Anomalous Spin-Related Quantum Phase in Mesoscopic Hole Rings
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Anomalous spin-related quantum phase in mesoscopic hole rings
M. J¨a¨askel¨ainen
Institute of Fundamental Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology,Massey University (Manawatu Campus), Private Bag 11 222, Palmerston North 4442, New Zealand
U. Z¨ulicke
Institute of Fundamental Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology,Massey University (Manawatu Campus), Private Bag 11 222, Palmerston North 4442, New Zealand andCentre for Theoretical Chemistry and Physics, Massey University (Albany Campus),Private Bag 102904, North Shore MSC, Auckland 0745, New Zealand (Dated: November 9, 2018)We have obtained numerically exact results for the spin-related geometric quantum phases that arise in p-typesemiconductor ring structures. The interplay between gate-controllable (Rashba) spin splitting and quantum-confinement-induced mixing between hole-spin states causes a much higher sensitivity of magnetoconductanceoscillations to external parameters than previously expected. Our results imply a much-enhanced functionalityof hole-ring spin-interference devices and shed new light on recent experimental findings.
PACS numbers: 85.35.Ds, 03.65.Vf, 71.70.Ej, 73.23.Ad
I. INTRODUCTION, MOTIVATION & SUMMARY
Quantum-interference effects dominate electric transportthrough conductors that are mesoscopic , i.e., have a smallersize than the decoherence length set by inelastic interac-tions of charge carriers with other degrees of freedom (e.g.,phonons, disorder). In particular, mesoscopic ring structuresexhibit magnetoconductance oscillations that reveal geomet-ric quantum (Berry, Aharonov-Anandan ) phases acquiredby charge carriers propagating quantum-coherently througha multiply connected geometry. Coupling of orbital motionto the spin of charge carriers affects the quantum interfer-ence and geometric phases manifested in charge transportthrough rings. Such spin-dependent electronic interferenceeffects could form the operational basis for novel transistordevices and quantum logic gates. Strong experimental efforts have been undertaken to iden-tify and measure spin-related geometric phases in magneto-transport through arrays of mesoscopic rings, single ringstructures, and anti-dot superlattices. Many recent ex-periments were performed in p-type semiconductor struc-tures because charge carriers from the valence band(holes) are expected to be subject to much larger momentum-dependent spin splittings than conduction-band electrons. In contrast, many theoretical works have considered spin-dependent interference in n-type semiconductor rings, while the electronic properties of p-type rings remain largelyunexplored. As conduction-band electrons and valence-bandholes are not merely distinguished by the sign of their chargebut are known to exhibit very different spin properties, es-pecially in quantum-confined structures, a careful analysisof tunable spin-related quantum phases in p-type mesoscopicrings is needed.Recent calculations of the magnetoconductance in holerings adopted a purely heavy-hole (HH) model where only thevalence-band states with spin projection ± / (heavy holes)and their effective Rashba-type spin splitting are taken into ac-count. It is tempting to follow such a route because the highest quasi-twodimensional (2D) valence subband is mostly of HHcharacter for typical hole sheet densities, and the HH modelbears resemblance to the one that applies to conduction-bandelectrons. However, such an approach neglects the hole-spinmixing induced by quantum confinement in ring structures.Here we report results of a theoretical study that fully accountsfor spin splitting and spin mixing in the valence band. Inter-estingly, we find a synergistic relation between gate-tunableRashba spin splitting, which arises from the structural inver-sion asymmetry (SIA) in the 2D semiconductor heterostruc-ture, and the hole-spin mixing due to the ring confinement. FIG. 1: (Color online) Dependence of hole-ring magnetoconduc-tance oscillations on the Fermi energy E . The latter is linearly relatedto the hole sheet density in the semiconductor heterostructure and canbe controlled by gate voltages. Φ denotes the magnetic flux penetrat-ing the ring, and Φ is the flux quantum. Panel a) shows results fromcalculations based on our more complete theory, whereas use of thesimplified heavy-hole model for the same ring device yields panel b).See text for more details. The range of energies shown correspondsto the situation where only the lowest ring subband is occupied. −50 −25 0 25 5012 mE / E FIG. 2: Hole-ring subbands in a quantum well without SIA. Param-eters: in-plane aspect ratio (radius/width) √ λ R = 20 , λ d = 0 . and ¯ γ = 0 . (value applies to GaAs). m is the eigenvalue of total angu-lar momentum component ˆ M z = ˆ L z + ˆ J z perpendicular to the ring. E = π ~ γ / (2 m d ) is the energy scale set by size quantizationin the 2D quantum well from which the ring is fabricated. This is illustrated in Fig. 1 where the dependence of magne-toconductance oscillations on the Fermi energy in the hole-ring structure is shown. The more complete theory under-lying our calculation predicts a much more frequent changebetween maxima and minima of the magnetoconductance asa function of the Fermi energy than is found for the samering geometry and materials parameters within the HH model.Hence, an analysis of experimental data based on the latterwould have to assume an unrealistically large SIA in the mea-sured sample. Conversely, our results suggest that moderatechanges in hole density and/or SIA, routinely achieved usinggate voltages, will be sufficient to operate a spin-interference-based nanoelectronic device.This article is organized as follows. Section II presents ourtheoretical model for mesoscopic hole rings. Zero-field spinsplitting of holes due to SIA is discussed in Sec. III. Sec-tion IV focuses on the spin-related Aharonov-Anandan phaseand how it is revealed in magneto-conductance oscillations. The frequently used heavy-hole model for p-type meso-scopic rings is introduced in Sec. V, and results obtained usingit are compared with those found within our more completetheory. The penultimate Sec. VI presents an interpretation ofrecent experiments in light of our new results, discussing alsopossible effects of spin splitting due to bulk inversion asym-metry (BIA). Our conclusions are given in Sec. VII. Somerelevant mathematical derivations are given in the Appendix.
II. MODEL FOR A MESOSCOPIC HOLE RING
Our calculations are based on the × Luttinger modelfor the uppermost valence band in typical semiconductors, which takes both the heavy-hole and the light-hole states intoaccount. For simplicity, we neglect band warping due to thecubic crystal symmetry. The ring confinement is assumed tobe due to a quantum-well potential in z direction (width d ) anda singular-oscillator potential V ⊥ ( r ) = m ω ( r − [ R /r ]) / for the radial coordinate r in the xy plane. Here R is the ef-fective ring radius, and the oscillator potential defines a lengthscale ℓ ω = p √ γ ~ / ( m ω ) that is a measure of the in-planering width. Only the lowest 2D quantum-well bound stateis taken into account, hence our theory applies in the (typi-cally realistic) case d ≪ ℓ ω . The energy splitting between2D heavy-hole and light-hole subband edges is accounted forby the Hamiltonian (we use the hole picture for the valenceband, counting energies as positive from the bulk valence-band edge) H qw = (cid:18) − γ (cid:20) ˆ J z − (cid:21)(cid:19) E , (1)where E = π ~ γ / (2 m d ) , ˆ J z is the operator for the spin-3/2 angular-momentum component perpendicular to the ringplane, and ¯ γ = (2 γ + 3 γ ) / (5 γ ) in terms of Luttinger pa-rameters. The in-plane hole motion is governed by H rg = λ d ((cid:18) γ (cid:20) ˆ J z − (cid:21)(cid:19) ℓ ω ˆ k ⊥ − ¯ γℓ ω (cid:16) ˆ k − ˆ J + ˆ k ˆ J − (cid:17) + (cid:18) rℓ ω − λ R ℓ ω r (cid:19) ) E , (2)with λ d = (2 d/ [ πℓ ω ]) and λ R = ( R/ℓ ω ) . Here ˆk ⊥ =(ˆ k x , ˆ k y ) is the in-plane hole wave vector, ˆ k ± = ˆ k x ± i ˆ k y , and ˆ J ± = ( ˆ J x ± i ˆ J y ) / √ .The hole-ring Hamiltonian H qw + H rg commutes with ˆ M