Observation of orientation- and $k$-dependent Zeeman spin-splitting in hole quantum wires on (100)-oriented AlGaAs/GaAs heterostructures

We study the Zeeman spin-splitting in hole quantum wires oriented along the $[011]$ and $[01\bar{1}]$ crystallographic axes of a high mobility undoped (100)-oriented AlGaAs/GaAs heterostructure. Our data shows that the spin-splitting can be switched `on' (finite $g^{*}$) or `off' (zero $g^{*}$) by rotating the field from a parallel to a perpendicular orientation with respect to the wire, and the properties of the wire are identical for the two orientations with respect to the crystallographic axes. We also find that the $g$-factor in the parallel orientation decreases as the wire is narrowed. This is in contrast to electron quantum wires, where the $g$-factor is enhanced by exchange effects as the wire is narrowed. This is evidence for a $k$-dependent Zeeman splitting that arises from the spin-3/2 nature of holes.


I. INTRODUCTION
The use of spin instead of charge to carry information is a central goal in the fields of spintronics and quantum information, generating significant interest in routes to efficient spin manipulation in semiconductor devices 1,2 . Low dimensional hole systems in p-type AlGaAs/GaAs heterostructures hold considerable potential because the much stronger spin-orbit coupling in holes 3 may lead to devices where spin can be manipulated electrostatically 4,5 . The strong spin-orbit coupling also presents some important fundamental physics questions, including how the peculiar spin-3/2 nature of holes 6 is manifested in the experimentally observable properties of lowdimensional GaAs hole devices 7,8,9,10,11,12,13,14,15,16,17 .
Experiments to date have focussed almost solely on devices fabricated in (311)-oriented AlGaAs/GaAs heterostructures. The Zeeman spin-splitting in twodimensional (2D) hole systems formed in these heterostructures is highly anisotropic 7,8 , due to spin-orbit coupling and the low symmetry of (311) surface. Recent studies have also revealed a significant anisotropy in the Zeeman spin-splitting in one-dimensional (1D) hole systems fabricated on the (311) heterostructures 15,16,17 , but it is not trivial to separate the competing influences of 1D confinement and 2D crystallographic anisotropy on the spin-splitting 17 . Hole systems fabricated on higher-symmetry planes such as (100) are not subject to such complex crystallographic effects, and are therefore a much better candidate for studying the spin physics of 1D hole systems. To achieve high quality 1D hole systems we use semiconductor-insulatorsemiconductor field-effect transistor (SISFET) devices, where a 2D hole system is 'induced' using a voltage applied to a degenerately-doped semiconductor gate rather than through modulation doping 18,19 . Klochan et al.
have used this approach to fabricate 1D hole systems with highly stable gate characteristics and clear conductance quantization 20 , and recently extended it to study the Zeeman spin-splitting anisotropy in 1D hole systems in (311)-oriented heterostructures 17 .
