Tailoring hole spin splitting and polarization in nanowires
Spin splitting in p-type semiconductor nanowires is strongly affected by the interplay between quantum confinement and spin-orbit coupling in the valence band. The latter’s particular importance is revealed in our systematic theoretical study presented here, which has mapped the range of spin-orbit coupling strengths realized in typical semiconductors. Large controllable variations of the -factor with associated characteristic spin polarization are shown to exist for nanowire subband edges, which therefore turn out to be a versatile laboratory for investigating the complex spin properties exhibited by quantum-confined holes.
Engineering spin splitting of charge carriers in semiconductor nanostructures may open up intriguing possibilities for realizing spin-based electronics Wolf et al. (2001) and quantum information processing Awschalom et al. (2002). Due to the generally strong dependence of -factors on band structure Roth et al. (1959), it is expected that spatial confinement will have an important effect on Zeeman splitting when bound-state quantization energies are no longer negligible compared with the separation of bulk-material energy bands. The degeneracy of heavy-hole (HH) and light-hole (LH) bulk dispersions at the zone center makes the spin properties of valence-band states especially susceptible to such confinement engineering Winkler et al. (2000); Danneau et al. (2006a); Pryor and Flatté (2006); Haendel et al. (2006). Recent advances in fabrication technology Samuelson (2003); Lu and Lieber (2006); Martelli et al. (2006); Janik et al. (2006); Johansson et al. (2006); Dick et al. (2007); Pfeiffer et al. (2005); Danneau et al. (2006b); Klochan et al. (2006) have created opportunities to investigate hole spin physics in semiconductor nanowires made from a range of different materials.
|0.28111From Ref. Dietl et al.,2001||0.31222From Ref. Vurgaftman et al.,2001||0.32||0.34||0.36||0.37|
In contrast to previous theoretical work Kyrychenko and Kossut (2000); Harada et al. (2006); Zhang and Xia (2007a); Csontos and Zülicke (2007) on hole spin splitting in quantum wires, we focus here on the influence of the spin-orbit coupling strength on Zeeman splitting of wire-subband edges. A suitable parameter quantifying spin-orbit coupling in the valence band can be defined in terms of the effective masses and associated with the HH and LH bands warpNote (), respectively: . Table 1 lists values for in common semiconductors and states its relation to basic band-structure parameters Luttinger (1956). A large part of the theoretically possible range is covered by available materials realNote (), enabling a detailed study of the interplay between spin-orbit coupling in the valence band and nanowire confinement. Our theoretical investigation reveals surprising qualitative differences in the hole spin properties of nanowires depending on the value of , showing that spin splitting (and polarization) of zone-center valence-band edges in nanowires is highly tunable and has a complex materials dependence. A detailed understanding of these properties is vital for proper interpretation of optical and transport measurements as well as for the design of spintronic applications involving p-doped semiconductor nanowires.
We use the Luttinger model Luttinger (1956) in the spherical approximation Lipari and Baldereschi (1970) for the top-most bulk valence bands. Including the bulk Zeeman term , the Hamiltonian is given by
Here is the linear orbital momentum, the vector of spin-3/2 matrices, the electron mass in vacuum, in terms of the Luttinger parameters Luttinger (1956), is the Bohr magneton and the bulk hole -factor. We neglect the small anisotropic part of the bulk-hole Zeeman splitting. A hard-wall confinement in the plane defines the quantum wire with either cylindrical or square cross-section. Our method for finding the zone-center subband edges and calculating their -factor in a magnetic field parallel to the wire axis has been described elsewhere Csontos and Zülicke (2007, ). An intriguing universal behavior of wire-subband spin splittings emerges when the bulk-Zeeman term dominates the orbital effects which, in principle, also contribute to the effective -factor. This universal regime, which is characterized by scaling with and being independent of wire diameter, is accessible in real nanowire systems Martelli et al. (2006) where is enhanced by the p-d exchange interaction with magnetic acceptor ions Dietl et al. (2001). Figure 1 illustrates that, for the highest (i.e., closest to the top of the valence band) GaAs hole-wire levels, only a moderate enhancement of is needed to quench orbital contributions to the -factor. Similar results are obtained for other materials. In the following, we focus entirely on the properties of hole-wire subband-edge -factors in the universal regime where orbital contributions can be neglected.
