1. Course code: EE4209 Instructor: Md. Nur Kutubul Alam Department of EEE KUET Energy band (valance & Conduction band) Introduction to heterostructures

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.

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.

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

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

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=ħK, where ħ=h/2π k = wave vector =2π/λ [λ is the wave length of electronic wave.] Hence, E = (ħK) 2 /2m * + E 0 [m * is called effective mass] E0E0 wave vector, k (Quantum mechanical quantity) Classical momentum, p

7. Energy from quantum mechanics Energy [= (ħK) 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

8. Compare hole & electron energy Energy [= (ħK) 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!

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!

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

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

12. Relative position of band minima x Energy, E Material-1 Energy, E x Material-2 Some common reference ∆E C ∆E V

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

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

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

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

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.