In this paper, we extend this SISFET-based approach to study the Zeeman spin-splitting in hole quantum wires oriented along the [011] and [011] directions of a (100)-oriented heterostructure. The crystallographic anisotropy that complicates transport studies of quantum wires on (311)-oriented heterostructures 16,17 does not occur in these devices. Instead, we find that the Zeeman spin-splitting is finite when the applied magnetic field B is oriented parallel to the wire, and nearly zero when B is oriented perpendicular to the wire. This behaviour is almost identical for both orientations of the wire relative to the dominant in-plane crystallographic directions. The ability to switch the spin-splitting 'on' or 'off' simply by rotating the applied magnetic field through 90 • may have useful spintronic applications. Finally, for B parallel to the wire, we observe k-dependent spin-splitting, where g * decreases as the wire is made narrower, in marked contrast to 1D electron systems, where g * instead increases as the wire becomes more one-dimensional 21,22 . This finding is reminiscent of the absence of exchange enhancement effects for 2D hole systems in (100)-oriented heterostructures 10 . system induced at the AlGaAs/GaAs interface for topgate voltages V T G < −0.1 V. Measurements of a separate unpatterned Hall bar of the same heterostructure gave a peak mobility µ = 4.8 × 10 5 cm 2 /Vs at a density p = 1.3×10 11 cm −2 and a temperature T = 100 mK. The device studied here consists of two orthogonal 400 nm long quantum wires, as shown in Fig. 1 (inset), defined by electron-beam lithography and shallow wet etching of the cap layer. Each wire has three gates, a top-gate used to control the density, and two side-gates used to narrow the wire. The two wires can be measured independently and are oriented along the [011] and [011] crystallographic directions of a Hall bar running along the [011] direction. The two wires are denoted as QW011 and QW011, respectively. The quantum wires were measured in a dilution refrigerator with a base temperature of 20 mK using standard a.c. lock-in techniques with an excitation voltage of 50 − 100 µV at a frequency of 17 Hz. Measurements were obtained at a top-gate voltage V T G = −0.80 V, which corresponds to 2D hole density of 2.56 × 10 11 cm −2 .
The width of the wire and its conductance G can be gradually reduced by applying a positive voltage V SG to the two side-gates. For both wires, we observe the well known 'staircase' of quantized conductance plateaus 23 as the wire is narrowed by increasing V SG , with the wire 'pinching off' at V SG ∼ 1.5 V. The similar pinch-off voltages indicate that the two wires are almost identical, with similar dimensions and confining potentials. The accurate quantization of the plateaus at G = n×2e 2 /h, where n is the number of occupied 1D subbands, confirms that transport through the wires is ballistic 20,23 . Moving from left to right in Fig. 1 corresponds to strengthening the 1D confinement, taking the wire from being only quasi-1D (large n and G) towards the 1D limit (small n and G).
We study the spin properties of the hole quantum wires by measuring the Zeeman spin-splitting for different ori- entations of the wire and magnetic field with respect to the crystallographic axes. To obtain the g-factor for the various 1D subbands n, we use a technique that compares the 1D subband splitting due to an in-plane magnetic field 24 (see Fig. 2) and an applied d.c. source-drain bias 25 (see Fig. 3). These two sets of measurements are repeated in two cool-downs to allow for rotation of the sample with respect to the magnetic field, thus providing data for the four different combinations of wire and magnetic field orientation with respect to the crystallographic axes.

A. 1D subband spacings and source-drain bias measurements
The 1D subband spacing of the wires is obtained by adding a d.c. bias V SD to the 20 µV a.c. bias used to measure the conductance. In Fig. 2(a) we plot the transconductance dg/dV SG , where g = dI/dV is the differential conductance, as a colour-map against V SG and V SD using data obtained from QW011. Figure 2(b) shows the conductance G vs V SG measured at V SD = 0 V and corresponds to taking a vertical slice through the center of the colour-map in Fig. 2(a). The dark regions in Fig. 2(a) correspond to high transconductance (risers between plateaus) and the bright regions correspond to low transconductance (the plateaus themselves). Thus the dark regions indicate when a particular 1D subband crosses the Fermi energy. As V SD is increased, the plateaus at multiples of 2e 2 /h evolve into plateaus at odd multiples of e 2 /h. The subband spacing ∆E n,n+1 = eV SD is obtained from the source-drain bias where adjacent transconductance peaks cross (i.e., from the dark regions at non-zero V SD ). The subband spacings for the two wires are plotted in Fig. 2(c), and increase monotonically from ∼ 100 to ∼ 300 µeV as the wire is made narrower and more one-dimensional. The subband spacings for the two wires agree to within 10 µeV, again highlighting the similarity of the two wires fabricated along different crystallographic axes.

