Investigating Spin‑Split Peaks in Transverse Electron Focusing Across Temperatures

Background

Electron transport in quasi‑one‑dimensional (1D) systems, realized in the 2DEG of GaAs/AlGaAs heterostructures, has long served as a benchmark for studying both non‑interacting quantum mechanics and many‑body phenomena [1–3]. In such systems the conductance quantizes in steps of n·(2e²/h) (n=1,2,3,…), yet recent experiments have uncovered rich low‑density phases, including incipient Wigner crystallization [6,7,10]. A central open question remains the origin of the so‑called 0.7 anomaly—a conductance plateau near 0.7·(2e²/h)—whose temperature dependence and evolution under in‑plane magnetic fields suggest a spin‑related origin [4,11–15]. Directly probing spin polarization in a 1D channel is therefore essential for resolving this debate. Transverse electron focusing (TEF) offers a sensitive, spin‑resolved probe: spin‑splitting of the first focusing peak reflects the exchange‑driven polarization of the injector, as demonstrated in p‑type GaAs [18,19] and n‑type InSb [20]. In earlier work we showed that spatially separated, spin‑polarized 1D electrons give rise to a clear split in the first focusing peak in n‑GaAs [21]. Here, we extend this approach by studying how the split evolves with temperature, providing insights into the underlying spin‑gap dynamics.

Method

The devices were fabricated from a high‑mobility 2DEG at the GaAs/Al_0.33Ga_0.67As interface. At 1.5 K, the electron density and mobility were 1.80 × 10¹¹ cm⁻² and 2.17 × 10⁶ cm² V⁻¹ s⁻¹, respectively, yielding a mean free path exceeding 10 µm—far larger than the device dimensions. Measurements were performed in a cryofree dilution refrigerator, with a base lattice temperature of 20 mK (electron temperature ≈ 70 mK). Temperature‑dependent data were collected from 20 mK to 1.8 K using a standard lock‑in technique.

Results and discussion

Figure 1a illustrates the experimental layout and a typical focusing spectrum. The injector and detector are separated by 1.5 µm, each defined by a 500 nm‑wide, 800 nm‑long quantum wire. Independent gate control suppresses cross‑talk, ensuring that the observed peaks arise solely from TEF (Fig. 1b). The first focusing peak shows a pronounced split into sub‑peaks I and II; the second peak remains unsplit. The periodicity of the peaks matches the semiclassical cyclotron orbit condition

\[\displaystyle B_{\text{focus}}=\frac{\sqrt{2}\,\hbar k_{F}}{eL}\]

(1) where L is the injector‑detector separation along the diagonal. The observed 60 mT spacing agrees well with the calculated value, confirming the semiclassical picture. The split of the first peak is attributed to spin‑orbit interaction (SOI). In a Rashba‑type SOI, the two spin species follow different cyclotron radii, producing two distinct arrival times at the detector (Fig. 2a). For the second peak, a boundary‑scattering process reverses the momentum of the spin‑up electron while preserving its spin, causing the two spin species to recombine into a single peak (Fig. 2b). These mechanisms are illustrated in both coordinate and k‑space (Figs. 2c,d). Temperature dependence (Fig. 3a–c) reveals that the split is sharp at 20 mK but gradually blurs as the temperature rises. For injector conductance G_i = G₀, the two sub‑peaks persist up to ~1.2 K before merging at 1.8 K. At G_i = 1.8 G₀ the split is less pronounced, with sub‑peak I dominating and decreasing in amplitude with temperature. In contrast, the G_i = 0.5 G₀ case shows only a single peak, broadened and symmetric at higher temperatures, suggesting thermal population of both spin sub‑bands. To quantify the peak evolution, we fit each focusing trace with a sum of two Lorentzians:

\[\displaystyle A(B)=\sum_{i=1}^{2}A_{i}\,\frac{\gamma_{i}^{2}}{\gamma_{i}^{2}+(B-B_{i})^{2}}\]

(2) where A_i is the amplitude, γ_i the full width at half maximum (FWHM), and B_i the peak center. The fitted widths increase monotonically with temperature, indicating thermal broadening. The extracted spin polarization p = |A₁ − A₂|/(A₁ + A₂) shows a weak dependence on temperature for G_i = G₀ (≈ 0.6 %) but decays exponentially for G_i = 1.8 G₀:

\[\displaystyle p=\alpha\,e^{-k_{B}T/\Delta E}+c\]

(3) yielding ΔE ≈ 0.041 meV (≈ 0.5 K). This aligns with theoretical predictions that splitting should persist until k_BT ≈ 2ΔE (~1 K), consistent with our observation of peak merging near 1.2 K.

Conclusion

We have demonstrated that the spin‑resolved split of the first transverse electron‑focusing peak in n‑GaAs is robust from 20 mK up to ~1.2 K. Above this temperature the split smears out, and the peak becomes increasingly symmetric, indicative of thermal equilibration between the two spin branches. These findings provide direct evidence of exchange‑driven spin polarization in the injector and establish TEF as a powerful, temperature‑dependent probe of spin physics in low‑dimensional systems.

Funding: This work was supported by the Engineering and Physical Sciences Research Council (EPSRC), UK.