Magnetic monopole mechanism for localized electron pairing in HTS
MMagnetic monopole mechanism for localized electron pairing in HTS
M. C. Diamantini, C. A. Trugenberger, and V. M. Vinokur NiPS Laboratory, INFN and Dipartimento di Fisica e Geologia,University of Perugia, via A. Pascoli, I-06100 Perugia, Italy SwissScientific Technologies SA, rue du Rhone 59, CH-1204 Geneva, Switzerland Terra Quantum AG, St. Gallerstrasse 16A, CH-9400 Rorschach, Switzerland
Despite more than three decades of tireless e ff orts, the nature of high-temperature superconductivity (HTS)remains a mystery [1–3]. A recently proposed long-distance e ff ective field theory [4] accounting for all the uni-versal features of HTS and the equally mysterious pseudogap phase, related them to the coexistence of a chargecondensate with a condensate of dyons, particles carrying both magnetic and electric charges. Central to thispicture are magnetic monopoles emerging in the proximity of the topological quantum superconductor-insulatortransition (SIT) that dominates the HTS phase diagram. However, the mechanism responsible for spatiallylocalized electron pairing, characteristic of HTS [5], remains puzzling. Here we show that real-space, local-ized electron pairing is mediated by magnetic monopoles and occurs well above the superconducting transitiontemperature T c . Localized electron pairing promotes the formation of superconducting granules connected byJosephson links. Global superconductivity sets in when these granules form an infinite cluster at T c which isestimated to fall in the range from hundred to thousand Kelvins. Our findings pave the way to tailoring materialswith elevated superconducting transition temperatures. The discovery of HTS about 40 years ago promised a newera of industrial and technological advances, with power-lossfree electric grid lines spanning continents and superconduct-ing kitchen appliances in every household. Yet, materials su-perconducting at room temperature are still a dream. Thelack in understanding of the HTS pairing [1–3] hindered thedesired progress. So far, neither Anderson’s resonating va-lence bond (RVB) theory of HTS based on electron corre-lations [6] and its recent development [7], nor the alternativeroute, deriving HTS from Cooper pairing near a quantum crit-ical point (QCP) associated with an antiferromagnetic order-itinerant electron spin transition [8, 9], while beautifully cap-turing many important features [1, 2, 10–22] of the supercon-ducting and the related pseudogap phase, succeeded to pro-vide a general, unifying theory of HTS.From the relevance of quantum criticality for HTS [9] onecan derive the superconducting transition temperature as T c ∼ E F / √ g , with E F being the Fermi energy and g being the cou-pling constant, in remarkable contrast to the standard BCS be-havior, T c ∼ exp( − const /g ). Yet, a strong coupling has to beassumed, and the characteristic feature of HTS, a small su-perconducting coherence length ξ , implying non-overlappingCooper pairs, is not explained. It has been suggested thatthe path towards a better understanding of the emergence ofsuperconductivity lies in investigating its breakdown via thesuperconductor-insulator transition (SIT) [23] and the emer-gent phases around it [24, 25]. However, while the relevanceof the SIT is mostly accepted, there are still many aspects thatdo not fit the observed HTS phenomenology, mostly originat-ing from the fact that the SIT is a phenomenon well under-stood only in 2D.A field-theoretical approach establishing the long soughtunified theory of HTS and accounting for all universal fea-tures within the superconducting dome and in the pseudo-gap state in a consistent whole [4], stemmed from the topo-logical gauge theory of the SIT [24, 26] and its generaliza- tion to three dimensions [27]. A fundamental mechanism ofthe HTS was identified as the formation of a condensate ofdyons, electrically charged magnetic monopoles [4]. The su-perconducting state is a coexistence phase of dyon and purecharge condensates. The pseudogap state above T c is a newstate of matter, the oblique superinsulator, which harbors apure dyon condensate and exhibits magnetoelectric and ne-matic e ff ects. Symmetry-protected fermionic surface dyonsof charge 2 e , living on a percolation network, cause an elec-tric resistance proportional to the square of the temperature, ina full accord with the experiment [1]. The elevated T c resultsfrom the topological obstruction of splitting Cooper pairs intosingle electrons by magnetic monopoles.Crucial in this picture is the emergent self-inducedgranularity[4, 27] characterizing the SIT at the antiferromag-netic quantum critical point (AFM-QCP) [12, 20]. Supercon-ducting granules of the spatial scale of order of a few super-conducting coherence lengths ξ and connected by tunnellinglinks produce a medium supporting magnetic monopoles [27–29] and dyons [4]. The last missing piece to complete thispicture of HTS is a mechanism of localized electron pairingin real space.Here we demonstrate that this long-sought HTS pair-ing mechanism is the binding of electrons by magneticmonopoles. The minimal model of a HTS material is takenas two conducting planes separated by a distance s of theatomic scale. Magnetic monopoles emerging between theplanes bind two electrons, residing on the respective oppositeplanes, into a state of higher angular momentum. The radicaldi ff erence of the monopole-based binding mechanism fromother commonly considered mechanisms [7, 9, 22, 30] is thatelectron pairs are spatially localized around the monopoles.The heavy monopoles anchoring the electron pairs serve, thus,as nucleation points for the granular structure that emergesupon cooling the system from the temperature of pair for-mation, T pair down to T c . In a system supporting a su ffi - a r X i v : . [ c ond - m a t . s up r- c on ] F e b Conducting planes near the SITPaired electrons Monopole
FIG. 1.