B. Zeeman spin-splitting measurements
The effect of an in-plane magnetic field B on the 1D subbands is shown in Fig. 3(a-d) for different orientations of the quantum wire and magnetic field. In each case we plot a colour-map of the transconductance dg/dV SG versus B and V SG , with the dark regions marking the 1D subband edges (high transconductance corresponding to the risers between conductance plateaus). The superimposed white dashed lines in Fig. 3 are guides to the eye tracking the evolution of the 1D subbands with B. Figure 3 shows that there is only a Zeeman splitting of the 1D subbands if B is aligned along the wire, independent of the crystallographic orientation of the wire: If the field is aligned perpendicular to the wire, as in Figs. 3(a) and (d), then the Zeeman spin-splitting is extremely weak. In Fig. 3(d) no splitting is evident up to the highest fields available in the experiment B = 10 T, whilst in Fig. 3(a) some splitting is just apparent near B ∼ 10 T. In stark contrast, if B is aligned parallel to the wire, as in Figs. 3(b) and (c), then the Zeeman spinsplitting is quite strong with clear splitting evident at quite modest fields B ∼ 1 T, crossings between adjacent subbands at moderate fields B ∼ 5 T, and ultimately, crossings between subbands differing in n by two at high fields B ∼ 10 T. The directional-dependence of the Zeeman spin-splitting in these (100)-oriented quantum wires is much simpler than in wires fabricated on (311)-oriented heterostructures, where a complex interplay between 1D confinement and 2D crystallographic anisotropy is observed 12,16,17 .
C. Obtaining the g-factors for the four magnetic field and wire orientations We now extract the effective Landé g-factors 12,17 . When g * is relatively large, it can be obtained by measuring the field B C (n) at which the spin down level of the n th subband crosses the spin-up level of the n + 1 th subband in Fig. 3. This crossing field, combined with the corresponding d.c. bias V C SD where the n and n + 1 th subbands cross in Fig. 2, gives: Data obtained in this way are plotted as solid symbols at (n + 1)/2 in Fig. 4, since they represent the average g-factor for the two subbands. When the spin-splitting is small, as in Figs. 3(b) and (d), we can only measure an upper bound on g * -i.e., g * must sit between zero and this upper bound otherwise the spin-splitting would be resolvable. We determine this upper bound from the width ∆V SG of the transconductance peak in the colour-map at B max = 10 T, which would be the maximum possible splitting if it could be resolved. We convert this width into a splitting rate due to the field ∂V SG /∂B = ∆V SG /B max , and combine it with d.c. biasing data ∂V SG /∂V SD to obtain the upper bound as: These upper bounds are indicated by the hatched regions in Fig. 4.

IV. DISCUSSION
In order to discuss the two key results in Fig. 4, namely the g-factor anisotropy and the decrease of g * as the wire is made narrower, it is first necessary to review some of the complexities of Zeeman splitting in the presence of spin-orbit coupling.
For electrons in free space, an applied magnetic field causes the spins to align along B, with a spin splitting ∆E = ± 1 2 gµ B B, with g = 2.
In the presence of strong spin-orbit coupling, the projections of spin and orbital angular momenta L and S are no longer good quantum numbers. Only the total angular momentum J = L + S is conserved, and an applied magnetic field causes J to align along B. However if the electron is in a nonsymmetric environment, such as a polar GaAs crystal, a quantum well, or a quantum wire, the quantisation axis for J does not automatically align with the applied B and it is rarely possible to find eigenstates of both B and J. This complicates the theoretical analysis of spin splitting considerably, since the microscopic details of the host crystal and the confinement due to the quantum well/wire must all be taken into account.

A. Zeeman splitting in 2D and quasi-1D holes
The upper-most valence band in bulk GaAs consists of 'heavy-hole' (HH) and 'light-hole' (LH) branches that are degenerate at the valence band edge (k = 0). Confinement to a quantum well breaks this HH-LH degeneracy, such that only the lowest HH subband (m j = ± 3 2 ) is occupied in a 2D hole system. However a residual HH-LH coupling at finite wavevector not only results in highly non-parabolic bands, but also plays a significant role in determining the electronic properties in lowerdimensional hole structures.
