Aharonov-Bohm oscillations in p-type GaAs quantum rings
Boris Grbic, Renaud Leturcq, Thomas Ihn, Klaus Ensslin, Dirk Reuter, Andreas D. Wieck
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Aharonov-Bohm oscillations in p-type GaAs quantum rings
Boris Grbi´c ∗ , Renaud Leturcq ∗ , Thomas Ihn ∗ , Klaus Ensslin ∗ , Dirk Reuter + , and Andreas D. Wieck + ∗ Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland + Angewandte Festk¨orperphysik, Ruhr-Universit¨at Bochum, 44780 Bochum, Germany
We have explored phase coherent transport of holes in two p-type GaAs quantum rings with orbitalradii 420 nm and 160 nm fabricated with AFM oxidation lithography. Highly visible Aharonov-Bohm(AB) oscillations are measured in both rings, with an amplitude of the oscillations exceeding 10%of the total resistance in the case of the ring with a radius of 160 nm. Beside the h/e oscillations,we resolve the contributions from higher harmonics of the AB oscillations. The observation of alocal resistance minimum at B=0 T in both rings is a signature of the destructive interference ofthe holes’ spins. We show that this minimum is related to the minimum in the h/2e oscillations.
The Aharonov-Bohm (AB) phase [1], represents thegeometric phase acquired by the orbital wave function ofthe charged particle encircling a magnetic flux line. Thisphase is experimentally well established and manifests it-self through oscillations in the resistance of mesoscopicrings as a function of an external magnetic field. The spinpart of the particle’s wave-function can acquire an addi-tional geometric phase in the systems with strong spin-orbit (SO) interactions [2, 3]. The SO induced phase ad-ditionally modulates the resistance oscillations in meso-scopic rings. This SO induced phase in solid-state sys-tems has been recently the subject of a number of exper-imental investigations [4, 5, 6, 7, 8, 9, 10].SO interactions are particularly strong in p-type GaAsheterostructures, due to the p-like symmetry of the statesat the top of the valence band and the high effective massof holes. The presence of exceptionally strong SO inter-actions in carbon doped GaAs heterostructure used forfabrication of rings investigated in this work, is demon-strated by the simultaneous observation of the beatingin Shubnikov-de Haas (SdH) oscillations and weak anti-localization in the measured magnetoresistance. The holedensity in an unpatterned sample is 3.8 × cm − andthe mobility is 200 000 cm /Vs at a temperature of 60mK. The strength of the Rashba spin-orbit interaction isestimated to be ∆ SO ≈ . E F = 2 . pg = −
172 mV andV pg = −
188 mV of the large ring is shown in Fig. 1(a),while the corresponding splitting of the h/e Fourier peakis shown in Fig. 1(c). In addition, we observe a resis-tance minimum at B = 0 T in all gate configurations,and attribute its origin to the destructive interference ofthe hole spins propagating along time reversed paths [10].We now focus on the magnetotransport measurements
100 120 140 160 FT ( a . u . ) frequency (1/T) ∆ R ( Ω ) Vpg1=-172 mV, Vpg2=-188 mV (a) µ mpg1pg234 56 (b) (c)100-100 FIG. 1: (a) AB oscillations in the gate configuration V pg = −
172 mV and V pg = −
188 mV obtained after subtractionof the low-frequency background from the raw data. A clearbeating pattern is revealed in the AB oscillations. (b) AFMmicrograph of the large ring with an orbital radius of 420 nmwith designations of the in-plane gates. (c) Splitting of theh/e Fourier peak. performed on the smaller ring with an orbital radius of160 nm (Fig. 2(a)). Fig. 2(b) shows the magnetoresis-tance of the small ring (oscillating curve, blue online), to-gether with a low-frequency background resistance com-posed of the low frequency Fourier components of thesignal (smooth curve, red online) in the plunger gate con-figuration V pg = − . pg = −
222 mV. AB os-cillations, obtained after subtraction of the low-frequencybackground from the raw data, with a peak-to-peak am-plitude of ∼ − and 17 T − .Besides, h/2e and h/3e peaks are clearly visible (Fig.2(d)). The electronic radius of the holes’ orbit of 150 nm,obtained from the period of the oscillations of around 60mT, agrees well with a lithographic mean radius of theholes’ orbit.Due to the large period of the AB oscillations, only upto 10 oscillations are present in the magnetic field range (-0.3 T, +0.3 T) where SdH oscillations do not obscure thedata analysis and no beating can be seen in the raw data.Therefore, although the amplitude of the AB oscillationsin the case of the small ring is quite large, a detailedanalysis of the beating of the AB oscillations can not beperformed for this sample as it was done in the case ofthe larger ring [10]. Gate1 R ( k Ω ) ∆ R ( k Ω ) -20 -0.2 -0.1 0 0.1 0.2B(T) pg1pg2 pg1 =-78.5 mVV pg2 =-222 mV (b)(c) µ m (a) FT ( a . u . ) (d) FIG. 2: (a) AFM micrograph of the small ring with an orbitalradius of 160 nm with designation of the gates. (b) Measuredmagnetoresistance of the small ring (oscillating curve, blueonline) together with the low-frequency background resistance(smooth curve, red online) in the plunger gate configurationV pg = − . pg = −
222 mV. (c) AB oscillationsobtained after subtraction of the low-frequency backgroundfrom the raw data. (d) Fourier transform of the data withsplit h/e peak and well defined h/2e and h/3e peaks.
