In Chapter 2, we will review the physics and formalism of a charge RRD around a local scatterer. Concepts such as nonequilibrium distribution of incident electrons, obtained from the Boltzmann kinetic equation, scattering wave functions of these electrons, and Thomas-Fermi screening of these pile-up of charges to the electric potential field of the electric dipole. As a preview to the resonant behavior, we will also consider the resonant behavior of the RRD around a ring-shaped microstructure. No in-plane potential gradient SOI is included in this preview session.
In Chapter 3, we reconsider the ring-shaped microstructure, but with in-plane
po-tential gradient SOI included. The 2D quantum-mechanical scattering becomes spin-dependent and asymmetric. The resonant behavior in the charge RRD and the spin accumulation is presented.
In Chapter 4, the effect of the extrinsic SOI to the resonant spin accumulation around the ring-shaped microstructure is presented. Correction is shown to be small.
In Chapter 5, we consider a smooth potential profile of a ring-shaped microstructure.
The resonant behavior is shown to remain intact.
Finally in chapter 6, we present our conclusion and possible future work.
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Landauer’s residual resistivity dipole around a ring-shaped microstructure
In this chapter, we will present the physics of Landauer’s residual resistivity dipole (RRD) and formalism leading to such an entity. The distribution of the charge pile up of the RRD around a local scatterer is obtained by summing up the density distribution of electrons scattering from a target scatterer. The distribution of the electrons is set up by an external field. The coulomb interaction between electrons causes screening to occur and results in an induced potential variation around a scatterer. In the lowest order in 1/(kρ) where k is the Fermi wave number and and ρ the distance from the scatterer, the induced potential takes on a dipole field from which we can define a RRD strength. We present the energy dependence of the RRD which, as expected, exhibits resonant behavior in the case of a ring-shaped microstructure.
2.1 Quantum-mechanical scattering in two dimensions
For a 2D central potential, the cylindrical symmetry governs that the scattering wave func-tion be most conveniently expressed in polar coordinates. The scattering wave funcfunc-tion from an incident plane wave contains a scattered wave component representing cylindrical
outgoing wave co-centered with the central potential.
Without the central potential, the free particle Schr¨odinger equation is given by
(∇2+ k2)ψ = 0,
with the particle energy ε =2k2/2m∗. Expressed in polar coordinates, the equation is
1
Applying the factored form ψ(ρ) = R(ρ) Φ(φ), it can be decoupled separately into the radial and azimuthal coordinates,
d2R
where the radial equation is the Bessel equation. Therefore the free electron wave function in two dimensions can be written in the form
ψ(ρ, φ) = Jl(kρ) eilφ,
where Jl(x) is the Bessel function of the first kind with l = 0,±1, ±2, . . . the quantum number of the azimuthal motion, along the coordinate φ which is the angle between ρ and ˆx.
2.1.1 Method of partial waves
To perform the quantum scattering calculation from a cylindrical central potential, we first expand the incident plane wave, along ˆx, in terms of the cylindrical waves, the
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Jacobi-Anger expansion [27], given by
eikx = eikρ cos φ=
+∞
l=−∞
ilJl(kρ) eilφ. (2.2)
Since Jl(kρ) is a standing wave along the radial direction, it is convenient, for our purpose, to express it in terms of two radial propagating waves, the Hankel functions,
Jl(kρ) = 1
The radial propagating nature of these Hankel functions is most transparent in their asymptotic form, i.e. the region where kρ 1,
kρ→∞lim Hl(1)(kρ) =
It is clear that Hl(1), Hl(2) can be viewed as circular waves propagating radial outwards from, inwards towards the scattering center, respectively. The cylindrical symmetry of the scattering potential causes waves to be coupled only within the same l. This is essentially the conservation of orbital angular momentum, which is true for a central potential but without SOI. The total wave function, including the incident and the scattered waves, is written in the form
Ψ(ρ) =
+∞
l=−∞
ilRl(ρ) eilφ, (2.4)
where Rl is yet to be determined. Substituting Eq. (2.4) into the Schr¨odinger equation Eq. (2.1), the radial equation becomes
1
Boundary conditions will be established in the next section for the solving of Rl(ρ).
2.1.2 Phase shift and the unitarity relation
For a scattering center, whatever goes in should be eventually go out. Hence physically, the only thing that can be changed is the phase of the outgoing cylindrical waves. Therefore phase shift in the asymptotic outgoing cylindrical wave, where V (ρ) = 0, should carry information about V (ρ). Therefore, the asymptotic form of Rl(ρ) should be of the form
kρ→∞lim Rl(ρ)∝ expected. Note that δl is the sole quantity to be determined since the amplitude of the two cylindrical wave components in Eq. (2.6) should be the same, for unitary requirement.
On the same token, δl must be real.
We need to find a boundary condition for the determination of δl. For an arbitrary radial potential profile, we need to integrate the radial equation for Rl(ρ) and matches it with the expected form, Eq. (2.6), outside the potential. The radial wave function outside the range of the potential can be expressed in the form
Routl (ρ) = eiδl[cos δlJl(kρ)− sin δlYl(kρ)] , (2.7)
where the superscript ’out’ stands for ’outside’. Eq. (2.7) coincides with Eq. (2.6) in its asymptotic region. The boundary condition for the radial differential equation, via integrating Eq. (2.5), is that, the logarithmic derivative of Rl,
L [Rl(ρ)]≡ ρd ln Rl(ρ)
dρ , (2.8)
must be continuous at any ρ. Since the phase shift δl is defined in the potential-vanishing region, we have
where ρb is the ’outer edge’ of the scattering potential, beyond which the potential is vanishing, and Jl(x)≡ dJl(x)/dx. After some rearrangement, the phase shift is cast into the form
is the logarithmic derivative of the radial functions evaluated at ρ smaller but infinitesi-mally close to ρb. Thus far the discussion is general, without being limited to any radial potential profile inside ρb. Taking a unity amplitude for the plane wave, the total wave function outside the range of the scattering potential is given by
Ψoutk (ρ) =
+∞
l=−∞
ileiδl[cos δlJl(kρ)− sin δlYl(kρ)] eilφ. (2.10)
The incident wave is along ˆx, and the incident energy is ε, giving a free particle wave
√ ∗ .
2.1.3 Asymptotic expansion of scattering wave and the differ-ential cross section
The phase shift is related to the differential cross section. Separating out the plane wave, in the total wave function in Eq. (2.10), we have
Ψk(ρ) = eik·ρ+1 2
+∞
l=−∞
il
e2iδl− 1
Hl(1)(kρ) eil(φρ−φk), (2.11)
where incident wave direction k = (k, φk), rather than being restricted to ˆk = ˆx. In the asymptotic region, the above equation is in the form
Ψk(ρ) ≈ eikρ+eikρ
√ρf (φ),
with φ≡ φρ− φk, and
fk(φ) = 2i
πk
+∞
l=−∞
eiδlsin δleilφ, (2.12)
is the two dimensional scattering amplitude [28]. The scattering differential cross sections is givne by D(φ)≡ |f(φ)|2.