Characteristic molecular properties of one-electron double quantum rings under magnetic fields
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Characteristic molecular properties of one-electrondouble quantum rings under magnetic fields
JI Climente , and J Planelles CNR-INFM - National Research Center on nano-Structures and bio-Systems atSurfaces ( S Departament de Qu´ımica F´ısica i Anal´ıtica, Universitat Jaume I, Box 224, E-12080Castell´o, SpainE-mail: [email protected]
Abstract.
The molecular states of conduction electrons in laterally coupled quantum ringsare investigated theoretically. The states are shown to have a distinct magnetic fielddependence, which gives rise to periodic fluctuations of the tunnel splitting and ringangular momentum in the vicinity of the ground state crossings. The origin of theseeffects can be traced back to the Aharonov-Bohm oscillations of the energy levels, alongwith the quantum mechanical tunneling between the rings. We propose a setup usingdouble quantum rings which shows that Aharonov-Bohm effects can be observed evenif the net magnetic flux trapped by the carriers is zero.PACS numbers: 73.21.-b,75.75.+a,73.22.Dj,73.23.Ra haracteristic molecular properties of one-electron double quantum rings under magnetic fields
1. Introduction
When the energy levels of two tunnel-coupled semiconductor quantum dots are setinto resonance, the carriers localized in the individual nanostructures hybridize formingmolecular-like states, in good analogy with atomic molecular bonds[1, 2, 3, 4, 5, 6].The possibility to engineer the properties of these ’artificial molecules’, such as thebond length, the material and shape of the constituent ’atoms’ or the number andnature of delocalized carriers has opened new paths to learn basic physics of molecularsystems[7, 8, 9].Vertically-coupled[5, 6, 7, 8] and laterally-coupled[10, 11, 12, 13, 14, 15, 16]double quantum dots (DQDs) have received particular attention owing to additionaltechnological implications for the development of scalable two-qubit logic gates[17].Much less is known about other kinds of artificial molecules, such as double quantumrings (DQRs). This is nonetheless an interesting problem, as the remarkable magneticproperties of semiconductor quantum rings (QRs)[18], related to the Aharonov-Bohm(AB) effect[19], have been thoroughly studied at a single-ring level[20], but their effectsat a molecular level remain largely unexplored.Molecular states in vertically coupled QRs have been investigated[21, 22, 23, 24,25, 26, 27], but the hybridized orbitals in such systems are aligned along the verticaldirection. As a result, their response to vertical magnetic fields, which are responsiblefor AB effects, is very weak[23]. The electronic states of concentrically coupled QRshave also been studied[28, 29, 30, 31, 32, 33]. However, the markedly different verticalconfinement of the inner and outer rings leads to carrier localization in individual rings,preventing molecular hybridization[28, 29, 30, 33].Recently, we studied the molecular dissociation of yet another kind of structure,namely laterally-coupled QRs.[34] Interestingly, in such structures the two rings mayhave similar dimensions, which grants the formation of covalent molecular orbitals. Inaddition, tunnel-coupling takes place in the plane of the rings, thus rendering molecularorbitals sensitive to vertical magnetic fields. These two ingredients make laterally-coupled QRs ideal systems to attain magnetic modulation of molecular bonds and theirderived properties. This is the subject of research in the present paper. We investigatethe energy structure of DQRs under magnetic fields, and find that the AB-inducedground state crossings lead to sudden maxima of the tunnel splitting between bondingand antibonding orbitals of DQRs, as well as to periodic suppressions of the carrierrotation within the rings.Laterally-coupled DQRs also offer the possibility to extend the research of ABeffects, typically restricted to effectively isolated ring structures (see e.g. Refs. [35, 36,37, 38]), to composite systems, which may unveil subtleties of these purely quantummechanical phenomena. As a matter of fact, here we show that, unlike in singleQRs[19, 20], AB effects in composite systems need no finite net magnetic flux to takeplace.The paper is organized as follows. In Section 2 we describe the theoretical model haracteristic molecular properties of one-electron double quantum rings under magnetic fields D x=0 y = R in R out R in R out Figure 1.
