Anomalous magnetic response of a quasi-periodic mesoscopic ring in presence of Rashba and Dresselhaus spin-orbit interactions
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Anomalous magnetic response of a quasi-periodic mesoscopic ring in presence ofRashba and Dresselhaus spin-orbit interactions
Moumita Patra and Santanu K. Maiti ∗ Physics and Applied Mathematics Unit, Indian Statistical Institute,203 Barrackpore Trunk Road, Kolkata-700 108, India
We investigate the properties of persistent charge current driven by magnetic flux in a quasi-periodic mesoscopic Fibonacci ring with Rashba and Dresselhaus spin-orbit interactions. Within atight-binding framework we work out individual state currents together with net current based onsecond-quantized approach. A significant enhancement of current is observed in presence of spin-orbit coupling and sometimes it becomes orders of magnitude higher compared to the spin-orbitinteraction free Fibonacci ring. We also establish a scaling relation of persistent current with ringsize, associated with the Fibonacci generation, from which one can directly estimate current for anyarbitrary flux, even in presence of spin-orbit interaction, without doing numerical simulation. Thepresent analysis indeed gives a unique opportunity of determining persistent current and has notbeen discussed so far.
PACS numbers: 73.23.Ra, 71.23.Ft, 73.23.-b
I. INTRODUCTION
Over the last couple of decades the phenomenon of per-sistent charge current in mesoscopic ring structures hasdrawn a lot of attention due to its crucial role in under-standing quantum coherence in such interferometric ge-ometries. In the early 80’s B¨uttiker et al. first proposedtheoretically that a small conducting ring carries a netcirculating charge current in presence of magnetic flux φ .This is a pure quantum mechanical phenomenon and cansustain even in presence of disorder. The experimentalverification of persistent charge current came into real-ization during 1990 through the significant experiment done by Levy et al. considering 10 isolated mesoscopiccopper rings. Later many experimental verifications andtheoretical propositions have been made towards thisdirection.A large part of the literature reported so far de-scribes the phenomenon of persistent currents consider-ing perfect periodic rings as well as completely randomones . But a little less attention was paid to thequasi-periodic ring structures which actually bridgethe gap between these fully ordered and randomly dis-ordered phases. However, the studies involving persis-tent current in quasi-periodic ring geometries are mostlyconfined within non-interacting picture and, to the bestof our knowledge, no one has addressed its behavior inpresence of spin-orbit (SO) interaction which can bringsignificant new features into light. It is therefore worth-while to analyze the characteristics of persistent currentin a quasi-periodic Fibonacci ring considering the effectof spin-orbit interaction (SOI).Usually two different types of SO interactions ,namely Rashba and Dresselhaus, are encountered in solidstate materials depending on their sources. The RashbaSO coupling is originated by breaking the inversion sym-metry of the structure, which can be thus tuned via exter-nal gate electrodes placed in the vicinity of the sample. While, the other SO coupling cannot be controlled byexternal means as it is generated from the bulk inversionasymmetry.In the present paper we make a comprehensive anal-ysis of non-decaying circular current in a quasi-periodicmesoscopic Fibonacci ring subjected to Rashba and Dres- Φ AB XY
N21
FIG. 1: (Color online). Schematic diagram of a 5th generationquasi-periodic Fibonacci mesoscopic ring subjected to Rashbaand Dresselhaus spin-orbit interactions. The ring, composedof two different types of atomic sites A and B those are rep-resented by two distinct colored filled circles, carries a netcirculating charge current in presence of magnetic flux φ . selhaus SO couplings. Two primary lattices, viz, A and B are used to get a N -site Fibonacci chain followingthe generation rule F m ( m ≥
