Before our work, there are several previous papers that had considered similar systems.
The simplest channel confining potential, which is an infinite uniform 2D wire with hard wall, was considered in papers [66, 67, 69]. It is easier to extend the magnetic field to high magnetic field regime in the hard wall confinement. In the reference [67] of H.
Tamura and T. Ando in 1991, a delta-profile potential impurity is considered, and there exist bound states with an energy larger than each landau-level energy for a repulsive scatterer, and a quasi-bound state relate to the attractive potential is also formed below each subband bottom. And in the other reference [66] of Gurvitz in 1995, he introduce analytically the quasibound states of local and non-local potentials. The state relate to the repulsive potential is not a bound one, it is rather a quasibound (resonance) state.
And such a quasibound state can generate resonant transitions of carriers between the edges. As a result, repulsive impurities can produce direct interedge transitions inside the propagating modes (the inner-mode transitions), in contrast with attractive impurities, which generate interedge transitions via bound states in the evanescent modes (the inter-mode transitions).
Actually, for a wire, realistic narrow channel in split-gate devices can not be taken as uniform wires, but rather as a parabolic constructions in the propagating direction. The parabolic confining potential is used in the references [40, 68, 70, 71] with magnetic field.
In the references [40, 70], the applied magnetic field is not very large and the resonance states above the subband relate to the repulsive is not generated. In the reference [68]
of E.V. Sukhorukov et. al. in 1994, they consider a central short-range impurity in the wire with a higher magnetic field with approximate. They found that if the magnetic field is sufficiently strong bound states exist not only for attractive impurities but also for the repulsive ones. Bound states are found not only below any mode threshold in series, but also above. They showed that a series of N bound states exist above the N-th mode threshold.
We try to push our theoretical description of magnetoconduction in a narrow parabolic confining potential which is more realistic to the high magnetic field regime. We will investigate the transmission dip found at the threshold of subband for repulsive potential and the two transmission dips found for the incident energy lies below a subband threshold.
And our approach can consider a general condition raging form low magnetic field to high magnetic field.
→
B
incident wave
-2
0
2
x/a
* -20 2
y/a
* 04 8 E (meV)
Figure 2.1: The figure of our system.
Our system of interest in this work is basically a quantum wire formed out of a 2DEG.
The propagation direction o the wire is x whereas the confinement potential that define of quantum wire is given by Vc(y). Of particular interest is the effect of a magnetic field, pointing along z, on the transport characteristic in the presence of a transverse potential barrier.
OUR PHYSICAL MODEL
The confinement potential Vc(y) is chosen to be parabolic namely
Vc(y) = 1
2m∗ω2yy2, (2.1)
where m∗ is the effective mass of an electron in media and ωy is a potential parameter.
The unperturbed Hamiltonian H0 of the electron in the constriction is given by
H0 = ~2 2m∗
·³
−i∇ + e c~A
´2 + 1
2m∗ωy2y2
¸
(2.2)
where −e is the charge of the electron.
And in this work we focus on the scattering effect due to an impurity, which the potential of impurity is Vd(x, y).
The total Hamiltonian:
H = ~2
2m[−i∇ + e
c~A(r)]2+ Vc(y) + Vd(x, y) (2.3)
A(r) is the vector potential
A(r) = −Byˆi → B(r) = ∇ × A = Bˆk (2.4)
Vc(y) is the confinement potential in the y-direction Vc(y) = 12mωy2y2
H = ~2
2m[−i∇ −eB
c~yˆi]2+1
2mωy2y2+ Vd(x, y) (2.5)
And then we choose some units to obtain the dimensionless expression of Hamiltonian
a∗ = 1
kF, ε∗ = ~2k2F 2m∗ and hence ωc∗ = ~kF2
m∗ , ωy∗ = ~kF2
m∗ = 2ε∗
~, B∗ = ~c e k2F.
(~kF2/m∗)B = ω∗cB, and lB = (~c/eB)1/2 is the magnetic length.
In our numerical examples, the nano-channel(NC) is taken to be that in a high mobility GaAs/AlxGa1−xAs with a typical electron density n ∼ 2.5 × 1011cm−2, and m∗ = 0.067 meV. Correspondingly, our choice of energy unit E∗ = ~2k2F/(2m∗) = 5, 933 meV, length unit a∗ = 1/kF = 9.7937 × 10−9m = 97.937 ˙A, angular frequency unit ωc∗~ = ωy∗~ = Ω∗~ = 2E∗ = 11.866meV, and the magnetic field unit B∗ = 6.863 Tesla.
