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Semiconductor devices & technology

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Energy band (valance & Conduction band)

Introduction to heterostructures

Course code: EE4209

Semiconductor devices & technology

Instructor:

Md. NurKutubulAlam

Department of EEE

KUET

F

ma

If you apply a force “F” on a particle having mass “m”, then it will accelerate. Newtons 2nd law states that,

F=ma

From that, the kinetic energy, Ek= (1/2).mv2

= (mv)2/2m

= p2/2m

Where, “p” is the momentum of the particle.

F

ma

Total energy, E = kinetic energy + potential energy

= p2/2m + E0

Now, lets plot this equation in a graph.

F

ma

E = p2/2m + E0

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.

E0

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.

Energy(= p2/2m + E0)

E0

Position, x

Momentum, p

In quantum mechanics,

P=ħK,

where ħ=h/2π

k = wave vector =2π/λ

[λ is the wave length of electronic wave.]

Hence, E = (ħK)2/2m* + E0

[m* is called effective mass]

F=ma

E = p2/2m + E0

This equation is true for any particle irrespective of its charge. (i,e for electron & hole)

E0

E0

Classical momentum, p

wave vector, k

(Quantum mechanical quantity)

Here, we are using “K” as one of the independent variables rather than momentum “p”.

Energy[= (ħK)2/2m* + E0]

E0

Position, x

Wave vector, k

Energy[= (ħK)2/2m + E0]

It is the curve for the total energy of an electron

Wave vector, K

It is the curve of energy of a hole.

Note that, it is negative in the graph!

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.)

Energy

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!

Position, x

Wave vector, k

Energy, E

Energy, E

It is so called conduction band

Conduction band minima, EC

Position, x

Position, x

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.

Wave vector, k

Energy, E

It is so called conduction 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.

EC

Position, x

It is the valence band

Energy, E

Energy, E

Some common reference

∆EC

x

∆EV

x

Material-1

Material-2

Energy, E

Energy, E

Vacuum, as the reference level

Electron affinity, 2

Electron affinity, 1

∆EC

x

∆EV

x

Material-1

Material-2

Energy, E

Energy, E

Vacuum, as the reference level

Electron affinity, 2

Electron affinity, 1

Here,

2 + ∆EC = 1

or, ∆EC = 1 - 2

EC2

∆EC

EC1

Also,

1 +Eg1+ ∆EV= 2 +Eg2

or, ∆EV = 2 +Eg2 – (1 +Eg1)

or, = (2 - 1) + (Eg2- Eg1)

= -∆EC + ∆Eg

= ∆Eg -∆EC

or, ∆EC +∆EV = ∆Eg

Eg2

Eg1

x

∆EV

x

Material-2

Material-1

Energy, E

Energy, E

Vacuum, as the reference level

Electron affinity, 2

Electron affinity, 1

∆EC

Eg2

Eg1

x

∆EV

x

Material-2

Material-1

Vacuum, as the reference level

Straddling type

Staggered type

Broken gap

E c2

E c1

E c1

E c1

E c2

E v1

E v1

E c2

E v1

E v2

E v2

E v2

Minimum energy of a conducting electron is called “conduction band minima” or EC. It is actually the potential energy of it. Similarly, minimum energy of a hole is the “valance band minima”, EV.

For any particular material, we can choose either EC or EV to be “zero”. It does not matter! Because at the end of the day, everything will be independent of the reference level.

Value of EC or EV is different for different materials. So if we choose EC or EV of one material equal to “zero”, their value for the other may/may not be so.

Difference between EC ( or EV)between two different materials is constant, and it depends on material parameters. What ever the design/physical influence is, it will remain fixed.