This text treats electronic transport in the regime where conventional textbook models are no longer applicable, including the effect of electronic phase coherence, energy quantization, electron-electron interactions, and spin in the mesoscopic regime. The book also provides an overview of semiconductor processing technologies and actual experimental techniques. With a number of examples and problems with solutions, this is an ideal introduction for students and beginning researchers in the field.
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About the Author
Thomas Heinzel received his PhD from the University of Munich in 1994. He subsequently joined the University of Pennsylvania, Philadelphia, for two years. From 1996 to 2001, he worked at the ETH Zurich, where he received his habilitation. From 2001 to 2004, Professor Heinzel held a professorship in experimental physics at the University of Freiburg, after which he accepted a post as professor of experimental physics of condensed matter at the University of Düsseldorf, both in Germany. His current research interests are electrons in nanostructured as well as in self-organized materials.
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Mesoscopic Electronics in Solid State Nanostructures
By Thomas Heinzel
John Wiley & Sons
Copyright © 2003
All right reserved.
An Update of Solid State Physics
Mesoscopic systems are fabricated from various bulk materials, which are often, but not always,
semiconductors. Some basic knowledge of their bulk properties is important and represents
the major part of this chapter. Although this is in many respects just a polishing up
of solid state physics at an introductory level, we introduce many specifics of the materials of
interest along the way, in particular of Si and GaAs. Occasionally, conventional metals and
carbon crystals are mentioned as well, although the reader is supposed to know their basics.
We begin by a brief recapitulation of the most relevant crystal structures in section 2.1,
and proceed by looking at the corresponding electronic band structures of the materials in
section 2.2. Here, it is of particular importance to model the valence and the conduction
bands around their maximum and minimum, respectively. As always, we can approximate
the energy dispersions near the band extremal points by parabolas, which leads to the concept
of effective masses. We shall see that within this approximation, the crystal properties can
be "put aside" in many cases. Instead, the chargecarriers behave like free electrons with a
modified mass. The properties of electrons and holes within the effective mass approximation
are looked at in section 2.3. Also, the effective mass approximation allows us to work with
envelope wave functions. With this approach, superpotentials like those frequently met in
nanostructures, can be treated with a Schrodinger equation for just this superpotential. The
crystal potential enters only via the effective masses as well as via its dielectric constant.
This is a very elegant concept, which simplifies our life substantially in subsequent chapters.
Developing this approximation is the topic of section 2.4.
Doping is the standard way to fill the bands of a semiconductor with a significant and
temperature-independent carrier density. The important issues concerning doping are reviewed
in section 2.5. In the subsequent section, we look at the transport properties of electron
gases within the simplest version of the Boltzmann model. We will occasionally use
these results when looking at diffusive samples later on. Furthermore, it is of help to know
the approximations that enter this model, in order to appreciate the deviations we will look
at in subsequent chapters. A non-vanishing resistance indicates that some sort of scattering
mechanism must be present, which are the topic of section 2.7. Finally, we spend a few words
on screening in section 2.8.
Readers who discover that parts of this chapter are white spots on their map of solid
state physics knowledge are encouraged to consult one of several excellent introductory textbooks
for further information, e.g., [Ashcroft1985, Ziman1995]. If everything sounds familiar,
please consider this chapter as a warm-up exercise!
2.1 Crystal structures
Many elements and compounds crystallize in a face centered cubic (fcc) lattice. This is not
surprising, since this crystal structure represents one of the two closed packings possible,
which one might naively expect to occur when identical or very similar spheres are piled up.
Both Si and GaAs have this lattice structure. The lattice constant a is the length of one edge of
a unit cell. Si is composed of two fcc lattices shifted relative to each other by (a/4, a/4, a/4). This crystal
structure is also known as the diamond structure. GaAs also has a two-atom base,
except that here, the one base atom is Ga, the other one As. This is the zincblend lattice. The
lattice constants are 0.565 nm for Si and 0.543 nm for GaAs (both numbers hold for room
temperature). Fig. 2.1 shows the Si and the GaAs structure.
