Resonance of spin dipole

l=0sin[2(δ_−l⁺ − δ⁺_−(l+1))]; the total σ⁺_⊥ is denoted by the black curve.

3.4.3 Resonance of spin dipole

In this section, we present the resonant characteristics of the spin dipole. The spin dipole strength is shown to manifest large enhancement and sign reversal at resonance.

From the lower curve of Fig. 3.4, the resonant features in the spin dipole strength, or kσ_⊥, is shown to carry a peak-dip structure. To show that this resonant feature is indeed remarkable, we plot, for comparison, the spin dipole strength due to a potential disc with the same outer radius a, but having the inner radius b = 0. Denoted by the red line in the kσ_⊥ curve in Fig. 3.4, this line matches very well with the ring-shaped potential result in the non-resonant and low energy regions. From this we also see that quantum resonance has lead to very large enhancement in the spin dipole strength. Furthermore,

0 0.5 1 1.5 2 2.5 3

Figure 3.4: kσ_tr (the upper plot) and kσ_⊥ (the lower plot) versus the chemical potential μ with parameters of a ring-shaped step potential barrier: the potential height V₀ = 1, the inner radius a = 6, and the outer radius b = 9. The red line in the lower plot is kσ_⊥ for a disk-shaped potential barrier with its radius equal to 9.

Fig. 3.4 contains two series of resonant peak-dip pairs. Labeled by (n,l), the first series is the n = 1 series, and starts from l = 1 at energy μ = 0.295 (energy unit E^∗ = 77.1 meV).

The second series is the n = 2 series, and starts from l = 1 at energy μ = 0.94. There is no peak-dip resonant structures in kσ_⊥ for l = 0 because again, the spin dependent term in Eq. (3.6), vanishes for l = 0.

From definitions Eq. (3.16), Eq. (3.21) and the relation Eq. (3.23), we have

σ_⊥ = 1 dependent differential scattering cross section, we have a better physical understanding

Figure 3.5: (a) kσ_⊥ versus the Fermi energy in units of E∗ = 77.1 meV. Parameters of a ring-shaped step potential barrier: the potential height V₀ = 0.75, the inner radius a = 14, and the outer radius b = 20.5. (b)-(d) Blowups of the resonance pair labeled in the same series n = 1 and l = 4, 5, 6, respectively. The dot-dashed (red) curves are the partial summations of kσ_⊥ including only terms involving δ^σ₅. The dashed (blue) curves are kσ_⊥ for V₀ = 0.751. The smallest abscissa in (b)-(d) is 0.0001.

of the spin dipole, which strength is kσ_⊥.

In our numerical examples, physical parameters are chosen according to practical experimental situation and for the material InAs. Parameters units typical for InAs are: electron density n^∗_e = 7.4× 10¹¹cm⁻²; energy unit E^∗ = n^∗_eπ²/m^∗ = 77.1 meV;

m^∗ = 0.023 m_e; k^∗ = 2.16× 10⁸m⁻¹; length unit L = 1/k^∗ = 46.3 ˚A. In these units, the ring-shaped potential has radii a = 14 and b = 20.5, and potential barrier height V₀ = 0.75; mean free path l^∗₀ = 238. The magnitude of electric field is E₀ = 0.1 kV/cm.

The spin dipole strength is fully characterized by kσ_⊥ except for a factor −˜nE/(2π) =

−1.7 × 10¹⁰cm⁻² which is in units of density.

In Fig. 3.5, the enhancement of spin dipole strength is large. Away from resonance,

|kσ⊥| ≈ 10⁻¹, while at resonance its maximum magnitude can be reach up to 3.5. The factor of enhancement reaches to 70. The peak-dip series labeled by n is clearly shown here. Having the largest resonant enhancement, the n = 1 series starts from μ = 0.064, with l = 1, and up to l = 16. The n = 2 series starts from μ = 0.213 and n = 3 from μ = 0.44. There are two other series, with smaller resonant enhancement, the n = 2 and n = 3 series, which starts at μ = 0.213 and μ = 0.44, respectively.

