Fundamental concepts of semiconductors
 
 
 
 
1.5 Holes and effective mass concept
 
As we discussed before, semiconductors are different from metals and insulators because they contain "almost-empty" conduction band and "almost-full" valence band. This implies the transport of carriers must be seen from both bands. To make such arguments, we introduce holes in "almost-full" valence band. These are not present in the semiconductor but simply, they are missing electrons spaces during the electron transport. Here you must know the fact that the electrons are the only charge carriers exists in semiconductors. Holes behave as particles with the same

Fig2.19: Electron- Hole concept

Fig.2.20. Variation of effective mass with band gap energy of different semiconductors.
properties as the electrons would have same occupying states, but they carry a positive charge. When certain energy is given to an electron in almost filled valence band, the departure of electron creates a hole, giving chance for another electron to occupy. It means that the energy of the hole is associated with the rise of electron energy. The electron and hole concept is schematically shown in the fig 2.19.

Now it is clear here that the electron and hole numbers are equal. But if we make either one of them more, what happens? That we will see in our later parts.




Let us discuss the effective mass concept in light of electrons and holes. We already know that the electrons in energy bands, behave differently to that of free-electrons. For general purpose, we can write the conduction and valance effective masses at their band minimum as for electrons and for hole.


The conduction band Energy in k space is , where Ec is the conduction band edge and similarly valance band energy.

Here one important thing : the conduction band effective mass is directly related to the band gap energy Eg. The smaller is the band gap, the smaller is the effective mass (Fig.2.20). Here you can observe some interesting points: for example see band gap is decreasing in the alloys CdS, CdSe and CdTe ( You know why?). Similarly AlAs, GaAs and InAs.

The band gaps of few semiconductors at the room temperature are given in the table. However, with the temperature the band gap substantially varies. This is shown graphically in fig 2.21. The empirical relation of band gap vs temperature is


, where α and β are constants.
 

Fig. 2.21. Temperature dependence of Band gap of some of the important semiconductors.


Let us understand more about conduction and valance band.

As we know that the energy bands are a mix of energy levels of individual atoms of the semiconductor. With the help of Schrödinger’s equation (through sophisticated mathematical algorithms) electron motion can be solved and one can obtain energy–momentum relation or simply, E-k band diagram. Though, throughout our semiconductor physics class, we are mostly interested in the energy levels at the top of the valence band and bottom of the conduction band, which lie in the vicinity of the band edge. However, every semiconductor has uniquely had its own band structure. As we observed earlier, semiconductors can be broadly divided into direct band gap and indirect band gap semiconductors, depending on the conduction band minimum position with respect to valence band maximum. As schematically represented in figure 2.22, we can see two kinds of conduction minima ( one at k=0 and another at k≠0). Similarly, at valance maximum we can see two closely connected bands (or curves) namely, Heavy Hole (HH) Light Hole (LH), and one at little far called Split-off (SO) .



Fig. 2.22. Valance and conduction band type in typical semiconductor


 
 
 
 
1.5.1 Conduction band
 
The minimum of conduction band occurs at k=0 in some semiconductors and they're called direct band gap materials, such as GaAs, InP, InGaAs etc,. In other semiconductor conduction minima does not occur at k=0 but some other point in the k-space. Some examples are Si, Ge, AlGas etc,. Broadly, semiconductor conduction bands are composed either of s or p orbitals that are present in the outermost shells. Let us draw some remarks for conduction band here:


  • In the direct band gap materials central band of electrons at the conduction band is purely s-type.
  • In the indirect band gap materials conduction band is either s and p mixture (longitudinal) or p type (transverse).
  • In silicon, the bottom of the conduction band does not occur at k=0 but at six equivalent minima long, kx, ky and kz directions ( look at those BZ points in Fig 2.9). For Si, there are six equivalent points (2Π/a)(0.85,0,0), (2Π/a)(0,0.85,0), (2Π/a)(0,0,0.85), and inverse of these three, where a is the lattice constant of Si (=5.430A)
  • Simple way of representing direct band conduction energy( in parabolic representation) is . For indirect band, generalized conduction energy is along a particular axis is and m*t stands for longitudinal and transverse effective masses.



Figure: Notice six equivalent points for the conduction band minima in Si along the crystallographic axes
 
 
 
 
 
1.5.2 Valance band
 
Valance band is a mixture of few degenerates states. If you look at the valance band in Fig2.22 (or 2.18 for real), we see three bands at the top of the band. These are Heavy-Hole (HH), Light Hole (LH), Split-Off (SO) branches of bands. Generalized HH, LH and SO energies are , and respectively. Without going into further details, we can classify them as degenerate angular momentum states, which arise due to strong interaction between heavy hole and light hole states.
These energy states could be generally represented as



Where A, B, C and D are dimensionless parameters. +ve and –ve signs represent heavy hole (HH) and light hole (LH) states. Corresponding to HH and LH states, effective masses of electrons in the conduction band also vary. Unlike conduction band, for both direct and indirect bandgap materials, the effective mass of valance band is defined as for both kinds of materials.


 
Table 1.5.1
Property GaAs Silicon Germanium
Conduction band      
Band Gap (eV) 1.42* (d), 1.73(I) 3.2(d), 1.12*(I) 0.8(d), 0.66*(I)
Effective mass (/m0 ) 0.067d,
1.98(l) and 0.37(t)		 0.2d,
0.98(l) and 019(t)		 0.04d,
1.64(l) and 0.08(t)
Valance band      
HH Effective mass(/m 0) 0.45 0.49 0.28
LH Effective mass(/m 0) 0.08 0.16 0.44
SO Effective mass(/m0 ) 0.15 0.29 0.08
  • indicative of lowest band gap (d =direct and I=indirect band gap)


  • Note that indirect band gap contains longitudinal as well as transverse effective masses
 
Overall, the effective mass of an electron (or a hole) entirely differs from band to band and also depends on the nature of the band (indirect or direct). Some of the important materials with their effective masses and their band gap energies are given in the table.


Effective mass- Density of states

 
Effective masses are further classified into two categories: effective mass for the density of states (DOS) calculations and effective mass for the conductivity calculations. Effective masses corresponding to DOS for conduction and valance bands are as shown in the above figure.

 
Table 1.5.2
Property GaAs Silicon Germanium
Lowest band gap (eV) 1.42 1.12 0.66
Electron effective mass (for DOS)(/m0 ) 0.06 1.08 0.55
Hole effective mass (for DOS)(/m 0) 0.45 0.81 0.37
Electron effective mass (for Conductivity)(/m 0) 0.067 0.26 0.12
Hole effective mass (for Conductivity)(/m0 ) 0.34 0.39 0.21
 
 

Fig 2.23 Band diagram(E vs k) (A) Ge (B) Si, where high-lying energy bands also shown.