Coexistence of diffusive resistance and ballistic persistent current in disordered metallic rings with rough edges: Possible origin of puzzling experimental values
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Coexistence of diffusive resistance and ballistic persistent current in disordered metallic rings withrough edges: Possible origin of puzzling experimental values
J. Feilhauer and M. Moˇsko ∗ Institute of Electrical Engineering, Slovak Academy of Sciences, 841 04 Bratislava, Slovakia (Dated: July 26, 2018)Typical persistent current ( I typ ) in a mesoscopic normal metal ring with disorder due to rough edges andrandom grain boundaries is calculated by a scattering matrix method. In addition, resistance of a correspondingmetallic wire is obtained from the Landauer formula and the electron mean free path ( l ) is determined. Ifdisorder is due to the rough edges, a ballistic persistent current I typ ≃ ev F /L is found to coexist with thediffusive resistance ( ∝ L/l ), where v F is the Fermi velocity and L ≫ l is the ring length. This ballistic currentis due to a single electron that moves almost in parallel with the rough edges and thus hits them rarely (it is shownthat this parallel motion exists in the ring geometry owing to the Hartree-Fock interaction). Our finding agreeswith a puzzling experimental result I typ ≃ ev F /L , reported by Chandrasekhar et al. [Phys. Rev. Lett. , 3578(1991)] for metallic rings of length L ≃ l . If disorder is due to the grain boundaries, our data reproducetheoretical result I typ ≃ ( ev F /L )( l/L ) that holds for the white-noise-like disorder and has been observed inrecent experiments. Thus, result I typ ≃ ev F /L in a disordered metallic ring of length L ≫ l is as normal asresult I typ ≃ ( ev F /L )( l/L ) . Which result is observed depends on the nature of disorder. Experiments thatwould determine I typ and l in correlation with the nature of disorder can be instructive. PACS numbers: 73.23.-b, 73.23.Ra
I. INTRODUCTION
It is known that a conducting ring pierced by magnetic fluxcan support persistent electron current [1]. Persistent cur-rents exist in superconducting rings [2], in mesoscopic resis-tive metal rings [3, 5–8], in ballistic metallic rings [9], and innanorings made of band insulators [10].At zero temperature, the mesoscopic resistive metal ringpierced by magnetic flux Φ supports the persistent current I = P ∀ E j ≤ E F I j , where I j (Φ) = − dE j (Φ) /d Φ is the cur-rent carried by the electron with eigen-energy E j (Φ) , and E F is the Fermi level [3, 4, 9]. Function I (Φ) is periodic withperiod Φ ≡ h/e , which is an experimental signature of thepersistent current [4–9]. If the ring is clean and possesses oneconducting channel, the sum P I j changes its sign whenevera new occupied state j is added. Due to the sign cancelationmainly the electron at the Fermi level contributes to the sum,and the amplitude of the current is I = ev F /L [11], where v F is the Fermi velocity and L the ring circumference. If thering is disordered, the size and sign of the current fluctuatefrom sample to sample and a typical current per one ring is I typ = h I i / , where h . . . i means ensemble average.The number of the conducting channels ( N c ) in the disor-dered metallic rings is usually large ( N c ≫ ) and the ringsobey the diffusive limit, l ≪ L ≪ ξ , where l is the elec-tron mean free path and ξ ≃ N c l is the localization length.To estimate I typ , assume again that mainly the electron atthe Fermi level contributes to the sum P I j . Since L ≫ l ,the electron is expected to move around the ring by diffu-sion. Its transit time is τ D = L /D , where D = v F l/d isthe diffusion coefficient and d is the sample dimensionality.So I typ ≃ e/τ D = (1 /d )( ev F /L )( l/L ) . A similar resultfollows from the Green function theory [12, 13] which as-sumes the non-interacting electrons and emulates disorder by a random potential V ( r ) obeying the white-noise condition h V ( r ) V ( r ′ ) i ∝ δ ( r − r ′ ) . The theory [12, 13] gives I theortyp = 2 × (1 . /d )( ev F /L )( l/L ) , l ≪ L ≪ ξ, (1)where the factor of is due to the electron spin, d = 1 , , or , and the origin of the factor of . is explained in Ref. [14].The first observation of the persistent current in a sin-gle metallic ring was reported [5] for three Au rings of size L ∼ l . The measured currents showed the desired flux-periodicity Φ , but they were ten-to-hundred times larger thanresult (1); they ranged from ∼ . ev F /L to ∼ ev F /L . Thishuge discrepancy has not been explained yet [15, 16]. OtherAu rings showed [6] the currents slightly larger than the result(1) and recent experiments [7, 8] confirmed the result (1) well.Why did the similar measurements of diffusive Au rings[5, 7] show quite different results, I typ ≃ ev F /L and I typ ≃ ( ev F /L )( l/L ) ? A puzzle [5] is why a multichannel disor-dered ring of length L ≫ l carries the current ev F /L , typicalfor a one-channel ballistic ring? These questions are knownas unresolved problems of mesoscopic physics [8, 15, 16].This paper answers both questions theoretically. It is known[16] that there is disorder due to polycrystalline grains andrough edges even in a pure Au ring. Using a single-particlescattering-matrix method [14, 17], we calculate the typicalpersistent currents in the Au rings with grains and rough edges without the white-noise approximation . Another key pointof our single-particle approach is that our description of thesingle-electron states in the ring captures an essential effectof the Hartree-Fock interaction, the cancelation of the cen-trifugal force by an opposite oriented Hartree-Fock field .Our findings can be summarized as follows. If the disorderis due to the polycrystalline grains, our results agree with thewhite-noise-related formula (1) and experiments [7, 8]. How-ever, if the disorder is due to the rough edges, we find theballistic-like result I typ ≃ ev F /L albeit the resistance is dif-fusive ( ∝ L/l ) and L ≫ l , like in the experiment [5]. Thisballistic current is due to a single electron that moves (almost)in parallel with the rough edges and thus hits them rarely. Weshow that this parallel motion exists in the ring geometry ow-ing to the Hartree-Fock interaction. Our major message reads:result I typ ≃ ev F /L in a metal ring of length L ≫ l is as nor-mal as result I typ ≃ ( ev F /L )( l/L ) . Which result is observeddepends on the nature of disorder.We note that we focus us on the typical current rather thanon the mean current h I i . The sign and amplitude of the meancurrent measured in the experiment by Levy et al. [4] is an-other puzzling problem in the field. This problem has beenaddressed in reference [18] within the interacting electronmodel. On the other hand, reference [18] did not study thetypical current. It is tempting to think that the typical currentis not affected by electron-electron interaction; at least, ex-periments [7, 8] confirm result I typ ≃ ( ev F /L )( l/L ) whichhas been derived [12, 13] for non-interacting electrons. Weare thus motivated to study the typical current within a single-particle model. However, our single-particle model is not atruly non-interacting model because it captures a key effect ofthe Hartree-Fock interaction.Our paper is organized as follows. In section II, resistanceof wires with rough edges and wires with grains is calculatedby means of the scattering-matrix approach [14, 17, 19–21].In section III we focus us on the single-particle states in cleanmetal rings. We demonstrate the key role of the Hartree-Fockinteraction and we provide a simple intuitive argument aboutthe existence of ballistic current I typ ≃ ev F /L in rings withrough edges. Microscopic calculations of persistent currentsare presented in section IV. Finally, in section V a summaryof our work is given with a few concluding remarks. II. RESISTANCE OF WIRES WITH GRAIN BOUNDARIESAND WIRES WITH ROUGH EDGES