3) = { F m − , F m − } with F = A and F = AB , which is then bent and coupledat its two ends to form a ring. Alternatively, we canthink that m th generation Fibonacci sequence F m canbe constructed using two lattice sites A and B apply-ing m times the inflation rules A → AB and B → A recursively, starting with the lattice A or B . Here westart with the lattice A , for the sake of simplicity, andthus, A , AB , ABA , ABAAB , ABAABABA , . . . , etc.,are the first few generations of the Fibonacci sequence.Therefore, as an example, F = ABAAB forms a 5-site( N = 5) Fibonacci ring. This is one representation, theso-called site model , of a Fibonacci generation. An-other form of it is also conveniently used which is knownas bond model where long ( L ) and short ( S ) bondsare taken into account, setting identical lattice sites. Infew cases mixed model , a combination of site and bondmodels, is also used in studying electronic behavior. Forthe sake of simplicity here we restrict ourselves to thefirst configuration.Based on a tight-binding (TB) framework we computepersistent current using second-quantized approach .With this formalism one can find current carried by in-dividual energy levels, and, from that total current fora particular band filling can be easily estimated. Themajor advantage of this technique is that, it reduces nu-merical errors especially for larger rings by avoiding thederivative of ground state energy with respect to flux φ , as used in conventional current calculations .Most importantly, studying individual state currents con-ducting nature of different eigenstates can be determinedwhich is quite significant to understand the response ofa complete system. Thus, utilizing it, the crucial roleplayed by SO interactions on current carrying states canbe analyzed clearly, which is one key motivation behindthis work. We find that state currents get increased sig-nificantly with SO coupling, which thus provide a largenet current and sometimes it becomes orders of magni-tude higher than the SOI-free Fibonacci rings. Undoubt-edly this is an important observation and might throwsome light in the era of deep-rooted debate between theexperimental observations and theoretical estimates ofcurrent amplitudes.Apart from this, we also discuss the behavior of per-sistent current for different band fillings, and, on itswon merit, the quasi-periodic structure exhibits severalanomalous features which can have great signature, par-ticularly, in the aspect of controlling conducting natureof the full system.Finally, we make a detailed analysis to find a scalingrelation of persistent current with ring size N , associatedwith the generation F m . From our extensive numericalanalysis we establish that for a typical flux φ , the cur-rent obeys a relation CN − ξ , where ξ depends on the ratiobetween the site energy difference and nearest-neighborhopping integral. Thus keeping the ratio constant, siteenergies as well as hopping integral can be tuned andwith these changes ξ remains invariant. The pre-factor C strongly depends on both SO coupling and magneticflux, which is also reported here in detail for the com-pleteness. These results offer a unique opportunity todetermine persistent current in a Fibonacci ring, sub-jected to SO coupling, for any arbitrary flux φ withoutdoing any numerical simulation. This is another essentialmotivation for the present investigation.We organize the rest of the article as follows. In Sec.II we present the model and its Hamiltonian in tight-binding framework. The procedure for calculating per-sistent current carried by different eigenstates as well asthe net current for a particular electron filling is given inSec. III, and the numerical results are discussed in Sec.IV. Finally, in Sec. V we summarize our main results. II. MODEL AND TIGHT-BINDINGHAMILTONIAN
We start by referring to Fig. 1, where a quasi-periodicmesoscopic Fibonacci ring composed of two differenttypes of atomic sites A and B is given. The ring, sub-jected to both Rashba and Dresselhaus SO interactions,carries a net circulating charge current in presence of anAB flux φ .To illustrate this model quantum system we adopta tight-binding framework. In the absence of electron-electron interaction the TB Hamiltonian for a N -site Fi-bonacci ring can be described as following: H = H + H rashba + H dressl . (1)The first term, H , represents the Fibonacci ring in theabsence of SO interactions and it becomes H = X n c † n ǫc n + X n (cid:16) e iθ c † n +1 tc n + e − iθ c † n t † c n +1 (cid:17) (2)where θ = 2 πφ/N is the phase factor due to the flux φ which is measured in unit of the elementary flux quantum φ (= ch/e ), and n = 1, 2, 3 . . . . The other factors aredescribed as follows. c n = (cid:18) c n ↑ c n ↓ (cid:19) and c † n = (cid:16) c † n ↑ c † n ↓ (cid:17) , where c † nσ ( c nσ ) isthe creation (annihilation) operator for an electron at n -th site with spin σ ( ↑ , ↓ ). Considering the on-site potentialat n th site for an electron with spin σ as ǫ nσ we express ǫ n = (cid:18) ǫ n ↑ ǫ n ↓ (cid:19) . Depending on the atomic site A or B , ǫ nσ becomes ǫ Anσ or ǫ Bnσ . t is (2 ×
