1. Crystal structures

1.1 Primitive lattice Cell

An ideal crystal is constructed by the infinite repetition of identical structural units in space. In the simplest crystals the structural unit is a single atom, as in copper, silver, gold, iron, aluminium, and the alkali metals.

The structure of all crystals can be described in terms of a lattice, with a group of atoms attached to every lattice point. The group of atoms is called the basis; when repeated in space it forms the crystal structure. The basis consists of a primitive cell, containing one single lattice point. Arranging one cell at each lattice point will fill up the entire crystal.

1.2 Simple Crystal Structures

There are several types of crystal structures. The simplest one is the simple cubic lattice (sc). Two other cubic lattices are the body-centered (bcc) and the face-centered (fcc) cubic lattice.

Figure 1.2.1. Body-centered cubic (bcc) lattice	Figure 1.2.2. Face-centered cubic (fcc) lattice

1.3 Diamond and zinkblend lattice structures

The diamond lattice structure is very common in semiconductor materials, Si, Ge. GaAs and GaP has a zinkblende lattice structure which is similar to the diamond lattice structure. The difference between the face-centered lattice structure and the diamond lattice structure is four atoms (see pictures). In the GaAs these four atoms are Ga-atoms and the rest are As-atoms.

Figure 1.3.1. Diamond lattice structure

1.4 Reciprocal lattice

The electron density at each atom depends on the numder of electrons per atom, binding type and the lattice structure. The lattice structure is periodical in 3D and therefore is the electron density periodical in 3D. A periodic function can be expanded into a Fourier series. We are often familiar with the concept of Fourier series in one dimension (from signal theory), while Fourier series in 3D feels more abstract. However, the extention of the Fourier analysis to periodical functions in 3D is straightforward and it turns out that we can write the Fourier series in the same way as:

To proceed futher with the Fourier analysis of the electron concentration we must find the vectors G of the Fourier sum. G can be constructed from the axis vectors b1, b2, b3 of the reciprocal lattice:

The vectors a1, a2, a3 are primitive translation vectors or primitive basis for the real space lattice, while b1, b2, b3G is called a reciprocal lattice vector. All reciprocal lattice vectors can be expressed as a linear combination of b1, b2, b3 using integer coefficients. Exemples of reciprocal lattice vectors:


are primitive translation vectors or primitive basis for the reciprocal lattice.

G=b1+b2+b3

G=b1-2*b2+2*b3

G=-3*b1-10*b2-2*b3

Figure 1.4.1. Real space lattice	Figure 1.4.2. Reciprocal lattice

Q1.4.1 Which type of lattice structures are the real space lattice in figure 1.4.1 and the corresponding reciprocal lattice in figure 1.4.2 (the red atoms are the corners of the primitive cell)?

The reciprocal lattice is a lattice in the Fourier space associated with the crystal. Wavevectors are allways drawn in Fourier space, so that every position in Fourier space may have a meaning as a description of a wave, but there is a special significance to the points defined by the set of G's associated with a crystal structure. The set of reciprocal lattice vectors G determines the possible x-ray reflections (Bragg reflections). This makes it possible to study the reciprocal lattice structure using x-ray diffraction.

1.5 Brillouin zones

A Brillouin zone is defined as a Wigner-Seitz primitive cell in the reciprocal lattice. A wigner-Seitz primitive cell can be constructed following this procedure:

draw lines to connect a given lattice point to all nearby lattice points.

at the midpoint and normal to these lines, draw new lines or planes.

the smallest volyme enclosed in this way is the Wigner-Seits primitive cell.

In the following illustrations several Brillouin zones has been constructed including the first zone (marked as green) for two different reciprocal lattices.

Figure 1.5.1. The construction of Brillouin zones in a square reciprocal lattice in two dimensions	Figure 1.5.2. The construction of Brillouin zones in an oblique reciprocal lattice in two dimensions

This procedure gives the first Brillouin zone which plays an important roll in the field of solid state electronics. The allowed energy levels within the first Brillouin zone is directly related to the electrical properties of the material.

Q1.5.1 Find the corners of the polygon holding the first Brillouin zone for a 2D lattice with the primitive vectors of the crystal as a1=a*[10x+y]/sqrt(26), a2=a*[x+5y]/sqrt(26).

1.6 Origin of the Energy Gap in solids

In this section we will look upon an electron as a propagating wave, as first suggested by de Broglie. A wave propagating in a crystal can be disturbed by Bragg reflection. The propability to find an electron at a certain location can be calculated using the Schrödinger equation. At Bragg reflections wavelike solutions to the Schrödinger equation do not exist.

The Bragg condition is:

In one dimension the condition becomes:

where n=1,2,3..... and a is the lattice constant.

The first reflection and the first energy gap occurs at n=1.

