilt is given by the chiral angle. However, the states associated with
the circumference appear as straight lines along the direction of vector R. The points
on the vertices of the hexagon (the Fermi surface) are the only electrons allowed to
participate in conduction of the object. The states associated with the circumference
are the only ones that are allowed so that the wave function fits on the tube. The
angular position of the hexagon is determined by the angle with which the graphite
is oriented relative to the rolling axis. When the hexagon lies at an angle such that
the hexagonal vertex intersects one of the circumference states, the tube has electrons that are mobile and it is a rather good conductor. However, when the hexagon
vertex misses the circumference state, the electrons at the vertex energy must be given some energy to make a transition into a conducting state. Does that mean the
tube acts like a semiconductor? That is right, the nanotube can be either a semicon-
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3 One-dimensional Solid-State Physics
The Fermi points of graphene sit at the points of
the hexagon shown. However, this hexagon is oriented with
respect to the axis of the tube.
Figure 3-20
ductor or a metal depending on its chiral angle. Furthermore, the gap depends on
the angle and the diameter of the tube. Since we can know the chiral angle in terms
of (n,m), we can use a geometric argument to see that if (n,m) is not divisible by 3,
then the tube is a semiconductor, otherwise it is a metal. Furthermore, 2/3 of the
possible nanotubes are semiconductors and only one third metals. Though we do
not show it here, it is clear that as the tube diameter grows, the spacing between the
circumference state gets more narrow. Thus, the bandgap of a nanotube Eg ~ 1/DIA.
Last, our example above is clearly a semiconductor. Perhaps you can see why people
have acquired such a fascination for this beautifully symmetric system?
Is this system truly one-dimensional? In what way can we tell? As mentioned earlier, the electronic signature of truly one-dimensional behavior is the occurrence of
van Hove singular points in the electronic density of states. Do we in fact observe
these? There is a way to measure such density of states on nanoscale objects, that is,
by using scanning tunneling microscopy and spectroscopy. In tunneling microscopy, the image is collected by scanning the surface with an atomically sharp tunneling tip. The image contrast is generated through subtle variations in the tunneling
current. When one wants to know the electronic structure at a specific place in the
image, the tip is stopped from scanning, feedback systems are disengaged, the voltage is ramped and current is collected. This tunneling spectrum is then differentiated to yield a differential conductivity. The tunneling differential conductivity is
normalized by a term that varies as the tip height above the object, and the result of
this normalization is proportional to the density of electronic states in the area
where the spectrum was taken. In a stunning set of experiments two sets of
researchers, one at Delft, one at Harvard, correlated the electronic structure of single-walled carbon nanotubes with the atomic structure as determined using scanning tunneling microscopy. Since the tunneling microscope can image at atomic
resolution, the researchers were able to show that the set of van Hove singularities predicted for a given set of (n,m) were exactly seen in tunneling spectra (Figure 3-21).
In fact, van Hove singular points were identified in multiwalled carbon nanotubes earlier. These objects are simply increasingly larger diameter single-walled
3.4 A Simple One-dimensional System
Tunneling micrographs and density of electronic
states showing the occurrence of van Hove singularities in single-walled carbon nanotubes.
Figure 3-21
nanotubes placed in a concentric configuration, giving more than one wall. The
interesting point here is that it appears as though there is little cross-talk between
the shells. In other words, the outer shell of the multiwalled nanotube is a onedimensional object as well, without regard for what is inside (Figure 3-22).
This leaves very little doubt that carbon nanotubes can reflect a one-dimensional
electronic nature. How this one-dimensional nature has manifest itself in thermal,
optical, and transport properties of both single-walled and multiwalled carbon nanotubes has been intensely studied over the past 10 years [11]. Furthermore, the role of
symmetry-breaking in such objects through defects [12], kinks [13] and bends [14],
and dopants [15] has also been of significant interest to the scientific community. In
each case however, the one-dimensional nature of the nanotube system must be
modified with respect to the extra-dimensional degree of freedom around the waist
The tunneling spectra, converted to electronic
density of states, for a multiwalled carbon nanotube. On the
graph below the spectra, a most simple tight binding prediction
for this tube is plotted. (Courtesy of D. Tekleab.)
Figure 3-22
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3 One-dimensional Solid-State Physics
76
of the tube geometry. This topological modification is especially clear for thermal
transport, where the vibrational degrees of freedom must include the so called twiston’, or the twisting of the tube about the axis [16]. In some sense one might claim
that a polymer system could exhibit a twisting of the atoms about the axis, and this
would be analogous to the twiston. This serves to demonstrate the limited applicability of the concept of one-dimensional’ materials.
References and Notes
1 N.W. Ashcroft and N.D. Mermin. Solid State
2
3
4
5
6
7
8
9
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12
Physics. Sounders, New York, 1976.
K.H. Hellwege. Einfhrung in die Festkrperphysik. Springer Verlag, Heidelberg, 1976.
C. Kittel. Introduction to Solid State Physics.
6th Edition, John Wiley & Sons, New York,
1986.
R.C. Evans. An Introduction to Crystal Chemistry. 2nd Edition, Cambridge University
Press, Cambridge, 1966.
U. Mller. Anorganische Strukturchemie.
B.G. Teubner, Suttgart, 1982.
E.A. Silinsh. Organic Molecular Crystals. In:
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R. Comes, M. Lambert and H.R. Zeller. A lowtemperature phase transition in the onedimensional conductor K2Pt(CN)4Br0.30·H2O.
physica status solidi (b) 58, 587 (1973).
K. Seeger. Semiconductor Physics. Springer
Verlag, Wien, 1973.
K.W. Ber. Survey of Semiconductor Physics.
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S.M. Sze. Semiconductor Devices – Physics
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Saito, G. Dresselhaus, and M. S. Dresselhaus.
Physical Properties of Carbon Nanotubes.
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D.L. Carroll, Ph. Redlich, P.M. Ajayan, J.C.
Charlier, X. Blase, A. De Vita, and R. Car. Elec-
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13 D. Tekleab, R. Czerw, P.M. Ajayan, and D.L.
Carroll. Electronic structure of kinked multiwalled carbon nanotubes. Applied Physics
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14 D. Tekleab, D.L. Carroll, G.G. Samsonidze, and
B.I. Yakobson. Strain-induced electronic property heterogeneity of a carbon nanotube.
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15 D.L. Carroll, S. Curran, P. Redlich, P.M. Ajayan, S. Roth, and M. Rhle. Effects of nanodomain formation on the electronic structure of
doped carbon nanotubes. Physical Review
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X. Blase, J.-C. Charlier, A. De Vita, R. Car, Ph.
Redlich, M. Terrones, W.K. Hsu, H. Terrones,
D.L Carroll, and P.M. Ajayan. Boron-mediated growth of long helicity-selected carbon
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R. Czerw, M Terrones, J.-C. Charlier, R. Kamalakaran, N. Grobert, H. Terrones, B. Foley,
D. Tekleab, P.M. Ajayan, W. Blau, M. Rhle,
and D.L. Carroll. Identification of electron
donor states in n-doped carbon nanotubes.
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Petit, H. Dai, A. Thess, R.E. Smalley,
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