---
abstract: '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 $g$-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.'
author:
- 'D. Csontos'
- 'U. Z[ü]{}licke'
title: Tailoring hole spin splitting and polarization in nanowires
---

Engineering spin splitting of charge carriers in semiconductor nanostructures may open up intriguing possibilities for realizing spin-based electronics [@sciencerev] and quantum information processing [@lossbook]. Due to the generally strong dependence of $g$-factors on band structure [@roth:pr:59], 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 [@roland:prl:00; @uz:prl:06; @pryor:prl:06; @haug:prl:06]. Recent advances in fabrication technology [@nanoWireRev2; @nanoWireRev1; @defran:nlett:06; @janik:apl:06; @johan:natmat:06; @dick:jcg:07; @piccio:apl:05; @romain:apl:06; @oleh:apl:06] have created opportunities to investigate hole spin physics in semiconductor nanowires made from a range of different materials.

  ------------ ------------ ------------ ------------ ------------ ------------
    ZnTe/ZnS     AlAs/AlP       AlSb         CdTe       GaN/AlN      GaAs/InP
    0.28[^1]     0.31[^2]    0.32$^{b}$   0.34$^{a}$   0.36$^{b}$   0.37$^{b}$
       Ge          InN          GaSb         InAs         InSb         GaP
   0.38$^{a}$   0.40$^{b}$   0.41$^{b}$   0.45$^{b}$   0.46$^{b}$   0.48$^{b}$
  ------------ ------------ ------------ ------------ ------------ ------------

  : \[gammamatdep\] Relative spin-orbit coupling strength $\gamma=
  \gamma_{s}/\gamma_{1}$ in the valence band of common semiconductors. Here $\gamma_{s}=(2\gamma_{2}+3\gamma_{3})/5$, and $\gamma_{1,2,3}$ denote the Luttinger parameters [@luttham2].

In contrast to previous theoretical work [@kossut:prb:00; @kita:prb:06; @xia:jpd:07; @uz:prb:07b] 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 $\gamma$ quantifying spin-orbit coupling in the valence band can be defined in terms of the effective masses $m_{\text{HH}}$ and $m_{\text{LH}}$ associated with the HH and LH bands [@warpNote], respectively: $2 \gamma = (m_{\text{HH}} - m_{\text{LH}})/\left(m_{\text{HH}} + m_{\text{LH}}
\right)$. Table \[gammamatdep\] lists values for $\gamma$ in common semiconductors and states its relation to basic band-structure parameters [@luttham2]. A large part of the theoretically possible range $0\le \gamma \le 1/2$ 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 $\gamma$, 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 [@luttham2] in the spherical approximation [@lip:prl:70] for the top-most bulk valence bands. Including the bulk Zeeman term $H_{\text{Z}} =-2
\kappa \mu_{\text{B}} B \hat{J}_{z}$, the Hamiltonian is given by $$H=-\frac{\gamma_{1}}{2m_{0}}p^{2}+\frac{\gamma_{s}}{m_{0}}\left [ ({\mathbf p}\cdot \hat
{\mathbf J})^{2}-\frac{5}{4}p^{2}{\mathbf 1}_{4\times 4} \right ]+H_{\text{Z}} \quad .
\label{Hamiltonian}$$ Here ${\mathbf p}$ is the linear orbital momentum, $\hat{\mathbf J}$ the vector of spin-3/2 matrices, $m_{0}$ the electron mass in vacuum, $\gamma_{s}=(2\gamma_{2}+3 \gamma_{3})
/5$ in terms of the Luttinger parameters [@luttham2], $\mu_{\text{B}}$ is the Bohr magneton and $\kappa$ the bulk hole $g$-factor. We neglect the small anisotropic part of the bulk-hole Zeeman splitting. A hard-wall confinement in the $xy$ plane defines the quantum wire with either cylindrical or square cross-section. Our method for finding the zone-center subband edges and calculating their $g$-factor $g^\ast$ in a magnetic field parallel to the wire axis has been described elsewhere [@uz:prb:07b; @uz:physe:07b]. 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 $g$-factor. This universal regime, which is characterized by $g^\ast$ scaling with $\kappa$ and being independent of wire diameter, is accessible in real nanowire systems [@defran:nlett:06] where $\kappa$ is enhanced by the *p-d* exchange interaction with magnetic acceptor ions [@dietl:prb:01]. Figure \[orbVSkappa\] illustrates that, for the highest (i.e., closest to the top of the valence band) GaAs hole-wire levels, only a moderate enhancement of $\kappa$ is needed to quench orbital contributions to the $g$-factor. Similar results are obtained for other materials. In the following, we focus entirely on the properties of hole-wire subband-edge $g$-factors in the universal regime where orbital contributions can be neglected.