Our results are summarized in Figure 2 where we show -factors for the ten highest zone-center subband edges in cylindrical hole nanowires, calculated for various spin-orbit coupling strengths . A naïve assumption that the hole spin projection parallel to the wire axis should be quantized would lead us to expect to find only two possible values for the -factor; namely and for the HH and LH states, respectively. Evidently, our results are quite different. Firstly, for any given material, the -factor values vary strongly between the different wire-subband edges, some levels even displaying vanishing -factors. Such seemingly random fluctuations can be explained Csontos and Zülicke (2007, ) by nontrivial microscopic hole spin-polarization profiles of wire-subband bound states. Large -factors are found for subband edges with predominantly HH or LH character, whereas subbands with mixed HH-LH character or with vanishing hole-spin polarization have strongly suppressed -factors. We will see below that the intrinsic connection between hole spin splittings and polarizations holds for all materials considered. Secondly, focusing on individual wire levels, it is found that their -factor can vary substantially between different materials. For some subbands, e.g., the third and seventh, the -factors span almost the entire range of values between and . For other subbands, -factors cluster around certain values, as is the case of the first, sixth, and tenth levels. Yet other subbands display a seemingly random sequence of alternatingly increasing and decreasing values of as the relative spin-orbit coupling strength is varied.
The anomalous spin splittings in hole nanowires can be attributed to strong HH-LH mixing that is present even at the wire-subband edges. The relative spin-orbit coupling strength determines this mixing. To be able to characterize the spin properties of individual subband-edge bound states independent of any particular spin-projection basis, we utilize scalar invariants of the spin-3/2 density matrix. See Refs. Winkler, 2004; Csontos and Zülicke, 2007 for details of the formalism. In particular, we consider the radial variation of the normalized hole-spin dipole density, denoted by , which provides a measure of the local hole spin polarization. A uniform value of () indicates a HH (LH) state characterized by a -projection quantum number (). As previously discussed, Zeeman splitting for such a state in a magnetic field parallel to the axis arises with effective -factor () Luttinger (1956). Figure 3 shows the radial spin-polarization profiles , for the highest hole-wire subband edges with (a) , and (b) the second-highest subbands with , for different representative values of . Here, is the eigenvalue of , i.e., the sum of the components of spin and orbital angular momentum, which is the good quantum number labelling wire-subband bound states Sercel and Vahala (1990); Csontos and Zülicke (2007). Deviations of the hole-spin polarization from the values and is an indication of the, in principle, ever-present HH-LH mixing in hole wires.
Interestingly, states with that form the highest subband edge in systems with are quite close to a pure LH character, having across most of the wire radius. However, a continuously increasing trend to develop a HH-LH texture is exhibited for larger . As can be seen, this feature is concomitant with a drastic reduction of the -factor from its value close to that is expected for pure LH states. A related trend is exhibited by the highest subband edges with (not shown here) where, for small values of , the normalized dipole moment is close to the value corresponding to a pure HH state. With increasing , however, the dipole moment is increasingly suppressed. The -factors show a corresponding monotonous suppression, from values close to to values close to 0.
In contrast to the previous two examples, a very nonmonotonous behavior as a function of is observed for the second-highest subband edge with . See Fig. 3(b) where, for small -values, suppressed polarization profiles correlate with very small effective -factors. As is increased, the spin dipole moment of the state increases dramatically, approaching values associated with HH character. [See the dashed-dotted and dashed curves corresponding to in Fig. 3(b).] The corresponding values come close to . For yet higher values of , the polarization is again suppressed, with concomitantly vanishing -factors.
A general comparison of polarization profiles for various subband edges with their -factors shows that, as the hole-spin dipole moment vanishes and/or HH-LH mixing in the radial profile increases, is increasingly suppressed. Thus, a direct correlation emerges between the relative spin-orbit coupling strength , the hole-spin polarization, and the Zeeman spin splitting. However, on average, the hole-spin polarization and effective -factors decrease as the relative spin-orbit coupling strength is increased. This is illustrated by the calculated mean -factors shown in Fig. 4. Such mean values will describe Zeeman splitting in experimental situations where single wire subbands are not resolved. Extrapolating to , which corresponds to Ge, the value found is consistent with the hole -factor measured recently stefanoSUB () in rod-shaped quantum dots fabricated from Ge/Si core-shell nanowires.
DC acknowledges support from the Massey University Research Fund. The authors benefited from useful discussions with P. Brusheim, A. Führer, S. Roddaro, and H.Q. Xu.
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