A sketch of the minimal model for HTS.
The conductingplanes near the SIT are shown in gray. Heavy monopole appears inthe middle between the planes and paired electrons are located in theopposite planes, their motion being restricted to respective planes. cient monopole density, global superconductivity sets in whenthe droplets comprising the electron pairs and linked by tun-nelling junctions form an infinite cluster. It thus occurs asthe temperature compares with the coupling energy, and ourestimate gives T c = O (cid:16) (cid:17) K for a typical granule size of O (1) nm. To conclude here, we stress that these purely mag-netic monopoles are di ff erent from the dyons in the pseudogapcondensate [4]. The former are fluctuation-induced topologi-cal excitations which become stable since electron pairing re-duces their repulsive energy and hence the total energy of thesystem. The pseudogap-residing monopoles are the endpointsof Josephson-like vortices residing in the granular array.We start our derivation by considering two electrons inter-acting via the spherically symmetric repulsive 1 / r Coulombpotential (we use natural units c = (cid:126) = ε =
1) andthe short-range repulsion potential V R ( r ). Decomposing thewave function into spherical harmonics yields an additional (cid:96) ( (cid:96) + / r repulsive centrifugal barrier for states with the an-gular momentum (cid:96) >
0. If the short-range repulsion V R ( r ) isstronger than the centrifugal barrier, the electrons settle into astate with the su ffi ciently high angular momentum. Due to ro-tational symmetry, the z -component m , | m | ≤ (cid:96) , of the angularmomentum falls out of the Hamiltonian. What happens, how-ever, if a spherical-symmetry-breaking mechanism encoded ina vector potential makes the Hamiltonian explicitly dependenton the z -component of angular momentum? On a dimensionalground, this brings an additional 1 / r term that can have eithernegative or positive sign. If negative, it can counterbalancethe centrifugal barrier resulting in the attractive 1 / r potentialat intermediate distances before the Coulomb repulsion takesover. Then a potential well forms, resulting in a discrete spec-trum of bound states with finite angular momentum.Magnetic monopoles provide precisely such a spherical-symmetry-breaking mechanism [31]. In presence of a mag-netic monopole of strength g an electron acquires an addi-tional angular momentum L M = ( e g/ π )ˆ r , with ˆ r being the unit vector pointing from the monopole to the electron. TheDirac quantization, e g = π n , n ∈ Z , requires that this addi-tional angular momentum contribution, originating from theinterplay of the electric and magnetic fields of two point par-ticles matches the spectrum imposed by the rotation group.This is just a spherical symmetry breaking contribution to theangular momentum since it singles out the vector connectingthe monopole and electron. As we now show, this monopole-induced angular momentum can bind electrons, and the opti-mal angular momentum of the resulting pair depends on themonopole density. The lower densities favor higher angularmomenta and vice versa.Importantly, monopoles change the statistics of originalelectrons, which become bosons themselves for e g/ π odd butremain fermions for the even values of e g/ π , [32]. The cen-trifugal barrier cancellation takes place only for even valuesof e g/ π leaving the original fermionic statistics of the elec-trons unchanged. Magnetic monopoles, thus induce higher-angular-momentum pairing of electrons. Paired electrons canthen Bose condense into droplets localized near the monopole.Because of the Dirac quantization, magnetic monopoles areheavy excitations, with the mass m M ∝ /α , with α = e / (cid:126) c ≈ /
137 being the fine structure constant. The droplets are an-chored in space and mediate a pairing mechanism giving riseto high- T c superconductivity. Since the product of the electricand the magnetic charge is O(1) due to the Dirac quantiza-tion condition, this is automatically a strong-coupling pairingmechanism, without any further assumption.Note the dimensional dichotomy of the HTS materials.From the viewpoint of the charge- and magneto-transport,HTS materials exhibit a profoundly 2D behavior [33, 34].At the same time phenomena related to topological aspectsof the electronic spectrum, like the magnetoelectric e ff ectin the pseudogap state [35], require the full underlying 3Dmicroscopic nature. The proposed pairing mechanism isaligned with this dichotomy. The monopole pairing restson a 3D quantum mechanical structure comprising two con-ducting planes separated by an atomic scale distance so thatthe charges can tunnel between the planes, providing robustJosephson links between them. At the same time, as longas the thermal coherence length L T = √ π D / ( k B T ) exceeds theinterplane distance s , the system exhibits two-dimensionaltransport properties. Here D = ( π/ γ )( k B T c / eB c2 (0)) is the dif-fusion length and γ = .