![\[orbVSkappa\] (Color online) Effective $g$-factors for the six highest zone-center subband edges in a GaAs wire with square cross-section, plotted as a function of the bulk-hole $g$-factor $\kappa$. An order of magnitude enhancement in $\kappa$ leads to saturation, in effect quenching orbital contributions to the Zeeman splitting.](fig1){width="3in"}

![\[gFactResults\] (Color online) Effective $g$-factors for the ten highest zone-center subband edges in cylindrical hole nanowires, calculated for various spin-orbit coupling strengths.](fig2){width="3in"}

Our results are summarized in Figure \[gFactResults\] where we show $g$-factors for the ten highest zone-center subband edges in cylindrical hole nanowires, calculated for various spin-orbit coupling strengths $\gamma$. 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 $g$-factor; namely $6\kappa$ and $2\kappa$ for the HH and LH states, respectively. Evidently, our results are quite different. Firstly, for any given material, the $g$-factor values vary strongly between the different wire-subband edges, some levels even displaying vanishing $g$-factors. Such seemingly random fluctuations can be explained [@uz:prb:07b; @uz:physe:07b] by nontrivial microscopic hole spin-polarization profiles of wire-subband bound states. Large $g$-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 $g$-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 $g$-factor can vary substantially between different materials. For some subbands, e.g., the third and seventh, the $g$-factors span almost the entire range of values between $0$ and $6
\kappa$. For other subbands, $g$-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 $g^{\ast}$ as the relative spin-orbit coupling strength $\gamma$ is varied.

![\[SpinPolProfile\] (Color online) Squared normalized spin-3/2 dipole (spin-polarization) density, $\rho_{1}^{2}(r)/
\rho_{0}^{2}(r)$, for (a) the highest subband with $F_z=1/2$, and (b) the second-highest subband with $F_z=3/2$. The values of spin-orbit coupling parameter $\gamma$ and corresponding $g$-factor $g\equiv g^\ast/\kappa$ are indicated.](fig3){width="3in"}

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 $\gamma$ 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.   for details of the formalism. In particular, we consider the radial variation of the normalized hole-spin dipole density, denoted by $\rho_{1}^{2}/\rho_{0}^{2}$, which provides a measure of the local hole spin polarization. A uniform value of $\rho_{1}^{2}/\rho_{0}^{2}=9/5$ ($1/5$) indicates a HH (LH) state characterized by a ${\hat J}_z$-projection quantum number $\pm 3/2$ ($\pm 1/2$). As previously discussed, Zeeman splitting for such a state in a magnetic field parallel to the $z$ axis arises with effective $g$-factor $6\kappa$ ($2\kappa$) [@luttham2]. Figure \[SpinPolProfile\] shows the radial spin-polarization profiles $\rho_{1}^{2}(r)/\rho_{0}^{2}
(r)$, for the highest hole-wire subband edges with (a) $F_z=1/2$, and (b) the second-highest subbands with $F_z=3/2$, for different representative values of $0.28\leq \gamma \leq 0.48$. Here, $F_z$ is the eigenvalue of $\hat J_z + \hat L_z$, i.e., the sum of the $z$ components of spin and orbital angular momentum, which is the good quantum number labelling wire-subband bound states [@sercel:apl:90; @uz:prb:07b]. Deviations of the hole-spin polarization from the values $9/5$ and $1/5$ is an indication of the, in principle, ever-present HH-LH mixing in hole wires.