Schematic of the DQR structure under study and the relevant geometricalparameters. The confining potential is zero inside the rings and V c outside.
2. Physical system and theoretical model
We consider QRs in the Coulomb-blockade regime, charged with only one conductionelectron. QRs in this regime may be fabricated using either self-assembly[18] orlitographic[35] techniques. However, control of the nanostructure dimensions andpositioning is only accurate when using litographic methods. Therefore, for illustrationpurposes in this work we choose to simulate a DQR system built from etched QRs, asthose of Ref. [35].Since QRs have much stronger vertical than lateral confinement, we calculate thelow-lying states of the DQRs using a two-dimensional effective mass-envelope functionapproximation Hamiltonian which describes the in-plane ( x − y ) motion of the electronin the ring. In atomic units, the Hamiltonian may be written as: H = 12 m ∗ ( p + A ) + V ( x, y ) , (1)where m ∗ stands for the electron effective mass, p is the canonical moment, and V ( x, y )is a square-well potential confining the electron within the DQR structure shown inFig. 1. In polar coordinates it has the compact expression V ( ρ, θ ) = 0 if R in < ρ < R out haracteristic molecular properties of one-electron double quantum rings under magnetic fields V ( ρ, θ ) = V c elsewhere, with V c as the barrier confining potential. A is the vectorpotential. Unless otherwise stated, we use the symmetric gauge, A = B/ − y, x, B pointing along the growth direction z . Replacingthis vector potential into the Hamiltonian, one obtains: H = ˆ p k m ∗ + B m ∗ ( x + y ) − i B m ∗ ( x ∂∂y − y ∂∂x ) + V ( x, y ) , (2)where ˆ p k = ˆ p x + ˆ p y . The eigenvalue equation of Hamiltonian (2) is solved numericallyusing a finite-difference scheme on a two-dimensional grid ( x, y ) extended far beyondthe DQR limits.Following Ref. [35], the QRs we study are made of In . Ga . As and they aresurrounded by GaAs barriers. Reasonable material parameters for this heterostructureare barrier confinement potential V c = 50 meV[35] and effective mass m ∗ = 0 . R in = 15 nm, the outer one R out = 45 nm, and the two rings are separated horizontally by a D = 3 nm barrier.
3. Results
In this section we analyze the energy structure and tunnel-coupling of a DQR undermagnetic fields, and compare them to the well-known case of DQDs. The DQD hasinterdot barrier D = 3 nm and R out = 30 nm, which gives a similar area to that of theDQR. The energy structure of both kinds of artificial molecule are illustrated in the toppanels of Fig. 2. As is known, the DQD energy structure resembles the spectrum of a (a)(c) (b)(d) t unn e l − s p litti ng ( m e V ) e n e r gy ( m e V ) B (T) B (T)
Figure 2.