z = ˆ L z + ˆ J z , where ˆ L z = x ˆ k y − y ˆ k x . The eigenvalues m of ˆ M z can thus be used to label states within the quasi-onedimensional (1D) ring subbands. Adopting polar coordi-nates r , ϕ for the in-plane motion and making the Ansatz h r, ϕ | ψ i = e i ( m − ˆ J z ) ϕ χ m ( r ) (3) for the four-spinor hole wave function generates a purelyradial Schr¨odinger equation that we solve numerically us-ing a pseudospectral method tailored to our needs. Fig-ure 2 shows a representative result for ring subbands E ( n ) s ( m ) ,where s = ± distinguishes spin-split dispersions with eigen-values related via E ( n ) s ( m ) = E ( n ) − s ( − m ) , and n = 0 , , . . . labels the doublets starting with the lowest-lying one. FIG. 3: (Color online) Spin-related quantum (Aharonov-Anandan)phase for holes in a mesoscopic ring, plotted as a function of hole en-ergy E and the voltage V SIA associated with SIA in the semiconduc-tor heterostructure from which the ring is fabricated. Other parame-ters are the same as in Fig. 2. The energy range shown correspondsto the situation where only the lowest ring subband is populated. Theband-structure parameter V is 11.44 V in GaAs. III. EFFECTS OF SIA SPIN SPLITTING
To investigate spin-related geometric phases in hole rings,we include the dominant SIA contribution to the bulk-holeHamiltonian, which is given by H ( bulk ) SIA = r v v (cid:16) ˆk × E (cid:17) · ˆJ .In our case of interest, the SIA electric field E has a z com-ponent E z determined by the 2D quantum-well confinement.In addition, the radial in-plane (ring) confinement induces anSIA spin splitting. We find H ( rg ) SIA = H ( rg,qw ) SIA + H ( rg,rg ) SIA , with H ( rg,qw ) SIA = V SIA V r λ d ℓ ω i (cid:16) ˆ k + ˆ J − − ˆ k − ˆ J + (cid:17) E , (4a) H ( rg,rg ) SIA = λ λ d " − λ R (cid:18) ℓ ω r (cid:19) ˆ L z ˆ J z E . (4b)The voltage V SIA = E z d is a measure for SIA in the quan-tum well, and V = πγ ~ / ( m | r v v | ) is a materials pa-rameter (11.44 V in GaAs). λ = | r v v | / ( eℓ ω ) is typi-cally very small unless the ring becomes narrow on the scale p | r v v | /e , which is of order ˚A. We checked that matrix el-ements of (4b) are negligible for typical hole-ring device pa-rameters. Thus, SIA splitting due to the in-plane ring confine-ment can be disregarded.The hole-ring Hamiltonian including the SIA terms stillcommutes with ˆ M z . Using the same procedure as outlinedin Sec. II, we obtain the ring-subband dispersions includingthe SIA term H ( rg,qw ) SIA . FIG. 4: (Color online) Magnetoconductance oscillations for a meso-scopic hole ring attached to ideal leads, shown as a function of thevoltage V SIA associated with SIA in the semiconductor heterostruc-ture. Panel a) is obtained from our more complete theory, whereaspanel b) results from application of the HH model to the same ringdevice. The band-structure parameter V is 11.44 V in GaAs. Inthe calculation, the Fermi level was assumed to be at . E , and thevisibility A was set to . Other parameters are as for Fig. 2. IV. AHARONOV-ANANDAN PHASE ANDMAGNETO-CONDUCTANCE OSCILLATIONS
Knowledge of the ring-subband dispersions makes it pos-sible to extract the Aharonov-Anandan (AA) phase for holes traversing the ring at a particular energy E (gen-erally, the Fermi energy of holes in the 2D semiconductorheterostructure). In the following, we focus entirely on theenergy range where only the lowest spin-split ring subbandis relevant, but our results can easily be generalized. Interms of the two Fermi angular momenta m ± ( E ) defined by E = E (0) s ( ± s m ± ) , the AA phase is given by (see the Ap-pendix for details of the derivation) Θ G = π ( m + − m − − . (5)The dependence of this spin-related geometric phase on theFermi energy E and SIA strength V SIA is shown in Fig. 3 fora set of typical hole-ring parameters.At low-enough temperatures, the electric conductance G of a mesoscopic ring attached to ideal leads exhibits aquantum-interference contribution that makes it possible tomeasure the Aharonov-Anandan phase. Quite generally, it isgiven by the expression (see the Appendix for more details) G = G (cid:20) A