Spontaneous spin polarization

The notion of spin-polarization in QWs was first proposed by Thomas et al.[34] that the 0.7 anomaly evolves to the 0.5G₀ plateau when the spin degeneracy is lifted. The width of the ‘0.7’G₀ plateau is linearly dependent with the in-plane magnetic field. Later, Reilly et al. reported that the conductance of the anomaly is closer to 0.5G₀ in a long QW of 2 µm compared with shorter QWs (0.5 µm and quasi-zero in length).[35] The 0.7 anomaly was also reported to be dependent on the carrier density. With increasing the top gate voltage and correspondingly increasing the carrier density, the anomaly moves closer to 0.5G0 as shown in Figs. 3.2(a) and (b).[35, 36, 43]

(a) (b)

Figure 3.2: (a) Conductance of a ℓ=0.5 µm QW as a function of side gate voltage V_S for V_T(top gate voltage)=560–1500 mV (right to left). (b) Open symbols: anomaly conduc-tance as a function of top gate voltage for QWs with different lengths. Solid symbols:

N=1 plateaus near G₀. (After Reilly et al., Ref.[35].)

Spontaneous spin polarization can be realized in a GaAs/AlGaAs two-dimensional-hole-gas (2DHG). Rokhinson et al. adopted electron-focusing technique to inject a current from a injector QPC to the detector QPC as shown in Fig. 3.3(a). Due to spin-orbit inter-action, the spin degenerated subbands are split. As shown in Fig. 3.3(b), by monitoring the voltage across the detector QPC, two peaks corresponding to spin-up and spin-down are observed when both injector and detector are set at G₀. When the spin-up subband

Chap. 3 Zero-bias anomaly and conductance reduction in quantum wires 25 of the injector QPC is lifted above the Fermi-energy, e.g. the G_i=0.66G₀ trace, only the spin-down peak is observable.[58]

Figure 3.3: Polarization detection via magnetic focusing. (a) The voltage across the detector QPC as a function of the perpendicular magnetic field B_⊥. A current of 0.5 nA flows through the injector QPC. The positions of the first two magnetic focusing peaks are marked with vertical lines. The trajectories of the ballistic holes for positive and negative B_⊥ are shown schematically in the insets. (b)The first focusing peak is measured at different injector conductances with the detector tuned into the middle of the 2e²/h plateau. (c) The gate voltage characteristic of the injector QPC. Vertical lines mark the positions where the curves in (b) are taken. (After Rokhinson et al., Ref.[58].)

Many theoretical proposals have been made to explain the spontaneous spin polar-ization of 1D electron gas. Reilly et al. proposed that spin splits in the QW and the energy of split increases with increasing carrier density. Therefore, as the spin gap opens up with the increasing top gate voltage of a QW, the anomaly conductance evolves to 0.5G₀.[35, 36, 44] The density dependent spin gap may be described by a more micro-scopic scenario–the deviated tendency of populating the subbands with opposite spins.[59]

Graham et al. observed that there is energy rearrangement for σ ↓ energy levels when σ ↑ meets σ ↓ in strong in-plane magnetic fields.[60] The energy displacement manifests as discontinuities of these σ↑ subbands in the transconductance spectroscopies as shown in Fig. 3.4(a) where this features of 1 ↑ and 2 ↓ are highlighted by the dashed frame.

Later, the σ ↓ was suggested to be rapidly populating for the V-sahpe splitting of dc bias spectroscopy shifting to a finite dc bias.[61] For the dc bias spectroscopy in magnetic fields where spin energy rearrangement happens, one of the bias branches was found to

be missing for 1↑.[62] Level pining on σ ↓ was suggested. Both analogs of fast populating and branch missing were found in 0.7 anomaly at B=0. E.g., as shown in Fig. 3.4(b), the µs branch is missing for G(Vsd = 0) = 0.7G0 compared with the expected pattern shown in the right panel.

(a) (b)

Figure 3.4: (a) Grey-scale diagram of dG/dV_g as a function of V_gfor B = 0 to 16 T. White represents conductance plateaus, and dark lines correspond to a subband populating. (b) Left panel: Grey-scale of dG/dVg at B=0 as a function of Vsd. The left branch is absent.