That the resonance features in kσ_⊥ are contributed by the energy dependence of phase shifts δ_l^± associated with one l only can be seen in Fig. 3.5 (b)-(d), which are blowups of Fig. 3.5 (a) at resonance pairs (n = 1, l = 4, 5, 6). The solid curves are the full expression evaluation of Eq. (3.17); the dot-dashed (red) curves are partial summations including terms that only involves l = 5 in Eq. (3.17); and the dashed (blue) curves are full expression evaluation of Eq. (3.17) but for V₀ = 0.751. The dot-dashed (red) curve of partial summation matches very well with the (1,5) peak-dip structure while producing qualitative features for the peak-dip structures (1,4) and (1,6). The former confirms that each peak-dip resonant structure is contributed from terms in Eq. (3.17) associated with one l value only. The partial, or qualitative matches, in Fig. 3.5 (b) and (d) are due to the fact that the summation involving only l = 5 have included partial contribution from l = 4 and l = 6.

We comment on a general feature in a n-series of the peak-dip resonant structures.

Peak-dip pairs in an series trace out an envelope which strength is increasing in low l and decreasing in high l. This demonstrates the competition between the trend of stronger SOI for larger l, and the weakening of the resonance due to the wider resonance width at energies higher than the potential height.

In Fig. 3.1, we present the radial variation of the spin density distribution S_z(ρ), as is given in Eq. (3.13). Energies at (a) 0.3312, and (b) 0.3328, are the peak and dip energies, respectively, of the (1, 5) resonance in Fig. 3.5 (a). The dashed curves are the asymptotes, given by the first term in Eq. (3.18), and the dashed-dot curves are the adjusted ones,

including the second term. Apart from the Friedel oscillations, the dash-dotted curves traces the spin distribution outside the ring remarkably. Fig. 3.1 (a) and (b) reveal that the sign of the spin density can be reversed by a small change in energy Δμ = 0.0016 E^∗, or 0.12 meV. The spin density at ˜ρ = 50, or ρ = 0.23 μm, is (a) −9.276 μm⁻², and (b) 9.568 μm⁻², or in terms of percentage of the electron density n = 2.45× 10¹¹cm⁻², the spin density is (a)−3.79%, and (b) 0.39% of n, respectively. If we take the thickness of the QW to be 100 ˚A, the 3D spin density becomes (a)−928 μm⁻³, and (b) 957 μm⁻³, which is certainly large enough for observation [31]. The spin density at ˜ρ = 100, or ρ = 0.46 μm is half of the above results. Finally in Fig. 3.6, we present the full 2D dependence of the spin dipole S_z(ρ) for the case of μ = 0.3312.

The proposed ring-shaped potential barrier pattern is expected to be within reach of the present experimental capability. Such pattern can be fabricated by recently developed focused ion/molecular beam epitaxy technique for the pattering of the δ-doped layer [32].

The assumed step-like profile in the ring-shaped potential might look oversimplified, but it provide us a clear physical picture for the spin dipole formation. We have considered smooth profile situations. The resonant feature remain intact. This will be discussed in Chapter 5.

3.5 Brief summary

The effect of the in-plane potential gradient SOI has a relatively small effect on the Landauer RRD. The resonance gives rise to double dip structures in the RRD strength.

However for the spin accumulation, the effect of in-plane potential gradient SOI has a large impact on it. There is no spin accumulation without the presence of SOI. With the SOI, quantum resonance leads to large spin dipole strength enhancement and also sign changes. The spin dipole orients perpendicularly to the applied electric field.

Figure 3.6: Spatial distribution S_z(ρ) at the a resonant energy ε = 0.3312 with the geometry of the ring: the potential height V₀ = 0.75, the inner radius a = 14, and the outer radius b = 20.5. The electric field E is applied along positive x-direction.

Spin dipole correction due to effect of extrinsic spin-orbit interaction

In this chapter we explore the correction to the spin dipole picture obtained in the previous chapter, if there are background spin-orbit scatterers (extrinsic SOI effect) in the system.

To obtain the electron distribution in the presence of these background scatterers, our procedure follows closely to that detailed in a paper by H.-A. Engel et al. [8] except that our derivation is 2D whereas the reference [8] is for 3D and they had mentioned 2D results in their footnote. Using this spin dependent nonequilibrium electron distribution, we obtain the spin dipole correction.

4.1 Differential cross section in terms of spin density

operator