781 is Euler’s constant. Since thedephasing length L φ = (cid:112) D τ φ (cid:29) L T as long as k B T (cid:29) /τ φ ,the quasiparticle description holds well in this 2D electricresponse regime and the 3D quantum mechanical consid-eration of electron binding applies. This explains why acuprate monolayer consisting of two conducting planes re-tains the same high transition temperature [36] as a 3D sam-ple, whereas 2D films of conventional superconductors have T c much lower than the bulk of the same material.Let us consider a heavy magnetic monopole of charge g formed in the middle between the two conducting planes.Next, using the remarkable result of [37] that the quasiparti-cle lifetime within the layer is proportional to the intraplanescattering rate, we conclude that electrons are bound to in-traplane motion with rare interplane hops, and that this intra-plane electron motion becomes even more pronounced withincreasing disorder and doping. We obtain a three-body quan-tum mechanical problem involving two electrons of charge e restricted to respective parallel planes and with short-range re-pulsion, and a Dirac magnetic monopole of magnetic charge g in between. We proceed with the simpler formulation ofan infinitely heavy magnetic monopole located at the centerof mass of the two electron system. This reduces to the one-body problem of an electron of reduced mass m / z -axis, thevector potential A u of the monopole is [31] A u = f u ( r , θ ) ˆ ϕ , f u ( r , θ ) = g π r − cos( θ )sin( θ ) , (1)which has a singularity at θ = π . Here r , θ and ϕ denote theusual spherical coordinates and ˆ ϕ is the unit vector in the ϕ direction. To solve the eigenstate problem in the field of aDirac monopole one cannot use a single set of coordinates forthe whole sphere but must use the Wu-Yang formalism [38] tocover the sphere with a so-called atlas of maps, supplementedby gauge transformation conditions on the overlap regions be-tween the di ff erent maps. The simplest atlas comprises twomaps, the northern hemisphere 0 ≤ θ ≤ π/ + (cid:15) , with thegauge potential (1) and the lower hemishpere π/ − (cid:15) ≤ θ ≤ π ,with the gauge-transformed potential A l = f l ( r , θ ) ˆ ϕ , f l ( r , θ ) = − g π r + cos( θ )sin( θ ) , (2)corresponding to the same magnetic monopole at the originbut with the Dirac string now along the positive z -axis A l = A u − ∇ (cid:18) g π ϕ (cid:19) . (3)In both hemispheres the gauge potential is now regular and thecorresponding Pauli equation can be solved. The price to payis gauge transformation between wave functions in the overlapregion [ π/ − (cid:15), π/ + (cid:15) ].The Hamiltonian for two electrons with charges e andmasses m and a fixed magnetic monopole with the magneticcharge g at the origin is H = m ( p − e A ( x )) + m ( p − e A ( x )) − em s · B ( x ) − em s · B ( x ) + V R ( | x − x | ) + V C ( | x − x | ) , (4)where A is the monopole gauge potential defined by Eqs. (1)and (2), s , denote the spin vectors of two electrons, V C ( r ) = e / πε r is the repulsive Coulomb potential, with ε the relative dielectric permittivity of the material and V R ( r ) is the short-range repulsion. We introduce the center of mass and relativecoordinates as R = ( x + x ) / r = ( x − x ) and we set R =
0. To make the model amenable to an analytical solutionwe make the further simplifying assumption that the infinitelyheavy, external magnetic monopole sits exactly at the centerof mass of the two-electron system. The time-independentPauli equation becomes then (cid:20) − m (cid:18) ∇ − ie A (cid:18) r (cid:19)(cid:19) − m (cid:18) ∇ + ie A (cid:18) − r (cid:19)(cid:19) − | e g | π m | r | + V R ( | r | ) + V C ( | r | ) (cid:21) ψ = E ψ , (5)where we have specialized to a total spin 0 state in which thespin of each electron has a hedgehog configuration parallel orantiparallel to the monopole magnetic field, depending on thesign of g .Suppose now the motion of the electrons is constrained tothe two conducting planes at z = ± s , with orbital angular mo-mentum ± (cid:96) on the upper and lower plane respectively. If themonopole charge