ELECTRONIC STRUCTURE OF SURFACES - …leung.uwaterloo.ca/CHEM/750/Lectures 2007/SSNT-5...Before we...

ELECTRONIC STRUCTURE OF SURFACES

Before we discuss the electronic structure of surfaces (i.e. the selvedge of the solid), we shall first review some of the basics about the electronic structure and electrical properties of the bulk.

1. Classification of bulk materials by their resistivity: insulators, semiconductors, semi-metal and metals.

2. Origin of the band gap and band structure of a solid: General (qualitative) consideration using simple MO theory of clusters.

3. Free electron gas model of a solid: Particle-in-a-1D-box; Particles-in-a-double-1D-box.

4. Periodic potentials and Bloch functions.

5. Energy band structure.

6. Surface states.

7. Properties and transitions involving these surface states.

8. Other surface related phenomena.

1. Classification of bulk materials by their resistivity: insulators, semiconductors, semi-metal and metals

A common way of classifying different types of bulk solids is via resistivity measurement. For example, the electrical resistivity of a pure metal can be as low as 10-10 ohm-cm at

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about 1 K, while the resisitivity of a good insulator may be as high as 1022 ohm-cm (at the other extreme). This range of 1032 may be the widest of any common physical property of solids.

This periodic table gives the resistivities of most metals at room temperature. The conductivity is the inverse of resistivity. The resistivity of metals decreases (or conductivitiy increases) as temperature is decreased. For example, for Cu, the resistivity at 4 K (liquid He temperature) is 10-5 times that at room temperature. In non-metallic solids, on the other hand, the resistivity increases as the temperature is lowered.



If we jump ahead a bit (see below) and picture the electronic structure of the solids to be consisted of energy bands separated by band gaps (all of these can be derived from quantum mechanics), then we may classify the solids according to how these bands are filled.



Briefly, there are three kinds of bands: the atomic-like core bands located deep down near the ionic cores; the valence bands (corresponding to bonding, nonbonding orbitals) and conduction bands (corresponding to empty antibonding orbitals and other “virtual” orbitals, and/or other Hydrogen-like Rydberg orbitals).

• Insulator - the allowed energy bands are either completely filled or completely empty such that no electrons can move in an electric field;

• Metal - one or more bands are partly (10% to 90%) filled;

• Semiconductor or semi-metal - all the bands are entirely filled except for one or two bands slightly filled or slightly empty.

The “block” schematic representation of the band is in fact



not correct. An energy level of a solid, as we shall see, changes with respect to the location in the Brillouin zone.

This gives rise to the energy band. Here, we can see a more realistic picture of the electronic structures of metals, insulators and semi-metals.



This “band” arrangement also leads to the so-called vertical and non-vertical transitions. If the minimum of the conduction band and the maximum of the valence band are located at the same k point, then we have a so-called directband gap. Vertical transitions across this direct band gap are known to be extremely important because they can be used to generate various “lasing” actions. (This is the basic principle of solid-state lasers, constructed for instance from GaAs/GasAlAs superstructures, from porous Si, or from superstructures of Si/Ge/Si, etc.)

ASIDE: We have used resistivity or conductivity to classify different kinds of solids. Beside this “electrical conduction” properties, different solids may also have different optical properties [the emission and absorption of light and photochemical reactions], magnetic properties [paramagnetism, ferro- and anti-ferromagnetic ordering], and surface properties [those involved in the transfer of



electrons to another phase as in electrochemistry].

Note also in solid state chemistry, we generally use the type of interactions or intermolecular forces to classify the solids. There are four important classes of simple solids:

• molecular [e.g. Xe, N2, benzene, molecules held together by van der Waals forces];

• ionic [e.g. NaCl, MgO, CaF2];

• covalent [e.g. diamond, P, SiO2, GaAs, Group IV to VI]; and

• metallic solids [e.g. Na, Fe, Cu].

Finally, we have complex solids which combine different types of bonding [e.g. graphite (covalent, metallic and van der Waals), CdS, CdSe, TiO2 (intermediate ionic-covalent), etc.]

2. Origin of the band gap and band structure of a solid: General (qualitative) consideration using simple MO theory of clusters.

As chemists, we tend to think of chemical bonding of molecules and solids in terms of orbitals. We can generally distinguish between core and valence atomic orbitals. The core (or deep lying) orbitals are centered close to the nuclear cores and therefore atomic-like; i.e. they have little bonding influence. The valence orbitals



have large spatial extents and will overlap with neighbouring atomic orbitals, forming (valence) molecular orbitals. There are three kinds: bonding (more stable), non-bonding and antibonding (less stable).