The reciprocal lattice of an fcc lattice is a body-centered cubic (bcc) lattice. Since the crystal
momentum is invariant under translations by reciprocal lattice vectors, we can represent the
behavior of electrons and phonons within one elementary cell of the reciprocal lattice, which
is always chosen as the first Brillouin zone. For an fcc lattice, this is a truncated octahedron,
composed of 6 squares and 8 hexagons, see Fig. 2.1. The center of the first Brillouin zone is
labelled the [GAMMA]-point, while the centers of the hexagons and squares are referred to as L- and
X-points, respectively. Occasionally, one hits upon more exotic directions of lower symmetry,
such as K,U, and W, which are located at the center of the edges and at the corners of the first
Germanium crystallize in a diamond structure like silicon. This is also the case for many
compound semiconductors. Any binary combination of Al, Ga, or In with As, Sb or P (the so-called
III-V compounds) will result in a zincblend lattice. Combining these group III elements
with nitrogen can lead to both a fcc lattice or a hexagonal lattice, depending on the crystallization
process and the subsequent treatment. This is also the case for most II-VI compounds,
such as CdSe or ZnS (which gave the zincblend structure its name, after all). Thus, when
working with semiconductors, you will barely ever meet any further crystal structures. To
finish this section, let us have a look at a particular simple lattice, namely a sheet of graphite,
the second crystal structure carbon forms besides diamond. It consists of a hexagonal, lattice
of [sp.sup.2] - hybridized carbon atoms (Fig. 2.2).
Question 2.1: Calculate the reciprocal lattice of the graphite sheet and construct its first
The reciprocal lattice is again hexagonal. The center of the first Brillouin zone is denoted
by [GAMMA], the corners by K, and the centers of the edges are labelled L, respectively (Fig. 2.2).
2.2 Electronic energy bands
An electronic energy band is an energy interval in which electronic states are allowed in the
crystal. The bands are separated by band gaps. This energy structure is obtained by solving
the Schrodinger equation for electrons in the crystal
(2.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Here, the electronic wave functions depend on both the wave vector k and the spatial coordinates
r. They are denoted by [phi](k,r), while [V.sub.crystal](r) is the crystal potential.
Elementary solid state physics tells us that the wave functions have to obey Bloch's theorem, which states
that they are of the form
(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [u.sub.k](r) has the periodicity of the crystal lattice. Such wave functions are Bloch functions.
The task is to determine the eigenvalues [epsilon](k) and eigenvectors, which is usually done by transforming
the differential equation into an algebraic equation. An exact solution, though, is only
possible for some special cases. Some reasonable approximation is therefore called for. How
eq. (2.2) is then solved in detail depends on the model. The nearly free electron model starts
from a free electron gas and treats a weak periodic crystal potential within perturbation theory.
Here, the band gaps emerge from interferences of the electronic waves that get scattered at the
crystal potential, which results in standing waves at the edges of the Brillouin zones. The
reader is referred to the extensive literature on solid state physics for details. Here, we look at
a different approach, which constructs the electronic eigenstates from those of the individual
atoms that form the crystal. This approach is known as the tight binding model. Within this
picture, the energy bands and the band gaps are remainders of the discrete energy spectrum of
The tight-binding model is based on the assumption that the atomic orbitals [[xi].sub.j](r) are a
good starting point for constructing Bloch waves [[xi].sub.j](k,r). Let us assume there is only one
atom per unit cell. We can obtain Bloch functions via
(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Here, the lattice vectors are denoted by [R.sub.n]. The crystal wave functions can be expanded in
these Bloch functions, such that
(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The Schrodinger equation for the Bloch functions is now multiplied by [[xi].sup.*.sub.i] (k,r) and integrated
over space. The emerging algebraic equation has a nontrivial solution only for
(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Here, [T.sub.ij](k) and [O.sub.ij](k) denote the "transfer matrix elements" and the "overlap matrix
elements", respectively. They are defined as
These matrix elements are often approximated by inserting the known atomic orbitals, and
choosing a suitable crystal potential, which can be used as a parameter to fit the experimentally
determined properties of the crystal.
Question 2.2: Determine the energy dispersion for the simplest case, namely for a single
band in one dimension, with a constant (and negative) transfer integral [gamma], and a vanishing
overlap integral. Show that the energy dispersion in that case reads E(k) = [E.sub.0] + 2[gamma] cos(ka)!
As an example, we consider the graphite sheet, in which atomic s - and p - orbitals generate
the bands of relevance, shown in Fig. 2.3. For the [p.sub.z] orbitals of the carbon atoms arranged in
a honeycomb configuration, a bonding and an antibonding [pi] band results [Wallace1947]. To
a first approximation, its tight-binding energy dispersion is
(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Solids are usually classified as metals, semiconductors, and insulators. In a metal, at least
one of the bands is partly occupied with electrons. These bands are called conduction bands
in metals. In semiconductors and insulators, all bands are either full or empty at zero temperature.
