Transcript Semiconductors

SEMICONDUCTORS
 Semiconductors
 Semiconductor devices
Electronic Properties
Robert M Rose, Lawrence A Shepart, John Wulff
Wiley Eastern Limited, New Delhi (1987)
Energy gap in solids
 In the free electron theory a constant potential was assumed inside the solid
 In reality the presence of the positive ion cores gives rise to a varying
potential field
 The travelling electron wave interacts with this periodic potential
(for a crystalline solid)
 The electron wave can be Bragg diffracted
Bragg diffraction from a 1D solid
n = 2d Sin
1D  =90o
n = 2d
n
k
d Sin 
2d 

λCritical  2d, d, , ...
3


2
3
 

k Critical   , 
,
, ...
d
d
 d

 The Velocity of electrons for the above values of k are zero
 These values of k and the corresponding E are forbidden in the solid
 The waveform of the electron wave is two standing waves
 The standing waves have a periodic variation in amplitude and hence the
electron probability density in the crystal
 The potential energy of the electron becomes a function of its position
(cannot be assumed to be constant (and zero) as was done in the
free electron model)
E →
2
k →
2
h k
E
8 2 m
Band gap
2

d


d


d
2

d
 The magnitude of the Energy gap between two bands is the difference
in the potential energy of two electron locations
K.E of the electron increasing
Decreasing velocity of the electron
ve effective mass (m*) of the electron
E →
Within a band
k →


d
Effective energy gap → Forbidden gap → Band gap
n
k
d Sin 
k

d Sin 90
k
o

d Sin 45o
[110]
[100]
E →
E →
Effective gap

 k →
d
2

d
k →
 The effective gap for all directions of motion is called the forbidden gap
 There is no forbidden gap if the maximum of a band for one direction of
motion is higher than the minimum for the higher band for another
direction of motion  this happens if the potential energy of the electron
is not a strong function of the position in the crystal
Energy band diagram: METALS
Divalent metals
Monovalent metals
 Monovalent metals: Ag, Cu, Au → 1 e in the outermost orbital
 outermost energy band is only half filled
 Divalent metals: Mg, Be → overlapping conduction and valence bands
 they conduct even if the valence band is full
 Trivalent metals: Al → similar to monovalent metals!!!
 outermost energy band is only half filled !!!
Energy band diagram: SEMICONDUCTORS
2-3 eV
 Elements of the 4th column (C, Si, Ge, Sn, Pb) → valence band full but no
overlap of valence and conduction bands
 Diamond → PE as strong function of the position in the crystal
 Band gap is 5.4 eV
 Down the 4th column the outermost orbital is farther away from the nucleus
and less bound  the electron is less strong a function of the position
in the crystal  reducing band gap down the column
Energy band diagram: INSULATORS
> 3 eV
Intrinsic semiconductors
 At zero K very high field strengths (~ 1010 V/m) are required to move an
electron from the top of the valence band to the bottom of the
conduction band
  Thermal excitation is an easier route
E →
P(E) →
EF
Eg
Eg/2
0
0.5
1
T>0K
1
P( E ) 
 E  EF 
1  exp 

 kT 
 Eg
E  E F Silicon  
 2


 0.55 eV
 Silicon
E  E F  
kT  0.026 eV
Eg
2
E  EF
 1
kT
 Unity in denominator can be ignored
 Eg 
P( E )  exp 

2
kT


 Eg 
ne  N exp 

2
kT


 ne → Number of electrons promoted
across the gap
(= no. of holes in the valence band)
 N → Number of electrons available
at the top of the valance band
for excitation
Conduction in an intrinsic semiconductor
 Under applied field the electrons (thermally excited into the conduction
band) can move using the vacant sites in the conduction band
 Holes move in the opposite direction in the valence band
 The conductivity of a semiconductor depends on the concentration of
these charge carriers (ne & nh)
 Similar to drift velocity of electrons under an applied field in metals in
semiconductors the concept of mobility is used to calculate conductivity
drift velocity
Mobility 
field gradient

 m/s 
2

m
/V / s
V / m 

  ne e e  nh e h
Mobility of electrons and holes in Si & Ge (at room temperature)
Species
Mobility (m2 / V / s)
Si
Ge
Electrons
0.14
0.39
Holes
0.05
0.19
Conductivity as a function of temperature
  ne e e  nh e h
 Eg 
  N e (  e   h ) exp 

 2kT 
 Eg 
  C exp 

2
kT


2kT
Ln()→
ln   C1 
Eg

Eg
2k
1/T (/K) →
Extrinsic semiconductors
 The addition of doping elements significantly increases the conductivity
of a semiconductor
 Doping of Si
 V column element (P, As, Sb) → the extra unbonded electron
is practically free (with a radius of motion of ~ 80 Å)
 Energy level near the conduction band
 n- type semiconductor
 III column element (Al, Ga, In) → the extra electron for bonding
supplied by a neighbouring Si atom → leaves a hole in Si.
 Energy level near the valence band
 p- type semiconductor
 Ionization Energy→
n-type
EF
EIonization
Eg
Donor level
Energy required to promote an
electron from the Donor level to
conduction band
 EIonization < Eg
 even at RT large fraction of
the donor electrons are exited
into the conduction band
 Electrons in the conduction band are the majority charge carriers
 The fraction of the donor level electrons excited into the conduction band
is much larger than the number of electrons excited from the valence band
 Law of mass action: (ne)conduction band x (nh)valence band = Constant
 The number of holes is very small in an n-type semiconductor
  Number of electrons ≠ Number of holes
p-type
Acceptor level
EF
Eg
EIonization
 At zero K the holes are bound to the dopant atom
 As T↑ the holes gain thermal energy and break away from the dopant atom
 available for conduction
 The level of the bound holes are called the acceptor level (which can accept
and electron) and acceptor level is close to the valance band
 Holes are the majority charge carriers
 Intrinsically excited electrons are small in number
  Number of electrons ≠ Number of holes
Ionization energies for dopants in Si & Ge (eV)
Type
n-type
Element
In Si
In Ge
P
0.044
0.012
As
0.049
0.013
Sb
0.039
0.010
B
0.045
0.010
Al
0.057
0.010
Ga
0.065
0.011
In
0.16
0.011
p-type
10 4
Eg
Intrinsic
slope
All dopant atoms have been excited
2k
10 3
 (/ Ohm / K)→

Exhaustion
10 2
Exponential
function
101
10
0
50 K
+ve slope due to
Temperature dependent
mobility term
0.02
0.04 0.06 0.08 0.1
1/T (/K) →
Slope can be used
for the calculation
of EIonization
10 K
 Semiconductor device  chose the flat region where the conductivity does
not change much with temperature
 Thermistor (for measuring temperature)  maximum sensitivity is
required