Top row: energy levels vs magnetic field for a DQR (a) and a DQD (b).Bottom row: tunnel splitting vs magnetic field for a DQR (c) and a DQD (d). Theinstets provide schematic representations of the structures under study. Note the non-monotonic evolution of the tunnel splitting for DQRs. haracteristic molecular properties of one-electron double quantum rings under magnetic fields B . Clearly,the values of the field where such peaks occur correspond to the level crossings in theenergy structure (Fig. 1(a)), i.e. to integer number of AB periods[19]. This tunnelingenhancement, whose origin we explain below, suggests that DQRs enable a strongermagnetic field-induced modulation of the molecular strength than DQDs, and largertunnel splittings may be achieved when operating at finite magnetic fields, which maybe of interest for spin qubit systems, where magnetic fields are used to manipulate spinstates[13, 14, 15, 17]. We point out that this behaviour is characteristic for laterallycoupled DQRs. In vertically coupled QRs, tunnel splitting is constant against thefield[23, 24].To understand the large values of the DQR tunnel splitting in the vicinity of levelcrossings, in Fig. 3 (right panel) we zoom in on the lowest-energy levels around thefirst crossing point. At this point, in a single QR one would expect a series of levelcrossings[20]. However, Fig. 3 reveals a crossing between the second and third levelsonly, the first and fourth states being pushed away by apparent anticrossings. Thisis because the point group symmetry of a QR under vertical magnetic fields, C ∞ , islowered to C in a DQR. The DQR electron states are then classified by the irreduciblerepresentations A and B, indicating even and odd symmetry under a rotation of π degrees, respectively. The symmetry of each level can be ellucidated by inspecting thewavefunctions before the anticrossing (left side of Fig. 3). The energy increases frombottom (ground state) to top (fourth state). Only the real part of the wavefunction isillustrated, as it suffices to capture the relevant aspects of the symmetry. It is clear thatthe first and second (third and fourth) levels form bonding and antibonding molecularstates built from the same ’atomic’ orbitals. On the other hand, it can be seen that thefirst and third levels have symmetry A (even) while the second and fourth ones havesymmetry B (odd). Therefore, the symmetry sequence of the four lowest-lying levelsshown in Fig. 3 is A B A B before the anticrossing, and A A B B after it. haracteristic molecular properties of one-electron double quantum rings under magnetic fields B (T) ene r g y ( m e V ) ABAB AABB
Figure 3.
Right side: zoom of the four lowest-energy levels of Fig. 2 DQR inthe vicinity of the first anticrossing. Solid (dashed) lines are used for states withirreducible representation A (B). Point group symmetry is C . Left side: contours ofthe wavefunction real part for the first (bottom panel) to the fourth (top panel) energylevel at B = 0 . Since the second and third states have different symmetry, they simply cross eachother. By contrast, the first and third (second and fourth) states have the samesymmetry, so that they undergo anticrossings. The anticrossings prevent a smoothmagnetic field dependece of the tunnel splitting. Hence the non-monotonic evolution ofFig. 2(b).Insight into the tunnel-coupling strenght can be obtained by observing Fig. 4, wherewe plot the charge density of the ground state before, during and after the anticrossing.At the anticrossing point, where the interaction between the first and third levels isat its maximum, the charge is pushed towards the inter-ring barrier, leading to thetunnel-coupling enhancement reported in Fig. 2(b).
Figure 4. (Colour online). Contour of the ground state charge density before (leftpanel), during (central panel) and after (right panel) the anticrossing. Dotted linesindicate the DQR limits. Note the enhanced tunnel-coupling of the ground state atthe anticrossing point ( B = 0 . haracteristic molecular properties of one-electron double quantum rings under magnetic fields An intrinsic property of QRs, closely related to the AB effect, is the appearance of one-electron ground states with finite angular momenta, which give rise to an equilibriumcurrent arising from carrier rotation within the structure[20]. In isolated QRs, thiscurrent is proportional to the ground state azimuthal angular momentum m z . In DQRs,due to the lowered symmetry, the angular momentum is no longer a good quantumnumber. However, the expectation value of the angular momenta within each of theconstituent rings can still be taken as a measurement of the carrier rotation within thenanostructure. Thus, in