This chapter is divided to two sections. In the first section, the complex band structures of GaAs and AlAs are presented. The second section is the mixing effect on the subband structure, where anticrossing gap is an important parameter. The logarithmic plots of transmission coefficients as a function of electron energy with
Γ − Χ
(0,0)
k& = are presented
as well. The anticrossing gap is defined as the minimum energy separation between the mixed Γ − Χ states. Since we want intense
interaction so that the electrons of
Γ − Χ Γ-valley can easily transfer to
-valley under the applying of electric field in the parallel direction, we want to obtain a structure with a large anticrossing gap. In this chapter, GaAs is defined as the well material and AlAs is defined as the barrier material in the name of the
Χ
Γ-valley electrons.
3.1 Complex band structures
The empirical pseudopotential method is used to calculate the bulk complex band structure. The pseudopotential form factors can be adjusted as well to fit the particular band parameters. Note that a little change in the form factors would lead to large change in the band structure. Fig.1 and Fig.2 are the complex band structure of GaAs and AlAs respectively.
The energy of conduction Γ-valley in GaAs is shifted to zero. The lowest conduction band in AlAs lies at the Χ-valley.
Fig 3 Complex band structure of GaAs along [001] direction
Fig 4 Complex band structure of AlAs along [001] direction
From Fig1. and Fig2., we can see that GaAs is a direct band gap material, and AlAs is an indirect band gap material. The X-valley of AlAs is at 0.1492 eV and the -valley of GaAs is at 1.1048 eV. Therefore, AlAs is the barrier for the -valley electrons, and GaAs is the barrier for the X-valley electrons. Thus, with specific length of each material, the
-valley electrons and X-valley electrons will confined at different material. The above phenomenon is the origin of
Γ Γ Γ
Γ -X mixing effect.
3.2
Γ Χ- mixing effect on the subband structure
Since only at the interface can electrons transfer from -valley to X-valley, the interfaces play an important role in the
Γ
Γ -X mixing effect.
Therefore, we will discuss the effect of interface in two ways. The first one is that the total length of the heterostructure is fixed, and the length of one pair of a barrier and a well is changed. The second one is that the length of one pair of a barrier and a well is fixed, and the total length of the heterostructure is changed.
Case 1. Total length of the heterostructure is fixed.
1. Total Length L=130 Å (1) Single barrier
Fig 5a The band profile of L=130 Å, single barrier
Fig 5b The subband structure of L=130 Å, single barrier
In this structure, since there is no GaAs well to constitute well, there is no quasi-bound states to provide
Γ
Γ Γ -X mixing effect. Therefore,
the subband structure monotonic increases with kx.
(2) 2 barriers, 1 well
Fig 6a The band profile of L=130 Å, 2 barriers, 1 well
Fig 6b The logarithmic plot of transmission coefficient of L=130 Å, 2 barriers, 1 well. The incident state is the state with k& =0
Fig 6c The real part of subband structure of L=130 Å, 2 barriers, 1 well
Fig 6d The imaginary part of subband structure of L=130 Å, 2 barriers, 1 well
From Fig 4b, we can see four peaks, which correspond to four energy states, the quasi-bound states. The lowest energy state is the state, which is confined at GaAs well and corresponds to the lowest subband in Fig 4c and has smaller curvature. The other three energy states correspond to X states, which correspond to the other three subbands in Fig4c and have the larger curvature. In Fig 4c, the lowest Γ state increases monotonically with
Γ
kx, and when it meets with the second X state, the -X mixing effect occurs, meaning that at this energy, the 2 wave functions have both the
Γ
Γ and X part. After that, the original Γ state becomes the X state, and the original X state becomes the state.
We can see that the second anticrossing gap is larger than the first anticrossing gap. This is because when the energy is higher, the wave function is less localized; thus the overlap of the
Γ
Γ and X wave functions is larger.
(3) 3 barriers, 2 wells
Fig 7a The band profile of L=130 Å, 3 barriers, 2 wells
Fig 7b The logarithmic plot of transmission coefficient of L=130 Å, 3 barriers, 2 wells. The incident state is the state with k& =0
Fig 7c The real part of subband structure of L=130 Å, 3 barriers, 2 well
Fig 7d The imaginary part of subband structure of L=130 Å, 3 barriers, 2 wells
From Fig 5c, we can see that the lowest subband is the X subband. The second subband is the subband, and when it meets with the higher X subband, -X mixing effect occurs.
