Chapter 3
Energy Bands and Charge Carriers in Semiconductors
• What holds atoms together in a solid?
• Why are some materials good conductors, whereas others are not?
• How does current flow in a solid?
3.1 Bonding Forces and Energy Bands in Solids
Bonding Forces in Solids
Ionic bonding Alkali halides such as NaCl, LiF, KBr, KCl are ionic solids
formed by ionic bonding. stable and hard crystals; high vaporization
temperatures; good insulators.
complete transfer of valence electrons
Covalent bonding Semiconductors such Si, Ge, ZnS and insulators such as diamond
are formed by covalent bonding, where each atom share its valence
electrons with its neighboring atoms. hard; high melting points; insulators.
sharing of valence electrons among neighboring atoms
Metallic bonding Valence electrons are contributed to the crystal as a whole. The bonding
force is the attractive force between the positive ions and the electron gas.
Less strong as ionic and covalent bonding; good conductors.
Sharing of valence electrons within the whole crystal
Molecular bonding e.g. organic solids, ice, inert gas crystals. The bonding force is
van de Waals forces. Weak bonds, low melting and boiling points.
NOTE:
• Only valence electrons participate in bond formation!
• When solids are formed from isolated atoms or molecules, the total energy of the
system is reduced!
Energy Bands
—What will happen when two isolated atoms (e.g., H) are brought together?
Wave functions
Energy levels
• The formation of new bonding and antibonding orbitals.
• Energy degeneracy is brokenthe splitting of energy level 1s and 2s
• The lowering of energy of the bonding state gives rise to the cohesion of the system.
• These results can be obtained by solving the Schrödinger equation with the LCAO
approximation. LCAO liner combination of atomic orbitals.
• What is wrong with the shape of the wave function in Fig. 3-2?
—What will happen when many (N) Si atoms are brought together to form a solid?
• Energy bands are formed
• Conduction band
• Valence band
• Forbidden band (band gap Eg)
Electronic configuration of Si
1s22s22p63s23p2
Metals, Semiconductors, and Insulators
—Every solid has its own characteristic energy band structure.
—In order for a material to be conductive, both free electrons and empty states must
be available.
• Metals have free electrons and partially filled valence
bands, therefore they are highly conductive (a).
• Semimetals have their highest band filled. This filled
band, however, overlaps with the next higher band,
therefore they are conductive but with slightly higher
resistivity than normal metals (b). Examples: arsenic,
bismuth, and antimony.
• Insulators have filled valence bands and empty conduction
bands, separated by a large band gap Eg(typically >4eV),
they have high resistivity (c ).
• Semiconductors have similar band structure as insulators
but with a much smaller band gap. Some electrons can
jump to the empty conduction band by thermal or optical
excitation (d). Eg=1.1 eV for Si, 0.67 eV for Ge and 1.43
eV for GaAs
Direct and Indirect Semiconductors
• The real band structure in 3D is calculated
with various numerical methods, plotted as
E vs k. k is called wave vector
p is momentum see example -31
• p k
• For electron transition, both E and p (k) must
be conserved.

• A semiconductor is direct if the maximum
of the conduction band and the minimum
of the valence band has the same k value
• A semiconductor is indirect if the …do
not have the same k value
• Direct semiconductors are suitable for
making light-emitting devices, whereas
the indirect semiconductors are not.
See Appendix III for more
data on semiconductor materials
3.2 Charge Carriers in Semiconductors
Electrons and Holes
Ec the bottom of the conduction band
Ev the top of the valence band
EHP an electron-hole pair
• At 0K, a semiconductor is an insulator with no free charge carriers
• At T > 0K, some electrons in the valence band are excited to the conduction band
• The electrons in the conduction band are free to move about via many available states
• An empty state in the valence band is referred as a hole
The concept of hole
The total current in a volume with N electrons
N
J  (q) v i  0
i
The total current with the jth electron missing
N

A valence band (E vs k ) diagram
with all states filled
J  (q) v i  (q)v j  qv j
i
The net result: a positive charge moving with velocity vj

• A hole is an imaginary positive charge moving in the valence band
• The energy of a hole increases downward in a normal band diagram
• The total current flow in a semiconductor is the sum of electron current and hole current
Energy Band Diagram under Electrical Field
What is the direction of the
electrical field?
Left  or
Right 

• (E, k) band diagram vs (E, x) simplified band diagram
• Total energy is the sum of potential energy and kinetic energy
• Band edges Ec and Ev correspond to electron potential energy
• Energies higher in the band correspond to additional kinetic energy of the electron
• Electron and hole lose kinetic energy to heat by scattering (section 3.4.3)
Effective Mass
—The effective mass of an electron in a band with a given (E, k) relationship is defined as
m 
2
*
(3-3)
d 2 E /dk 2
2
1 2 1 p2

k2
For free electrons, E  mv 
2
2 m 2m
 m* = m

• The effective mass is inversely proportional to the curvature of the band

• The electrons near the top of the valence band have negative effective mass
• In general m* is different in each direction and is a tensor; appropriate averages are
needed for various calculation purposes (e.g. density of state effective mass vs
conductivity effective mass, section 3.4.1)
• The introduction of m* will simplify calculations
• electron effective mass is denoted by me* ; hole effective mass is denoted by me*
Realistic Band Structures in Semiconductors
• GaAs is a direct semiconductor
• For holes we have light hole band,
heavy hole band and split-off band
• Si is an indirect semiconductor
• Si has six equivalent conduction
band minima at X along six
equivalent <100> directions
• The constant energy surface for
silicon in one of the six conduction
bands is a ellipsoid
ml is the longitudinal effective mass
mt is the traverse effective mass
Intrinsic Semiconductor
—a perfect semiconductor crystal with no impurities or lattice defects
n  conduction band electron concentration
(electrons per cm3)
p  valence band hole concentration
n=p=ni
(3-6)
ri  recombination rate of EHP; gi  generation rate
n0 , p0  concentrations at equilibrium; r  constant
ri=rn0p0 = rni2 =gi
EHP generation in an
intrinsic semiconductor
(3-7)
Extrinsic Semiconductor
 a doped semiconductor crystal whose equilibrium carrier concentrations n0 and
p0 are different from the intrinsic carrier concentration ni
The consequences of doping
• new donor or acceptor levels are created in the band
gap
• conductivities can be vastly increased
(n0 or p0 >> ni )
• semiconductor becomes either n-type or p-type
(either n0 >> p0 or p0 >> n0 )
For Si and Ge
• Group V elements such as As, P, Sb are donor
impurities
• Group III elements such as B, Al, Ga and In are
acceptor impurities
The donor binding energy for GaAs—an example
From Bohr model, the ground state energy of an “extra” electron of the donor is
m*q 4
E
,
2K 2 2
where K  4 r0
(3-8)

Compare with the room temperature (300K) thermal energy E=kT≈26meV
 All donor electrons are freed to the conduction band (ionized)
Compare with the intrinsic carrier concentration in GaAs (ni=1.1 x 106 /cm3)
 We will have an increase in conduction electron concentration by 1010 if we dope
GaAs with 1016 S atoms/cm3
“Band Gap Engineering” ( 3.1.5 & 3.2.5 )