Direct Observation of the Aharonov-Casher Phase

Abstract

Ring structures fabricated from HgTe/HgCdTe quantum wells have been used to study AharonovBohm type conductance oscillations as a function of Rashba spin-orbit splitting strength. We observe non-monotonic phase changes indicating that an additional phase factor modifies the electron wave function. We associate these observations with the Aharonov-Casher effect. This is confirmed by comparison with numerical calculations of the magneto-conductance for a multichannel ring structure within the Landauer-Büttiker formalism. In the early 1980s it was shown that a quantum mechanical system acquires a geometric phase for a cyclic motion in parameter space. This geometric phase under adiabatic motion is called Berry phase [1], while its later generalization to include non-adiabatic motion is known as Aharonov-Anandan phase [2]. A manifestation of the Berry phase is the well known AharonovBohm (AB) phase [3] of an electrical charge which cycles around a magnetic flux. Aside from the AB effect, the first experimental observation of the Berry phase was reported in 1986 for photons in a wound optical fiber [4]. Another important Berry phase effect is the AharonovCasher (AC) effect [5], which has been proposed to occur when an electron propagates in a ring structure in an external magnetic field perpendicular to the ring plane in the presence of SO interaction [6].This AC effect can be seen when two partial waves move around the ring in different directions. They will acquire a phase difference which depends on the spin orientation with respect to the total magnetic field B tot = B ext + B ef f and the path of each partial wave. B ef f is the effective field induced by the SO interaction. The phase difference is approximately [6] where s =↑ and ↓ denote parallel and anti-parallel orientation to B tot , b = +1 for s =↑ and b = −1 for s =↓, and the superscript −(+) denotes a clockwise (counterclockwise) evolution, respectively. In the above equations, α is the SO parameter, r the ring radius, m * the effective electron mass and θ the angle between the external ( B ext ) and the total magnetic field B tot . For both equations, the first term on the right hand side can be identified with the AB phase and the second term of Eq. (1) with the geometric Berry or Aharonov-Anandan phase. The second term in Eq. (2) represents the dynamic part of the AC phase, i.e. the phase of a particle with a magnetic moment that moves around an electric field. From the expressions above, it can be seen that an increase of the AC phase will lead to a phase change that increases continuously with α, whereas the contribution due to the geometric phase results in a phase shift limited to ∆ϕ geom ≤ π.Both the AC phase [7] and the geometric phase [8, 9] depend on the SO interaction. As a result, one expects a complicated non-monotonic interference pattern as a function of magnetic field and SO interaction strength. So far, to our knowledge, apart from the AB effect no direct observation of phase related effects in solid state systems has been reported. Rec...

Spin nutation and polarization in ballistic semiconductor nanostructures

“…1 It may enable the realization of quantum spintronics [2][3][4] by manipulating the spin degree of freedom of electrons without destroying their phase coherence. A spin-dependent transport has been studied in the context of the Aharonov-Casher phase [5][6][7][8] and spin Berry phase. [9][10][11][12] Various types of interesting phenomena have been found in spin transport systems.…”

Anomalous spin-related quantum phase in mesoscopic hole rings

“…Applied gate voltages have been observed to shift magnetoconductance oscillations. 13,15,36 Comparison with Shubnikov-de Haas data measured in the unstructured 2D HH system enabled experimentalists to quantify the change in SIA strength required for a -phase shift. The HH model predicts ⌬V SIA = V 41 / ͑n F 2 d R ͒ 1/2 for a ring with n F occupied 1D subbands.…”

“…6 Such spin-dependent electronic interference effects could form the operational basis for novel transistor devices 7 and quantum logic gates. 8 Strong experimental efforts have been undertaken to identify and measure spin-related geometric phases in magnetotransport through arrays of mesoscopic rings, 9,10 singlering structures, [11][12][13][14][15] and antidot superlattices. 16 Many recent experiments were performed in p-type semiconductor structures 11,[14][15][16] because charge carriers from the valence band ͑holes͒ are expected to be subject to much larger momentum-dependent spin splittings than conduction-band electrons.…”

Spin interference and the Fano effect in electron transport through a mesoscopic ring side-coupled with a quantum dot

“…Recently, the AC effect has been observed experimentally in electron transport through semiconductor mesoscopic rings [7][8][9][10][11]. The modulation of the conductance can be explained theoretically in terms of noninteracting electrons [9,12,13]. The electron interaction effect on conductance in this Rashba SO coupling ring has also been studied based on the Hubbard model [14].…”

“…In the presence of Rashba SO coupling, aside from the AB effect, another important Berry phase effect called the Aharonov-Casher(AC) effect [6] occurs, it is due to the electron magnetic moment moving in an effective magnetic field caused by the Rashba SO coupling. Recently, the AC effect has been observed experimentally in electron transport through semiconductor mesoscopic rings [7][8][9][10][11]. The modulation of the conductance can be explained theoretically in terms of noninteracting electrons [9,12,13].…”