Here, the full band with the highest energy is the valence band, while the conduction
band is the empty band with the lowest energy. In a semiconductor, a significant density of
electrons can be transferred from the valence band into the conduction band by thermal excitation,
which requires a band gap of less than 4 eV. Consequently, insulators have larger band
It turns out that the graphite sheet is a very special case in this classification scheme. The
bonding [pi] band is in fact the valence band, while its antibonding counterpart is the conduction
band. As can be seen from eq. (2.6), the valence band can be mapped onto the conduction
band by a reflection at the planes defined by the K-points. The conduction band and the
valence band of a graphite sheet, represented by bold lines in Fig. 2.3, touch each other at the
K-points. It can thus be regarded as a semiconductor with zero band gap.
By adopting the tight-binding method appropriately, the band structure of other materials,
like Si and GaAs, can be calculated. Naively, one might assume that due to the similar crystal
structures, the band structures of the two semiconductors should be very similar as well.
However, this is not the case, mainly because the Ga-As base is polar, while the Si base is
covalent. Fig. 2.4 shows the structures of the valence and conduction bands of both crystals.
The extremal points of the bands shown here dominate both the electronic and optical properties.
The number of electrons in the conduction band, as well as that one of the holes in the
valence bands, is small compared to the number of available electronic states in all cases of
relevance, and the few carriers will find themselves in close proximity to the band extremal
points. Around these extremal points, we can expand the energy dispersion in a Taylor series
up to second order:
(2.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
By comparing this expression with the energy dispersion of the free electron gas E(k) =
[h.sup.2][k.sup.2]/2m, we see that the tensor of second derivatives of the energy can be identified with
(2.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
which is therefore also known as the effective mass tensor. It can be diagonalized, such that the
extremal points of energy bands can be characterized by its three effective masses along the
principal axes. Carriers in semiconductors therefore usually behave free-electron like, except
that their masses have been changed by the crystal structure. Throughout the rest of the book,
we will use effective masses to describe the behavior of carriers.
Question 2.3: What is the effective mass around the minimum of the the energy band obtained
in Question 2.2?
Let us have a somewhat closer look at these band structures. Si has a conduction band
minimum at 0.85 [GAMMA] X. Around this minimum, two different effective masses exist, a transverse
mass in all directions perpendicular to the [GAMMA] X - direction, [m.sub.e,t] = 0.19 m, and a
longitudinal mass along the [GAMMA] X - direction, [m.sub.e,l] = 0.92 m. Since there are 6
X-points, the conduction band minimum in Si shows a 6- fold degeneracy known as "valley degeneracy". In GaAs,
the conduction band minimum is located at the [GAMMA]-point. Here, the 3 effective electron masses
are identical: [m.sup.*.sub.e,1](GaAs) = [m.sup.*.sub.e,2](GaAs) = [m.sup.*.sub.e,3](GaAs) = 0.067m. In both
materials, there are two (nearly) degenerate valence bands at the [GAMMA]-point. As in most semiconductors
of interest, the valence band emerges from atomic p - states, which have a threefold orbital
degeneracy and a spin degeneracy of 2. Typically, the corresponding [sigma] - band formed by the
atomic s-orbitals has its maximum well below the maximum of the p-bands and do not have
to be taken into account for transport considerations. In the crystal, the degeneracy of the p
- orbitals is removed, and 3 different, spin degenerate bands are obtained. Two of them are
shown in Fig. 2.4, while the third one is split off and shifted to lower energies. This splitting
has its origin in the spin-orbit interaction. The spin-orbit Hamiltonian is given by
(2.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where V is the electrostatic potential, and [sigma] are the Pauli matrices. This is a relativistic term,
which means we have to replace the Schrodinger equation by the Dirac equation, and the wave
function becomes a two-component spinor. In a spherical symmetric potential, the spin-orbit
Hamiltonian becomes proportional to the scalar product of the angular momentum and the
spin L·S. To get an idea what the spin-orbit Hamiltonian does to the energies, we assume that
the interaction in the solid can be approximated by that one in the individual atoms.
Excerpted from Mesoscopic Electronics in Solid State Nanostructures
by Thomas Heinzel
Copyright © 2003 by Thomas Heinzel.
Excerpted by permission.
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Table of Contents
An Update of Solid State Physics
Surfaces, Interfaces, and Layered Devices
Important Quantities in Mesoscopic Transport
Magnetotransport Properties of Quantum Films
Quantum Wires and Quantum Point Contacts
Modeling of Ballistic transport in mesoscopic structures
Electronic Phase Coherence
Single Electron Tunneling