SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

2.5. METALS AND INSULATORS 45

+

ke

kh

k

Eleremovedctron

aWssoaveveciated with ctor

the missing electron

Missing electron hole

Filled Σk= 0band

• –
  
  
  • –
    
    
    • –
      
      
      • ––––
        
        
        • –
          
          
          • –
            
            
            • –
              Valence band
            
            
            
            
          
          
          
          
        
        
        
        
      
      
      
      
    
    
    
    
  
  
  
  

Electron removed

Total momentum becomes –ke

hole kvector is –ke

• –
  
  
  • –
    
    
    • ––––––
      
      
      • –
        
        
        • 
          

          –

          
          
        
        
        
        
      
      
      
      
    
    
    
    
  
  
  
  

Figure 2.10: Illustration of the wavevector of a filled valence band with a missing electronke.
The wavevector is−ke, which is associated with the hole.

Semiconductors have zero conductivity at 0 K and quite low conductivity at finite temper-
atures, but it is possible to alter their conductivity by orders of magnitude through doping or
applied electric potentials. This makes semiconductors useful for active devices.

2.5.1 ElectronsandHoles

In semiconductors the valence band is full of electrons and the conduction band is empty
at 0 K. At finite temperatures some of the electrons leave the valence band and occupy the
conduction band. Electrons in the conduction band can carry current. When electrons leave the
valence band there are unoccupied states. Consider the situation as shown in figure 2.10, where
an electron with momentumkeis missing from the valence band. When all of the valence band
states are occupied, the sum of the total momentum is zero; i.e.
∑
ki=0=

∑

ki=ke

ki+ke (2.5.1)

This result is just an indication that there are as many positivekstates occupied as there are
negative ones. Now, in the situation where the electron at wavevectorkeis missing, the total
wavevector is ∑

ki=ke

ki=−ke (2.5.2)

The missing state is called a hole and the wavevector of the system−keis attributed to it. It is
important to note that the electron is missing from the statekeand the momentum associated
with the hole is at−ke. The position of the hole is depicted as that of the missing electron. But
in reality the hole wavevectorkhis−ke, as shown in figure 2.10 and we have

kh=−ke (2.5.3)

If an electric field is applied, all the electrons move in the direction opposite to the electric field.
This results in the unoccupied state moving in the field direction.Theholethusrespondsasif