Robert+Wexler_Final

=A review of the theory and modeling of surface charge transport=

Robert Wexler Drexel University Chemistry Information Retrieval - CHEM367 Submitted December 3, 2011

Contents
(I) Introduction (II) Theory (III) An empirical approach (IV) A physics approach

**Introduction**
Surface charge transport has vital applications to the health and transportation of human beings. For example, semiconducting compounds such as tin (II) oxide can be used to detect hydrogen, hydrogen sulfide, nitrous oxide, carbon monoxide, and various hydrocarbons to name a few (1). Thus if the exhaust from an automobile or some other gasoline powered machine is allowed to accumulate in a closed setting, the carbon monoxide produced from incomplete combustion can be detected through its surface interactions with a tin (II) oxide gas sensor. Tin (II) oxide and other semiconducting compounds such a titanium dioxide, gallium (III) oxide, cerium (IV) oxide, etc. can be used to detect the ratio of air (oxygen) to fuel at an engine's point of ignition using surface conductivity measurements (1). Surface charge transport also has important applications in organic synthesis. For example, insulating compounds such as gamma-alumina can be used for heterogeneous catalysis and as drying agents due to surface charge mobility and water storage respectively (2, 3). Currently, the full potential of surface charge transport and its applications has not yet been realized because its mechanism of action is still largely unknown (4). In this minireview, the available surface charge transport mechanisms will be discussed and separated into two main categories: empirical and physically based formulae.

**Theory**
In order to understand surface charge transport, it is important to develop a basic understand of quantum mechanics, modes of conductivity, and band theory. Additionally, it would be beneficial to the reader to assume that the terms surface charge transport and surface conductivity are interchangeable.

**A Brief History of Quantum Mechanics**
Prior to the development of quantum mechanics, it was widely known that electromagnetic radiation was a wave phenomena. 19th century chemists and physicists found that a description of black body radiators using classical Newtonian mechanics predicted infinitely intense radiation. This was coined the 'ultraviolet catastrophe' (5). In 1887, Heinrich Hertz discovered that the amount of electrons released from a metal exposed to electromagnetic radiation was not dependent of the intensity of the radiation but rather the wavelength. This phenomena is known as the photoelectric effect. In 1900, Max Planck hypothesized that the oscillating centers in black body radiators cannot absorb and emit any energy value but rather quantized bits called quanta (6). Combining all of these discoveries, Albert Einstein clairvoyantly proposed that all electromagnetic radiation comes in packets of energy quanta, which later were referred to as photons (7).

In 1909, Rutherford performed his famous gold foil experiment affirming the existence of electrons and a positively charged atomic nucleus (8). In 1913, the hydrogen line spectra mystery was finally solved by Neils Bohr who suggested that the wavefunction of hydrogen must have an integer number of wavelengths (9). In 1925, Louis de Broglie discovered that the momentum (p) of a given particle could be determined by Planck's constant and the wavelength of the particle. This discovered suggested wave-particle duality; light can act both like a wave and a particle (7). Combining all of these discoveries, Erwin Schrodinger proposed that the wavefunctions for a particle (eigenfuctions) and their associated energies (eigenenergies) can be related using the following expression

[1]

where E is the total energy of the allowed wavefunctions, psi is the wavefunction, and H is the Hamiltonian which operates on the wavefunction to give the eigenenergies. The Hamiltonian is the sum of the potential (V) and kinetic energy (T) of the particle. The kinetic energy term can be determined using the formula T = p^2 / (2 * m) where the m is the mass of the particle and p is the momentum operator, which is defined by a quantum mechanical postulate

[2]

where h bar is Planck's constant divided by 2 * Pi, i is the square root of negative one, and the upside down triangle (nabla symbol) is the differential of the spatial coordinates. The potential energy function can be and has been described as many things such as an infinite well, a parabolic potential, etc. The combination of equations [1] and [2] gives a more complete form of the time-independent Schrodinger equation.

[3]

Schrodinger's equation can be applied to any atom or molecule and, when solved, gives its energy levels; this greatly builds upon Bohr's model that only gives the energy levels of hydrogen (10).

