The application of SIA in spintronics

Insights into the spin-splitting mechanisms due to SIA are fundamental for spintron- ics [51]. As defined by ˇZuti´c and Das: “spintronics is a multidisciplinary field whose central theme is the active manipulation of spin degrees of freedom in solid-state systems” [28]. The following outlines qualitatively the main aims of spintronics: i) finding the right way to generate and control the spin polarisation of carriers; ii) being able to sustain or suppress this polarisation over a certain distance inside materials; iii) detecting the spin-polarisation.

While the spin of carriers is already used in realising magnetic hard disk drives (commercially developed by IBM and based on the giant magnetoresistance effect [28]), one of the most famous ideas in spintronics is the spin field-effect tran- sistor (spin-FET) proposed by Datta and Das in 1990 [62]. A spin-FET scheme is depicted in Figure 2.17 and is reminiscent of the structure of MOSFETs [Figure 2.9(a)] or QW-FETs [Figure 2.10(a)]. The differences in a spin-FET are: i) ferro- magnetic source- and drain-contacts and ii) channels which allow the propagation of a spin-polarised current together with the chance to manipulate it by applying a gate voltage. As in ferromagnets (like iron) the density of states for carriers with a certain spin is, atEF, higher than the one of carriers with opposite spin, they can be

used to inject and detect a spin-polarised current. Apart from the existence of spin- split states, by referring to Figure 2.17, the main requirement for spin-FET channels

is the possibility of sustaining a spin-polarised current (top part of the channel) or depressing it by applying a gate voltage (bottom). This is the case of the SIA spin- splitting in 2D-systems outlined in Section 2.3. As shown in Figure 2.13(b,c) the SIA provides an effective magnetic field which would result in a precession of the carrier’s spin along a certain distanceL, that is a phase-shift ∆θ(L) =π in Figure 2.17. From SIA assumptions, Datta and Das obtained the following relation [62]:

∆θ= 2m

∗_φ

B(VG)L

~ , (2.58)

where φB(VG) ≡ α for electrons, βLH for LHs and βHH for HHs, is the electrically tunable Rashba coefficient. Almost 20 years after the idea of Datta and Das, Kooet al. demonstrated the control of the spin-precession of a 2DEG within an InAs-QW by means of a gate voltage [63]. As expected from Equation (2.58), at a certain VG the spin-precession resulted in a π-rotation of the spin orientation of a 2DEG

and a voltage difference, measured between the two ferromagnetic electrodes, was observed.

As spintronic devices are based on the chance to inject and detect spin polarised carriers, materials are characterised in terms of spin lifetime and diffusion coefficient which define the spin diffusion length. This indicates the scale over which the spin information can be transmitted with negligible loss of spin polarisation [64]. Gen- erally, the range of spin lifetime is ps toµs and is related to the transport lifetime which, in turn, depends on scattering processes with phonon and defects [28].

Elliott-Yafet mechanism (EY). This mechanism assumes that spin-relaxation is due to the SO-coupling with ions in the crystal, owing to a matrix termVSO,ions = ~2/mc2∇V0 ×p·σ in the Hamiltonian [65], where V0 is the lattice potential. By

considering this further perturbation, carrier’s wave functions are described by a mix of spin-up and spin-down states. The VSO,ions couples states at the same k

but with different spin and band indexes, and its combination with the momentum scattering leads to: [28, 59]:

τs = τtr

(∆g)2. (2.59)

The relation between the spin lifetimeτs and the transport lifetimeτtrscales there-

fore with ∆g = g∗ −g0, where g∗ is the effective g-factor and g0 = 2.022 is the

electron g-factor.

Figure 2.18: Variation of the spin life- time and transport lifetime for 2DHG in a Ge/Si0.5Ge0.5-QW as a function of the hole density (τSO corresponds toτs in the text). Reproduced from [59].

metry assumptions [66] and considers the precession of carriers as a further Hamil- tonian term related to the effective magnetic fields in Figure 2.13. Generally, the DP-mechanism is considered in the regime τsωeff ≤ 1, i.e. when carriers are scat- tered before completing a full precession around the effective magnetic field Beff with intrinsic effective Larmor frequency ωeff [28]. The main result is the inverse proportionality in respect of the SIA, or BIA, spin-splitting energy ∆, as well as to τtr, i.e. [59]:

τs∝

1 ∆τtr

. (2.60)

Spin lifetime for 2DHG in a sGe-QW. In Section 1.2 experimental values of diffusion and spin lifetime were summarised for bulk Ge which have been linked to the EY-mechanism due to the centrosymmetric structure of Ge. Differently from the bulk case, as the DP-mechanism is related to the SIA [28], in MOD Ge-QWs both mechanisms can contribute to the spin lifetime. From the WAL measurements outlined in the previous section for 2DHG in a sGe-QW, Moriyaet al. obtained a comparison betweenτs and τtr which is shown in Figure 2.18. Since the change of

the two lifetimes had an opposite behaviour with increasing carrier density, which also enhances ∆, the authors indicated the DP-mechanism as dominant [59].

After estimatingτtrfor HHs within the characterised sGe-QWs in Chapter 4, Equa-

tions (2.59) and (2.60) will be used in Chapter 4 to qualitatively assign the dominant spin-relaxation mechanism.