In the simplest approximation, the 2D confinement forces the quantisation axis for J to point out of the 2D plane. To lowest order there is only a spin-splitting of the HH states if B is applied perpendicular to the quantum well, since B.J = 0 for in-plane magnetic fields 7 . In practise however, the cubic crystalline anisotropy terms 6 , as well as higher order terms in the in-plane wave-vector k , can result in a finite in-plane Zeeman splitting. For quantum wells on the (311) GaAs surface, the cubic anisotropy terms result in a linear in B spin splitting at k = 0. For (100) oriented quantum wells the zerothorder contributions due to cubic crystalline anisotropies are absent 7 , but a substantial linear spin-splitting can still be achieved due to LH-HH mixing at k = 0, as discussed in §7.4 of Ref. 6 . Because the Zeeman splitting on (100) surfaces arises from k dependent LH-HH mixing, it is hard to define a g-factor for (100) 2D holes since g * must be averaged over all occupied states, and is strongly dependent on carrier density. One of the main advantages of quasi-1D systems compared to 2D systems is the ability to perform energy spectroscopy, and thereby measure the g-factor directly 21 . (blue hatched region). In the latter two cases, hatching is presented because at best we can determine the upper bound on g * as minimal spin-splitting is observed up to B = 10 T (see text).
We can predict the expected spin splitting in our quantum wire using a quasi-1D model in which we take the 2D results in Ref. 6 and add on the effects of quantisation of the transverse wave-vector by the 1D confinement. We define the components of the wavevector k = (k l , k t ) with respect to the axis of the quasi-1D wire, where k l and k t are the in-plane wave-vector components parallel (longitudinal) and perpendicular (transverse) to the quantum wire. In the experiments on 1D holes, the spinsplitting is measured at the 1D subband edges, where k l = 0. Since k t is quantised by the lateral 1D confinement, we can express the g-factor of the n th 1D subband as: Here k n is the quantised transverse wavevector k t of the n th 1D subband, and we have rotated the expressions in Eqn 7.22 of Ref. 6 to align along the [011] and [011] axes. The subscripts on g * indicate the direction of the wire relative to the crystal and the field relative to the wire, respectively. The terms γ 1 , γ 2 and γ 3 are Luttinger parameters 26 and Z 1,2 are LH-HH coupling terms (see p. 147 of Ref. 6 ). The first inference we can draw from Eqns 3 and 4 is that the g-factor for both [011] and [011] quantum wires should exhibit the same anisotropy with respect to the magnetic field, i.e., g * /g * ⊥ is the same for both wires. This is evident in Figs. 3 and 4. For both QW011 and QW011, g * is the same for B parallel to the wire (see Fig. 3(b) and (c)) and very small for B perpendicular to the wire (see Fig. 3(a) and (d)). This behaviour is quite different to quantum wires on (311) surfaces, where the anisotropy depends both on the orientation of the quantum wire with respect to the magnetic field and on the orientation of the field with respect to the crystal axes 17 .
However the quasi-1D theory disagrees with experiment on whether g * > g * ⊥ or g * < g * ⊥ . Using expressions for Z 1,2 for square quantum wells 6 and GaAs bandstructure parameters, Eqns 3 and 4 predict g ⊥ > g . The experimental data exhibits exactly the opposite trend, g ⊥ > g , as shown in Fig. 4. This is a surprising result, and we have repeated our experiment to confirm that this is indeed the case, obtaining identical results (to within 10 %). We can only surmise that this discrepancy lies in the dependence of Z 1,2 on the quantum-well confinement, as our 2D holes are confined in a triangular potential well at a single heterojunction, not in a square quantum well. Unfortunately Z 1,2 are not available for a self-consistent triangular quantum well.