Fig. 5 we compare h m z i for a single QR, a DQR with high( V c = 500 meV) inter-ring barrier and a DQR with regular (experimental-like) barrierheight. ‡ In the single QR case, Fig. 5(a), h m z i is a step function, decreasing in one unit ofangular momentum with every ground state crossing[20]. A similar behaviour is foundin the DQR case with high inter-ring barrier, solid line in Fig. 5(b), as the QRs arealmost isolated. However, a new feature appears in the vicinity of the level crossings( B = 2 . , . . h m z i rapidly goes to zero, which implies as a suddenquenching of the carrier rotation. The introduction of full tunnel-coupling in the DQR,solid line in Fig. 5(c), further modifies the magnetic field dependence of h m z i . First, h m z i no longer takes integer values. Instead, it takes fractional values, reduced as comparedto the case of weakly-coupled rings. This is because the charge density in the tunnelbarrier, which is now increased, does not contribute to the rotation within the rings.Second, h m z i goes to zero every time an AB period is completed, in such a way thatthe periodic quenching of the ring current starts from weak magnetic fields.The origin of the h m z i peaks in Fig. 5(b) and (c) is the interaction between thefirst and third levels of the DQR at the anticrossing points, discussed above. Awayfrom the anticrossing, the first and third levels have angular momenta h m z i and h m z i ,solid and dashed lines in Fig. 5. Since these levels are the two lowest bonding orbitals,their angular momenta are similar to those of the two lowest levels in a single QR.At the anticrossing, however, the strong interaction couples h m z i and h m z i in such away that the ground state tends to h m z i gs = h m z i − h m z i and the excited state to h m z i ex = h m z i + h m z i . Thus, at the first anticrossing h m z i ex ≈ −
1) = −
1, atthe second one h m z i ex ≈ − −
2) = −
3, at the third one h m z i ex ≈ − −
3) = − h m z i − h m z i value,because it tends to deposit a large amount of charge density in the tunneling barrier(recall Fig. 4 center). This provides maximum stabilization for the (bonding) groundstate, but it also leads to h m z i gs ≈ ‡ For the DQR structures, the h m z i is calculated as the sum of the left and right QR local angularmomenta, i.e. the expectation values defined from the origin of each QR. haracteristic molecular properties of one-electron double quantum rings under magnetic fields zzz V = 0.5 eV c V = 0.05 eV c V = 0.05 eV c < m >< m >< m > −4−2 0−4−2 0−6−4−2 0 0 1 2 3 4 5 6 B (T)
Figure 5.
Expectation value of the ground state angular momentum vs. magneticfield in (a) a single QR, (b) a DQR with high inter-ring barrier, and (c) a DQRwith regular inter-ring barrier height. The dashed lines in (b) and (c) represent theexpectation value of the third level. The insets are schematics of the structures. is proportional to the magnetization[43]: M = − ∂E gs ∂B . (3)Here E gs is the ground state energy. The persistent current includes not only thecurrent arising from the electron angular momentum, but also that coming from the haracteristic molecular properties of one-electron double quantum rings under magnetic fields −1−0.8−0.6−0.4−0.2 0 0.2 0 1 2 3 4 5 6 M ( m e V / T ) B (T)
Figure 6.
Magnetization of DQR. The C symmetry is responsible for the roundededges of the curve. The energy level oscillations of charged carriers confined in QRs under magnetic fields, asthose shown in Fig. 2(a), constitute a manifestation of the AB effect, i.e. an action uponthe quantum system exerted by the vector potential A [19]. The action is induced by themagnetic flux enclosed in the trajectory described by the carrier, Φ = H A d l = R B d S .Every time this flux is a multiple of the unit flux quantum, Φ = 2 π ¯ h/e , the energyspectrum retrieves the zero field structure[48] and the system is said to accumulate oneAB phase unit[20]. Experimental evidence of such AB oscillations have been reportedin different types of mesoscopic[37] and nanoscopic semiconductor QRs.[35, 36].In this section, we report on the (to our knowledge) first study of AB effect incomposite systems, and reveal a new aspect of this phenomenon, namely that ABoscillations can be found even when the total flux trapped by the carriers is Φ = 0,provided the flux threading the individual rings is finite. To this end, we consider twomagnetic field configurations:(i) a uniform positive magnetic field goes through the entire system. Hereafter we referto this as parallel field, B p . haracteristic molecular properties of one-electron double quantum rings under magnetic fields B a .The two cases can be modeled using a Coulomb gauge vector potential A = B (0 , x, A = ± B (0 , x,