2) diagonal matrix withthe diagonal elements t = t = t , where t representsthe nearest-neighbor hopping integral.The second term, H rashba , describes the Hamiltonianassociated with Rashba SO coupling and it becomes H rashba = − X n α h c † n +1 ( i σ x cos ϕ n,n +1 + i σ y sin ϕ n,n +1 ) e iθ c n + h.c. (cid:3) (3)where α measures the Rashba SO coupling strength and ϕ n,n +1 = ( ϕ n + ϕ n +1 ) / ϕ n = 2 π ( n − /N . σ x and σ y are the Pauli spin matrices in σ z diagonal repre-sentation.In a quite similar way we can write the last term of thetotal Hamiltonian Eq. 1 which is related to DresselhausSO coupling as, H dressl = X n β h c † n +1 ( i σ y cos ϕ n,n +1 + i σ x sin ϕ n,n +1 ) e iθ c n + h.c. (cid:3) (4)where β is the Dresselhaus coefficient. III. THEORETICAL FORMULATION
In this section, we calculate persistent charge cur-rent carried by individual eigenstates using the second-quantized approach and from these individual state cur-rents we determine the net current for a particular elec-tron filling.We start with the current operator I = e ˙ x / ( N a ),where a is the lattice spacing and ˙ x is the velocity oper-ator written in the form, ˙ x = 1 i ~ [ x , H ] (5)where x = a P n c † n n c n denotes the position operator.Thus we can write the current operator as I = eN a i ~ [ x , H ] = 2 πieN ah [ H , x ] . (6)Substituting x and H into Eq. 6 and doing quite lengthybut straightforward calculations we eventually reach tothe expression I = 2 πieN h X n (cid:16) c † n t † n,n +1 ϕ c n +1 e − iθ − c † n +1 t n,n +1 ϕ c n e iθ (cid:17) (7)where t n,n +1 ϕ is a (2 ×
2) matrix whose elements are asfollows: t n,n +1 ϕ, = t n,n +1 ϕ, = t , t n,n +1 ϕ, = − iαe − iϕ n,n +1 + βe iϕ n,n +1 , t n,n +1 ϕ, = − iαe iϕ n,n +1 − βe iϕ n,n +1 .Once I is established, the current carried by any energyeigenstate | ψ m i (say) can be calculated by the relation I m = h ψ m | I | ψ m i (8)where | ψ m i = P n (cid:16) a mn ↑ | n ↑i + a mn ↓ | n ↓i (cid:17) . | nσ i ’s are theWannier states and a mnσ ’s are the coefficients. After sim-plification we reach to the following I m = 2 πieN h X n (cid:0) ta m ∗ n, ↑ a mn +1 , ↑ e − iθ − ta m ∗ n +1 , ↑ a mn, ↑ e iθ (cid:1) + 2 πieN h X n (cid:0) ta m ∗ n, ↓ a mn +1 , ↓ e − iθ − ta m ∗ n +1 , ↓ a mn, ↓ e iθ (cid:1) + 2 πieN h X n (cid:8)(cid:0) iαe − iφ n,n +1 + βe iφ n,n +1 (cid:1) × a m ∗ n, ↑ a mn +1 , ↓ e − iθ + (cid:0) iαe iφ n,n +1 + βe − iφ n,n +1 (cid:1) a m ∗ n +1 , ↓ a mn, ↑ e iθ (cid:9) + 2 πieN h X n (cid:8)(cid:0) iαe iφ n,n +1 + βe − iφ n,n +1 (cid:1) × a m ∗ n, ↓ a mn +1 , ↑ e − iθ + (cid:0) iαe − iφ n,n +1 − βe iφ n,n +1 (cid:1) a m ∗ n +1 , ↑ a mn, ↓ e iθ (cid:9) (9)This is the general expression of persistent charge cur-rent carried by an eigenstate | ψ m i in presence of Rashba and Dresselhaus SO interactions. With this relation totalcharge current at absolute zero temperature ( T = 0 K)for a N e -electron system becomes I = N e X m =1 I m (10)where the contributions from the lowest N e states aretaken into account.This is one way (viz, the second-quantized approach)of calculating persistent charge current which we use inthis work due its potentiality for our present analysis.But, there exists another method, the so-called derivativemethod , where net circulating current is evaluatedby taking a first order derivative of ground state energy E (say) with respect to AB flux φ . IV. NUMERICAL RESULTS AND DISCUSSION
According to the theoretical formulation introduced inSec. III we are now ready to analyze numerical results,computed in the limit of zero temperature, for chargecurrent carried by individual energy levels, net currentfor a particular electron filling and its scaling behaviorwith system size in presence of Rashba and DresselhausSO interactions. In our model since the sites are non-magnetic we can write ǫ Anσ simply as ǫ A for all A -typeatomic sites, and similarly, for B -type sites ǫ Bnσ = ǫ B .When ǫ A = ǫ B , the system becomes a perfect ring as on-site energies are independent of site index n , and thus, wecan set them to zero without loss of any generality. Allthe energies used in our calculations are scaled with re-spect to the nearest-neighbor hopping integral t which isfixed at 1 eV throughout the presentation, and, we mea-sure the current in unit of et/h .Before addressing the central results of persistent cur-rent, let us have a look at the energy band spectrum forboth perfectly ordered and Fibonacci rings to make thepresent work a self contained one. A. Energy Spectrum
