1. Course code: EE4209 Md. Nur Kutubul Alam Department of EEE KUET High Electron Mobility Transistor (HEMT)

2. Types of HFET/HEMT In terms of doping, HEMTs are of two types- 1. Modulation doped: Whole wide band gap material is doped 2. Delta-doped: Only a portion of wide band gap material is doped

3. Modulation doped FET Spacer layer

4. Two dimensional electron Gas (2DEG): Here electrons are not allowed to move along ‘z’, but they obviously can move along ‘x’ or ‘y’ direction! The two dimensional movement on “x-y” plane, which is attached at the interface of two materials is very much interesting. This type of electrons are called “Two Dimensional Electron Gas” or “2DEG”.

5. Effect of Gate bias (MODFET)

6. Delta-doped FET

7. Difference between MODFET and Delta-doped FET

8. How to obtain band diagram? Band diagram means the potential energy of “electron” in conduction band. Electrostatic potential at some position is defined as- “Energy required to bring +1C charge at that position.” That means, if electrostatic potential at some position is V volt, energy required to bring +1C charge at that position will be V jule. Or, energy required to bring +Q charge at that position will be QV jule. Or, energy required to bring -q charge at that position will be -qV jule. If, “-q” is the charge of an electron, then potential energy for an electron =“-qV” Volt, or equivalently “-V” electron-volt. So, conduction band = - electrostatic potential in “electron-volt” unit.

9. How to obtain band diagram? Electrostatic potential is given by Poisson’s equation. If “V” is the electrostatic potential, then the Poisson’s equation is- And conduction band edge potential (By not considering any band offset)- Electric field “E” is given by negative of the gradient of the electrostatic potential. Or,

10. How to obtain band diagram? Therefore, we need the following equations- Steps: 1.Integrate the charge density to obtain the electric field. 2.Integrate the electric field to obtain the “Conduction band”.

11. Conduction band of HEMT: (Calculation)

12. Conduction band of HEMT: (Step 1) Step 1: Integrate the charge density to obtain the “Electric field”

13. Conduction band of HEMT: (Step 2) Step 2: Integrate the “Electric field”, to get the conduction band. But, here we need to add the band offsets at every junctions. At z= - d, we need to use the boundary condition. At that boundary, band edge potential

14. Conduction band of HEMT: (Step 2) Step 2: Integrate the “Electric field”, to get the conduction band. But, here we need to add the band offsets at every junctions. Integral of Negative field leads to decrease in Integral of Positive field leads to Increase in At the interface of two material, we add the band offset Finally, Positive field increases the

15. Charge control model Charge at the Channel (2DEG), n s = CV G

16. Calculation of n s The band diagram at the interface of wide band gap and narrow band gap material looks like a triangle. Therefore, 2DEG inside this triangular quantum well is confined and can not move along the “z” direction. Any confinement discretize the allowed energy of electron. Therefore, z-confinement allows enegy E 0, E 1 … E i. Each discrete energy can accommodate two electron with opposite spin.

17. Calculation of n s Along x, and y direction, there is no confinement. Therefore in (x,y) plane, energy of electron possess an parabolic relation with the wave vector “k(x,y)”. When energy is not discrete, we use “Density of states” or “DOS” to calculate the carrier concentration. Number of electron Therefore, in case of 2DEG, for each eigen level (produced by z confinement),

18. Calculation of n s Here notice the limit of integral! I did not calculate the limit, just let it to be * Again limit changes! =

19. Calculation of n s = = = =

20. Thickness of 2DEG Although E 0 is not at the bottom of the conduction band, we are assuming so for the sake of simplicity. Then, we can say, Whenever we need to relate energy with number of particle/charge concentration, we use statistical mechanics. In this case, we need the relation derived from Fermi Dirac statistics.

21. Thickness of 2DEG From Fermi Dirac statistics, we got-

22. Thickness of 2DEG Now, from Fermi Dirac statistics- From Electrostatistics, i,e. Poisson’s Equation-

23. 2DEG as a function of Gate bias E eV 1 EcEc eV - di eV 2

24. 2DEG as a function of Gate bias E eV 1 EcEc eV - di eV 2

25. 2DEG as a function of Gate bias E eV 1 EcEc eV - di eV 2

26. 2DEG as a function of Gate bias E eV 1 EcEc eV - di eV 2

27. Charge control model (Again) Finally This is exactly the definition of charge control model.

28. Pinch-off voltage Definition: The value of gate voltage at which charge in the channel is “zero”. Put n S (V G )=0 and find the V G …………………………………………………

29. Modulation Efficiency: From the definition of charge control model: But it happens only in ideal case. At low temperature, it might not be true.

30. Modulation Efficiency:

31. Polar material & polarization Two types of polarization  Spontaneous Polarization  Piezoelectric Polarization

32. Spontaneous Polarization Each unit cell of GaN (or other polar material) can be thought as a charged capacitor. Since, there is always a voltage between two plates of a charged capacitor, GaN unit cell can also be thought as a voltage source.

33. Spontaneous Polarization Now, what happens if we grow more and more atomic layer of GaN on a substrate epitaxially? Only one unit cell Two unit cell One more unit cell Thicker the epilayer, greater the charge separation & higher the internal/built-in voltage…..

34. Spontaneous Polarization Now, what happens if we grow more and more atomic layer of GaN on a substrate epitaxially? No current flows in the growth process. Therefore, Fermi level must be flat.

35. Spontaneous Polarization Now, what happens if we grow more and more atomic layer of GaN on a substrate epitaxially? No current flows in the growth process. Therefore, Fermi level must be flat. In respond to built in electric field, or internal voltage, both conduction band and valence band bends.

36. Spontaneous Polarization Now, what happens if we grow more and more atomic layer of GaN on a substrate epitaxially? No current flows in the growth process. Therefore, Fermi level must be flat. In respond to built in electric field, or internal voltage, both conduction band and valence band bends.

37. Spontaneous Polarization Now, what happens if we grow more and more atomic layer of GaN on a substrate epitaxially? No current flows in the growth process. Therefore, Fermi level must be flat. More voltage, more bending in the bands.

38. Spontaneous Polarization Now, what happens if we grow more and more atomic layer of GaN on a substrate epitaxially? No current flows in the growth process. Therefore, Fermi level must be flat. At some critical thickness, valence band will touch (or tend to go above) Fermi level. It creates a hole But, hole actually means an electron liberated from that position.

39. Spontaneous Polarization Now, what happens if we grow more and more atomic layer of GaN on a substrate epitaxially? No current flows in the growth process. Therefore, Fermi level must be flat. At some critical thickness, valence band will touch (or tend to go above) Fermi level. It creates a hole But, hole actually means an electron liberated from that position. This thickness of epilayer is called critical thickness

40. Spontaneous Polarization Above critical thickness, we get polarization charge at the substrate-epilayer interface. No current flows in the growth process. Therefore, Fermi level must be flat. At some critical thickness, valence band will touch (or tend to go above) Fermi level. It creates a hole

41. Spontaneous Polarization Now, lets calculate the value of critical thickness. No current flows in the growth process. Therefore, Fermi level must be flat. In the figure, total band bending is almost equal to the band gap. And, band bending = e*applied voltage….. Total voltage across epilayer, Voltage at critical thickness,