cos (cid:18) π ΦΦ (cid:19) cos Θ G (cid:21) , (6)where A ≤ measures the visibility of quantum interferencein the ring. The first cosine term contains the magnetic flux Φ penetrating the ring’s area, measured in units of the fluxquantum Φ = 2 π ~ /e . It gives rise to magnetoconductanceoscillations that are the electric analogue of the Aharonov-Bohm effect. The modulation of the magnetoconductance asa function of SIA strength achieved, e.g., by external gate volt-ages, reveals the presence of the spin-related quantum phase Θ G . This is illustrated in Fig. 4a.Experimentally, a change in the strength of SIA whilekeeping all other parameters (in particular, the hole density)constant can be achieved by simultaneously applied frontand back-gate voltages. However, in the majority of sam-ples, only a single (front or back) gate is available. Insuch a situation, both SIA and the density of charge carriers inthe semiconductor heterostructure are changed by a gate volt-age. For holes, changing the density (i.e., the Fermi energy)has a profound effect on the spin-related quantum phase. Thiscan be inferred from the strong dependence of Θ G on E forconstant V SIA seen in Fig. 3. The modulation of magneto-conductance oscillations when changing only the hole density(keeping V SIA = 0 . V constant, and with A = 1 ) is illus-trated in Fig. 1a. Upto a constant shift, E is directly propor-tional to the 2D sheet density of holes in the heterostructure. V. COMPARISON WITH THE HEAVY-HOLE MODEL
The necessity to fully account for valence-band mixingin hole rings can be illustrated by a direct comparison withthe simpler HH model. The latter results from a perturba-tive (L¨owdin-partitioning) treatment of valence-band mix-ing and SIA splitting for the lowest 2D HH subband, ne-glecting further spin splitting and mixing due to the in-planeconfinement. The Hamiltonian of the HH-model ring is H (HH)qw + H (HH)rg + H (HH)SIA , with H (HH)qw = (1 − γ ) E , (7a) H (HH)rg = λ d ( (1 + ¯ γ ) ℓ ω ˆ k ⊥ + (cid:18) rℓ ω − λ R ℓ ω r (cid:19) ) E , (7b) H (HH)SIA = V SIA V (cid:18) λ d (cid:19) ℓ ω i (cid:16) ˆ k ˆ J − − ˆ k − ˆ J (cid:17) E , (7c) ≡ V SIA V (cid:18) λ d (cid:19) ℓ ω i (cid:16) ˆ k ˆ σ − − ˆ k − ˆ σ + (cid:17) E . (7d)The second line defining H (HH)SIA applies when the HH ( J z = ± / ) amplitudes are treated as an effective spin-1/2 degreeof freedom; this is the way SIA spin splitting for heavy holesis usually written. No coupling to light-hole amplitudes ispresent in the HH model, even after the in-plane ring confine-ment is introduced.Using the same numerical method as for the full spin-3/2Luttinger theory of hole rings outlined above, we find thesubbands of the HH-model ring and the spin-related quantumphase associated with the lowest one. Its dependence on bothenergy E and strength of SIA turns out to be much weakerthan in the more complete theory. For the energy dependenceof magnetoconductance oscillations, this is illustrated in Fig. 1for a ring with V SIA = 0 . V and all other parameters as inFig. 2. A similar result is obtained when energy E is fixed and V SIA is varied. See Fig. 4. The different behavior exhibited
TABLE I: Phase shift of magnetoconductance oscillations inducedby varying V SIA /V between 0.527 and 0.551, as measured for a1D GaAs hole ring (Ref. 15) and calculated using the full Luttingermodel and the simpler HH model, respectively. Parameters used inthe calculations are ¯ γ = 0 . , λ d = 1 . , λ R = 28 , and E =1 . E (Luttinger model), n F = 1 (HH model). In general, E can bedetermined from the 2D hole sheet density and details of the sample’squantum-well confinement.experiment Luttinger model HH model π . π . π by the full Luttinger model as compared with the HH modelarises from HH-LH mixing induced by the in-plane ring con-finement. Hence, differences in quantitative predictions fromthe two models scale with λ d and thus vanish in the 2D limit. VI. APPLICATION TO REAL HOLE-RING SAMPLES
Our theory enables a more detailed quantitative interpreta-tion of experimental results. Applied gate voltages have beenobserved to shift magnetoconductance oscillations.