in between satisfies e g/ π = − (cid:96) , Eq. (5)reduces to a single 2D radial equation (see Methods) (cid:20) − m (cid:32) x ∂∂ x (cid:32) x ∂∂ x (cid:33)(cid:33) + V R ( x ) − | e g/ π | m ( s + x ) + V C ( x ) (cid:21) F ( x ) = EF ( x ) , (6)where x = r sin( θ ) is the radial distance on the planes and V C ( x ) = α/ε √ s + x . The short-range repulsion modelsthe quantum statistical pressure of electrons when they aresqueezed by the two conducting planes. Its exact form doesnot matter; however, it forces electrons to fall into non-zeroorbital angular momentum states (cid:96) > F ( x ) ∝ x (cid:96) for x (cid:28) s and thehigher (cid:96) , the more the wave function is suppressed at the ori-gin. Yet, for e g/ π = − (cid:96) , the resulting repulsive centrifugalbarrier gets completely canceled by the additional, monopole-induced angular momentum, and only an attractive interac-tion due to the electron magnetic moments survives. Notethat these are exactly the values of the monopole charge thatleave the individual electron statistics unchanged. In this case,the potential well forms between the two relevant scales s and a = ε/ m α , where the Coulomb repulsion takes over, and theelectrons form pairs that can Bose condense in a droplet local-ized around the positions of the heavy monopole. To estimatethe optimal value of (cid:96) , note that, on one side, increasing (cid:96) makes the magnetic moment attraction stronger, and on theother side, the higher the monopole charge, the heavier theyare and the higher the energy cost of creating them betweenthe planes. Given the large mass of monopoles, one expectsthat at small monopole densities larger values of (cid:96) are favoredand vice versa.The construction of the potential well result again from theinterplay of dimensionality. The angular momentum of theelectrons constrained to two planes is a 2D e ff ect. The addi-tional angular momentum due to the monopole and the mag-netic moment interactions, however, are the 3D e ff ects, sincethey are directed from the monopole at the centre to the loca-tions of the electrons on the planes. At su ffi ciently large dis-tances, the additional angular momentum cancels out the cen-trifugal barrier and the magnetic moment interaction causesthe overall attraction.The potential well is determined by the combination of themagnetic moment attraction with the short-range repulsionrepresenting quantum statistical pressure of electrons of inter-layer atoms squeezed between the two conducting planes. Assuch, it should be a scale-free, i.e., 1 / x potential. Therefore,in general, both the position of the minimum of the poten-tial well and the bound state energy, are functions of the twospatial scales s and a . If they are comparable, there remainsonly a single spatial and, accordingly, a single energy scale.To estimate it, one can neglect V R ( x ) and V C ( x ) in eq. (6). Bymultiplying the whole equation by m we see that the scale mE in the right-hand side is determined by the unique remainingscale s in the left hand-side. Therefore E = O (1 / ms ). Asshown in [39], the numerical coe ffi cient is of order one so that E (cid:39) (cid:126) ms , (7)where we restore physical units. Since the interplane spac-ing s (cid:39) k F , with k F being the Fermi wavevector, E = O ( E F ). Thecorresponding localization size of the bound state within theplane is (cid:96) (cid:107) (cid:39) s , and as discussed above is the same orthog-onal to the plane, (cid:96) ⊥ (cid:39) s . At finite monopole density ρ , whenmonopoles are brought near each other, the binding energy E first keeps increasing as long as the two-dimensional paral-lel to the plane inter-monopole distance d = ( ρ s ) − / > (cid:96) (cid:107) ≈ s [40]. This is the e ff ect of the 2D Lifshitz localization [41].However, at d < s , the “flat” bottoms of the potential wells inthe many-monopole generalization of Eq. (6) overlap. Thenthe pairs are localized by fluctuations in the monopole den-sity rather than by single potential wells, see [41], where asimilar problem was discussed, and E starts to decrease