In the Fig. 6.4, we plot the total wavefunction and the current density patterns which we can see the edge states clearly in the wire of the repulsive barrier is 20, which is really strong and the transmission is almost zero, the amplitude of the magnetic field is B = 1.0B∗, and the incident energy is X = 1.7 and incident from the first subband. And the magnetic length is 1.0a∗, and the cyclotron radius is 2.8a∗ in this case. In the figure of the transport particle current density, the electrons are seen to describe the edge state current along the edge of the wire, the edge states is clearly generated in the Fig. 6.4;
we are in the quantum Hall regime. The particle current density flow into the wire on the topside form the left side, and almost total reflect to the left on the underside of the
REPULSIVE BARRIER
0.7 0.8 0.9 1
1 1.5 2 2.5 3 3.5
T
X
V=0.0 V=0.2 V=0.6 V=1.0 V=1.4
Figure 6.3: The transmission versus the incident energy and incident from the first subband for various strength of repulsive barrier from 0.0E∗ to 1.4E∗ and the amplitude of the magnetic field is 2.0B∗.
wire and before the barrier at x = 0. This phenomenon and the path of the edge state is suitable to the classical picture that the charged particle move on the edge when it is applied a large magnetic field.
In the Fig. 6.5, Fig. 6.6, and Fig. 6.7, we plot the wavefunction and the current density patterns of the strength of repulsive barrier is V0 = 1.0E∗, the amplitude of the magnetic field is 0.5B∗ and the magnetic length is lB = 1.41a∗ , the the incident wave come form the first subband which the incident energy is X = 1.8, X = 2.2, and X = 2.8 in the three figures in sequence. And the cyclotron radius are rc = 5.98a∗, rc = 7.33a∗ and rc = 8.97a∗ in the three figures. The small curve on the right of the top of each figure is the transmission versus the incident energy and the wave incident from the first subband.
We want to find out the reason why there is a resonance valley near X = 2.2 in the curve of transmission. In the Fig. 6.5, the edge state had generated and come from the left
x
y
S20Vp20dX1d7000B1d00
−8 −6 −4 −2 0 2 4
−4
−3
−2
−1 0 1 2 3 4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.5 2 0
0.5 1
Figure 6.4: The total wavefunction and the current density patterns in the wire with a strong repulsive barrier V0 = 20E∗, the amplitude of the magnetic field is 1.0B∗, and the incident energy is X = 1.7 from the first subband.
edge state on the underside of the wire. We don’t think the edge strongly interact with the evanescent modes because the amplitude of the evanescent mode is two order smaller then the propagating mode. In the Fig. 6.6, the incident energy is X = 2.2 and there is a valley on the transmission here. We can find that the propagating mode interact with the second propagating mode and have back scattering in both first and second propagating modes. And in the Fig. 6.7, the incident energy is X = 2.8 which is the highest one of these three figures. The first propagating mode had shift to the edge, and the center shift of the second propagating mode is also larger then the shift in Fig. 6.6. The interference of each propagating mode itself or between each other is small and have less back reflection.
We find that the resonance valley near X = 2.2 is a resonance with the propagating
REPULSIVE BARRIER
Figure 6.5: Top: The total wavefunction and the current density patterns of the strength of the repulsive barrier is V0 = 1.0E∗, the amplitude of magnetic field is B = 0.5B∗, and the incident energy is X = 1.8 from the first subband; Left of bottom: The wave-function and the current density components of the first subband; Right of bottom: The wavefunction and the current density components of the second subband.
modes themselves but not have much interference with the evanescent modes because we can see the amplitude of the evanescent mode is not very large. In order to have
x
Figure 6.6: Top: The total wavefunction and the current density patterns of the strength of the repulsive barrier is V0 = 1.0E∗, the amplitude of magnetic field is B = 0.5B∗, and the incident energy is X = 2.2 from the first subband; Left of bottom: The wave-function and the current density components of the first subband; Right of bottom: The wavefunction and the current density components of the second subband.
the resonance, the shifted wavefunction of propagating modes must have some overlap to each other and then the resonance can have interference between each other. Due to the
REPULSIVE BARRIER
Figure 6.7: Top: The total wavefunction and the current density patterns of the strength of the repulsive barrier is V0 = 1.0E∗, the amplitude of magnetic field is B = 0.5B∗, and the incident energy is X = 2.8 from the first subband; Left of bottom: The wave-function and the current density components of the first subband; Right of bottom: The wavefunction and the current density components of the second subband.
necessary overlap of each wavefunction, the resonance easier happen at the energy which is a little large then each subband bottom.