In Figs. 2 and 3 flux dependent energy spectra areshown for a 8-site perfectly ordered (viz, ǫ A = ǫ B = 0)and Fibonacci ( ǫ A = − ǫ B = 1 eV) rings, respectively.From the spectra it is clearly observed that the corre-lated disorder removes the energy level crossings noticedin the perfect case and also it reduces the slope of theenergy levels. Most importantly we see that the num-ber of energy levels gets twice when the ring is subjectedto both AB flux φ and SO interaction compared to theSOI-free ring. From the fundamental principle of quan-tum mechanics it is well known that if the Hamiltonianis symmetric under time-reversal operation the Kramer’sdegeneracy gets preserved, resulting degenerate energylevels. For our model, the two physical parameters, mag-netic flux and SO coupling, affect the degeneracy. In - - H a L Φ E H Φ L - - H c L Φ E H Φ L - - H b L Φ E H Φ L - - H d L Φ E H Φ L FIG. 2: (Color online). Electronic energy levels of a 8-sitering (5th generation) with ǫ A = ǫ B = 0 as a function of flux φ for different values of α and β , where (a) α = β = 0; (b) α = 1 eV, β = 0; (c) α = 0, β = 1 eV and (d) α = 0 . β = 1 . - - H a L Φ E H Φ L - - H c L Φ E H Φ L - - H b L Φ E H Φ L - - H d L Φ E H Φ L FIG. 3: (Color online). Electronic energy levels of a 8-sitering (5th generation) with ǫ A = − ǫ B = 1 eV as a function offlux φ for different values of α and β , where (a), (b), (c) and(d) correspond to the identical meaning as given in Fig. 2. presence of φ two-fold degenerate energy levels are ob-tained from the SOI-free (viz, α = β = 0) ring. Similarkind of two-fold degenerate energy states are also notedunder time-reversal symmetry condition (i.e., φ = 0)when the ring is subjected to SO coupling. For this sit-uation we can write E ( k, ↑ ) = E ( − k, ↓ ) following theKramer’s degeneracy, where k represents the wave vec- tor. But, it disappears completely as long as the mag-netic flux is introduced ( E ( k + φ, ↑ ) = E ( − k + φ, ↓ )), andtherefore, we get twice distinct energy levels comparedto the SOI-free AB ring. In addition it is also crucial - H a L Α E - H c L Α E - H b L Α E - H d L Α E FIG. 4: (Color online). Electronic energy levels as a functionof Rashba SO coupling α of a 8-site ring (5th generation)with ǫ A = ǫ B = 0 for different values of φ and β , where (a) φ = β = 0; (b) φ = 0 . β = 0; (c) φ = 0, β = 1 . φ = 0 . β = 1 . - H a L Α E - H c L Α E - H b L Α E - H d L Α E FIG. 5: (Color online). Electronic energy levels as a functionof Rashba SO coupling α of a 8-site ring (5th generation) with ǫ A = − ǫ B = 1 eV for different values of φ and β , where (a),(b), (c) and (d) correspond to the identical meaning as givenin Fig. 4. to note that even for perfectly ordered ring finite gapsappear near the two edges of the energy band spectrumwhen both the Rashba and Dresselhaus SO interactionsare present (see Fig. 2(d)). The origin of such gaps in aring system with α and β has been described elaboratelyby Chang et al in 2006 and they have shown how thegap is sensitive with these parameter values.In order to understand the precise role of SO cou-pling on energy levels in Figs. 4 and 5 we present the SOcoupling dependent spectra for perfectly ordered and Fi-bonacci rings, respectively, considering the identical ringsize as taken in Figs. 2 and 3, for different values of φ and β . With increasing the SO interaction strength splittingof the energy levels gets wider, while the degeneracy fac-tors in different diagrams remains identical as discussedin the spectra Figs. 2 and 3. In these SOI dependent spec-tra (Figs. 4 and 5) eigenenergies are plotted as functionof Rashba SO coupling setting some typical values of β .Exactly similar feature is also obtained under swappingthe parameters α and β (not shown here to save space),and its origin can be understood from the forthcomingsub-section. B. Enhancement of persistent current
Let us start with discussing the influence of SO cou-plings on the behavior of persistent current carried byindividual energy eigenstates for a typical flux φ . The H a L Α=Β= - - - E m I m H d L Α=Β= - - E m I m H b L Α= Β=
03 0 3 - E m I m H e L Α= Β= - E m I m H c L Α= Β=
13 0 3 - E m I m H f L Α= Β= - E m I m FIG. 6: (Color online). Variation of persistent current I m carried by individual energy eigenstates | ψ m i having eigenen-ergy E m of a 8th generation Fibonacci ring at the typical flux φ = 0 . ǫ A = ǫ B = 0, while inthe right column we choose ǫ A = − ǫ B = 1 eV. results of a 8th generation Fibonacci ring are shown inFig. 6 considering φ = 0 .