0) for the left one, where the positive(negative) sign applies for parallel (antiparallel) fields. Note that this gauge choicegrants continuity of A and ∇ A in the whole domain The Hamiltonian now reads: H = ˆ p k m ∗ + B m ∗ x ± i Bm ∗ x ∂∂y + V ( x, y ) , (4)where the negative sign of the linear term in B applies for B a and x < R in = 15 nm and R out = 35 nm ( D = 3 nm for the DQR), are shown inFig. 7. For a single QR, while B p (gray lines) yields usual AB oscillations, B a (blacklines) yields a featureless spectrum, affected by the diamagnetic shift only. This is theexpected difference between the cases where finite and null magnetic flux is trapped bythe carrier. A strikingly different response is however obtained for the DQR, as theenergy structure looks almost the same regardless of the magnetic field configuration.This is surprising, because in the antiparallel case, the magnetic flux penetrating theleft and right rings cancels out, so that the net flux enclosed by the electron is againzero, and one may not expect AB manifestations.To gain some insight into the different behavior of the single and double QR, inFig. 8 we depict the angular momentum expectation value for the left ( h m z i l ) and right( h m z i r ) halves of the each structure, as a function of B a . Clearly, the antiparallelfield induces opposite left and right angular momenta for both structures. Therefore, h m z i = h m z i l + h m z i r = 0, which is consistent with the systems picking a net zero ABphase[20]. Yet, only the single QR energy spectrum shows no AB oscillations.This can be interpreted as follows. The antiparallel field induces clockwise andanticlockwise carrier rotation in each half of the structure. For a single QR, the twocurrents cancel each other out, the angular momentum of the ring is always zero andthen the states are insensitive to the linear term of the magnetic field (responsible forthe AB effects). By contrast, for a DQR, the are finite (though opposite) currents ineach of the rings. Thus, if the left and right rings were uncoupled, it is immediate thatboth would trap magnetic flux and hence show AB oscillations. Moreover, from the C symmetry of Hamiltonian (4), it is easy to show that the energy spectrum of the twouncoupled QRs would be identical but with reversed sign of the angular momenta (inother words, the currents are identical in magnitude but opposite in sign).Switching on the tunnel-coupling enables the electron to delocalize over the tworings while keeping the net trapped magnetic flux zero. Still, as can be seen inthe bottom panel of Fig. 7, the spectra under parallel or antiparallel fields remainsimilar. This is because the tunnel-coupling for the two magnetic field configurations isqualitatively similar (although not identical) as can be seen in Fig. 9, where gray lines haracteristic molecular properties of one-electron double quantum rings under magnetic fields e n e r gy ( m e V )
17 27 e n e r gy ( m e V ) B (T)
Figure 7.
Energy levels of a single QR (top panel) and a DQR (bottom panel)vs. absolute value of the magnetic field. Black (gray) lines stand for the case ofantiparallel (parallel) fields applied on the left and right halves of the structure (seetext). The insets are schematics of the structures under study. Note that paralleland antiparallel fields yield dramatically different responses in a single QR, but almostidentical in a DQR. represent the tunnel splitting under B p and black lines that under B a . Note also that, inthe antiparallel field case, the tunnel splitting increases with | B | , this being responsiblefor the differences in the energy spectra of Fig. 7. The same behaviour is found in DQRswith stronger tunnel-coupling, as shown in Fig. 9 inset, where the tunnel splitting ofDQRs with a thin ( D = 1 nm) barrier is plotted.We then conclude that both the individual ring energies and the tunnel-couplingare similar for B p and B a , which explains the appearence of similar spectra in Fig. 7.Last, we comment on the fact that the tunnel splitting increases with | B | in theDQR subject to antiparallel field, Fig. 9. This is an anomalous behaviour, because theincreasing magnetic confinement should lead to reduced charge density in the barrierand hence to decreasing tunnel splitting, as in the parallel field case[41]. This can beexplained in terms of the sense of the carrier circulation in each ring. For parallel field, haracteristic molecular properties of one-electron double quantum rings under magnetic fields z z lz rz lz r z a < m > < m >< m > < m >< m > −0.4 0 0.4 0.8 < m > −0.8−0.4 0 0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 B (T)
Figure 8.