For simplicity, we study two-dimensional (2D) rings anddiscuss the 3D effects briefly at the end of the paper. Ex-perimentally [4–9], persistent currents in rings were studiedtogether with the resistance of the co-deposited wires in or-der to determine the mean free path l . In this section westudy the wire resistance and mean free path. Sections II.Aand II.B describe our transport model and our results, respec-tively. Of special importance is section II.C. It shows that ouredge-roughness model gives the transport results which areuniversal - independent on the choice of the roughness model. A. Transport model
We consider a stripe-shaped 2D wire (Fig. 1) described byHamiltonian [14, 17] H = − ~ m ∗ (cid:18) ∂ ∂x + ∂ ∂y (cid:19) + U ( x, y ) + V ( x, y ) , (2) h(x) d(x) LW xy 2 ∆ ∆ xc (a) (b) FIG. 1: Our models of disordered wires: (a) wire with grain bound-aries, (b) wire with rough edges. The meaning of all symbols used inthe figure is described in the main text. where m ∗ is the electron effective mass, U is the grain bound-ary potential, and V is the potential due to the wire edges.To simulate the electron transport in wires with grain bound-aries, we will rely on the scattering matrix approach developedin the works [17, 19, 20]. Similarly, to simulate the electrontransport in wires with rough edges, we will rely on the scat-tering matrix approach described in the works [14, 21]. Herewe review both approaches briefly by means of figure 1.Let d ( x ) and h ( x ) be the y -coordinates of the edges. Then V ( x, y ) = (cid:26) , d ( x ) < y < h ( x ) ∞ , elsewhere . (3)For smooth edges one has d ( x ) = 0 and h ( x ) = W , while inthe case of the rough edges d ( x ) and h ( x ) fluctuate randomlyin the intervals h− ∆ , ∆ i and h W − ∆ , W + ∆ i , respectively.It can be shown [14] that the RMS of such random fluctuations( δ ) is simply δ = ∆ / √ . The fluctuations are assumed to ap-pear along the edges abruptly with a constant step ∆ x whichplays (within this model) the role of the roughness correlationlength [14]. The parameters of our roughness model are thus δ and ∆ x . The grain boundaries are modeled as a randomly-oriented mutually non-intersecting lines, where the angle be-tween the line and x -axis is random [17]. Each line consistsof equidistant repulsive dots (depicted by the plus signs) withpotentials γδ ( x − x i ) δ ( y − y i ) , where ( x i , y i ) is the positionof the i -th dot. Thus U ( x, y ) = P ∀ i γδ ( x − x i ) δ ( y − y i ) .If the inter-dot distance c approaches zero and the ratio γ/c is fixed, a grain boundary scatters electrons as a structure-less line-shaped barrier independent on the choice of c . If a2D electron impinges on such a barrier perpendicularly withFermi wave vector k F , it is reflected with probability [17] R G = (¯ γ/c ) / [ k F + (¯ γ/c ) ] , (4)where ¯ γ = m ∗ γ/ ~ . The parameters of our grain boundarymodel are the reflection probability R G (typically [22] R G ∼ . − . ) and the mean inter-boundary distance d G .We connect the wire to two ideal leads - clean long wiresof width W . The spectrum of the electron wave functions ψ ( x, y ) and electron energies E in the leads is given by ψ ( x, y ) = e ikx χ n ( y ) , n = 1 , , . . . ∞ , (5)and E = ǫ n + ~ m ∗ k , ǫ n ≡ ~ π m ∗ W n , (6)where k is the electron wave vector in the x direction, ǫ n isthe eigen-energy of motion in the y -direction, and χ n ( y ) = ( q W sin (cid:0) πnW y (cid:1) , < y < W , elsewhere (7)is the wave function in direction y . Thus, in the leads we havefor the electron energy E a general wave function [14, 17, 23] ψ ( x, y ) = P N n =1 [ A + n ( x ) + A − n ( x )] sin( nπyW ) , x ≤ ψ ( x, y ) = P N n =1 [ B + n ( x ) + B − n ( x )] sin( nπyW ) , x ≥ L (8)where N is the considered number of channels (ideally N = ∞ ), A ± n ( x ) ≡ a ± n e ± ik n x , B ± n ( x ) ≡ b ± n e ± ik n x , and k n ( E ) is the wave vector given by equation ~ k m ∗ + ~ π n m ∗ W = E .Vectors A ± (0) and B ± ( L ) with components A ± n =1 ,...N (0) and B ± n =1 ,...N ( L ) obey the matrix equation [14, 17, 19–21, 23] (cid:18) A − (0) B + ( L ) (cid:19) = (cid:20) r t ′ t r ′ (cid:21) (cid:18) A + (0) B − ( L ) (cid:19) , S ≡ (cid:20) r t ′ t r ′ (cid:21) , (9)where S is the scattering matrix [23]. Its elements t ( E ) , r ( E ) , t ′ ( E ) , and r ′ ( E ) are matrices with dimensions N × N . Ma-trices t and t ′ are the transmission amplitudes of the waves A + and B − , respectively, and matrices r and r ′ are the corre-sponding reflection amplitudes. In particular, the matrix ele-ment t mn ( E ) is the transmission amplitude from channel n inthe left lead into the channel m in the right lead. We evaluate S ( E ) for disorder in figure 1 by methods of papers [14, 17].At zero temperature, the wire conductance g (in units e /h ) is given by the Landauer formula g = P N c n =1 T n ,where T n ( E F ) = N c X m =1 | t mn ( E F ) | k m ( E F ) k n ( E F ) (10)is the transmission probability of channel n . We evaluate t mn for a large statistical ensemble of samples [14, 17] and obtainthe mean transmission h T n i and mean resistance h ρ i = h /g i . B. Transport results
Our results are shown in figure 2. Note that the wireswith grain boundaries exhibit the features typical of the white-noise-like disorder. First, h ρ i follows the usual diffusive de-pendence (the full line in the top left panel) in the form h ρ i = 1 /N c + (2 /k F l )( L/W ) , (11)where /N c is the fundamental contact resistance and themean free path l is a fitting parameter. Second, all h T n i areequivalent in the sense that h T n i ∝ /L for all n [24].The wires with rough edges exhibit a fundamentally differ-ent behavior. Specifically, the data for h ρ i follow the diffusivedependence (the full line in the top right panel) in the form h ρ i = 1 /N effc + (2 /k F l )( L/W ) , (12) < ρ > - / N C
9 34 10 0.1 3430 115 50 0.2 6890 347 500 0.2 56990 347 100 0.2 114 9 34 0.5 0.87 2120 77 2.0 1.9 6570 270 5.0 6.1 28790 347 5.0 8.7 3510 0.1 0.2 0.3 0.4 0.5
L / ξ -3 -2 -1 < T n > L / ξ W N C d G R G l [nm] [nm] [nm] n = 1n = 347 n = 347 n = 1W N C ∆ x δ l [nm] [nm] [nm] [nm] ξ _ ~ C l ξ _ ~ C l Wire with grain boundaries Wire with rough edges (a)(b)
FIG. 2: Transport in disordered Au wires. Parameters of Au are m ∗ = 9 . × − kg and E F = 5 . eV, other parameters are listed.Not to affect the results, in our calculations N is usually kept largerthan N c . Figure (a) shows the mean resistance h ρ i versus L . Notethat h ρ i is reduced by resistance /N c and L scaled by ξ . The local-ization length ξ is obtained [14, 17] from numerical data for h ln g i byusing the fit h ln g i = − L/ξ at L ≫ ξ . The full lines show the linearfit of the diffusive regime (see text) from which we obtain the meanfree path l (the results for l are listed in the figure). In the right panelone should see four slightly different full lines for different N c ; weshow only one of them for simplicity. Figure (b) shows h T n i versus L/ξ for parameters indicated by bold arrows. For n = 1 , , . . . N c the resulting curves are ordered decreasingly. where /N effc is the effective contact resistance due to the N effc open channels and both l and N effc are the fitting pa-rameters. The obtained values of N effc are universal ( ≃ ≪ N c ) for large N c and small ∆ x (see the discussion below).The existence of the N effc open channels reflect also the trans-missions in the right panel of figure 2(b). Specifically, channel n = 1 is almost ballistic ( h T i ≃ ) even for L = 0 . ξ ≃ l and a few channels with low n show h T n i ∼ . . Unlike theopen channels, for all other channels one sees that h T n i de-cays with L rapidly; these channels are in the diffusive regimeor even in the localization regime [14, 25].Figure 3 shows in detail how l and N effc in the wires withrough edges depend on the roughness correlation length ∆ x .Indeed, the N effc versus ∆ x dependence shows that N effc isa universal ( N c -independent) number of the order of forsmall enough ∆ x and large enough N c . The universal N effc has been discovered in Ref. [14], here it is demonstrated for N c as large as . Further, the l versus ∆ x dependenceshows clearly that the minimum mean free path due to theedge roughness scattering is always a few times larger thanthe wire width W . This means that the edge roughness alonecannot explain the experimental [5, 7] observation l . W . Wewill return to this point later on. ∆ x / λ F N e ff C ∆ x [nm] l [ n m ]