4, where the left column corre-sponds to ǫ A = ǫ B = 0, while for the right column wechoose ǫ A = − ǫ B = 1. Several interesting features areobtained those are analyzed as follows.At a first glance one can see that in the absence of SOcoupling all distinct energy levels carry finite currents forthe perfectly ordered ring ( ǫ A = ǫ B = 0), whereas these H a L Α= Α= Α= - - Φ I H Φ L H b L Α= Α= Α= - - Φ I H Φ L FIG. 7: (Color online). Net current at a particular elec-tron filling as a function of flux φ for some Fibonacci rings( ǫ A = − ǫ B = 1 eV) with different values of α , where the red,orange and cyan curves correspond to α = 0, 1 and 2 eV,respectively. The other physical parameters are: (a) N = 610(14th generation), N e = 400 and (b) N = 377 (13th genera-tion), N e = 300. Here we set β = 0. currents almost cease to zero in the case of correlateddisordered ring ( ǫ A = − ǫ B = 1 eV). This is quite ob-vious since a pure ring provides extended states whichcarry finite currents, while almost localized states ob-tained from the Fibonacci ring yield vanishingly smallcurrents. These currents even more decrease with in-creasing the correlation strength ( | ǫ A ∼ ǫ B | ) (which arenot shown here in the figure). This fact has alreadybeen discussed in literature in connection with the lo-calization aspects of different aperiodic crystal classes.But one of the major issues of our present investigationi.e., the interplay between SO interactions and quasiperi-odic Fibonacci sequence on electronic localization has notbeen addressed earlier. To illustrate it, in the middleand last rows of Fig. 6 we show the dependence of statecurrents on α and β , respectively. A large number ofdiscrete states of the Fibonacci ring, those were almostlocalized in absence of SO coupling (Fig. 6(d)), providessufficiently large current in presence of non-zero SO cou-pling. This enhancement of current in presence of SOcoupling can be elucidated in terms of quantum interfer-ence as it is directly related to the localization process.In presence of disorder, quantum interference gets dom-inated which gives rise to the electronic localized states,while this effect becomes weakened as a results of SOcoupling as it involves spin-flipping, resulting enhancedcharge currents. Naturally, this effect will be reflectedinto the net current for a particular band filling as dis-cussed later. In addition, it is important to note that H a L Α= - - Φ I H Φ L H b L Α= - - Φ I H Φ L FIG. 8: (Color online). Filling dependent current-flux char-acteristics of a 11th generation Fibonacci ring ( ǫ A = − ǫ B =1 eV) where (a) α = 0 and (b) α = 1 eV. The solid, dashedand dot-dashed curves correspond to N e = 18, 34, and 56,respectively. The Dresselhaus SO coupling is fixed at zero. though the perfect ring exhibits extended states, theyeven carry higher currents in presence of finite SO cou-pling which is clearly spotted from the spectra given inthe left column of Fig. 6.Figure 6 also depicts that the nature (viz, magnitudeand phase) of current carrying states remain unchangedunder swapping the parameters α and β . This invariantnature can be understood through a simple mathematicalargument. Inspecting carefully the Rashba and Dressel-haus Hamiltonians one can see that they are connectedby a unitary transformation U † H rashba U = H dressl , where U = ( σ x + σ y ) / √ | ψ p i (say) of the Rashba ring can be writtenin terms of the eigenstate | ψ ′ p i of the Dresselhaus ringwhere | ψ p i = U | ψ ′ p i . This immediately gives the currentfor the Dresselhaus ring: I p (for H dressl ) = h ψ ′ p | I | ψ ′ p i = h ψ p | U † I U | ψ p i = h ψ p | I | ψ p i = I p (for H rashba ). Hence, itis clearly observed that the nature of the current car-rying states for the Rashba ring is exactly identical tothat of the Dresselhaus ring. Using SU (2) spin rotationtransformation mechanism Sheng and Chang have es-tablished that the Hamiltonian of the Rashba SOI aloneis mathematically equivalent to that of the DresselhausSOI alone, and thus, our findings regarding the invari-ant nature of current carrying states under swapping theSOIs are consistent with their analysis.Following the above characteristics of different current carrying states now we discuss the behavior of net currentfor a particular electron filling. The results are shown in H a L - - Α I H Φ t yp L H b L - - Α I H Φ t yp L FIG. 9: Current, evaluated at a particular flux φ = 0 . φ , asa function of α ( β is fixed at zero) for (a) ordered and (b)Fibonacci ( ǫ A = − ǫ B = 1 eV) rings considering N = 34 (8thgeneration) and N e = 20. H a L - Α E H Φ t yp + D Φ L - E H Φ t yp L H b L - Α E H Φ t yp + D Φ L - E H Φ t yp L FIG. 10: (Color online). Variation of ∆ E with α at φ typ =0 . φ for the same parameter values as taken in Fig. 9, where(a) and (b) represent the identical meaning as given in Fig. 9.We choose ∆ φ = 0 . / Fig. 7, where we show the variation of net persistent cur-rent as a function of φ for some typical Fibonacci ringsconsidering different values of Rashba SO coupling. In(a) the currents are computed for N = 610 (14th gener-ation) and N e = 400, while in (b) these are performedfor N = 377 (13th generation) and N e = 300. From thespectra it is observed that the current almost vanishesfor the entire flux window when the ring is free from SOcoupling (red curves). This is solely due to the aperiodicnature of the site potentials. Introducing the SO cou-pling one can achieve higher current (orange lines), and,for a moderate SO coupling a dramatic change is ob-served (cyan curves), reflecting the I m - E m spectra givenin the right column of Fig. 6. C. Effect of electron filling
To test the dependence of persistent current on elec-tron filling, in Fig. 8 we show the current-flux charac-teristics of a 11th generation Fibonacci ring considering H a L Φ®
00 800 16000.03.57.0 N I t yp H b L Φ= - - I t yp - - FIG. 11: (Color online). Variation of current with ring size N considering ǫ A = − ǫ B = 0 . β = 0 where (a) φ → φ = 0 .