This kind of resonance is very different with the resonance of the finite range potential.
The resonance of the finite range potential is that one propagating mode has interference with itself and depend on the phase of the path difference, this kind of resonance could be a large one; but the edge state resonance is due to the interference of the propagating modes which are always not only one propagating mode, and the resonances are not a large one. And another specially character of the edge state resonance could be generated for a δ-type potential which is a localized potential on the longitudinal direction. That is never happened for a coordinate resonance without magnetic field.
And then we also plot the wavefunction and the current density patterns in Fig. 6.8 of the case of strength of repulsive barrier is V0 = 1.0E∗ and the amplitude of magnetic field is 0.5, like the parameter of the three figures Fig. 6.5, Fig. 6.6, and Fig. 6.7. But change the incident energy and incident subband to the second propagating mode. In the Fig. 6.8, the incident energy is X = 2.2 and the cyclotron radius is rc= 2.99a∗. The physical insight is similar to the above discuss, but in this figure we can easily see the particle current cycle and the diameter of the cycle is match to the magnetic length and the cyclotron radio (but the cyclotron radius is a radius) in order.
6.4 Summary
In this chapter, we find that the resonance valleys is according to the interference of the propagating modes themselves and not only one propagating mode. In order to have the resonance and the interference between different propagating modes, the shift wavefunction must have some overlap to each other and the impurity potential. Because our potential barrier is uniform of the transverse direction, the more strength of the impurity potential, the larger of the overlap. Sometimes, the wavefunction have not only one peak or one node, due to the overlap of the wavefunctions, they may have two or more valleys on the transmission. Besides, Due to the necessary overlap between each subbands, the resonance easier happen at the energy which is a little larger then each
REPULSIVE BARRIER
Figure 6.8: Top: The total wavefunction and the current density patterns of the strength of the repulsive barrier is 1.0, the amplitude of magnetic field is 0.5, and the incident energy is X = 2.2 and incident from the second subband; Left of bottom: The wavefunction and the current density components of the first subband; Right of bottom: The wavefunction and the current density components of the second subband.
subband bottom. In other words, they should have a wavefunction with small kinetic energy or small cyclotron radius to be the media to mix the propagating modes together.
This kind of resonance will not happen above the first subband bottom, because there is only one propagating mode here.
This kind of resonance is very different between the resonances due to the multiple scattering in a final range of potential. The resonance of the multiple scattering of a final range potential is that one propagating mode has interference with itself and depend on the phase difference of the multiple scattering and the position difference; this kind of resonance could be a large one. The edge state resonance is due to the interference or the overlap of different propagating modes and the impurity potential, the amplitude of the overlap are often small and the resonance will not be a large one.
And another specially character of the edge state resonance could be generated for a δ-type potential which is a localized potential on the longitudinal direction. The edge state resonance is a kind of resonance on the transverse direction, and differ with the resonance on the longitudinal direction without magnetic field. The physical insight is close to the classical picture, which the edge states mean the charged particle move along the edge and the potential barrier when it is applied a magnetic field. When the edge state move along the potential barrier, there are two boundaries which are the edge of the wire around the edge states, and the edge states can have the resonance between the edge of the wire on the transverse direction.
Chapter 7
Magnetoconduction in quantum channel with an attractive barrier
We discuss the phenomena with an attractive barrier in the wire by tuning the magnetic field and the barrier strength in this chapter. And we will investigate the effective strength of the impurity barrier with the applied magnetic field, the two quasi-bound states of the evanescent modes in pair caused from the complex property of eigen-function, and we also compare the magnetic length and the cyclotron radius with the classical width of the wire.
7.1 Tuning of the magnetic field
In the Fig. 7.1, we plot the transmission as a function of X for various B and fix V0 =
−1.0E∗. And in the Fig. 7.2, we change the amplitude of the applied magnetic field from 0.0B∗ to 5.0B∗ and mark the position in energy of the dip structures, and the strength of the impurity barrier is −1.0E∗ in (a) and −1.4E∗ in (b).
In the first curve of the Fig. 7.1(a), because the impurity barrier is uniform on the transverse direction, the system should be symmetry without magnetic field and could be reduced to a one dimension problem which T ∼ 1−2V10/ik, and there are not subband
0 0.2 0.4 0.6 0.8 1
1 1.2 1.4 1.6 1.8 2
T
X (a)
B=0.0 B=0.4 B=0.8 B=1.2 B=1.6 B=2.0
0 0.2 0.4 0.6 0.8 1
1.75 1.8 1.85 1.9 1.95 2
T
X (b)
B=0.0 B=0.4 B=0.8 B=1.2 B=1.6 B=2.0
Figure 7.1: The transmission versus the incident for various amplitudes of magnetic field from 0.0 to 2.0, and the strength of the impurity barrier is 1.0.