withincreasing ρ . Therefore, one expects that the optimal bindingis achieved at d (cid:39) s . This prediction for the optimal monopoledensity is in accord with the recent experimental data of [42]showing that the highest transition temperature in carbona-ceous sulfur is indeed achieved at some optimal distance s be-tween the conducting planes. Exact calculating of E requiresmicroscopic treatment, but for present purposes the estimate(7) is su ffi cient.The localized pairs with the binding energy E are the nu-cleation centers for superconducting droplets. As we have de-rived above, the T = s . Athigher temperature the droplet dimension ξ ( T ) will typicallybe larger. Global superconductivity sets in at the temperaturewhen su ffi cient monopoles have formed so that these dropletsform an infinite three-dimensional cluster and is given by theIo ff e-Larkin [43] formula for the transition temperature T c inhighly inhomogeneous superconductors k B T c = ω e − . d /ξ ( T c ) , (8)where ξ ( T ) is the characteristic size of a localized pair and ω is the attempt frequency in the matrix element t = ω exp( − d /ξ ) describing tunneling of a bound pair between ad-jacent monopoles. The mean inter-monopole distance is itselfa function of temperature and material characteristics, but forthe estimate we take the optimal monopole density providingthe strongest binding, i.e. d (cid:39) s . To favor the factors that canlower the expected T c , however, we take the smallest possi-ble size of the pair, i.e. ξ (cid:39) s . The tunneling matrix elementbetween monopoles at the distance d is found following [41],and gives ω = √ /π ( (cid:126) / ms ) √ s / d . The resulting estimatefor the transition temperature is T c ≈ . · [ (cid:126) / ( ms k B )] , (9)where we have restored physical units. Taking the interplanedistance s as 1 nm, one obtains T c = O (10 ) K .The proposed real-space monopole pairing mechanism re-veals the microscopic nature of HTS. It solves the puzzlewhy the pairing size in HTS does not exceed the inter-pairdistance; this follows from the fact that since the optimal d (cid:39) s , then, in general, d (cid:38) s . Therefore the distance be-tween paired electrons is larger than the pair size in HTS. Ourfindings pave the way for tailoring superconducting materialswith enhanced T c . To that end, one has to maximally increasethe electron density harbored within the adjacent conductingplanes, while minimizing the inter-plane distance s . The ele-vated density, by increasing the electronic repulsion, promotesthe enhanced generation of monopoles, thereby lowering thesystem’s overall energy. Determining the optimal monopoledensity and finding the corresponding optimal material pa-rameters requires a detailed self-consistent microscopic treat-ment of the intertwined electron-monopole ensemble and thederivation of the superconducting transition temperature T c .This self-consistent microscopic theory will be the subject ofa forthcoming publication. APPENDIXDerivation of the Pauli equation on the planes
Using that the monopole gauge potentials (1) or (2) are di-vergenceless, we can simplify equation (5) of the main textto (cid:20) − m ∇ + ie m (cid:18) A (cid:18) r (cid:19) · ∇ − A (cid:18) − r (cid:19) · ∇ (cid:19) + e m (cid:18) A (cid:18) r (cid:19) + A (cid:18) − r (cid:19)(cid:19) − | e g | π m | r | + V R ( | r | ) + V C ( | r | ) (cid:21) ψ = E ψ . (10)Starting from this generic equation, one has to formulate twoPauli equations, one for each hemisphere, as explained in themain text. Let us begin with the upper hemisphere, denotedby the subscript “ u ”. Since we restrict to values 0 ≤ θ ≤ π/ + (cid:15) , the arguments of the second gauge potentials A in(10) relate to the lower hemisphere, denoted by subscripts “ l ”.Therefore, we have to use Eq. (1) for the first instance of thegauge potential and Eq. (2) for the second one. This gives (cid:20) − m ∇ + ie m (cid:18) f u (cid:18) r , θ (cid:19) − f l (cid:18) r , π − θ (cid:19)(cid:19) r sin( θ ) ∂∂ϕ + e m (cid:18) f u (cid:18) r , θ (cid:19) + f l (cid:18) r , π − θ (cid:19)(cid:19) − | e g | π mr + V R ( r ) + V C ( r ) (cid:21) ψ u = E ψ u . (11)Repeating the same reasoning for the lower hemisphere π/ − (cid:15) ≤ θ ≤ π , we obtain the second Pauli equation (cid:20) − m ∇ + ie m (cid:18) f l (cid:18) r , θ (cid:19) − f u (cid:18) r , π − θ (cid:19)(cid:19) r sin( θ ) ∂∂ϕ + e m (cid:18) f l (cid:18) r , θ (cid:19) + f u (cid:18) r , π − θ (cid:19)(cid:19) − | e g | π mr + V R ( r ) + V C ( r ) (cid:21) ψ l = E ψ l . (12)Using (1) and (2), we obtain, finally, the explicit expressionsof our pair of Pauli equations (cid:20) − m ∇ + i ( e g/ π ) mr − cos( θ )sin ( θ ) ∂∂ϕ + ( e g/ π ) mr (cid:32) − cos( θ )sin( θ ) (cid:33) − | e g | π mr + V R ( r ) + V C ( r ) (cid:21) ψ u = E ψ u , (cid:20) − m ∇ − i ( e g/ π ) mr + cos( θ )sin ( θ ) ∂∂ϕ + ( e g/ π ) mr (cid:32) + cos( θ )sin( θ ) (cid:33) − | e g | π mr + V R R ( r ) + V C ( r ) (cid:21) ψ l = E ψ l . (13)The presence of the magnetic monopole is reflected in threenew terms, the first two of which, as anticipated, break thespherical symmetry of the original Coulomb problem. Thefirst embodies the monopole-induced additional contributionto the z-axis component of the angular momentum: it has adi ff erent sign in the upper and lower hemispheres since, as wediscussed above, it points from the monopole to the electronsand the monopole sits exactly in the middle. Its coe ffi cientis only weakly dependent on θ since it varies from 1 on theequator to 1 / π/ − (cid:15), π/ + (cid:15) ] of the maps of theatlas [38], ψ l = e − i e g π ϕ ψ u . (14)We can thus make the Ansatz ψ u ( r , θ, ϕ ) = e + i e g π ϕ F u ( r , θ, ϕ ) ,ψ l ( r , θ, ϕ ) = e − i e g π ϕ F l ( r , θ, ϕ ) . (15) Because an exchange of the two electrons involves necessar-ily also a swap of hemispheres, the exchange operator on thewave function must take into account the Wu-Yang gaugetransformation [38]. Therefore, for e g/ π an odd integer theexchange implies a factor (-1) and the statistics of the elec-trons is changed to bosons. For e g/ π an even integer, insteadthere is no additional (-1) factor and the statistics of the in-dividual electrons remains fermionic. Correspondingly, for e g/ π an odd integer the additional gauge factor is a dou-ble covering representation of 2 π rotations, while it is single-valued for e g/ π an even integer. This is the statistical trans-mutation induced by magnetic monopoles (for a review see[31]) . Independently of this statistical transmutation of theindividual components, however, the total spin 0 pair is a bo-son.We now constrain the electron motion to two parallel hori-zontal planes at z = ± s with the monopole at the origin. As aconsequence, the Laplace operator reduces to ∇ = ∂ ∂ x + x ∂∂ x + x ∂ ∂ϕ , (16)where x denotes the radial distance on the two planes. We areinterested primarily in small values of x and the second newterm in (13), O ( θ ), is subdominant with respect to the firstone, O (1) near the poles: we will henceforth neglect it. In ad-dition, since the electrons are forced to move on the two hori-zontal planes, we do not have to use the usual monopole har-monics [38] but we can make use of the much simpler cylin-drical harmonics decomposition by making the Anstaz F u ( r , θ, ϕ ) = e + i (cid:96)ϕ F ( x ) , F l ( r , θ, ϕ ) = e − i (cid:96)ϕ F ( x ) , (17)where r and θ are bound by the condition x = r sin θ and (cid:96) is theangular momentum. The z -components of the angular mo-mentum of the electrons on the two planes cancel out, but thetotal angular momentum | (cid:96) | can well be di ff erent from zero. Itis this value that indicates how much the axis of the compositewave function is tilted with respect to the z -axis. Combining(17) with (15) gives the announced result. When e g/ π = − (cid:96) the total angular momentum and the ensuing centrifugal bar-rier vanish altogether. We obtain thus a single radial equationfor both planes, Eq. (6) of the main text. DATA AVAILABILITY
Data sharing not applicable to this article as no datasets weregenerated or analyzed during the current study.
Acknowledgements
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