3. The red and black dots, correspondingto α = 0 and 1 . I typ and N which produces continuous curvesdepending on the scaling factors. three different values of N e . The results are shown forboth zero (Fig. 8(a)) and finite (Fig. 8(b)) values of α ,where the solid, dashed and dot-dashed lines correspondto N e = 18, 34 and 56, respectively. For the ring with-out any SO coupling, currents are less fluctuating with N e , while the fluctuation becomes significant in the pres-ence of SO coupling. This is due to the irregular pat-tern of current amplitudes for different current carryingstates (Fig. 6(e)). It is clearly observed from the spec-trum Fig. 6(e) that one or more states those carry smallercurrents reside among the higher current carrying states,and accordingly, when we set N e to a particular value,depending on the top most filled energy level higher orsmaller current is obtained since the net current essen-tially depends on the contributions from the neighboringstates of this highest filled level. D. Anomalous oscillation of current with SOcoupling
The results analyzed so far are worked out only forsome typical values of Rashba SO interaction. In or-der to establish the critical role played by SO interactionmore precisely on persistent current now we focus on thebehavior given in Fig. 9, where we plot persistent cur- H a L Α=
00 0.25 0.5 - Φ C H b L Α= - Φ C H c L Α= - - Φ C FIG. 12: (Color online). Dependence of C on φ at three dis-tinct values of α . The dotted points are evaluated by exactlycalculating currents for a wide variation of ring size N con-sidering ǫ A = − ǫ B = 0 . β = 0. Fitting these datasets we generate functional forms which provide continuouscurves. rent as a function of α for a particular flux φ = 0 . φ .Both the perfect and Fibonacci rings reflect the fact thattypical current gets increased with increasing α provid-ing anomalous oscillations. Interestingly we see that inthe impurity-free ring the typical current changes its signalternately from positive to negative for a wide windowof α and the window widths get broadened (Fig. 9(a))for higher values of α . On the other hand, a continuousvariation with smaller current (Fig. 9(b)) is obtained inthe Fibonacci ring. These features can be substantiatedfrom the spectra shown in Fig. 10. Here we plot the dif-ference ∆ E of ground state energies, determined at twotypical fluxes ( φ typ , φ typ + ∆ φ (∆ φ → α considering the same parameter values as taken inFig. 9. The factor − ∆ E / ∆ φ gives the persistent currentat φ typ , as used in conventional method, and thus fromthe nature of ∆ E - α characteristics (Fig. 10) we can es-timate the oscillating behavior of current with α as ∆ φ is always positive. This is exactly what we present inFig. 9. E. Scaling behavior
Finally, in this sub-section, we discuss size-dependentpersistent current in presence of SO interaction and fromthat we try to find the scaling behavior.Figure 11 demonstrates the variation of typical current I typ with system size N for two different values of α inthe half-filled limit. Two cases are analyzed depending onthe flux φ , one is for φ → φ = H a L Φ®
00 0.2 0.40.02.505.00 Α C H b L Φ= - - - Α C FIG. 13: (Color online). Dependence of C on α for two dif-ferent values of φ . The colored dots and the continuous linescorrespond to the similar meaning as given in Fig. 12. Theother parameters are: ǫ A = − ǫ B = 0 . β = 0. .
3, and they are presented in (a) and (b), respectively.The dots in the spectra are computed from the second-quantized approach and they obey a scaling relation ofthe form: I typ = CN − ξ where ξ = 1 and 1 .
03 for α = 0and 1 . C depends on both α and φ . In the limit φ → C becomes 0 .
504 and 19 . α = 0 and 1 . − .
862 and − . φ = 0 .