Figure 1.6.1 Left: Plot of energy versus wavevector k for a free electron. Right: Plot of energy versus wavevector k for an electron in a monatomic linear lattice of lattice constant a.

The energy gap Eg is assosiated with the first Bragg reflection at the first Brillouin zone boundary (n=1). The Bragg reflections at the zone boundaries will make standing waves in the crystal. A wave that travels neither to the left nor to the right is a standing wave: it does not go anywhere. There are two different standing waves that can occure: one representing the difference of a right and a left-directed wave and one representing the sum of a right and a left-directed wave. In figure 1.6.3 the first is called standing wave 1 and the second standing wave 2.

Figure 1.6.2 One dimensional periodic potential

Figure 1.6.3 Distribution of probability density in the periodic potential for standing wave 1 and 2. The standing wave 1 piles up charges in the region between the ion cores while standing wave 2 piles up charges around the core points.

The standing wave 2 piles up electrons around the postitive ion cores, which means that the average potential energy will be lower than for a free traveling wave (constant probability density). The potential energy corresponding to standing wave 1 will have higher potential energy than a free traveling wave, since it piles up electrons between the ion cores (not compensated by positive ions). The energy difference between the standing waves is the origin to the energy gap Eg.

The relation between the electron energy and the electron wave vector is called the band structure. The band structure is directly related to the crystall structure of the material. In the following section we will calculate the band structure for a square-well periodic potential. Band structure calculation of a real lattice is much more complicated and this exemple should be looked upon as a simple demonstration.

Figure 1.6.4 One dimesional periodic potential model.

We will use the limit where U0 approaching positive infinity and b approaches zero (a series of delta functions). Solving the Schrödinger equation in for this potential structure gives the following band structure.

Figure 1.6.5 Normalized energy versus normalized wave vector for the potential structure shown in figure 1.6.4. Note that the energy gap is always at the zone boundaries.

The band structure in figure 1.6.5 is plotted in several Brillouin zones. Usually a band structure is only plotted in the first Brillouin zone. We can transform figure 1.6.5 into a first brillouin zone plot by adding reciprocal lattice vectors until we get the proper format as in figure 1.6.6.

Figure 1.6.6 Normalized energy versus normalized wave vector in the first Brillouin zone for the potential structure shown in figure 1.6.4.

Real crystals are three dimensional, which means that there are energy values associated with each wave vector k=(kx,ky,kz). The band structure is in general divided into several bands, band 1, band 2, band 3 and so on. Figure 1.6.7a and figure 1.6.7b shows two of the Silicon band.

Figure 1.6.7a The first conduction band in Silicon.

Figure 1.6.7b The second conduction band in Silicon.

The band structure will deside how many wave vector states that are available for each energy level. If we consider all the available wave vectors for a specific energy level, we will be able to construct a 3D surface for each energy level. If we integrate all the available wave vectors in the 3D surface and multiply it with two (including spin up and spin down), we will get the total available wave vector states associated with that perticular energy level (density of states). In figure 1.6.8 to 1.6.10, constant energy surfaces has been plotted for the first conduction band and the first and second valence band in silicon.

Figure 1.6.8 Constant energy surface for the first conduction band in Silicon. Notice that the energy minima is not located at the first Brillouin zone. The energy value is 0.25eV above the conduction band minima. (The MPEG-movie starts at 0.0eV and stops at 0.25eV above the minima.)

Figure 1.6.9 Constant energy surface for the first valence band in Silicon. The wraped sphere is located at the center of the first Brillouin zone. The energy value is 0.5eV below the valence band maxima. (The MPEG-movie starts at 0.01eV and stops at 0.5eV below the maxima.)

Figure 1.6.10 Constant energy surface for the second valence band in Silicon. The wraped sphere is located at the center of the first Brillouin zone. The energy value is 0.5eV below the valence band maxima. (The MPEG-movie starts at 0.01eV and stops at 0.5eV below the maxima.)

Newtons equations for an electron in a semiconductor crystal can be written as:

Notice that the electric field will change the electron position in the reciprocal space and the corresponding movement in real space depends on the gradient of the band structure.

Q1.6.1 In figure 1.6.8 to 1.6.10 constant energy surfaces has ben plotted in the first Brillouin zone. How can we see that the band gap between valence and conduction band is indirect?

Q1.6.2 The effective mass approximation assumes that we can consider the electron or hole as a free particle. What is the difference between figure 1.6.8 to 1.6.10 and the corresponding energy bands using an effective mass appoach?

Hint: The band structure for an effective mass model is:

Q1.6.3 An electron is located at r0=[0,0,0], k0=[0,0,0] and a constant electric field of 10KV/cm is applied in the [1,1,0] direction. Use an effective mass band structure to find the final position of the electron in k-space and in r-space after 0.1 ps.

Q1.6.3 Which figure (1.6.9 or 1.6.10) correspondes to so called heavy holes?