Tuning the magnetic moment of charge carriers

The continuing drive to miniaturize device features on integrated circuits is fast approaching a realm where the microscopic quantum-mechanical properties of charge carriers determine electric transport. Spintronics (short for spin-electronics) aims to capitalize on quantum effects by using the intrinsic spin of electrons, instead of their charge, as the principal carrier of information.¹ Spin is one of the most counterintuitive quantum degrees of freedom, and makes the electron appear like a spinning top of vanishing size. Spintronics researchers anticipate advantages such as smaller device features, lower power dissipation, and faster operation compared with conventional electronics. Future progress depends on better understanding of how spin, and its associated microscopic magnetic moment (i.e., the tiny bar magnet carried around by elementary particles), can be addressed and possibly tailored in semiconductor nanostructures.

Interestingly, holes—the positive mobile charge carriers in p-type semiconductors—currently appear to hold the greatest promise for spintronics applications. For one thing, most of the recently fabricated magnetic semiconductors are p-type.² More important, the holes' spin degree of freedom is very sensitive to quantum engineering. When confined to move in just one spatial dimension, as in a nanowire (such as indicated schematically in the right panel of Figure 1), the holes' effective magnetic moments have starkly different values depending on their quantized-motional state, as well as on the wire material.³ Our theoretical results suggest the possibility of engineering hole magnetic moments for spintronics applications,⁴ using as building blocks, for example, nanowires grown by self-assembly.⁵

Electrons in vacuum are spin-1/2 particles: the projection of their spin angular momentum along any axis can take only one of the two quantized values ±ħ/2, where, ħ is Planck's constant. The binary nature of spin makes it attractive for information encoding, transport, processing and storage, but these applications require deterministic control of the spin state of the electron. In a solid, the electrons move in a periodic crystal potential, and the angular momentum of their atomic orbitals mixes with the intrinsic spin-1/2 degree of freedom. As a result of this ‘spin-orbit’ coupling, holes in the bulk of typical semiconductors turn out to behave like spin-3/2 particles and, in the simplest of cases, assume only four quantized values, ±3ħ/2 and ±ħ/2 (schematically indicated in the left panel of Figure 2). When holes are confined to move in a nanowire, the possible hole states are always quantum superpositions of the spin states with ±3ħ/2 and ±ħ/2, thus realizing a continuous range of values for their average spin (indicated in Figure 2 by the blue-shaded region). We also found that the superposition in each quantized-motional state is very sensitive to the nanowire material, as well as the strength and shape of the wire confinement. This implies that confinement and material engineering may provide a versatile tool for controlling the holes' effective magnetic moment.

Figure 1. (left) The g-factor of holes in the lowest three nanowire subband-edge levels, plotted as a function of cross-sectional aspect ratio A≡L_y/L_x. The calculation was performed using gallium arsenide materials parameters. Note the strikingly different values for the three levels (labelled 1, 2, 3) and strong variation with A. (right) Rectangular nanowire with dimensions L_x and L_yconsidered in our work, with applied magnetic field B_z. The wire-induced mixture of spin ±3ħ/2and ±ħ/2components in hole states can be visualized by cross-sectional plots of the spin magnetic moment's spatial profile.

Figure 2. (left) States of a spin-3/2 particle, such as a hole in a semiconductor. States corresponding to quantized spin projection in the z direction, J_z, of ±3ħ/2[±ħ/2] are called heavy holes [light holes]. In a nanowire, the quantized-motional states are superpositions of ±3ħ/2, ±ħ/2spin states, indicated by the blue-shaded region. (right) Cross-sectional spatial profile of hole magnetic moment in nanowire subband levels n=1, 2, 3, and 8, for A=1.

We solved the equivalent of the Schrödinger equation for holes confined to a nanowire and subject to a magnetic field parallel to the wire axis. The states at the edge of the subbands, related to the quantized motion of the confined holes, often determine the optical and transport properties of the wire. We used the levels for these states to calculate the g-factor, which is a measure for the holes' effective magnetic moment. The obtained g-factor varies strongly between different hole subband levels, for different materials used to realize the nanowire,³ and as a function of the wire's cross-sectional aspect ratio (height to width). The latter behavior is illustrated, for the lowest three hole-subband levels, in the left panel of Figure 1. The results predict that straightforward confinement engineering can tune the sign and magnitude of the g-factor very sensitively, and even realize situations with zero g-factor value. These characteristics may be used in future spintronics applications where deterministic spin injection, manipulation, and detection depend on such tunability.

We identified the superposition between ±3ħ/2 spin states and ±ħ/2 spin states, which in a nanowire is unavoidable, as the microscopic origin of g-factor variability. A close correspondence between hole magnetic moments and the degree of quantum superposition can be illustrated by comparing the g-factor and the spatial profile of the hole spin-dipole moment for the lowest three levels. This is shown in the right panel of Figure 2, for a square-cross-section wire. For the second level (2), a highly suppressed g arises in conjunction with an almost uniformly vanishing spin-dipole moment: a situation which arises when there is a ‘perfectly balanced’ superposition of spin states with ±3ħ/2 and ±ħ/2. The first (1) level exhibits a large component of ±ħ/2 states, explaining its g-factor value close to that expected for such states in the bulk. The third (3) and eighth (8) levels have ±3ħ/2 and ±ħ/2 components being dominant in different regions of the wire's cross-section.

Functional spintronics devices will require addressing and manipulating the magnetic dipole moment of charge carriers in semiconductor nanostructures. Our results suggest a promising way to tailor the spin-dipole moment of holes in nanowires. Comparison of our theoretical predictions with measurements will decide whether this approach could be successfully applied in the future.