2.2.3 Energy bands

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Energy bands

2.2.3 Energy bands

Energy bands occur in solids where the discreet energy levels of the individual atoms merge into bands which contain a large number of closely spaced energy levels.

In this section, we first discuss the crystal structure of common semiconductors to illustrate the fact that most semiconductors have an ordered structure in which atoms are placed in a periodic lattice. We then consider the Kronig-Penney model. This one dimensional model illustrates how a periodic potential yields a set of energy bands and energy band gaps. It is the detailed band structure of a given material which can be directly linked to its conducting, insulating or semiconducting behavior.

2.2.3.1 Crystals and crystal structures

Semiconductors consist of atoms which are placed in an ordered form which is called a crystal. Crystals are identified based on their lattice structure. For instance the crystal structure of silicon is like that of diamond and refered to as the diamond lattice, shown in the figure below. Each atom in the diamond lattice has a covalent bond with four adjacent atoms which together form a tetrahedron. This lattice can also be formed from two face-centered-cubic lattices which are displaced along the body diagonal of the larger cube in the figure by one quarter of that body diagonal.

diamond.gif

Fig.2.2.4 The diamond lattice of silicon and germanium

Compound semiconductors such as GaAs and InP have a lattice structure which is similar to that of diamond. However the lattice contains two different types of atoms. Each atom still has four covalent bonds, but they are bonds with atoms of the other type. This structure is refered to as the zinc-blende lattice as shown below.

zincblen.gif

Fig.2.2.5 The zinc-blende lattice of GaAs and InP

2.2.3.2 The Kronig-Penney model

The Kronig-Penney model demonstrates that a simple one-dimensional periodic potential yields energy bands as well as energy band gaps. The periodic potential is shown in the figure below:

kropot.gif

Fig.2.2.6

The periodic potential assumed in the Kronig-Penney model. The potential barriers with width b are spaced by a distance a+b

The analysis requires the use of Bloch functions, traveling wave solutions multiplied with a periodic function which has the same periodicity as the potential.

Solutions are obtained only if the following equation is satisfied:

This transcendental equation can be solved graphically as illustrated with the figure below. Solutions are only obtained if F is between -1 and 1 since it has to equal cos(ka).

kronigp.xls - kronigf.gif

Fig.2.2.7

Graphical solution to the Kronig-Penney model. Shown is the function F which has to equal cos(ka)

The corresponding band structure is shown below (black curve) as well as the energy for a free electron (green curve).

kronigp.xls - kronigp.gif

Fig.2.2.8

Energy versus k corresponding to the previous figure. Shown are the reduced band diagram of electrons in the periodic potential with rectangular barriers (black curve) and that of a free electron (green curve)

The corresponding wavefunctions are shown below.

kronigpw.xls - kronigpw.gif

Fig.2.2.9

Wavefunction of an electron in a periodic potential consisting of delta functions (yellow curve). Shown are the real part (red curve) and the imaginary part (green curve) of the wavefunction as well as the square of the absolute value (blue curve).

Note that the wavefunction resembles that of a traveling wave which is distorted at the delta functions.

2.2.3.3 Metals, insulators and semiconductors

Once we know the bandstructure of a given material we still need to find out which energy levels are actually occupied and whether specific bands are empty, partially filled or completely filled.

Empty bands do not contain electrons and therefore are not expected to contribute to the electrical conductivity of the material. Partially filled bands do contain electrons as well as unoccupied energy levels which have a slightly higher energy. These unoccupied energy levels enable carriers to gain energy when moving in an applied electric field. Electrons in a partially filled band therefore do contribute to the electrical conductivity of the material.

Completely filled bands do contain plenty of electrons but do not contribute to the conductivity of the material. This is due to the fact that the electrons can not gain energy since all energy levels are already filled.

In order to find the filled and empty bands we must find out how many electrons can be fit in one band and how many electrons are available: Since one band is due to one ore more atomic energy levels we can conclude that the minimum number of states in a band equals twice the number of atoms in the material. The reason for the factor of two is that even a single energy level can contain two electrons with opposite spin.

To further simplify the analysis we assume that only the valence electrons (the electrons in the outer shell) are of interest, while the core electrons are assumed to be tightly bound to the atom and are not allowed to wander around in the material.

Four different possible scenarios are shown in the figure below:

band2.gif

Fig.2.2.10

Possible energy band diagrams containing one filled or partially filled band and one empty or partially empty band. Shown are a) a half filled band, b) two overlapping bands, c) an almost full band separated by a small bandgap from an almost empty band and d) a full band separated by a large bandgap from an empty band.

A half-filled band is shown in a). This situation occurs in materials consisting of atoms which contain only one valence electron each. A lot of highly conducting metals satisfy this condition. Materials consisting of atoms which contain two valence electrons can still be highly conducting provided that the resulting filled band overlaps with an empty band. This scenario is shown in b). No conduction is expected for scenario d) where a completely filled band is separated from the next higher empty band by a larger energy gap. Such materials behave as insulators. Finally scenario c) depicts the situation in a semiconductor. The completely filled band is now close to the next higher empty band so that electrons can make it into the next higher band yielding an almost full band below an almost empty band. We will call the almost full band the valence band since it is occupied by valence electrons and the almost empty band will be called the conduction band as electrons are free to move in this band and contribute to the conduction of the material.

2.2.3.4 Energy bands of semiconductors

As semiconductors are of primary interest in this text, we now introduce a simplified energy band diagram for semiconductors and define some key parameters. The diagram is shown in the figure below:

band.gif

Fig.2.2.11

A simplified energy band diagram used to describe semiconductors. Shown are the valence and conduction band as indicated by the valence band edge E_v and the conduction band edge E_c. The vacuum level, E_VACUUM and the electron affinity, c are also indicated on the figure.

The diagram identifies the almost-empty conduction band simply by a line which indicates the bottom of the conduction band and is labeled E_c. Similarly the top of the valence band is indicated with a line labeled E_v.The energy band gap is between the two lines which are separated by the bandgap energy E_g. The distance between the conduction band edge E_c and the energy of a free electron (called the vacuum level labeled E_vacuum) is quantified by the electron affinity, c multiplied with the electronic charge q.

Note: The actual bandstructures of semiconductors is more complex than the reader is lead to believe by the discussion above. More details regarding the bandstructure of germanium, silicon and gallium arsenide together with the effective masses for the different bands can be found in section 2.2.3.5.

2.2.3.5 Effective mass of electrons and holes in semiconductors

See 2.2.3.5.

2.2.2 Ź Ž 2.2.4

Š Bart J. Van Zeghbroeck, 1996, 1997