3. Usingthis scaling relation we generate the continuous curves,where the black and red lines correspond to α = 1 . α and φ .In analyzing Fig. 11 two important aspects should benoted. Firstly, the reduction of current with ring size N . The reason behind this reduction can be easily un-derstood in terms of the coherence of electronic wavefunction. For smaller rings wave function becomes co- herent throughout the ring yielding larger current, whilethe phase coherence gets reduced with increasing N providing lesser current. Secondly, the current ampli-tudes of different Fibonacci rings satisfy a specific scal-ing law. This scaling behavior essentially comes fromthe quasiperiodicity of the system. The quasiperiodicitynotably affects, as well, the energy band spectra of a Fi-bonacci ring (not shown here to save space). For instance,total energy bandwidth ∆ E (= P n | E n ( φ = 0) − E n ( φ = φ / | , where E n ’s are the eigenvalues) sharply decreaseswith the Fibonacci generation F m and it satisfies a sim-ilar kind of scaling relation with F m . This is exactlyreflected in I typ - N characteristics. Similar type of scalingnature is also obtained in other different quasiperiodicsystems those have been described elsewhere .Following this analysis one question naturally ariseshow the coefficient C depends on α and φ , so that chargecurrent can be estimated at arbitrary values of these Φ Α- C FIG. 14: (Color online). Simultaneous variation of C with α and φ for the Fibonacci rings described with ǫ A = − ǫ B =0 . β = 0. parameters for any generation of the quasi-periodic Fi-bonacci ring. The answer is given in Figs. 12-14.Focusing on the spectra given in Fig. 12, we see thatfor lower values of α , C - φ data exhibits sinusoidal-likepattern, though we cannot find a simple functional rela-tionship with these data sets, and accordingly, here we donot present that functional form. On the other hand, for α = 1 . C - φ data can be fitted well through a sim-ple relation: C = 20 . − φ . and it gives a linear-likevariation with φ (Fig. 12(c)).In Fig. 13 we demonstrate C - α characteristics for twotypical values of flux φ . Two different functional formsare obtained for these fluxes and they are: C ( α ) = 0 . . α . (for φ →
0) and C ( α ) = − . − . α . (for φ = 0 . C on both α and φ simultane-ously. The result is given in Fig. 14 which clearly reflectsthe above scaling analysis as presented in Figs. 12-13.For the complete analysis of scaling behavior now wediscuss the interplay of Rashba and Dresselhaus SOIs onpersistent current. The results are shown in Fig. 15 wherethe typical current is calculated for two different valuesof Rashba SO coupling, like Fig. 11, at two distinct ABfluxes ( φ → φ = 0 .
3) considering the DresselhausSO coupling β = 0 . I typ = CN − ξ is obtained where ξ becomes 1 and 1 .
08 for α = 0 and 1 . C dependson both the SO couplings and flux φ as well. To have a H a L Φ®
00 800 16000.03.57.0 N I t yp H b L Φ= - - I t yp - - FIG. 15: (Color online). Variation of current with ring size N considering ǫ A = − ǫ B = 0 . β = 0 . φ → φ = 0 .
3. The colored dots and the continuous linescorrespond to the similar meaning as described in Fig. 11. Φ Α- FIG. 16: (Color online). Simultaneous variation of C with α and φ for the Fibonacci rings described with ǫ A = − ǫ B =0 . β = 0 . complete idea about the variation of C for a wide range of φ and α in presence of finite Dresselhaus SO coupling, inFig. 16 we present a 3D diagram (like Fig. 14), and, from this spectrum we can easily determine the pre-factor C at the desired parameter values.With these scaling results (Figs. 11-16) one can eas-ily determine persistent current in any quasi-periodic Fi-bonacci ring in presence of SO couplings without do-ing detailed numerical calculations. Certainly this is aunique opportunity and has not been discussed before. • Application Perspective:
All the results describedabove are worked out only for isolated rings. Now itis interesting and significant as well to know how sucha system can be utilized in possible spintronic devicessince SO interaction in low-dimensional geometries hasattracted much attention due to its potential applicationsin diverse directions. For example, to enhance quantuminformation processing as well as quantum computationcontrolling electron’s spin degree of freedom is highly im-portant . The SO interaction can provide much deeperinsight for generating spin current and also its manip-ulation rather than conventional methodologies. Some-times the interplay between Rashba and Dresselhaus SOinteractions may give a significant change in electronictransport, as discussed by several groups . In orderto reveal these facts the ring has to be connected withexternal electrodes, viz, source and drain. In presence ofsuch electrodes one can study different aspects of spin-dependent transport like, spin currents, spin resoled con-ductances, spin polarization to name a few. One