Measuring hole g-factor anisotropies using transverse magnetic focusing

TL;DR

This paper proposes a method using transverse magnetic focusing to measure the anisotropy of the g factor in heavy hole systems, providing a practical approach to quantify anisotropic spin properties.

Contribution

It introduces a novel measurement technique for g factor anisotropy in heavy hole gases using TMF, demonstrating its feasibility and insensitivity to spin-orbit interactions.

Findings

g factor anisotropy causes measurable variations in TMF spectra

The method is effective at accessible magnetic fields

The approach is robust against additional spin-orbit effects

Abstract

Recent theoretical and experimental results from quasi-one dimensional heavy hole systems have suggested that heavy hole gases have a strongly anisotropic g factor. In this theoretical paper, we propose a method for measuring this anisotropy using transverse magnetic focusing (TMF). We demonstrate that for experimentally accessible fields, the g factor anisotropy leads to a relative variation in the characteristic of spin-splitting of the TMF spectrum which allows for the measurement of the anisotropy of the g factor. We show that this variation is insensitive to additional spin-orbit interactions, and is resolvable with current devices.

Figures

Equations

$${\mathcal{H}}_1=\frac{g_1{\mu }_B}{2}\left(B_{+}{p}_{+}^2{\sigma }_{-}+B_{-}{p}_{-}^2{\sigma }_{+}\right)$$

$${\mathcal{H}}_2=\frac{g_2{\mu }_B}{2}\left(B_{-}{p}_{+}^4{\sigma }_{-}+B_{+}{p}_{-}^4{\sigma }_{+}\right)$$

$${\tilde{g}}_1=g_1{k}_F^2{\tilde{g}}_2=g_2{k}_F^4$$

$$\mathcal{H}$$

$$\dot{\hat{\mathbf{\pi }}}=i\left[\mathcal{H},\hat{\mathbf{\pi }}\right]=m{\omega }_c\hat{\mathbf{v}}\times \mathbf{n}$$

$$\dot{\mathbf{\sigma }}=i\left[\mathcal{H},\mathbf{\sigma }\right]=-\mathcal{B}\left(\hat{\mathbf{\pi }}\right)\times \mathbf{\sigma }$$

$$\mathbf{r}(\theta (t))=\frac{\mathbf{\pi }(\theta (t))\times \mathbf{n}}{e{B}_z}-\frac{\mathbf{\pi }(\theta (0))\times \mathbf{n}}{e{B}_z}$$

$$\mathbf{\pi }=\hslash k_{F,s}(\cos \theta (t),\sin \theta (t),0)$$

$$k_{F,s}=k_F\left(1+\frac{s|\mathcal{B}|}{2{\epsilon }_F}\right)$$

$$|\mathcal{B}|={\mu }_B{B}_{||}\sqrt{{\tilde{g}}_1^2+{\tilde{g}}_2^2+2{\tilde{g}}_1{\tilde{g}}_2\cos (2\theta -2\phi )}$$

$$\mathbf{B}=\left(B_{||}\cos \phi ,B_{||}\sin \phi ,B_z\right)$$

$$|\mathcal{B}|\gg \frac{1}{|\mathcal{B}|}\frac{\partial \mathcal{B}}{\partial t}\sim 2{\omega }_c$$

$$l=y(\theta =\pi )\approx \frac{\hslash k_{F,s}(\theta =0)+\hslash k_{F,s}(\theta =\pi )}{e{B}_z}$$

$$B_z\approx \frac{\hslash k_{F,s}(\theta =0)+\hslash k_{F,s}(\theta =\pi )}{eL}$$

$$\frac{\delta B_z}{B_z}\approx \frac{k_{+}(\phi )-k_{-}(\phi )}{k_F}$$

$$\frac{\delta B_z}{B_z{\Delta }_0}\approx \left(1+\frac{{\tilde{g}}_2}{{\tilde{g}}_1}\cos (2\phi )\right)$$

$$|{\mathcal{B}}_{min}|=\left(|{\tilde{g}}_1|-|{\tilde{g}}_2|\right){\mu }_B{B}_{||}$$

$$4\frac{m}{m^{\ast }}\frac{B_z}{B_{||}}<|{\tilde{g}}_1|-|{\tilde{g}}_2|\approx 1$$