ATTRACTIVE BARRIER
transition between each subbands.
As we apply the magnetic field to B = 0.2B∗, there is a fano profile happened suddenly at X = 1.79. And then we increase the amplitude of magnetic field to 0.4B∗, the fano has a blue shift to the higher energy at X = 1.85. we can find that the first dips move to the higher energy when we increase the applied magnetic field till B = 0.8B∗(which is figured in the Fig. 7.2), and the fano become more and more sharp. As we increase the amplitude of magnetic field to 0.6B∗, the second dip appear below the subband bottom of the second subband. The larger of the magnetic field, the second dips have red shift to the lower energy till B > 2.8B∗.
And then we trace out the relation of the energy of the first two dips and the amplitude of the applied magnetic field. In the Fig. 7.2, we and find that in the small magnetic field regime, the first dips move to the higher energy and the second dips move the the lower energy. As increasing the amplitude of magnetic field, the two dips mix together and the minimum of the transmission of the dips will not touch zero in the regime of the magnetic filed near 0.7B∗; the Fano structures is gone in this regime. And in the high magnetic field regime, the third and the fourth dips appear below the bottom of second subband.
As long as the magnetic field is large enough, the fifth and sixth dips or the 7-th and 8-th dips will be appear.
And in the Fig. 7.2, we find that the larger of the applied magnetic field, the smaller of the energy difference of the first two dips, and in the high magnetic field regime, the first two dip degenerate into together.
7.2 Tuning of the barrier strength
In the Fig. 7.3 and 7.4, we fix the amplitudes of the magnetic field of these two figures to 0.2B∗ and 1.0B∗, and plot the transmission as a function of X for various V0 from 0.0E∗ to −1.4E∗.
We find that the energy of the dip structures move to the lower energy as the strength
1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2
0 1 2 3 4 5
X
B (a)
First dip Second dip Third dip Fourth dip
1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2
0 1 2 3 4 5
X
B (b)
First dip Second dip Third dip Fourth dip fifth dip sixth dip
Figure 7.2: The position in energy X of the dip structure in the transmission curve versus the amplitude of magnetic field; the strength of the impurity barrier is 1.0E∗ in (a) and 1.4E∗ in (b).
of the attractive barrier become larger. In the Fig. 7.3, we find that the energy spacing between one and the next data is getting larger when we increase the strength of the
ATTRACTIVE BARRIER
0 0.2 0.4 0.6 0.8 1
1 1.2 1.4 1.6 1.8 2
T
X V=-0.0
V=-0.2 V=-0.4 V=-0.6 V=-0.8 V=-1.0 V=-1.2 V=-1.4
Figure 7.3: The transmission as a function of X for various strength of impurity barrier from 0.0 to −2.0E∗. The magnetic field is 0.2B∗.
impurity barrier, but in the Fig. 7.4, the difference of the energy spacing does not change so obviously. The energy spacings or the position of the dips have fewer dependence in the higher magnetic field regime.
In the regime of low magnetic field, the edge states are not generated and the wave-function is spread in the center of the wire. The effective impurity potential should be proportional to the impurity barrier in the wire below the Fermi energy. In the Fig. 7.5, we define δE is the energy difference between the first dips and the subband bottom of second mode, and we find that the slope of the curve approach to 0.5 as the magnetic field approach to 0.0B∗, which mean that the relation of the δE and the strength of the impurity barrier is V0 ∝ √
δE or δE ∝ V02 in the small magnetic field regime. In the regime of high magnetic field, the edge states are generated and shift to the edge of the wire. In this situation, the transport of the two dimensional problem can be reduced to a one dimensional like problem, and the electrons moving in the wire only see the barrier in
0 0.2 0.4 0.6 0.8 1
1 1.2 1.4 1.6 1.8 2
T
X V=-0.0
V=-0.2 V=-0.6 V=-1.0 V=-1.4
0 0.2 0.4 0.6 0.8 1
1.8 1.85 1.9 1.95 2
T
X V=-0.0
V=-0.2 V=-0.4 V=-0.6 V=-0.8 V=-1.0 V=-1.2 V=-1.4
Figure 7.4: The transmission as a function of X for various strength of impurity barrier from 0.0 to −2.0E∗. The magnetic field is 1.0B∗. Plotting Fig(a) again of energy range at 1.7 to 2.1 in Fig(b).