We also take ωy = 0.5 of which ωyωy∗~ = 5.933 meV, such that the effective NC width is of the order of 102A. In the following, in presenting the dependence of transmission˙ on µ, it is more convenient to plot transmission(T) as a function of X instead, where X = µ/2ωy + 12. The integral value of X is the number of propagating channels. The conservation of current condition is better represented by the function CSV (n) defined as CSV (n) = log|1 −P
n0(|tn0n|2+ |rn0n|2)|, where n is the incident channel and n0 is the outgoing channel. We thus obtain the dimensionless Schr¨odinger equation
⇒ {−∇2+ Ω2y2+ 2iωcy ∂
∂x + Vd(x, y)}ψ(x, y) = Eψ(x, y) (2.6)
where Ω2 = ωc2+ ωy2 = ω2y+ B2
In this work, we consider now electron scattering from an barrier potential of the form
Vd(x, y) = V0Vs(y)δ(x − x0), (2.7)
where Vs(y) is an arbitrary function of the coordinate y, but we use it to be a uniform function of the transversal coordinate y and equal to 1 for simplify. x0 is the longitudinal position of the barrier, and the magnitude of V0sets the magnitude of the barrier potential, which may be repulsive (V0 > 0) or attractive (V0 < 0).
In Ch.3, we keep the scattering potential Vd(x, y) = V0Vs(y)δ(x − x0) and Vs(y) is still an arbitrary function of the coordinate y in the analytical calculation, and set Vs(y) = 1 in the numerical process for a simpler system.
OUR PHYSICAL MODEL
Finally we can have the dimensionless Schr¨odinger equation of our physical model
{−∇2+ Ω2y2+ 2iωcy ∂
∂x + V0δ(x − x0)}ψ(x, y) = Eψ(x, y) (2.8)
Mode-matching (MM) method
In this chapter, we use the mode-matching approach to solve our physical model. After the formalism, we find that the eigen-function of the wire with magnetic field is not a or-thogonal and complete basis set .The propagating mode with real wave vector has a center shift on y-direction and the evanescent mode become a highly oscillating complex func-tion. We choose another orthogonal basis set φon(y) to expend the eigen-function φ±n(y, kn) which is the better one in our three choices. We also find out a special normalization con-stant for the evanescent modes which are complex functions. The normalization concon-stant of propagating modes and evanescent modes are different and the normalization constant of evanescent modes depend on the center shift αn.
3.1 Formalism
We first solve the unperturbed Hamiltonian in this section and obtain the eigen-function of the confinement potential.
In the previous Ch.2, we have obtained the dimensionless Schr¨odinger equation, Eq. (2.8)
{−∇2+ Ω2y2+ 2iωcy ∂
∂x + Vd(x, y)}ψ(x, y) = Eψ(x, y) (3.1)
Firstable, let’s solve the wavefunction of unperturbed Hamiltonian:
{−∇2+ Ω2y2+ 2iωcy ∂
∂x}Ψ(x, y) = EΨ(x, y) (3.2)
Because the asymptotic form of wavefunction at x → ±∞ can be expanded as plane wave, we can assume the eigenfunction of this form
Ψ(x, y) ∼ e±ikxφ±(y) (3.3)
where k is a wavevector.
Substituting Eq. (3.3) the above wavefunction into Eq. (3.2), we obtain
⇒
½ ∂2
∂y2 − Ω2(y ∓ωck
Ω2 )2+ ωc2k2
Ω2 − k2 + E
¾
φ±(y) = 0 (3.4)
let α = ωck
Ω2 , u± = y ∓ α,
which the superscript ± of u denotes the right (left) going wave, and K2 = ωc2k2
Ω2 − k2+ E = E − ωy2 Ω2k2
⇒
½ ∂2
∂u±2 − Ω2u±2+ K2
¾
φ±(y) = 0 let u±0 =√
Ωu±,
⇒
½ ∂2
∂u±02 − u±02+ K2 Ω
¾
φ±(y) = 0 (3.5)
Based on the definition of Hermite function, we can obtain the discrete energy identity:
(2n + 1)Ω = ω2ck2
Ω2 − k2+ E = E − ωy2
Ω2k2 (3.6)
and φn(y) ∝ e−u±0n
2/2Hn(u±n0) = e−Ω2u±n2Hn(√
Ωu±n) (3.7)
In Eq. (3.6), the energy is quantized, and then we change our variables to have the quantized physical quantities labeled by the subband index n:
k → kn, α → αn= ωckn
Ω2 , u± → u±n = y ∓ αn, φ±(y) → φ±n(y, kn) = Nn× e−Ωu±n2/2Hn(√
Ωu±n)
We can write down the total wavefunction:
ψn(x, y) = Nn e±iknxe−Ω(y∓αn)2/2Hn(√
Ω(y ∓ αn)) (3.8)
εn = E − ω2y
Ω2k2n= (2n + 1)Ω, u±n = y ∓ αn, αn= ωckn
Ω2 kn = Ω
ωy
pE − (2n + 1)Ω, Ω2 = ωy2+ ω2c.
and φ±n(y, kn) is the eigenfunction of this equation
½ ∂2
∂y2 − Ω2(y ∓ ωckn
Ω2 )2+ K2
¾
φ±n(y, kn) = 0 (3.9)
which is a shifted harmonic oscillator of frequency Ω, The center of the transverse eigen-function φ±n(y, kn) is at y = ±ωckn/Ω2. Hence the larger the momentum ±~kn along x-direction, the more the center of wave function is shifted, and φ±n(y, kn) is not longer a complete set. The center shift of wavefunction may be real or pure imaginary because it depends on the momentum kn along x-direction.