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Semiconductor Physics and Devices: Basic Principles, 4
th
 edition Chapter 3 
By D. A. Neamen Problem Solutions 
______________________________________________________________________________________ 
 
Chapter 3 
 
3.1 
 If oa were to increase, the bandgap energy 
 would decrease and the material would begin 
 to behave less like a semiconductor and more 
 like a metal. If oa were to decrease, the 
 bandgap energy would increase and the 
 material would begin to behave more like an 
 insulator. 
_______________________________________ 
 
3.2 
 Schrodinger's wave equation is: 
 
 
   txxV
x
tx
m
,
,
2 2
22



 
 
 
t
tx
j



,
 
 Assume the solution is of the form: 
    




















 t
E
kxjxutx

exp, 
 Region I:   0xV . Substituting the 
 assumed solution into the wave equation, we 
 obtain: 
  


























t
E
kxjxjku
xm 

exp
2
2
 
 
 



























 t
E
kxj
x
xu

exp 
  

























 
 t
E
kxjxu
jE
j

 exp 
 which becomes 
    
























t
E
kxjxujk
m 

exp
2
2
2
 
 
 























 t
E
kxj
x
xu
jk

exp2 
 
 



























 t
E
kxj
x
xu

exp
2
2
 
  




















 t
E
kxjxEu

exp 
 This equation may be written as 
  
   
  0
2
2
22
2
2 





 xu
mE
x
xu
x
xu
jkxuk

 
 
 
 Setting    xuxu 1 for region I, the equation 
 becomes: 
 
        02 1
221
2
1
2
 xuk
dx
xdu
jk
dx
xud
 
 where 
 
2
2 2

mE
 Q.E.D. 
 In Region II,   OVxV  . Assume the same 
 form of the solution: 
    




















 t
E
kxjxutx

exp, 
 Substituting into Schrodinger's wave 
 equation, we find: 
    
























t
E
kxjxujk
m 

exp
2
2
2
 
 
 























 t
E
kxj
x
xu
jk

exp2 
 
 



























 t
E
kxj
x
xu

exp
2
2
 
  




















 t
E
kxjxuVO

exp 
  




















 t
E
kxjxEu

exp 
 This equation can be written as: 
  
   
2
2
2 2
x
xu
x
xu
jkxuk





 
     0
22
22
 xu
mE
xu
mVO

 
 Setting    xuxu 2 for region II, this 
 equation becomes 
 
   
dx
xdu
jk
dx
xud 2
2
2
2
2 
   0
2
22
22 





 xu
mV
k O

 
 where again 
 
2
2 2

mE
 Q.E.D. 
_______________________________________