A second conclusion we can draw from Eqns 3 and 4 is that in the quasi-1D limit, the g-factor of the wires should decrease with decreasing k 2 n . In the 1D constriction k n is given by the difference between the Fermi energy in the 2D reservoirs E 2D F and the bottom of of the 1D saddlepoint potential 27 . At large subband index k n approaches k 2D F , and g * should saturate to a constant value (as seen in Fig. 4). At small subband index the wire becomes narrower, the saddle-point rises up in energy, k n decreases and so does g * . Additionally, the increase in 1D confinement increases the LH-HH separation ∆E LH,HH , which reduces the magnitude of the higher order Zeeman terms, and thereby reduces g * . The decrease in g * with decreasing subband index is consistent with the data shown in Fig. 4, but is different to almost all other studies of 1D systems, where a strong exchange enhancement of g * is observed at low subband index 12,21,22,28,29 . It is also different to previous studies of 1D holes in (311) quantum wells, where the Zeeman splitting is believed to be due to a combination of crystal anisotropies at large n and re-orientation of the quantisation axis for J at small n.

B. Zeeman splitting in the 1D limit
In the quasi-1D description, the 1D confinement is a weak perturbation, so thatĴ, the quantisation axis for J, remains perpendicular to the 2D system. The lowest order terms for the spin-splitting are zero, since B.J = 0, and g * is only finite due to the higher order k terms. It is thus interesting to consider what happens in the 1D limit where the wire width becomes equal to the width of the 2D confinement. In this caseĴ is aligned with the wire axis and the lowest order spin-splitting is large and positive for B applied along the wire, but is zero for B perpendicular to the wire. This is consistent with the anisotropy measured in Fig. 4, where g * > g * ⊥ . If the 1D confinement is causing a re-orientation ofĴ, then one might expect that g * would increase as the system is made more 1D, as seen in previous experiments on (311) based hole wires 12,17 . Furthermore, it is predicted theoretically that the sign of g * is opposite for wires in the [011] and [011] orientations for a square 2D confinement, so one would expect the measured g-factors for the two quantum wires to show different behaviour as we go from the quasi-1D to the 1D limit.
Thus the quasi-1D model can explain the observed dependence of g * on k , but not the anisotropy of g * , whereas the 1D-limit model can explain the observed anisotropy of g * , but not the dependence on k , since the latter depends on the quasi-1D model. To be able to resolve this conundrum it will be essential to perform more detailed calculations in the quasi-1D limit for realistic 2D confining potentials.

V. CONCLUSION
In conclusion, we have studied the Zeeman spinsplitting in hole quantum wires fabricated in (100)oriented AlGaAs/GaAs heterostructures, and find two new results: Firstly, if the applied in-plane magnetic field B is aligned along the wire, we see strong spin-splitting, and if it is perpendicular to the wire, then we observe negligible spin-splitting up to B = 10 T. This behaviour is independent of the orientation of the wire on the heterostructure surface. Although this latter finding is consistent with theoretical predictions, our finding that the spin-splitting is maximized for B aligned along the wire is at odds with a quasi-1D theory, which predicts maximum splitting instead for B perpendicular to the wire. At present the only solution to this disagreement may lie in the sensitivity of the theoretical calculations on the 2D confining potential -theoretical results have only been obtained for a square potential well so far, whereas the single heterojunction in our device leads to a more triangular confinement. Secondly, we report a decreasing g * as the 1D confinement is increased, which is at odds with previous experiments of both 1D electron systems in GaAs and InGaAs 21,22,29 , and 1D hole systems in (311)-oriented GaAs heterostructures 12,28 . This suggests that despite the strong hole-hole interactions there is no exchange enhancement in our 1D wires, consistent with recent measurements of (100)-oriented 2D hole systems 10 . These results highlight the complex and interesting spin-physics associated with j = 3 2 hole systems, and suggest that much more theoretical work is needed before we understand the physics of holes, even on 'simple' (100) surfaces.