Right panel: Schematic illustration for the expected pattern of the spectroscopy. (After Graham et al., Ref.[62].)

The 0.7 anomaly was suggested to originate from the combination of the aforemen-tioned effects and thermal depopulation. This proposal was supported later by Chen et al. by showing that G(V_g) and _∂V^∂G

g (V_g) as a function of temperature and V_sd present odd-even behavior in magnetic fields.[53]

‘Spin density functional theory’ (SDFT) calculation was widely used to numerically investigate the local electron density and self-consistent potential of QWs for the two opposite spins.[63, chap. 7] Exchange and correlation of electrons were usually included in the calculation.[38, 39, 40, 41, 43, 64] Density of one spin was proposed to be larger than the other in the center of a QW as shown in Fig. 3.5(a) by Starikov et al. although there are still disagreements among works. The energy potential for σ ↑ is higher than σ ↓ correspondingly as shown in Fig. 3.5(b), and the 0.7 anomaly feature was reproduced due to deviation of transmission between spins, Fig. 3.5(c).

Spin polarization implies two discrete subbands below the Fermi energy for G <

G₀. Kristensen et al. reported that the 0.7 anomaly develops an ‘anomalous subband’

in the transconductance spectrum _∂V^∂G

g (Vsd).[55] The linear conductance (Vsd = 0) in

Chap. 3 Zero-bias anomaly and conductance reduction in quantum wires 27

(a) (b) (c)

Figure 3.5: (a) Local spin density along the longitudinal axis in the middle of a QW. (b) Self consistent total potential for σ =↑ (solid) and σ =↓ (dashed) along longitudinal axis.

(c) Numerical results for conductance vs. gate voltage. (solid line) (After Starikov et al., Ref.[39].)

the vicinity of the first plateau decreases with increasing temperature corresponding to thermal excitation. The authors suggested that an extra level exists in addition to the integer plateau. The reducing conductance with increasing temperature as well as the observed conductance shoulder in the non-linear conductance was associated with electron depopulating from the extra energy level. The scenario was explored in another way by studying the shot noise. Besides the 1/f , telegraph and amplifier noises, the measured noises S_I(V_sd) are dominant by thermal noise 4k_BT_eg(V_sd) and partition noise S_I^P(V_sd), where T_eis the electron temperature. The noise factor N = ¹₂ ∑

τ_n,σ(1−τn,σ)∝ SI^P(V_sd), where τ_n,σ is the transmission of n-th subband for spin σ. For spin-degenerate case, the shot noise is expected to minimize to zero on a plateau, and maximize as 0.25 for odd integer multiples of ¹₂G₀. Roche et al. and Dicarlo et al. observed that the shot noise was reduced at the 0.7 anomaly.[65, 66] As shown in Fig. 3.6(a),N reaches zero for G = G0

and 2G0 and N is suppressed at G ≈ 0.7G0. Additionally, the asymmetric single dome of N becomes symmetric double domes with respect to 0.5G0 at B_∥ = 7.5T as shown in Fig. 3.6(b) when spin degeneracy is lifted.

Contradictorily, Thomas et al. and Pyshkin et al. observed that the 0.7 anomaly approaches 0.5G0 also at low density regime.[67, 68] In low density regime, the coulomb interaction dominates over kinetic energy. To reach a minimal total energy for the sys-tem (QW), the electrons become crystalized forming the prominent ‘Wigner Crystal’.[69]

Klironomos et al. proposed that electrons form a zig-zag crystal in a realistic devices.

(a) (b)

Figure 3.6: (a) Experimental N as a function of averaged conductance gavg at B_∥ = 0 along with model curves (solid and dashed lines). (b) Experimental N as a function of gavg in the range 0–1(2e²/h) at various B_∥. (After Dicarlo et al., Ref.[65].)

The distance between two particle chains depends on carrier density as well as the type of exchange interaction. When the QWs form a ferromagnetic ground state, the system is spin polarized.[70]