suchwork has been done Wang and Chang towards this direc-tion . They have studied two-terminal spin-dependenttransport through a 1D ring subjected to both Rashbaand Dresselhaus SO couplings. The interplay betweenthese two SOIs leads to a significant change in electronictransmission, localization of electrons and also spin polar-ization of the current. They have also shown how conduc-tance is sensitive to the ring-electrode interface geometry,and these results certainly give a great impact in design-ing future spintronic devices. Several other works have also been put forward along this direction to exploremany interesting features of spin transport in a bridgesetup. Before the end, we would like to state that sincethe study in open system (viz, source-ring-drain system)requires a complete separate theoretical approach, herein the present manuscript we do not go for this. We willanalyze these aspects in our future work. V. CLOSING REMARKS
In conclusion, we have investigated the critical rolesplayed by Rashba and Dresselhaus SO couplings on per-sistent charge current in a quasi-periodic Fibonacci ringthreaded by a magnetic flux φ . Using a tight-bindingframework we have computed individual state currentsas well as net current for a particular band filling basedon second-quantized approach. Analyzing state currentswe can predict the conducting nature of individual energylevels, which on the other hand, provides an important0tool in understanding the net response of a complete sys-tem. From the calculation of net current we have foundthat SO interaction can enhance the current significantlyand sometimes it becomes orders of magnitude highercompared to the SOI-free Fibonacci ring. This observa-tion might throw some light in the era of deep-rooteddoubt between the experimental observations and theo-retical predictions of persistent current.In the rest of our work, we have essentially focusedon the scaling behavior of persistent current with ringsize N , associated with the Fibonacci generation, and es-tablished a unique way of determining persistent chargecurrent without going through detailed numerical calcu-lations. In the analysis of scaling properties we have re-stricted ourselves to the half-filled band limit consideringodd electron filling. But, these scaling relations can be well applied to the even electron filling for the half-filledband case, expect the small rings (viz, F and F ) wherethe current deviates slightly from our fitting curve. In-deed, the establishment of scaling relation for any generalelectron filling and for any disordered ring, be it randomor made of any kind of quasi-periodic lattices, will behighly interesting and important too. These issues willbe available in our next work and it is the first step to-wards this direction.Finally, it should be important to note that throughoutthe analysis we have presented the results only for the sitemodel. But almost identical features are also obtainedfor the bond model and even for the mixed model, whichwe verify through our exhaustive numerical analysis, andaccordingly, here we do not present those results. ∗ Electronic address: [email protected] M. B¨uttiker, Y. Imry, and R. Landauer, Phys. Lett. A ,365 (1983). L. P. L´evy, G. Dolan, J. Dunsmuir, and H. Bouchiat, Phys.Rev. Lett. , 2074 (1990). E. M. Q. Jariwala, P. Mohanty, M. B. Ketchen, and R. A.Webb, Phys. Rev. Lett. , 1594 (2001). N. O. Birge, Science , 244 (2009). V. Chandrasekhar, R. A. Webb, M. J. Brady, M. B.Ketchen, W. J. Gallagher, and A. Kleinsasser, Phys. Rev.Lett. , 3578 (1991). H. Bluhm, N. C. Koshnick, J. A. Bert, M. E. Huber, andK. A. Moler, Phys. Rev. Lett. , 136802 (2009). V. Ambegaokar and U. Eckern, Phys. Rev. Lett. , 381(1990). A. Schmid, Phys. Rev. Lett. , 80 (1991). U. Eckern and A. Schmid, Europhys. Lett. , 457 (1992). L. K. Castelano, G.-Q. Hai, B. Partoens, and F. M.Peeters, Phys. Rev. B , 195315 (2008). J. Splettstoesser, M. Governale, and U. Z¨ulicke, Phys. Rev.B , 165341 (2003). G.-H. Ding and B. Dong, Phys. Rev. B , 125301 (2007). H. F. Cheung, Y. Gefen, E. K. Reidel, and W. H. Shih,Phys. Rev. B , 6050 (1988). R. A. Smith and V. Ambegaokar, Europhys. Lett. , 161(1992). H. Bouchiat and G. Montambaux, J. Phys. (Paris) ,2695 (1989). E. Gambetti-C´esare, D. Weinmann, R. A. Jalabert, andPh. Brune, Europhys. Lett. , 120 (2002). S. K. Maiti, M. Dey, S. Sil, A. Chakrabarti, and S. N.Karmakar, Europhys. Lett. , 57008 (2011). Y. M. Liu, R. W. Peng, G. J. Jin, X. Q. Huang, M. Wang,A. Hu, and S. S. Jiang, J. Phys.: Condens. Matter ,7253 (2002). X. F. Hu, R. W. Peng, L. S. Cao, X. Q. Huang, M. Wang,A. Hu, and S. S. Jiang, J. Appl. Phys. , 10B308 (2005). R. Z. Qiu and W. J. Hsueh, Phys. Lett. A , 851 (2014). G. J. Jin, Z. D. Wang, A. Hu, and S. S. Jiang, Phys. Rev.B , 9302 (1997). Y. A. Bychkov and E. I. Rashba, JETP Lett. , 78 (1984). G. Dresselhaus, Phys. Rev. , 580 (1955). R. Winkler,
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