$$3{\omega }_c<{\Delta }_R$$

$$\frac{12}{k_{F}L}<{\Delta }_R$$

Full text

The strength of the coupling of an electron to a magnetic field in free space is defined by the Bohr magneton, $\mu_{B}$, and the electron $g$ factor. Like free electrons, quasi-particles in condensed matter systems couple to an applied magnetic field, however, the form and magnitude of the $g$ factor is strongly influenced by the surrounding material. The “renormalisation” of the $g$ factor can lead to effective $g$ factors for charge carriers in semiconductors orders of magnitude larger than the free space value. With such large values, an applied magnetic field can result in a significant change to transport properties, even in relatively weak magnetic fields. This effect persists in reduced dimensional systems, and can be enhanced or suppressed, and develop asymmetries depending on the confinement. In low dimensional hole systems additional kinematic structures are possible, due to the holes angular momentum being $J=3/2$, with the Zeeman interaction in heavy hole systems depending on both momentum and in-plane magnetic fieldWinkler (2003); Li et al. (2016). Recent experimental and theoretical results suggest a strongly anisotropic in-plane $g$-factor for two and quasi-one dimensional heavy holes systemsMiserev and Sushkov (2017); Miserev et al. (2017).

Measuring the $g$ factor anisotropy is difficult with typical transport techniques. For instance, magnetic (Shubnikov de Haas) oscillations measure the total size of the Fermi surface and therefore have no first order dependence on the anisotropy. Instead Shubnikov de Haas oscillations measure the total Fermi surface area. In this work we propose a method to measure the relative $g$ factors in hole systems, based around transverse magnetic focusing (TMF), which has a long history of use in the measurement of the shape of the Fermi surface in both metals and semiconductors (see Fig. 1)Sharvin (1965); Sharvin and Fisher (1965); Tsoi (1974); V. S. Tsoi and Wyder (1999); Van Houten et al. (1989). When employed in systems with strong spin-orbit coupling, the first magnetic focusing peak is spin split resulting in a “double” focusing spectrumRokhinson et al. (2004); Zülicke et al. (2007); Usaj and Balseiro (2004); Reynoso et al. (2007); Schliemann (2008); Kormányos (2010). This spin-splitting can be directly translated to the strength of the spin-orbit interaction. We make use of this feature, combined with a unique dependence on magnetic field rotations in TMF resulting from the anisotropy in the hole $g$ factor to determine the magnitude of the anisotropy. We then demonstrate that the effect is robust in the presence surface or bulk inversion asymmetry weaker that the Zeeman interaction due to the in-plane field, and provides a straightforward method of determining $g_{1}$ and $g_{2}$.

We begin with kinematic structure leading to this $g$ factor anisotropy in two dimensional hole systems. Holes have a total angular momentum of ${J}=3/2$. At $k=0$, there are four degenerate states, typically denoted as “light”, $\pm 1/2$, and “heavy”, $\pm 3/2$, holes due to the difference in effective mass. When confined to two dimensions only the heavy holes lie below the chemical potential. Coupling $J_{z}=3/2$ to $J_{z}=-3/2$ requires $J_{\pm}^{3}$, which is obtained with the combined action of the Luttinger term, $({\bf P\cdot J})^{2}$,Luttinger and Kohn (1955); Winkler (2003) and Zeeman interaction, ${\bf B\cdot J}$. Two kinematic structures are possible, $P_{+}^{2}B_{+}J_{-}^{3}$, or $P_{+}^{4}B_{-}J_{-}^{3}$. Since only the heavy holes lie below the chemical potential, it is convenient to work in the subspace $\pm 3/2$, spanned by the Pauli matrices, with $J_{\pm}^{3}\rightarrow\sigma_{\pm}$. The kinematic structure is then

[TABLE]

for the $g_{1}$ interaction and

[TABLE]

for the $g_{2}$ interaction. Here $p_{\pm}=p_{x}\pm ip_{y}$ and $\sigma_{\pm}=\sigma_{x}+i\sigma_{y}$. Due to the momentum dependence of the interactions, the coefficients $g_{1}$ and $g_{2}$ are not dimensionless. For the following analytical calculations it is useful to consider the dimensionless coefficients,

[TABLE]

where $k_{F}=\sqrt{2m\varepsilon_{F}}$ is the Fermi momentum, and $\varepsilon_{F}$ is the Fermi energy. Importantly, recent theoretical and experimental work has shown that at experimental accessible densities, $\tilde{g}_{2}$ can be comparable to $\tilde{g}_{1}$Miserev and Sushkov (2017); Miserev et al. (2017).