**What is Conductivity?**
Resistance measures how easily a current moves through a material and has units of ohms. The inverse of resistance, measured in mhos (how whimsical), is conductivity; lower resistance means higher conductivity. Concerning conductivity, there are four main types of materials: metals, semiconductors, insulators, and superconductors (7). Conductors have low resistance to the flow of charged particles. Some examples of conductors are silver, gold, and copper. Copper is commonly used for electrical wiring because of its excellent conductivity and affordability. Semiconductors have medium resistance to the flow of charged particles. Some examples of semiconductors are silicon, germanium, and other group IV elements (7). Silicon is widely-know for its use in integrated circuits, an essential component of computers, cell phones, CDs, DVDs, and many more (11). Semiconductors have two main subgroups: intrinsic and extrinsic. Intrinsic semiconductors are pure, natural semiconductors (silicon, germanium, gallium arsenide) whereas extrinsic semiconductors use impurities called dopants to improve the conductivity of intrinsic semiconductors (7). Insulators have high resistance to the flow of charged particles. Some examples of insulators are wood, glass, plastics, and ceramics. Lastly, superconductors have zero resistance to the flow of charged particles. Some examples of superconductors are yttrium barium copper oxide, bismuth strontium calcium copper oxide, among others. Superconductors could be used to improve the efficiency of energy transmission and levitated transportation (12). At the moment, superconductor applications are few because superconducting materials must be supercooled to obtain zero resistance. Thus, an exciting avenue of research today pertains to developing higher temperature superconductors. The reason for the conductive behavior of metals, semiconductors, and insulators will be discussed in the next section.

**A Taste of Band Theory**
Quantum mechanics can be used to describe the conductive behavior of metals semiconductors, and insulators. Let's go back to the very basics: all matter is made up of protons, neutrons, and electrons thanks to Ernest Rutherford and J.J. Thomson respectively (13). An electron can be considered both a wave and a particle due to de Broglie's wave-particle duality. With a wavefunction, Schrodinger's equation can now be solved for the possible energy levels of the electrons. These energy levels can be described by the quantum numbers n, l, ml, and ms organizing the electron configuration of each atom into atomic orbitals that portray the probabilistically favored location of an electron. When two atoms react to form a diatomic molecule, the atomic orbitals combine to form a molecular orbital. This is called linear combination of atomic orbitals (LCAO) and the number of atomic orbitals is equal to the number of molecular orbitals. If the wavefunctions reinforce each other, a bonding molecular orbital is formed. However if the wavefunctions cancel each other, a node of zero electron density or antibonding orbital is formed (14). Figure 1 shows wavefunction reinforcement and canceling in the hydrogen atom.

Figure 1. The molecular orbital of hydrogen showing the atomic orbitals (far right and left) and the bonding (bottom) and antibonding (top) molecular orbitals

As the number of atoms in the molecule increases to a solid, the energy levels overlap to form bands. The highest occupied and lowest unoccupied energy band are known as the valence and conduction band respectively; the energy gap between these bands are called band gaps. Metals, semiconductors, and insulators have characteristic band gaps allowing one to determine a compound's conductive nature using its band diagram. Figure 2 shows an example of a band diagram for a metal, semiconductor, and insulator.

Figure 2. Depiction of the band diagrams for metals, semiconductors, and insulators

Figure 2 shows that the energy bands of metals overlap. Basically, this could mean either one of two things: the bandgap between the valence and conduction bands is very small allowing electrons to jump to higher energy levels or there are less electrons than available energy levels (14). Figure 2 also depicts that there is a large band gap between the valence and conduction band for insulators. At room temperature, an electron does not have enough energy to overcome the band gap's energy barrier. However, the conductivity of insulators is known to increase as the temperature of the system increases (2). In between metals and insulators, semiconductors have a much smaller band gap between the valence and conduction band allowing some conductivity at room temperature. However, the conductivity of semiconductors can be greatly improved by doping, which is adding small amounts of impurities; the dopant only makes up a very small percentage of the solid in order to maintain the integrity of the original crystal lattice. For example if the conductive properties of elemental silicon want to be improved, it can be doped with a group III or V element such as Boron or Arsenic respectively (7). If the three-valence electron element Boron is added, there is a mobile electron deficiency, a positive hole, in the lattice which increases the conductivity. Doped semiconductors with holes are known as p-type (positive-type) and those with extra electrons are n-type (negative-type). Figure 3 shows n and p-type Silicon semiconductors.