Interestingly, states with $F_z=1/2$ that form the highest subband edge in systems with $\gamma\le 0.37$ are quite close to a pure LH character, having $\rho_{1}^{2}(r)/\rho_{0}^{2}(r)
\approx 0.2$ across most of the wire radius. However, a continuously increasing trend to develop a HH-LH texture is exhibited for larger $\gamma$. As can be seen, this feature is concomitant with a drastic reduction of the $g$-factor from its value close to $2\kappa$ that is expected for pure LH states. A related trend is exhibited by the highest subband edges with $F_z=3/2$ (not shown here) where, for small values of $\gamma$, the normalized dipole moment is close to the value $9/5$ corresponding to a pure HH state. With increasing $\gamma$, however, the dipole moment is increasingly suppressed. The $g$-factors show a corresponding monotonous suppression, from values close to $6\kappa$ to values close to 0.

![\[avGfacts\] (Color online) Mean g-factors $g_{\text{av}}^{\ast}=\frac{1}{N}\sum_{i=1}^{N} g^\ast_{i}$, obtained by averaging over the $N$ highest wire levels, plotted as a function of relative spin-orbit coupling strength $\gamma$. Inset: Wire geometry and orientation of the magnetic field.](fig4){width="3in"}

In contrast to the previous two examples, a very nonmonotonous behavior as a function of $\gamma$ is observed for the second-highest subband edge with $F_z=3/2$. See Fig. \[SpinPolProfile\](b) where, for small $\gamma$-values, suppressed polarization profiles correlate with very small effective $g$-factors. As $\gamma$ 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 $\gamma=0.37,0.41$ in Fig. \[SpinPolProfile\](b).\] The corresponding $g^\ast$ values come close to $6\kappa$. For yet higher values of $\gamma$, the polarization is again suppressed, with concomitantly vanishing $g$-factors.

A general comparison of polarization profiles for various subband edges with their $g$-factors shows that, as the hole-spin dipole moment vanishes and/or HH-LH mixing in the radial profile increases, $g^\ast$ is increasingly suppressed. Thus, a direct correlation emerges between the relative spin-orbit coupling strength $\gamma$, the hole-spin polarization, and the Zeeman spin splitting. However, on average, the hole-spin polarization and effective $g$-factors decrease as the relative spin-orbit coupling strength $\gamma$ is increased. This is illustrated by the calculated mean $g$-factors shown in Fig. \[avGfacts\]. Such mean values will describe Zeeman splitting in experimental situations where single wire subbands are not resolved. Extrapolating to $\gamma=0.38$, which corresponds to Ge, the value found is consistent with the hole $g$-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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In general, the HH and LH effective masses have a slight dependence on the [*direction*]{} of wave vector $\mathbf k$. We neglect this, typically very small, band warping in the following.

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In the following, we present results for $0.28 \le \gamma \le 0.48$ because nanowires have recently been grown from ZnTe [@janik:apl:06] ($\gamma=0.28$), GaAs [@defran:nlett:06] ($\gamma=0.37$), Ge [@stefanoSUB] ($\gamma=0.38$), InAs [@dick:jcg:07] ($\gamma=0.45$), and GaP [@johan:natmat:06] ($\gamma=0.48$).

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S. Roddaro, A. Fuhrer, C. Fasth, L. Samuelson, J. Xiang, and C. M. Lieber, arXiv:0706.2883 (unpublished).

[^1]: From Ref.  

[^2]: From Ref.