The Hamiltonian for a hole system subject to these two respective Zeeman interactions due to an in plane magnetic field, and some significantly weaker transverse focusing field is

[TABLE]

where ${\bm{A}}$ is the vector potential. The equations of motion are

[TABLE]

where $\omega_{c}=eB_{z}/m$. Solving these equations of motion requires some approximation of spin-dynamics. In the case of TMF, the appropriate approximation for the resolution of a double focusing peak is adiabatic spin dynamics, where the spin follows the instantaneous effective magnetic field, $\left<{\bm{\sigma}}\right>=s{\cal B}/{|{\cal B}|}$, where $s=\pm 1$ is a pseudo scalar defining the spin projectionZülicke et al. (2007). The resulting semi-classical equation is obtained using the method of Ref. Bladwell and Sushkov (2015); Bladwell and Sushkov (2017a),

[TABLE]

where the momentum $\bm{\pi}$ depends on the spin-state of the hole, and $\theta(t)$ is the polar (running) angle and is a function of time. Physically, this corresponds to the classical cyclotron motion of a quasi particles with different cyclotron radii depending on the spin projection. We present both cyclotron orbits and spin orientations in Fig. 2.

To explore the dynamics analytically, we consider the following approximation for the spin split momentum, $\bm{\pi}$,

[TABLE]

where the total effective magnetic field is

[TABLE]

with an in-plane field angle $\varphi$, and

[TABLE]

being the magnetic field applied to the sample. To satisfy the requirement for adiabatic spin dynamics, we need to ensure that the magnetic field remains sufficiently large. If $\tilde{g}_{1}>\tilde{g}_{2}$ the approximate condition of adiabatic spin dynamics is

[TABLE]

where the factor of $2$ comes for the two rotations of the spin-orbit field for each rotation in momentum space, see Fig. 2. In this adiabatic regime, the classical focusing peak where interference effects are neglected corresponds to an injection angle of $\theta=0$Bladwell and Sushkov (2017b). The focusing length, l, from Eqs. (6) and (Measuring hole $g$-factor anisotropies using transverse magnetic focusing) is

[TABLE]

which is analogous to the case of classical TMF in metals Sharvin and Fisher (1965). We have cast the above result in terms of spatial variation of the peaks; in TMF the detector and collector are fixed, and the focusing field, $B_{z}$ is varied instead,

[TABLE]

For $2\omega_{c}\ll|{\cal B}|\ll\varepsilon_{F}$, using Eqs. (12) and (Measuring hole $g$-factor anisotropies using transverse magnetic focusing),

[TABLE]

where $\delta B_{z}$ denotes the splitting between the spin-split focusing peaks. Fig. 1 corresponds to a fixed $B_{z}$ and a varied focusing length $l$; in a real experiment $l$ is fixed and $B_{z}$ is varied, see Fig. 3.

We are now in a position to explore the angular dependence of in-plane field response in the TMF spectrum. We consider the case of a relatively short focusing length, of $l=1000$nm, with a hole density $n=1.65\times 10^{11}$cm*-2*. Let us start with the case where all other spin-orbit interactions have been tuned to be small. For quantum wells grown along high symmetry axes, this is a reasonable approximation. For the case where $\tilde{g}_{1}\gg\tilde{g}_{2}$, by expanding in terms of $\tilde{g}_{2}/\tilde{g}_{1}$ we obtain the following approximate analytical expression for the angular dependence,

[TABLE]

where we have introduced the dimensionless splitting, ${\Delta_{0}}=\tilde{g}_{1}\mu_{B}B_{||}/\varepsilon_{F}$. In Fig. 4 we plot the fractional focusing field splitting, $\delta B_{z}/B_{z}\Delta_{0}$, as a function of the in plane field angle, $\varphi$. Here $\tilde{g}_{1}=-2.5$ and $\tilde{g}_{2}=1$. For GaAs quantum wells, $|\tilde{g}_{1}|,|\tilde{g}_{2}|<3$, dependent on the manner of the confinementMiserev et al. (2017).

