THE k-SPACE, BLOCH’S THEOREM AND BRILLOUIN ZONES

In nature there are two types of solids: the crystalline and the amorphous. We will be dealing with only the crystalline phase because the vast majority of electronic devices are made from crystalline materials. There are more applications for crystalline materials than

Three examples of Bravais lattices

FIGURE 2.3 Three examples of Bravais lattices: the cubic (a), the face-centred (b), and the hexagonal (c).

for amorphous materials. Crystalline materials are those in which the atoms inside them are arranged periodically. Three particular examples, the simple cubic, the face-centred cubic, and the hexagonal, are shown in figure 2.3a-c. The vast majority of semiconductors belong to the face-centred cubic type, usually abbreviated as FCC. Now we must differentiate between what is called a Bravais lattice and the real lattice of atoms. A Bravais lattice is one whose points JRf are given by the relation

where mj, mf, mf are integers and the av a2, a3 are unit vectors in three dimensions, not necessarily orthogonal to each other. Figure 2.4 illustrates this non-orthogonality in two dimensions for the hexagonal lattice. There are 14 types of such lattices. Obviously equation 2.1 defines a periodic arrangement of the 2?,. So in what way is a real lattice of atoms different from a Bravais lattice? A real lattice coincides with a Bravais lattice if the mathematical points defined by 2.1 are each occupied by one atom.

The planar hexagonal lattice with an non-orthogonal basis (ai,«2 )•

FIGURE 2.4 The planar hexagonal lattice with an non-orthogonal basis (ai,«2 )•

The zinc-blende lattice is a FCC Bravais lattice with each Bravais lattice point occupied by a diatomic unit as shown selectively in

FIGURE 2.5 The zinc-blende lattice is a FCC Bravais lattice with each Bravais lattice point occupied by a diatomic unit as shown selectively in (a). The resulting structure has each atom surrounded by 4 others of the opposite type, in a tetrahedral arrangement, as shown again selectively in (b).

However, each mathematical point defined by 2.1 may not be occupied by one atom but by a many-atom molecular unit. Then we have the real lattice of a solid substance. In GaAs, for example, each Bravais point is occupied by one Ga and one As atom. Hence the real solids are composed by as many interpenetrating Bravais lattices as the number of atoms in their molecular unit. In figure 2.5a, we show the FCC lattice of the As atoms (black) and selectively the Ga atoms that each accompany an As atom. The end result is the so-called zinc blende or diamond lattice that is shown in figure 2.5b. The volume assigned to this molecular unit, which if repeated according to 2.1 fills the entire space, is called the unit cell. Often in the literature the unit cell may contain more than one unit. This is done for a better pictorial representation.

The Schroedinger equation for the crystalline solid reads

where the crystalline potential energy Vcr(r) includes all the electrostatic interactions of a given electron with all the remaining ones and with all the positive nuclei. Now, given that the Bravais lattice underlying any real lattice of atoms has the same molecular unit on each of its points and given that is periodic, we deduce that the crystalline potential energy Vcr(r) must also be periodic with the same period, that is

for all the R{ given by 2.1. The unit vectors apj=l,3 in 2.1 are of the order of Angstroms so that Vcr(r) is macroscopically constant. It may vary greatly within a unit cell but it is the same from unit cell to unit cell, so we expect the wavefunctions not to differ much from the plane waves of vacuum where the potential is constant. (Whether the potential is constant or zero in space makes little difference since any constant potential can be made zero by a proper choice of energy).

In fact, Bloch’s theorem says that the wavefunctions are of the form

where к is a wavevector in 3-dimensions and Uk(r) is the same in each unit cell, i.e. the ВД are modulated plane waves. Figure 2.6 illustrates how Ч(г) and its components vary in space. It should now be obvious that the wavefunctions in a solid are not simple atomic orbitals but wavefunctions that extend from one end of the solid to the other, something we hinted at in the first section of this chapter. However, the wavefunctions in a solid can be constructed from the atomic orbitals of their constituent atoms and then the connection between the simplified picture of section 2.1 and к space will become evident in section 2.3. The drawing of V(x) in Figure 2.1 gives an adequate 1-dimensional picture of the behaviour of the crystalline potential.

We will initially give a simple 1-dimensional proof of Bloch’s theorem. However, this simple proof contains all the physical insight that lies behind the 3-dimensional proof that requires group theory to be complete. Let us imagine a linear chain of atoms like that of figure 2.1 consisting of N atoms with a distance a between the atoms and of total length L. Periodic boundary conditions on the chain are assumed, i.e. the wavefunction is the same

The total (b), cellular (c), and plane wave (d) parts of the wavefunction ЧЛ Figure reproduced from the book “Solid State Theory” by W. Harrison, Dover publications, 1980

FIGURE 2.6 The total (b), cellular (c), and plane wave (d) parts of the wavefunction ЧЛ Figure reproduced from the book “Solid State Theory” by W. Harrison, Dover publications, 1980.

at the end atoms. This can be achieved by assuming that the chain is bent on itself and becomes a ring.

