1. SEMICONDUCTORS Semiconductors Semiconductor devices
Electronic Properties
Robert M Rose, Lawrence A Shepart, John Wulff
Wiley Eastern Limited, New Delhi (1987)

2. 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. In a simple model the potential as in Fig.1 can be assumed (‘a’ is the lattice spacing and ‘w’ is the width of the potential). If ‘w’  0, we get ‘’ functions.
The travelling electron wave interacts with this periodic potential (for a crystalline solid).
The electron wave can be Bragg diffracted.
1D   = 90o
Bragg diffraction from a 1D solid
n = 2d Sin
n = 2a
n2/k = 2a

4. k →
E →
Band gap

5. 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).

6. The magnitude of the Energy gap between two bands is the difference in the potential energy of two electron locations.
The effective mass of an electron (m*) in a solid could be different from the rest mass (m0). The effective mass can be larger or smaller than the rest mass.
Collisions with atoms increases the rest mass, while enhanced propagation in the crystal gives a reduced effective mass.
From the concept of group velocity the effective mass can be deduced. It is related to the curvature of the E-k curve. Close to band edges, m* can be negative.
K.E of the electron increasing Decreasing velocity of the electron ve effective mass (m*) of the electron
E →
k →
Within a band

7. If Fermi energy lies in this region then the conductivity would be high

8. [100] [110] Effective gap E → E → k → k →
Effective energy gap → Forbidden gap → Band gap
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.
[100]
[110]
Effective gap
E →
E →
k →
k →

9. Divalent metals Monovalent metals Energy band diagram: 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 !!!

10. Energy band diagram: SEMICONDUCTORS
Conduction Band
2-3 eV
Valence Band
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

11. > 3 eV Energy band diagram: INSULATORS Insulator Conduction Band
Valence Band

12. P(E) → E → Eg 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
P(E) →
E → Eg
Eg/2

13. T > 0 K
 Unity in denominator can be ignored
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

14. 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
Mobility
Species
Mobility (m2 / V / s) at RT Si Ge
Electrons
0.14
0.39
Holes
0.05
0.19

15. Conductivity as a function of temperature
Ln()→
1/T (/K) →

16. Extrinsic semiconductors
The addition of doping elements significantly increases the conductivity of a semiconductor. In fact impurity is accidentally present in semiconductors (even in low concentrations like 1 atom in 1012 atoms), which make it extrinsic.
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

17. Ionization Energy→ 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.
n-type
EIonization~0.01 eV
Eg ~1 eV
Donor level (ED)
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

18. p-type Eg Acceptor level (EA) EIonization~0.01 eV
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

19. Ionization energies for dopants in Si & Ge (eV)
Type
Element
In Si
In Ge
n-type P
0.044
0.012 As
0.049
0.013 Sb
0.039
0.010
p-type B
0.045 Al
0.057 Ga
0.065
0.011 In
0.16

20. +ve slope due to Temperature dependent mobility term
Intrinsic
slope
All dopant atoms have been excited
Exhaustion R4 R2 R3
Exponential function R1
 (/ Ohm / K) →
Extrinsic
+ve slope due to Temperature dependent mobility term
Slope can be used for the calculation of EIonization
0.02
0.04
0.06
0.08
0.1
10 K
50 K
1/T (/K) →

21. Semiconductor device  chose the flat region where the conductivity does not change much with temperature
Thermistor (for measuring temperature)  maximum sensitivity is required