Comparison with Shubnikov-de Haas data measured in theunstructured 2D HH system enabled experimentalists to quan-tify the change in SIA strength required for a π phase shift.The HH model predicts ∆ V SIA = V / ( n F λ d λ R ) / for a ringwith n F occupied 1D subbands. Interestingly, the experi-ment reported in Ref. 15 observed an order-of-magnitude dis-crepancy between the measured value and that expected fromapplication of the HH model. As the comparison given inTable I shows, taking into account the enhancement due toHH-LH mixing within the full Luttinger model markedly re-duces this discrepancy. We suspect that even better agreementcould be reached if (a) more details about the ring structurewere known, thus facilitating a more realistic modeling of thequantum-well and in-plane confinement potentials, and (b) theeffect of band-warping corrections were included. Finally,typical ring devices are fabricated in semiconductors whoseunit cell lacks inversion symmetry and, thus, are subject to anadditional spin splitting due to BIA. We will briefly discussBIA effects before concluding.The most important BIA spin-splitting term in the bulk-holeHamiltonian is H (bulk)BIA = b v v (cid:0)(cid:8) k x , k y − k z (cid:9) J x + c.p. (cid:1) .Introducing the 2D quantum-well confinement by replacing ˆ k z → h ˆ k z i ≡ and ˆ k z → h ˆ k z i ≡ π /d yields theBIA contribution to the model-ring Hamiltonian as H (rg)BIA = H (rg,qw)BIA + H (rg,rg)BIA , where H (rg,qw)BIA = ℓ BIA d r λ d ℓ ω (cid:16) ˆ k + ˆ J + + ˆ k − ˆ J − (cid:17) , (8a) H (rg,rg)BIA = ℓ BIA d (cid:18) λ d (cid:19) ℓ ω (cid:16) ˆ k − ˆ k − (cid:17) (cid:16) ˆ k + ˆ J − − ˆ k − ˆ J + (cid:17) . (8b)The length scale ℓ BIA = πm | b v v | / ( γ ~ ) ≡ . ˚A inGaAs. As typical quantum-well widths are of the order of nm, we have ℓ BIA /d ∼ . . This value is an order of mag-nitude smaller than V SIA /V measured in GaAs ring sampleswith the strongest Rashba splitting. Hence, as a first approx-imation, it is admissible to neglect BIA spin splitting whendiscussing this experiment.Formally, the BIA terms do not commute with ˆ M z , and the Ansatz given in Eq. (3) will not eliminate the ϕ -dependencefrom the BIA part of the ring Hamiltonian. In essence, previ-ous eigenstates with quantum number m are coupled via theBIA term to those with m ± . A reduced-band model may beadequate to explore BIA effects in the lowest ring subband. VII. CONCLUSIONS
We have obtained numerically exact results for electronicsubbands and spin-related geometric phases for holes in meso-scopic rings. Unlike previous models, we account fully forspin splitting and mixing arising in the quantum-confined va-lence band. For quasi-onedimensional ring structures, a muchstronger modulation of magnetoconductance oscillations (as afunction of Fermi energy and/or SIA spin-splitting strength) isfound as compared with simplified (purely HH) models. Thiseffect arises due to HH-LH mixing induced by the in-planering confinement, and the magnitude of the enhanced depen-dence is quantified by the parameter λ d , which is related tothe ratio of quantum-well width and in-plane ring width.We have applied our model to discuss a recent experi-ment where an anomalously strong modulation of Aharonov-Bohm oscillations was observed. A sizable enhancementof magnetoconductance-oscillation modulations is obtainedwithin our (on some level still idealized) model, but its mag-nitude is smaller than the observed value. A more realisticmodelling of the ring structure may be needed to reach fullagreement. We also ascertained the effect of BIA spin split-ting. The parameter quantifying its importance is the ratio ofa length scale ℓ BIA ( = 4 . ˚A in GaAs) and the quantum-wellwidth, which was negligible compared to the strength of SIAsplitting present in the experiment under consideration. Ourtheory, possibly with further refinement, should be useful forguiding efforts aimed at realizing novel electronic devicesbased on spin-dependent quantum interference. Acknowledgments
This work is supported by the Marsden Fund Council (con-tract MAU0702) from Government funding, administered bythe Royal Society of New Zealand.