Recall the simple Linear Combination Atomic Orbital - Molecular Orbital (or LCAO-MO) theory. For diatomics, we can write the molecular orbitals as:

ψ1 = a1χA + b1χB

ψ2 = a2χA - b2χB

From the two atomic orbitals, we now have a bonding orbital and an antibonding orbital (and in some other cases, non-bonding orbitals as well). The number of MO’s therefore depends on the number of AO’s used to make them.



NOTE: The MO theory emphasizes the delocalized nature of the electron density so that the MO’s are generally extended over all the constituent atoms.

Now, suppose we have more and more atoms (a cluster, for example), there will correspondingly be more MO’s. If we now have a solid (with a very, very large number of atoms), the separations between energy levels of the MO’s become correspondingly smaller and smaller. The original bonding and antibonding orbitals of a diatomic molecule now evolve respectively into the valence and conduction bands in the case of a solid. The energy separation between the top of the valence (bonding) band and the bottom of the conduction (antibonding) band becomes the band gap.



Furthermore, the MO’s of a solid will not be distributed evenly in different crystallographic axes over all energies (we may think of this as a form of degeneracy corresponding to different types and symmetries of the orbitals). This leads to the concept of the density of states N(E), which is defined as:

N(E)dE is the number of allowed energy levels per unit volume of the solid in the energy range E to E+dE.

ASIDE (a) Optical properties of metals, insulators and semiconductors.

The interaction of a solid with visible and UV radiation causes excitation of valence electrons from filled to empty



bands (as in the case of insulators) or within partially filled bands (as in the case of metals). The visible appearance of a solid depends on the degree of absorption and reflectance in the visible region of the electromagnetic spectrum (1.5 to 3.0 eV). Solids with an energy gap in this range will appear coloured (yellow to deeper orange and red, as the band gap moves down thru the visible). If the band gap is below 1.5 eV (i.e. infrared), the solid may appear dark in colour or shiny metallic, depending on the reflectivity. If the band gap is greater than 3 eV (i.e. ultraviolet), no absorption occurs in the visible and a good quality crystal will appear transparent. Furthermore, light may be scattered at defects and crystallite surfaces in the case of a powder sample, which leads to white appearance (e.g. TiO2used in white paint).

In the case of insulators and semiconductors (with band gap ranging from 12 eV to 0.1 eV), the wavelength on-set of the absorption often indicates the band gap. In the case of metals (with partially filled band), there is no band gap and so the range of electronic excitation extends down to zero energy.



Measuring the threshold of optical absorption is therefore a simple method for estimating the band gap. In the case of non-metals (insulators and semiconductors), however, thereare additional peaks at the absorption threshold that are caused by an exciton.



An exciton is an excited state of the solid where the electron does not escape fully into the conduction band but remains trapped in the electrostatic potential of the hole left behind in the valence band. Not all solids show such excitons.

ASIDE (b) Thermal excitation of electrons.

Thermal excitation of electrons from the ground state is important because this governs many properties of solids (e.g. specific heat). Recall that the number of atoms or molecules ni in excited states at a given temperature T is given by the Boltzmann distribution:

where Ei is the energy of the state and k is the Boltzmann constant. For electrons, we must take into account of the Pauli exclusion principle (i.e. each spin-orbital can only hold one electron) and of indistinguishability under exchange. These two properties give rise to the Fermi-Dirac distribution:

kTEi i

en 1∝



where f(E) is the fraction of allowed level with energy E that are occupied. When T approaches absolute zero, the distribution gives a step function. EF is called the Fermi energy or Fermi level and it corresponds to the top-filled level in the band in the case of metals. As we increase the temperature, the distribution is smeared out, indicating that some electrons have been excited to higher energies.

NOTE: The total width of the distribution (from an energy where all energy levels are filled to one where they are nearly empty) is about 4kT. At room temperature, this is about 0.1 eV.

kTEE

kTEE FF

eeEf )()(

1

1

1)( −− ≈+

=



The Fermi energy can be thought of as the thermodynamic chemical potential for electrons in the solid. For instance, when two solids are placed in electrical contact, electrons will flow from one solid to the other, causing a potential difference across the contact. This is necessary in order to make the chemical potentials to be the same.

Semiconductors are non-metallic solids that conduct electricity due, in effect, to thermal excitation of electrons across the band gap.

An n-type semiconductor has more electrons in the conduction band than holes in the valence band so that negative electrons are the predominant current carriers.

A p-type semiconductor has more positive holes in the valence band than electrons in the conduction band so that positive holes are the predominant current carriers.



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