A more formal statement of the periodic boundary condition is

where (assuming L is very large) L=Na. Now since the potential energy is periodic with period a, the charge density must also be periodic, that is

But given that the relation between the charge density of a single electron p'(x) and its wavefunction is p'(x) = e|vF(x)| , the above equation for the charge density of the whole crystal can only be true if for every wavefunction we have

When the magnitude of two complex numbers is the same, the complex numbers ^(x), У(х+д) can only differ by a phase factor. We then get

We can repeat the argument and we have

Repeating the argument N times we have By comparing now equations 2.4 and 2.7 we deduce Hence X must be the N roots of 1, that is

so that 0 in 2.6 is equal to

The term 2nv/L has the dimension of a wavevector (1/distance), so that we can finally write

where k = 2nv/L,v = 0,1,2,...,N-l and more generally

We have to generalize equation 2.10 to 2- and 3-dimensions. Before giving the proof in

  • 2- and 3-dimensions we can do a bit of guesswork as follows: na is the R„ position vector in this linear 1-dimensional chain, so that the phase factor in front of Ч'(х) can be written e'kR". To guess correctly the desired result in 3-dimensions, we expect that we only need to
  • 1. treat R as a vector R,
  • 2. turn the scalar x into the position vector r,
  • 3. turn the simple scalar wavevector к into the vector k, which also labels the wavefunction ЧУг) and finally
  • 4. turn the simple product into a dot product Then we expect to obtain

Equation 2.11 is indeed the correct relation and is called the Bloch condition. It is very easy to verify that equations 2.3 and 2.11 are equivalent. Simply substitute r+R„ for r in 2.3 and you will immediately get 2.11 given that Uk(r) is periodic. Although we correctly guessed the 3-dimensional generalization of 2.10, this guesswork has not helped us to specify the

3- dimensional wavevector k. We will now follow a more formal path in 2 dimensions which can be easily generalized to 3 dimensions.

Let us first rewrite the 1-dimensional к as

Now imagine a 2-dimensional rectangular lattice in the x and у directions with spacings av a2 between the atoms in the respective x, у directions as in figure 2.7. Let us also assume periodic boundary conditions along both the x and у directions as we did for the linear

A square lattice with an orthogonal basis (

FIGURE 2.7 A square lattice with an orthogonal basis (2 )•

chain. The series of arguments leading from 2.5 to 2.10 can be repeated here along each of the x and у directions separately leading to

or in vector form

where к is now a 2-dimensional vector given by

with Nv N2 the number of unit cells in the x and у directions and V[ = 0,1,2 -1 and

v2 = 0,1,2,...,ЛГ2 — 1. The above equation maybe written

where bx is a unit vector along the x direction and of magnitude — and b, is a unit vector

1

along they direction and of magnitude —. Equation 2.14 is identical to equation 2.11, so

й2

one would have thought that we have proven Bloch’s theorem in at least 2 dimensions. However, we have not, the reason being that in our proof we have used a rectangular lattice, which constitutes a simplification in that the unit vectors of such a lattice are at right angles to each other. However, as can be seen from figure 2.4, all lattices do not possess an orthogonal set of unit vectors.

The general proof is, as we stated earlier, a bit more complicated and we will not give it here. However, the simple proof has all the physical insight that we need so that the general case can be made transparent. The general case can be stated as follows. Let av a2, a3 be the unit vectors of the 3-dimensional Bravais lattice of the solid in question. Define the so called reciprocal lattice unit vectors by

Then the к vectors consistent with or obeying Bloch’s theorem (equation 2.11) are

This is the generalization of equation 2.15 for any of the 14 Bravais lattices where N,, /=1,3 (meaning 1 to 3) are the number of unit cells of the Bravais lattice in the ith direction and v, = 0,1,2,.. .,1V,-1. Note that for a rectangular (or a cubic) lattice, where the b, are orthogonal to each other, equation 2.15 forms the 2D part of 2.17.

Equations 2.16 and 2.17 have a simple geometrical interpretation. Just as the three a„ /=1,3 define a lattice of isolated points in real r space in 3 dimensions, so do the three b„ /=1,3 in к space. This lattice is called the reciprocal lattice The relation between the a, and bj is

The reciprocal lattice points K, are obviously defined by the equation

where the nf,j=1,3 are integers. In fact, equations 2.16 constitute the solution of 2.18 for the b,. Then the к vectors which are given by equation 2.17 and which label the are all contained in the unit cell of this reciprocal lattice. The inverse lattice of a cubic lattice of size a is also a cubic lattice of size l/a, the inverse of an FCC lattice of unit length a is a BCC lattice of unit length 1/я1/а and likewise that of a BCC lattice is an FCC, hence the name reciprocal or inverse.

So given the direct lattice, the inverse lattice (which amounts to computing the b, in 2.16) is a lattice as equally simple as the direct. At any rate, the inverse lattices for all the 14 Bravais lattices have been computed and are known but, as has been stated before, most semiconductors relevant to electronics like Si, Ge, and GaAs crystallize in the FCC lattice and

The Brillouin zone of the FCC lattice

FIGURE 2.8 The Brillouin zone of the FCC lattice.

have a BCC inverse lattice. Figure 2.8 portrays the unit cell of the reciprocal lattice of the FCC lattice. The latter is called Brillouin zone (or first Brillouin zone) and this name stands for the unit cell of all reciprocal lattices, not just for the FCC. The number of к vectors in the first Brillouin zone is enormous: it is exactly IV) xN2 xN3 which is of the order of Avocadro’s number=1023/mole. We can therefore treat к as a continuous variable.

A brief summary is worthwhile before closing this section.

  • 1. A crystalline solid is a solid with a periodic arrangement of the atoms
  • 2. The crystalline lattice of a crystalline solid has an underlying lattice called a Bravais lattice. The Bravais lattice is defined as the set of points on each of which a molecular unit of the solid “sits” and, if it is repeated according to the translations of equation 2.1, it reproduces the whole crystal
  • 3. The periodicity of the crystalline potential imposes a condition on the allowed wave- functions, the Bloch condition, i.e. equation 2.11 which we repeat here

Consequently, the T^r) are waves that fill the entire crystal

4. The к vectors labeling the wavefunctions are defined as follows: given the Bravais lattice of the solid, the reciprocal lattice is obtained using the unit vectors defined by 2.16. Then the allowed к are all the vectors (defined by 2.17) that lie in the unit cell of the reciprocal lattice, which is called the “Brillouin zone”

 
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