Figure 3. n and p-type Silicon semiconductors on the right and left respectively

Band theory is an excellent model to describe the conductivity of bulk metals, semiconductors, and insulators. However, when the material is made into thin films, the surface interactions dominate and conductivity does not follow the patterns of bulk samples often times performing better (15). As was stated in the introduction, the mechanism of surface charge transport has yet to be sufficiently explained. In the next two sections, I will discuss two different methods of experimentation that hope to elucidate this mechanism.

An empirical approach
Currently, there are many empirical investigations into the surface conductivity of the semiconductor tin oxide because it is commonly used in gas detectors (1, 4, 16-18). Sberveglieri splits gas sensors into two main categories: air to fuel ratio sensors for engines and toxic gas sensors (1). The latter sensor commonly employs doped and un-doped thin film tin oxide. Experimental observation shows the following relationship between conductivity and the partial pressure of detectable gas

[4]

where G is the conductivity, G0 is the initial conductivity, gamma is constant characteristic of the semiconducting material, Pgas is the partial pressure of the detectable gas, and m is an exponential parameter that depends on the gas being detected (1). For hydrogen, carbon monoxide, and methane, the m value is positive and thus the surface conductivity of thin film tin oxide increases as partial pressure the gas increases.

Kissine et al. show that different gases have different effects on the surface conductivity of thin film tin oxide. Data suggests that a basic power law can be used to approximate the surface conductivity of tin oxide

[5]

where the variables are the same as in Sberveglieri's paper (18). Utilizing some surface physics, a more involved equation is produced

[6]

where sigma is the surface conductivity, ni and n are parameters of tin oxide, mu_n is the electron mobility, and mu_p is the hole mobility. The n parameter is dependent upon the partial pressures of the detectable gases present. Results show that increasing oxygen pressure decreases surface conductivity and increasing ethanol pressure increases surface conductivity. When the two gases are combined, the surface conductivity is even lower than that of pure oxygen (18). Kissine attempts to describe the latter trend by defining the ethanol and oxygen donor-like and acceptor-like impurities respectively. It is possible that semiconductor activity between ethanol and oxygen take away from the surface conductivity of thin film tin oxide (18).

Clarke shows that the presence of oxygen has an opposite effect on the surface conductivity of the semiconductor Germanium (19). In the experiment, Germanium was exposed to different component gases of the atmosphere; oxygen showed a sizable increase in the surface conductivity. This unexpected phenomena could be explained by oxygen adding energy levels to the surface through which conduction is allowed (19).

A physics approach
Rather than fit surface conductivity to an empirical function, some scientist seek to describe its mechanism using solid state and surface physics (2, 3, 20-24). One model used to describe surface charge transport in gamma-alumina suggests a dual charge-carrier mechanism (2). At lower temperatures, the surface conductivity of gamma-alumina is controlled by water molecules whereas, at higher temperatures, it is controlled by mobile protons. The general trend is that as gamma-alumina is heated from room temperature to 450K, the surface conductivity decreases because absorbed and adsorbed water molecules evaporate (3). As gamma-alumina is heated from 450 to 673K, the surface conductivity increases because protons become mobile (20). Using solid state and surface physics, the following expression is derived to describe this trend in surface conductivity

[7]

where G is the surface conductivity, B is a constant, v is velocity of the charge-carrier, and n is the charge-carrier concentration. This model is in good agreement with experimental data by suggesting that, at lower temperature, hydrogen bonded water is the charge-carrier whereas, at higher temperature, protons becomes a charge-carrier migrating along a water scaffold (2). Cai also determined the entropic parameters of the dual charge-carrier model in gamma-alumina by treating equation [7] with thermodynamics.

Using a different approach, Geistlinger describes surface charge transport for gallium (III) oxide using quantum mechanical treatments. Ultimately, it is suggested that the Volkenstein theory is an excellent basis to describe chemisorption (22, 23).

Conclusions
There are many different methods that seek to describe the trends and mechanism of surface charge transport. Some are empirical in nature and others are derived directly from solid state and surface physics. Nonetheless, development in this field is crucial in order to systematically develop gas detectors, catalysts, computer chips, and, in general, surface conductivity-based applications.