In practice other spin-orbit interactions due to bulk and surface inversion symmetry may not be small. To examine the influence of additional spin-orbit interactions we consider a Rashba spin-orbit interaction, ${\cal H}_{R}=i\gamma_{R}p_{+}^{3}\sigma_{-}/2+h.c.$, resulting from an asymmetric confining potential. The Rashba induces a spin-splitting in the hole gas of $\Delta_{R}=\gamma_{3}\hbar^{3}k_{F}^{3}/\varepsilon_{F}$. We include the Rashba term in the Hamiltonian, Eq. (Measuring hole $g$-factor anisotropies using transverse magnetic focusing), and use the aforementioned method to determine the variation in the focusing field. In Fig. 4 we present the response to in-plane magnetic field rotations, with varying strength of the Rashba spin-orbit interaction. Provided $\Delta_{R}<\tilde{g}_{1}\mu_{B}B_{||}/\varepsilon_{F}$ and $\Delta_{R}<\tilde{g}_{2}\mu_{B}B_{||}/\varepsilon_{F}$, there is minimal variation in the magnetic focusing field splitting, $\delta B_{z}$. In general, spin-orbit interactions which are odd in momentum such as the Dresselhaus and Rashba interactions will only weakly perturb the variation in the TMF peak spacing.

The measurement of $\tilde{g}_{2}/\tilde{g}_{1}$ depends only on the classical focusing field, and is therefore independent of the source and detector. However, the effect must be larger that the spread of the focusing peaks to be observed, $\delta B_{z}/B_{z}\geq B_{FWHM}/B_{z}$, where $B_{FWHM}$ is the full width half maximum of the focusing peak. Recent hole experiments with spin-splitting induced via a large Rashba type interaction have $B_{FWHM}\sim 0.02$TRokhinson et al. (2004), giving a ratio of $B_{FWHM}/B_{z}\sim 0.1$. To make a direct comparison between this and our results, we consider an in-plane field of $B_{||}=3.5$T and $\tilde{g}_{1}=-2.5$. with $\Delta_{0}\approx 0.2$. Comparing to Fig. 4, the minimum value at $\theta=\pi$, corresponds to an effective splitting $\delta B_{z}/B_{z}>0.1$. Hence the two peaks are resolvable over the full range of $\varphi$ at $B_{||}=3.5$T.

Finally we turn our attention to the assumption of adiabatic spin dynamics that we have employed in preceding calculations. As has been noted, $2\omega_{c}\ll|{\cal B}_{min}|$. The minimum value of ${\cal B}$,

[TABLE]

which results in the following condition,

[TABLE]

The fraction $B_{z}/B_{||}$ is the ratio between the in-plane field, and the focusing field, while $m^{*}$ is the effective mass. For a typical device, $B_{z}\sim 0.1$T, while $B_{||}$ can be several Tesla, and in quantum wells, $m^{*}\sim 0.2m$. We can compare this to some recent results in GaAs heavy hole quantum wells. The commensurate criterion to (16) is

[TABLE]

where $\Delta_{R}$ is the strength of the Rashba splitting, $\Delta_{R}\sim 0.2\varepsilon_{F}$. Which can be converted to an expression in terms of the Fermi momentum, $k_{F}$ and focusing length $L$,

[TABLE]

For this Rashba hole system, $k_{F}L\sim 100$, so $\Delta_{R}k_{F}L/12\sim 1/2$, while the two spin-split peaks are still clearly observable. Comparing this to Eq. (16), $B_{||}\sim 4$T is sufficient to satisfy this condition. Taken together with considerations of the spread of the focusing peak, we can conclude that it is possible to measure anisotropies in the in-plane $g$ factor in hole systems using TMF.

In summary, we have shown TMF can be used to determine the relative magnitude of $\tilde{g}_{1}$ and $\tilde{g}_{2}$, via the unique dependence on the in plane magnetic field angle, $\varphi$. Futhermore, this dependence is robust with the addition of residual spin-orbit interactions. Based on results from TMF in heavy hole gases, variation the focusing field is significantly larger than the broadening due to both scattering and the finite size of injectors and detectors at experimentally accessible in plane fields. This work was supported by the Australian Research Council Centre of Excellence in Future Low-Energy Electronics Technologies (project number CE170100039) and funded by the Australian Government. The authors would like to thank Dima Miserev for his valuable discussions.