ATTRACTIVE BARRIER
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5
-2 -1.5 -1 -0.5 0 0.5
Log(Energy difference)
Log(V) B=0.01, slope: 0.4987
B=0.1, slope: 0.5083 B=0.2, slope: 0.5272 B=1.0, slope: 0.7630 B=2.0, slope: 0.7936 B=5.0, slope: 0.8671
Figure 7.5: The relation of the position of the first dips with V0.
the edge regime which is a small area. And the effective impurity potential is no longer the barrier in the wire and below the fermi energy, but the part of the edge regime, the effective strength of the effective impurity potential will be proportional to the strength of the impurity barrier more linearly.
7.3 Analyses of numerical results and physical inter-pretations
In the Figs.7.6 - 7.10, we plot the wavefunction and the current density patterns of the first two dips for V0 = 1.4E∗ and the amplitude of the magnetic field is 0.4B∗, 0.7B∗, and 1.0B∗ correspondingly, which the first two dips have mixed together at B = 0.7B∗.
We first determine the generation of the edge states depend on the amplitude of the magnetic field. In the Fig. 7.6 and 7.7, the applied magnetic fields are both 0.4B∗ and
x
Figure 7.6: The wavefunction and the current density patterns at the first dip with B = 0.4, Top: the total wavefunction and current density patterns in the wire; Left of bottom: the contribution of the propagating mode; Right of bottom: the contribution of the evanescent mode.
figures. The cyclotron are both larger then the classical width of the wire, which are rc = 4.89a∗ and rc= 5.62a∗. And we can also see the figures, the interference between the upside and downside edge states is still viewable in the center part of the wire, and the edge states are not clearly generated. And in the Fig. 7.8, the amplitude of the applied
ATTRACTIVE BARRIER
Figure 7.7: The wavefunction and the current density patterns at the second dip with B = 0.4, Top: the total wavefunction and current density patterns in the wire; Left of bottom: the contribution of the propagating mode; Right of bottom: the contribution of the evanescent mode.
magnetic field is 0.7B∗ and the magnetic length is lB = 1.20a∗. The cyclotron radius is rc = 4.52a∗ in this figure and the classical width of the wire is 5.34a∗. The length scales of cyclotron radius and classical width are more comparable and a little smaller then the case of B = 0.4B∗, and we can also see the figure and find that the interference is less
x
Figure 7.8: The wavefunction and the current density patterns at the dip structure with B = 0.7, Top: the total wavefunction and current density patterns in the wire; Left of bottom: the contribution of the propagating mode; Right of bottom: the contribution of the evanescent mode.
in the center part of the wire. The edge states are roughly generated in this magnetic field regime. In the Fig. 7.9 and 7.10, the amplitude of the magnetic field is B = 1.0B∗ and the magnetic field is 1.0B∗ in these two figures. The cyclotron radius are 3.1a∗ and 3.2a∗ and the classical width of the wire are 5.26a∗ and 5.42a∗, and it is allowed to have
ATTRACTIVE BARRIER
Figure 7.9: The wavefunction and the current density patterns at the first dip with B = 1.0, Top: the total wavefunction and current density patterns in the wire; Left of bottom: the contribution of the propagating mode; Right of bottom: the contribution of the evanescent mode.
the skipping orbit current in the wire. The cyclotron radius are smaller then the classical width and we can also see the figures, the amplitude of the interference of the edge states in the center part of the wire is very small and the edge state had been already generated.
Due to the comparison, we know that we can compare the cyclotron radius and the width
x
Figure 7.10: The wavefunction and the current density patterns at the second dip with B = 1.0, Top: the total wavefunction and current density patterns in the wire; Left of bottom: the contribution of the propagating mode; Right of bottom: the contribution of the evanescent mode.
of the wire to know does the edge states are generated or not, although it is not really correct and roughly.
In the system with magnetic field, we have two length scales about magnetic interaction scale, magnetic length and cyclotron radius. Cyclotron radius is more classical-like then
ATTRACTIVE BARRIER
the magnetic length and is better to describe the circular motion due to the Lorenz force.
And we can also use the scale, cyclotron radius, to separate the edge states and the traversing states in the Fig. 1.1.
The amplitude of the wavefunction in the Fig. 7.7 is much larger then the other figures, the maximum value is at the center of the x- and y-direction and about 37. The next
The amplitude of the wavefunction in the Fig. 7.7 is much larger then the other figures, the maximum value is at the center of the x- and y-direction and about 37. The next