Course code: EE4209 Instructor: Md. Nur Kutubul Alam Department
of EEE KUET Energy band (valance & Conduction band)
Introduction to heterostructures
Slide 2
Energy from classical newtonian mechanics: Fma If you apply a
force F on a particle having mass m, then it will accelerate.
Newtons 2 nd law states that, F=ma From that, the kinetic energy, E
k = (1/2).mv 2 = (mv) 2 /2m = p 2 /2m Where, p is the momentum of
the particle.
Slide 3
Classical newtonian mechanics Fma Total energy, E = kinetic
energy + potential energy = p 2 /2m + E 0 Now, lets plot this
equation in a graph.
Slide 4
Classical newtonian mechanics Fma E = p 2 /2m + E 0 Negative
value of p means velocity of particle is in the negative direction.
Another thing, this equation is independent of geometric position.
I,e if it is true in front of EEE building, it is also true in
front of DSW building. E0E0
Slide 5
Classical newtonian mechanics Energy (= p 2 /2m + E 0 ) Since
it is true at every geometric position x, so a three dimensional
plot of position dependent E-P plot would be like this one. E0E0
Momentum, p Position, x
Slide 6
Classical vs Quantum mechanics F=ma E = p 2 /2m + E 0 This
equation is true for any particle irrespective of its charge. (i,e
for electron & hole) E0E0 In quantum mechanics, P=K, where =h/2
k = wave vector =2/ [ is the wave length of electronic wave.]
Hence, E = (K) 2 /2m * + E 0 [m * is called effective mass] E0E0
wave vector, k (Quantum mechanical quantity) Classical momentum,
p
Slide 7
Energy from quantum mechanics Energy [= (K) 2 /2m * + E 0 ]
Here, we are using K as one of the independent variables rather
than momentum p. E0E0 Wave vector, k Position, x
Slide 8
Compare hole & electron energy Energy [= (K) 2 /2m + E 0 ]
Wave vector, K It is the curve for the total energy of an electron
Actually energy is not negative! The negative sign indicates the
particle moves at opposite direction to an electron upon
application of force. (Force is applied by electric field. For
hole, F=qE, for electron=-qE.) It is the curve of energy of a hole.
Note that, it is negative in the graph!
Slide 9
Energy, the important concept Position, x Energy Wave vector, k
Energy depends on the value of wave vector, k, which is
proportional to the momentum p. Positive k means particle is moving
in +ve x direction. Negative k means it is going in ve x direction.
Negative energy means it is the energy of hole. Value of energy is
not negative!
Slide 10
Energy, the important concept Position, x Energy, E Wave
vector, k Position, x Energy, E It is so called conduction band It
is the valence band Just rotate the graph so that you can see only
E-x plane. You will see the k axis just like a dot. Conduction band
minima, E C
Slide 11
Conduction band minima of heterojunction Position, x Energy, E
It is so called conduction band It is the valence band Conduction
band minima is also the potential energy of an electron inside a
system. Here, a system can be a material, like Si/Ge/GaAs. And
origin of this potential energy is the interaction of electron with
its surroundings. Like protons, electrons, of the same as well as
neighboring atoms. Hence, when system/material changes, so the
potential energy. ECEC
Slide 12
Relative position of band minima x Energy, E Material-1 Energy,
E x Material-2 Some common reference E C E V
Slide 13
Relative position of band minima x Energy, E Material-1 Energy,
E x Material-2 Vacuum, as the reference level E C E V Electron
affinity, 1 Electron affinity, 2
Slide 14
Relative position of band minima x Energy, E Material-1 Energy,
E x Material-2 Vacuum, as the reference level E C E V Electron
affinity, 1 Electron affinity, 2 Here, 2 + E C = 1 or, E C = 1 - 2
E g1 E g2 Also, 1 +E g1 + E V = 2 +E g2 or, E V = 2 +E g2 ( 1 +E g1
) or, = ( 2 - 1 ) + (E g2 - E g1 ) = -E C + E g = E g -E C or, E C
+ E V = E g E C1 E C2
Slide 15
Possible band alignments x Energy, E Material-1 Energy, E x
Material-2 Vacuum, as the reference level E C E V Electron
affinity, 1 Electron affinity, 2 E g1 E g2
Slide 16
Possible band alignments of two different materials Vacuum, as
the reference level Straddling typeStaggered type Broken gap E c1 E
v1 E c2 E v2 E c1 E v1 E c2 E v2 E c1 E v1 E c2 E v2
Slide 17
Summery: 1.Minimum energy of a conducting electron is called
conduction band minima or E C. It is actually the potential energy
of it. Similarly, minimum energy of a hole is the valance band
minima, E V. 2.For any particular material, we can choose either E
C or E V to be zero. It does not matter! Because at the end of the
day, everything will be independent of the reference level. 3.Value
of E C or E V is different for different materials. So if we choose
E C or E V of one material equal to zero, their value for the other
may/may not be so. 4.Difference between E C ( or E V ) between two
different materials is constant, and it depends on material
parameters. What ever the design/physical influence is, it will
remain fixed.