Γ Γ
(4) Unsymmetrical structure, 2 barriers, 1 well
Fig 8a The band profile of L=130 Å, unsymmetrical structure, 2 barriers, 1 well
Fig 8b The logarithmic plot of transmission coefficient of L=130 Å, unsymmetrical structure, 2 barriers, 1 well. The incident state is the state with k& =0
Fig 8c The real part of subband structure of L=130 Å, 2 barriers, 1 well
Fig 8d The imaginary part of subband structure of L=130 Å, 2 barriers, 1 well
Compared with the result in (2), although the number of interfaces is the same, the anticrossing gap in (4) is smaller due to two reasons. The first one is that at the energy which Γ − Χ mixing occurs is lower in (4); thus the and X wave functions are more localized. Therefore, the overlap of the wave functions is weaker. The second one is that when Γ − Χ mixing effect occurs, the X wave function almost localizes at one AlAs barrier, since the two barriers are unsymmetrical. This is like the case with only one AlAs barrier.
Γ
2. Total Length L=215
Å
(1) 3 barriers, 2 wells
Fig 9a The band profile of L=215 Å, 3 barriers, 2 wells
Fig 9b The logarithmic plot of transmission coefficient of L=215 Å, 3 barriers, 2 wells. The incident state is the state with k& =0
Fig 9c The subband structure of L=215 Å, 3 barriers, 2 wells
(2) 4 barriers, 3 wells
Fig 10a The band profile of L=215 Å, 4 barriers, 3 wells
Fig 10b The logarithmic plots of transmission coefficient of L=215 Å, 4 barriers, 3 wells. The incident state is the state with k& =0
Fig 10c The subband structure of L=215 Å, 4 barriers, 3 wells
Case 2. Total length of one pair of a barrier and a well is fixed.
1. BL=30Å, WL=35Å (1) 2 barriers, 1 well
Fig 11a The band profile of BL=30 Å, WL=35 Å. (2 barriers, 1 well)
Fig 11b The subband structure of BL=30 Å, WL=35 Å. (2 barriers, 1
well)
(2) 4 barriers, 3 wells
Fig 12a The band profile of BL=30 Å, WL=35 Å. (4 barriers, 3 well)
Fig 12b The subband structure of BL=30 Å, WL=35 Å. (4 barriers, 3 wells)
2. BL=45Å, WL=45Å (1) 2 barriers, 1 well
Fig 13a The band profile of BL=45 Å, WL=45 Å. (2 barriers, 1 well)
Fig 13b The subband structure of BL=45 Å, WL=45 Å. (2 barriers, 1 well)
(1) 3 barriers, 2 wells
Fig 14a The band profile of BL=45 Å, WL=45 Å. (3 barriers, 2 wells)
Fig 14b The subband structure of BL=45 Å, WL=45 Å. (3 barriers, 2 wells)
Case 1: Total length of the structure is fixed.
Total Length L=130
Å
1st Anticrossing Gap 2nd Anticrossing Gap Structure
(meV) (meV)
1 barrier 0 0
2 barriers, 1 well 4.09 7.16
3 barriers, 2 wells 18.28 Unsymmetrical,
2 barriers, 1 well
2.47 Table 1 Anticorssing gap. Total length of the structure is fixed. L=130 Å
otal Length L=215
Å
1st Anticrossing Gap 2nd Anticrossing Gap T
Structure
(meV) (meV)
3 barriers, 2 wells 4.76 7.72
4 barriers, 3 wells 11.86
Table 2 Anticorssing gap length of the structure is fixed. L=215 Å
ase 2: The Length of one pair of a barrier and a well
fixed. BL=35 Å, WL=35 Å
First of all, let’s consider case 1. From Table 1, we can see that when the total length of the structure is fixed, the anticrossing gap increases with the number of the interfaces per unit length. This is because the interface dominates the Γ - Χ mixing effect. Therefore, when the total length of the structures xed, the increase of the number of the interface would lead to stronger
is fi
Γ - Χ mixing effect. In the same structure, the second anticrossing gap is larger than the first anticrossing gap, since the larger the energy state is; the less localized the wave function is. Compare the result in the second row and the fourth row of the Table 1, and the anticrossing gap of the unsymmetrical structure is smaller than that of the symmetric one, although the two structures have the same number of interfaces. This is because when Γ - Χ mixing effect occurs, the energy states of the two divided GaAs wells are different.
Therefore, this is like the case that only one AlAs well can contribute to the Γ - Χ mixing effect. The result of the Table 2 can be explained by the above discussion.
Second, let’s consider case 2. The length of the one pair of a barrier and a well fixed means that the number of interfaces per unit length is fixed. The change of the anticrossing gap is much smaller than that of the case 1. This is due to two reasons. First, the energy states of the first row and the second row in the Table 3 and Table 4 are almost the same.
Therefore, the overlap of the Γ and Χ wave functions is almost equal.
Second, the number of the inter ces p unit length is the same, which is the same as the discussion in case 1.
Finally, we can conclude that the
fa er
dominant factor of the anticrossing gap is the number of the interfaces per unit length, not the number of the interfaces. Consequently, with the purpose of large anticrossing gap, the increase of the number of the interfaces per unit length and a symmetric structure is necessary.