Ground state angular momentum expectation value of the left and righthalves of the structure vs absolute value of the antiparallel magnetic field. Top row:single QR. Bottom row: DQR. Dashed and dotted lines are used for h m z i l and h m z i r ,respectively. The insets are schematics of the structures. the electron rotates in the same sense (say clockwise) in the two rings. As a result,the current in the surroundings of the tunneling barrier is different for each ring (seeschematic representation in Fig. 10(a)). This hinders the charge sharing between thesubsystems. By contrast, for antiparallel field, the sense of rotation in the left andright rings is opposite. As a result, the current in the surroundings of the tunnelingbarrier is now the same (Fig. 10(b)). This in turn favours the charge density sharing.With increasing field, the current grows and these trends become more important[49].Indeed, for antiparallel field, the favoured charge sharing is able to compensate for thewavefunction squeezing.To illustrate this effect, in Fig. 10(c) and (d) we plot the difference in chargedensity between B a and B p ground states, for weak ( B = 0 . B = 3 . haracteristic molecular properties of one-electron double quantum rings under magnetic fields t unn e l s p litti ng ( m e V ) B (T)
Figure 9.
Tunnel splitting vs absolute value of the magnetic field for a DQR with D = 3 nm. Black and gray lines are for antiparallel and parallel magnetic fields. Theinset shows the results for a DQR with D = 1 nm (enhanced tunnel-coupling). apparent that the antiparallel ground state deposits much more density in the tunnelingbarrier, which results in its enhanced tunnel-coupling. (a) (b)(d)(c) B=0.5 T B=3.5 T
Figure 10. (a) and (b): schematics of the currents induced by parallel and antiparallelfields, respectively. (c) and (d): contour of the charge density difference betweenthe antiparallel field ground state and the parallel field one, under weak and strongmagnetic fields, respectively. White (black) regions indicate excess of antiparallel(parallel) field state charge. Dotted lines show the DQR limits. Note that antiparallelstates have larger charge density in the tunneling region for strong magnetic fields. haracteristic molecular properties of one-electron double quantum rings under magnetic fields
4. Conclusions
We have shown that the electron states in side-coupled coupled QRs display acharacteristic behaviour when subject to vertical magnetic fields, different from thatobserved in other QR and QD structures. In particular, the tunnel-coupling strengthis found to oscillate with the field, showing sharp maxima when the AB effect inducesground state crossings. This may be of interest e.g. for strong magnetic modulation ofthe transport probability between the nanostructures. In these tunnel-coupling maxima,the carrier rotation within the rings is abruptly supressed, owing to charge accumulationin the inter-ring barrier. This introduces a characteristic magnetic field dependence ofthe persistent currents which may be verified experimentally.[42]We have also shown that DQRs, as quadruply-connected systems, may reveal newfundamental aspects of quantum physics arising from the AB effect. In the single QRstructures (doubly-connected systems) investigated to date, a non-zero magnetic fluxpiercing the loop is required to produce AB effects, such as AB oscillations. From ourtheoretical prediction, it follows that this is not necessarily the case for DQRs, where ABoscillations are present even if the net magnetic flux piercing the two loops (and hencethe total accumulated AB phase) is zero, provided the flux going through the individualrings is finite. The experimental setup to prove this should consist of two tunnel-coupledQRs subject to antiparallel magnetic fields in the left and right rings. DQR structurescan be currently realized with remarkable precision using litographic techniques, as inRefs. [35, 37], but the antiparallel field realization may be more challenging. Laser-controlled currents might provide a feasible alternative[50].
Acknowledgments
We acknowledge support from MEC-DGI projects CTQ2004-02315/BQU, UJI-Bancaixaproject P1-1B2006-03 and Cineca Calcolo Parallelo 2007. One of us (J.I.C.) has beensupported by the EU under the Marie Curie IEF project MEIF-CT-2006-023797.
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