We further analyze the dependence of the AB oscilla-tions on plunger gates voltages. The evolution of the ABoscillations along the line V pg = 0 . · V pg − V pg , V pg ) is investigated in Fig. 3. Fig.3(a) shows the raw data, while Fig. 3(b) and 3(c) showfiltered h/e and h/2e oscillations, respectively. When V pg and V pg increase along the given line, both ringarms become narrower and the holes’ orbit inside thering shrinks, causing the resistance of the ring to increasecontinuously from 20 −
50 kΩ.We again observe a resistance minimum at B = 0T inall gate configurations (Fig. 3(a)) and find that it is re-lated to the minimum in the h/2e oscillations at B = 0T(Fig. 3(c)), consistent with the results from the largering sample [10, 11]. It should be emphasized that theobserved minimum is not caused by weak-antilocalizationin the ring leads, since the weak-antilocalization dip inbulk two-dimensional samples has a much smaller mag-nitude (less than 1Ω) than the minimum at B = 0T inthe rings [11]. The resistance minimum at B = 0T is aresult of the destructive interference of the holes’ spinsin the ring.The presence of phase jumps in the h/e oscillations(Fig. 3(b)) and their absence in the h/2e oscillations FIG. 3: (a) Evolution of the AB oscillations upon changingplunger gate voltages along the line V pg = 0 . · V pg − B = 0 T at all gate voltages. (Fig. 3(c)) is also observed. The fact that the phase ofthe AB oscillations can not change continuously, but onlyin discrete steps of π is a consequence of the Onsager re-lations G ij ( B ) = G ji ( − B ). For changes of the plungergates along the line V pg = 0 . · V pg − π in h/e oscillations at lower mag-netic fields up to 0.2 T, but at the higher fields, above0.2T, we find continuous monotonic shifts of the AB min-ima and maxima (Fig. 3(a) and 3(b)). We attribute thesecontinuous shifts of the AB minima and maxima towardshigher fields upon increasing plunger gate voltages V pg and V pg to an increase of the AB oscillation frequencyupon continuous shrinking of the holes’ orbit within thering. Continuous, but non-monotonic top-gate inducedshifts of the AB minima and maxima were recently ob-served in HgTe quantum rings [7], and this behavior isinterpreted as a manifestation of the Aharonov-Casherphase.In conclusion, we have measured highly visibleAharonov-Bohm oscillations in two quantum rings withradii 420 nm and 160 nm fabricated by AFM oxidationlithography on p-type GaAs heterostructure with strongspin-orbit interaction. The visibility of the AB oscilla-tions exceeds 3% in the larger ring and 10% in the smaller ring. Beside the h/e oscillations, the higher harmonics ofthe AB oscillations are resolved in both rings. A resis-tance minimum at B = 0T, present in both rings, pointsto the signature of destructive interference of the holes’spins propagating along time-reversed paths. [1] Y. Aharonov and D. Bohm, Phys. Rev. , 485 (1959).[2] H.A. Engel and D. Loss,
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