Appendix: Derivation of expressions given for theAharonov-Ananadan phase and the magnetoconductance
We assume a standard two-terminal transport geometry asshown, e.g., in Fig. 2 of Ref. 29. To keep the notation unclut-tered, we consider the situation where only the lowest spin-split subband is relevant, but all formulae can be straighfor-wardly generalized to the multi-subband case. Holes with en- ergy E are injected by an external lead at ϕ = 0 in chan-nel s in a superposition of ring-state amplitudes: | in i s = ξ (in) s + χ ( s ) sm + + ξ (in) s − χ ( s ) − sm − . Here χ ( s ) m denotes the radial four-spinor wave function [see Eq. (3)] for an eigenstate from sub-band s , and the coefficients ξ (in) s ± depend on details of the cou-pling between ring and injecting lead. At a draining lead lo-cated diametrically opposite to the injecting one, holes fromchannel s will enter with an amplitude | out i s = e iπ ( m + − s ˆ J z ) ξ (in) s + χ ( s ) sm + + e iπ ( m − + s ˆ J z ) ξ (in) s − χ ( s ) − sm − ≡ si M (cid:16) e iπm + ξ (in) s + χ ( s ) sm + − e iπm − ξ (in) s − χ ( s ) − sm − (cid:17) , (A.1)with the matrix M = diag { , − , , − } . Thus the phasedifference between forward- and backward-propagating am-plitudes is found to be s Θ G , with Θ G given by Eq. (5). Wehave used the freedom that phases are determined only mod-ulo integer multiples of π to adjust Θ G such that it vanishesin the limit where V SIA = 0 and HH-LH mixing is neglected.When the ring is penetrated by a magnetic flux Φ and cou-pled to ideal leads, the probability for transmission of holes isobtained as T = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X s (cid:20) e iπ (cid:16) m + + s ΦΦ0 (cid:17) ξ (in) s + ξ (out) s + − e iπ (cid:16) m − − s ΦΦ0 (cid:17) ξ (in) s − ξ (out) s − (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (A.2)where the factors ξ (out) s ± depend on the coupling between states s M χ ( s ) sm ± and the scattering state in the outgoing lead. Thetwo-terminal ring conductance G rg = e π ~ T is then given, infull generality, by G rg = G " X s A (1) s cos (cid:18) π ΦΦ + s Θ G (cid:19) + A (2) cos Θ G + A (3) cos (cid:18) π ΦΦ (cid:19)(cid:21) . (A.3)The familiar contributions proportional to A (1) s arise from in-terference between counter-propagating amplitudes from thesame channel and manifest the AA phase. Additional interfer-ence terms (those proportional to A (2 , ) are possible becausecoupling to leads may induce a mixing conductance betweenthe two channels. In practice, this happens when the Hilbertspace spanned by scattering states in the leads does not fullycontain the space spanned by ring eigenstates. Such a situa-tion could occur, in principle, because of the sensitive depen-dence of hole states on quantum confinement in the leads. Assuch effects will be small in typical situations and also de-pend strongly on the particular realizations of ring-lead cou-plings, we have not considered them further in the contextof this work. For similar reasons, we assume that the leadscouple symmetrically to states in the two subbands. Setting A (2) = A (3) = 0 and A (1)+ = A (1) − = A/ yields Eq. (6),after application of an addition theorem for cosine functions. L. L. Sohn, L. P. Kouwenhoven, and G. Sch¨on, eds.,
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