In Ch.3, we have discuss about the center shift of the eigen-function φ±n(y, kn), the center of the eigen-function shift to the edge in the wire as the wave vector is real, and the
THE LIPPMANN-SCHWINGER (PFTLS) EQUATION
center is back to the center of the wire as the wave vector is pure imaginary. In other words, the eigen-function φ±n(y, kn) is a propagating mode and a real function if the wave vector is real, and be an evanescent mode and a complex function if the wave vector is pure imaginary. Because the eigen-function φ±n(y, kn) is not a orthonormal basis and a complete set as the wave vector kn is a pure imaginary variable, we have to expand the eigen-function φ±n(y, kn) to a orthonormal basis φon(y) which we have done in Ch.3.
In this chapter, we use the approach of partial Fourier transformation to transform the x coordinate to momentum space and keep the wave vector real and be the Fourier transformation variable. According to this method, the eigen-function φ±n(y, kn) will be a orthonormal and complete basis. It is a good news to avert the expansion of eigen-function φ±n(y, kn) to another, but what we have to pay is the more heavily numerical calculation and the δ-type barrier will become a finite range potential.
After we transform the original Hamiltonian to partial momentum space and then transform back to the coordinate space, the original two-dimensional problem which is a wire and embedded a δ-type barrier will become a quasi-one-dimensional problem. In some sense, it is earlier to be solved then a two-dimensional problem here and we use the scattering matrix method.
In the next chapter, we will discuss what is the same and what is different between these two approaches, MM and PFTLS, and which one is better in our case. And we have also compare the numerical results of these two approach, and make sure our calculation is correct and believable.
Comparing the numerical results from the MM and PFTLS
approaches
We compare the numerical results from the MM and PFTLS approaches in this chapter.
We compare the curves of transmission ,the wavefunction and current density patterns, and find that the results of these two approaches are almost the same.
5.1 Transmission
In the Fig. 5.1, we plot the transmission versus the amplitude of magnetic field of the two approaches, MM and PFTLS. We fix the incident energy at E = 7ωyE∗ which X = 3 and the barrier strength is repulsive with 1.0E∗ in (a) and attractive with −1.0E∗ in (b). We find that the curves in both approaches in Fig. 5.1(a) and (b) does not overlay together but they have more overlap when we increase the numbers of subbands. In the Ch. 3 and 4, we know that the two approaches have their own regime which can saturate easier and the unsuitable regime which can not saturate in the calculations, e.q. the approach of MM is more easier to saturate in the lower magnetic field regime and can’t saturate in the
PFTLS APPROACHES
0 1 2 3
0 0.5 1 1.5 2 2.5 3 3.5
T
B (a)
MM, Nc= 17 PFTLS, Nc= 60
0 1 2 3
0 0.5 1 1.5 2 2.5 3 3.5
T
B (b)
MM, Nc= 17 PFTLS, Nc= 60
Figure 5.1: The transmission versus the amplitude of magnetic field of the two approaches, MM and PFTLS. we fix the incident energy at 7ωyE∗which X = 3 and the barrier strength is repulsive with 1.0E∗ in (a) and attractive with −1.0E∗ in (b).
higher magnetic field regime which we had discussed in the Sec. 3.6; and the approach of PFTLS is easier to saturate in the higher magnetic field regime and hard to saturate in
the lower magnetic field regime which we had also discussed in the Sec. 4.5. According to the different of these two approaches, the curves in Fig. 5.1 of these two approaches is reasonable that does not overlay together. But the the regime of lower magnetic field, the curves should be closer as we increase the numbers of subbands which used in the approach of PFTLS, and it does, e.q. the curves of approach of MM are roughly saturate below the amplitude of the magnetic field is B = 1.0B∗, and the curves of approach of PFTLS are saturate above the amplitude of the magnetic field is B = 1.0B∗ by using 80 subbands. The curves should overlay together near the amplitude of magnetic field is B = 1.0B∗ in the Fig. 5.1.
Besides the comparing of quantity, in qualitatively the character is the same in both approaches which showed in the Fig. 5.1.
In the Fig. 5.2, we compare the curves of transmission versus incident energy of the two approaches, MM and PFTLS. In this comparison, we fix the amplitude of magnetic field to B = 1.0B∗ and the strength of barrier to 1.0E∗ but repulsive in (a) and attractive in (b). According to the discussion of the Fig. 3.3 and Fig. 4.2, we use 30 subbands for the calculation of the approach of MM and the curve is saturate below X = 4, and use 80 subbands for the calculation of the approach of PFTLS and the curve is saturate. And we find that the curves in Fig. 5.2 overlay together in the lower incident energy, e.q. X < 3, and within a little space between two curves above X = 4. And the two curves roughly overlay together between 3 < X < 4.
In Fig. 5.3, we enlarge the regime of 0.95 < X < 2.5 and 1.8 < X < 2.01 in Fig. 5.2, and check how close of the curves of the two approaches. In the Fig. 5.3(a), it is still hard to separate the difference of curves of the two approaches, MM and PFTLS. In the Fig. 5.3(b), we find that the curves of ‘MM, Nc = 30’ and ‘PFTLS, Nc = 80’ does have a little space between them and the spacing depend on the numbers of subbands used in the calculation, the more numbers of subbands used, the less spacing between the two approaches. According to this, we believe that the two approaches will be the same if we can use “enough” subbands, but it is hard to do in the numerical calculation. And the
PFTLS APPROACHES
0 1 2 3 4
0.5 1 1.5 2 2.5 3 3.5 4 4.5
T
X
(a) MM, Nc=30 PFTLS, Nc=80
0 1 2 3 4
0.5 1 1.5 2 2.5 3 3.5 4 4.5
T
X
(b) MM, Nc=30 PFTLS, Nc=80
Figure 5.2: The transmission versus the incident energy X of the two approaches, MM and PFTLS. The magnetic field amplitude is 1.0B∗ and the strength of the repulsive barrier is V0 = 1.0E∗ in (a) and the attractive barrier is V0 = −1.0E∗ in (b).
physical insight must had be saturate down in both approaches.
0
Figure 5.3: Comparing the saturation of the two approaches by plotting the transmission versus the incident energy, the magnetic field amplitude is B = 0.5B∗ and the strength of the impurity barrier is V0 = −1.0E∗. And we enlarge one part of figure (a) from 1.8 to
(a) repulsive barrier MM, B=0.10
PFTLS, B=0.10
(b) attractive barrier MM, B=0.10
PFTLS, B=0.10
(c) repulsive barrier MM, B=0.20
PFTLS, B=0.20
(d) attractive barrier MM, B=0.20
PFTLS, B=0.20
Figure 5.4: Compare the wavefunction of the two approaches, MM and PFTLS, for
−10 < x < 10 and y = 0. The incident energy X = 1.6, and the strength of the barriers are repulsive and 1.0 in (a) and (c), attractive and −1.0 in (b) and (d), and the amplitude of the magnetic field are 0.1 in (a) and (b), 0.2 in (c) and (d).
PFTLS APPROACHES