1. Speaker: Sheng Horng Yen 2003/5/26
Theoretical Investigation on Band Structure of the BAlGaInN Semiconductor Materials
Speaker: Sheng Horng Yen 2003/5/26
Sheng Horng Yen 2003/5/26

2. Outline Crystal structure CASTEP theory introduction
Wurtzite and zinc-blende AlGaInN energy-band property
Zinc-blende BAlGaInN energy-band property
Sheng Horng Yen 2003/5/26

3. Crystal coordinate
In order to describe crystal structure, we choose some symmetric coordinate systems.
Three axes of these coordinates are represented by (a, b, c).
Angles between any two axes are represented by (, , ).
Sheng Horng Yen 2003/5/26

4. Lattice types in three dimensions
晶 系
必備對稱元素
a b c
α β γ
註 解
等軸晶系
四方晶系
六方晶系
三方晶系
斜方晶系
單斜晶系
三斜晶系
四個三重軸
一個四重軸
一個六重軸
一個三重軸
三個二重軸
一個二重軸 ≠ 無
a = b = c
a = b ≠ c
a ≠ b ≠ c **
(**為任意角)
** ** **
c軸=四重軸
c軸=六重軸
c軸=三重軸
a < b < c或
c < a < b
c軸=二重軸
a<b,γ鈍角或
b軸=二重軸
a > c,β鈍角
αβγ全為
銳角或鈍角
Sheng Horng Yen 2003/5/26

5. Common structures of semiconductor
(a) Cubic zinc-blende structure
(b) Hexagonal wurtzite structure
Sheng Horng Yen 2003/5/26

6. Crystal band diagram Wurtzite structure of GaN band diagram
Sheng Horng Yen 2003/5/26

7. CASTEP theory introduction
The calculating method is based on First Principle. (Physical fundamental principle )
What is First Principle
Physical principle is correct
Alter physical principle
Match experimental result
Not match experimental result
Physical principle calculation
Sheng Horng Yen 2003/5/26

8. Density Functional Theory
First principle method is according to DFT. (Density Functional Theory)
It is difficult to deal with a many-electron system because each electron does not interact only with nearest electrons.
Kohn and Sham use mean-field theory to deal with such system.
In Kohn-Sham method, the electron density plays a crucial role. So this function is so-called DFT.
Sheng Horng Yen 2003/5/26

9. Local Density Approximation
For purposes of practical calculation, Kohn-Sham theory function must be supplemented by an approximation.
But LDA will underestimate band-gap energy of semiconductor.
Now, some alter methods for LDA like electron-density gradient and electron self-energy can make band-gap energy approach experimental results.
Sheng Horng Yen 2003/5/26

10. Wurtzite AlGaInN energy-band property
InGaN energy-band property
1.strain-free
2.with strain
AlGaN energy-band property
AlInN energy-band property
Sheng Horng Yen 2003/5/26

11. InxGa1-xN parameter Lattice constant a(x)=3.501x+3.162(1-x)
b(x)=3.501x+3.162(1-x)
c(x)=5.669x+5.142(1-x)
Band-gap energy
Eg(x) = x · Eg,InN + (1-x) ·Eg,GaN - b · x · (1-x)
Sheng Horng Yen 2003/5/26

12. Eg versus Indium composition
Band-Gap Energy decreases with Indium composition increases. (b=1.210 eV)
Sheng Horng Yen 2003/5/26

13. Band-gap energy curves in different strain
Tensile strain makes the bowing parameter smaller.
Sheng Horng Yen 2003/5/26

14. AlxGa1-xN parameter Lattice constant a(x)=3.082x+3.162(1-x)
b(x)=3.082x+3.162(1-x)
c(x)=4.948x+5.142(1-x)
Band-gap energy
Eg(x) = x · Eg,AlN + (1-x) ·Eg,GaN - b · x · (1-x)
Sheng Horng Yen 2003/5/26

15. Eg versus Aluminum composition
Band-Gap Energy increases with Aluminum composition increases. (b=0.353 eV)
Sheng Horng Yen 2003/5/26

16. Band-gap energy curves in different strain
When tensile strain is about 2%, the bowing parameter has a minimum value.
Sheng Horng Yen 2003/5/26

17. AlxIn1-xN parameter Lattice constant a(x)=3.082x+3.501(1-x)
b(x)=3.082x+3.501(1-x)
c(x)=4.948x+5.669(1-x)
Band-gap energy
Eg(x) = x · Eg,AlN + (1-x) ·Eg,InN - b · x · (1-x)
Sheng Horng Yen 2003/5/26

18. Result 1
Band-gap energy curves of AlGaInN.
Sheng Horng Yen 2003/5/26

19. wurtzite InGaN、AlGaN、AlInN
Result 2
WZ-Ⅲ-nitride與基板之能帶間隙與晶格常數
wurtzite InGaN、AlGaN、AlInN
能帶間隙對晶格常數之關係圖
Sheng Horng Yen 2003/5/26

20. Result 3
Tensile strain makes the bowing parameter of AlxIn1-xN smaller.
Sheng Horng Yen 2003/5/26

21. Zinc-blende AlGaInN energy-band property
InGaN energy-band property
1.strain-free
2.with strain
AlGaN energy-band property
AlInN energy-band property
Sheng Horng Yen 2003/5/26

22. InxGa1-xN parameter Lattice constant a(x)=b(x)=c(x)=4.932x+4.537(1-x)
Band-gap energy
Eg(x) = x · Eg,InN + (1-x) ·Eg,GaN - b · x · (1-x)
Sheng Horng Yen 2003/5/26

23. Eg versus Indium composition
Direct bowing parameter is eV.
Indirect bowing parameter is eV.
Sheng Horng Yen 2003/5/26

24. Band-gap energy curves in different strain
Under different strain, the direct bowing parameters always smaller than indirect.
Sheng Horng Yen 2003/5/26

25. AlxGa1-xN parameter Lattice constant a(x)=b(x)=c(x)=4.376x+4.537(1-x)
Band-gap energy
Eg(x) = x · Eg,AlN + (1-x) ·Eg,GaN - b · x · (1-x)
Sheng Horng Yen 2003/5/26

26. Eg versus Aluminum composition
Direct bowing parameter is eV.
Indirect bowing parameter is eV.
Sheng Horng Yen 2003/5/26

27. Band-gap energy curves in different strain
Under different strain, the direct bowing parameters always larger than indirect.
Sheng Horng Yen 2003/5/26

28. AlxIn1-xN parameter Lattice constant a(x)=b(x)=c(x)=4.376x+4.932(1-x)
Band-gap energy
Eg(x) = x · Eg,AlN + (1-x) ·Eg,InN - b · x · (1-x)
Sheng Horng Yen 2003/5/26

29. Eg versus Aluminum composition
Direct bowing parameter is eV.
Indirect bowing parameter is eV.
Sheng Horng Yen 2003/5/26

30. Band-gap energy curves in different strain
When strain is about 2.7%, direct and indirect bowing parameters have a point of intersection.
Sheng Horng Yen 2003/5/26

31. Result
Band-gap energy and lattice constant curves of zinc-blende Nitride-compounds.
Sheng Horng Yen 2003/5/26

32. Zinc-blende BAlGaInN energy-band property
BAlN energy-band property
1.strain-free
2.with strain
BGaN energy-band property
BInN energy-band property
Sheng Horng Yen 2003/5/26

33. BxAl1-xN parameter Lattice constant a(x)=b(x)=c(x)=3.596x+4.376(1-x)
Band-gap energy
Eg(x) = x · Eg,BN + (1-x) ·Eg,AlN - b · x · (1-x)
Sheng Horng Yen 2003/5/26

34. Eg versus Boron composition
Direct bowing parameter is large.
Indirect bowing parameter is eV.
Sheng Horng Yen 2003/5/26

35. Band-gap energy curves in different strain
Under different strain, direct bowing parameter is much larger than indirect.
Sheng Horng Yen 2003/5/26

36. BxGa1-xN parameter Lattice constant a(x)=b(x)=c(x)=3.596x+4.537(1-x)
Band-gap energy
Eg(x) = x · Eg,BN + (1-x) ·Eg,GaN - b · x · (1-x)
Sheng Horng Yen 2003/5/26

37. Eg versus Boron composition
Direct bowing parameter is eV.
Indirect bowing parameter is eV.
Sheng Horng Yen 2003/5/26

38. Band-gap energy curves in different strain
Under different strain, direct bowing parameter is much larger than indirect.
Sheng Horng Yen 2003/5/26

39. BxIn1-xN parameter Lattice constant a(x)=b(x)=c(x)=3.596x+4.932(1-x)
Band-gap energy
Eg(x) = x · Eg,BN + (1-x) ·Eg,InN - b · x · (1-x)
Sheng Horng Yen 2003/5/26

40. Eg versus Boron composition
Direct bowing parameter is eV.
Indirect bowing parameter is eV.
Sheng Horng Yen 2003/5/26

41. Band-gap energy curves in different strain
Under different strain, direct bowing parameter is much larger than indirect.
Sheng Horng Yen 2003/5/26

42. Summary
Ⅲ-Ⅴ Nitride compounds still have many unknown physical properties.
There are two reasons for larger bowing parameter.
1.Nitride compounds have Indium atom.
2.Large lattice constant difference between atoms of column Ⅴ will result large bowing parameter.
Sheng Horng Yen 2003/5/26