9 0.87 3420 1.9 7770 6.1 27090 8.7 347W [nm] δ [nm] N C λ F = 0.52 nm (a) (b) FIG. 3: (a) The mean free path l and (b) effective number of the openchannels N effc in the wire with rough edges, both plotted in depen-dence on the roughness correlation length ∆ x for the parameters asindicated. These data were extracted from the numerical data for h ρ i versus L by means of the fit h ρ i = 1 /N effc + (2 /k F l )( L/W ) , as itis explained in the text and in Fig. 2(a). For simplicity, ratio δ/W iskept nearly the same ( ∼ / ) for each set of δ and W . C. Universality of the step-shaped-roughness model
Before we start to discuss the rings with rough edges (nextsections), we want to make an important remark. In this paper,all our transport results for the wires/rings with rough edgesare obtained for the step-shaped-roughness model in figure1(b). We wish to point out that all these results would remainthe same also for models with a smoothly varying roughness.Any smoothly varying roughness can be modeled by means ofthe step-shaped roughness in figure 1(b) if the latter is appliedas a discretization scheme with very small and very densesteps. Using this approach, all calculations presented in thispaper can be repeated in principle for any roughness model.We show below that the obtained transport results would agreewith the results presented in this paper, if they are comparedat the same value of
L/ξ .It is known for the impurity disorder [26, 27] that a sta-tistical ensemble of the macroscopically-identical mesoscopicconductors with a microscopically-different configuration ofimpurities exhibits the conductance distribution which is thesame (for a given value of
L/ξ ) for any choice of the impuritydisorder model. The weaker the disorder the better the accordof the conductance distributions for various models.A similar universality (the independence on the specificmodel of disorder) seems to exist also when disorder is due tothe rough edges. The conductance calculations in Ref. [25],performed for the same step-shaped-roughness model as ourmodel in figure 1(b), give a quite similar results as the conduc-tance calculations in paper [28], performed for the smoothlyvarying roughness with Gaussian-correlation function. Herewe demonstrate this universality by means of the direct com-parison. We calculate the conductance for the smoothly-varying roughness with Gaussian correlation (model of Ref.[28]), and compare it with the conductance obtained for thestep-shaped-roughness model in figure 1(b). ξ < ρ > - / N C roughness with Gaussian correlationsstep-shaped roughness ξ -1 < T n > roughness withGaussian correlations n = 1n = 10 (a)(b)(c) step-shaped roughness FIG. 4: (a) The top view on the 2D wire with the rough edges gen-erated numerically for two different roughness models. In this nu-merical example the Au wire of width W = 9 nm is considered,which implies that the number of the conducting channels ( N c ) is .For the step-shaped roughness we use the RMS roughness amplitude δ = 0 . nm and roughness-correlation length ∆ x = 0 . nm. Forthe roughness with the Gaussian correlation function we choose theRMS roughness amplitude of . nm and the roughness-correlationlength of . nm. In the former case we obtain the mean free path l = 21 nm and localization length ξ ≃ . N c l , and in the latter casewe find l = 20 . nm and ξ ≃ . N c l . (b) The mean resistance h ρ i versus L/ξ ; a comparison for the roughness models specified above.(c) The same comparative study as in figure (b), but for the channeltransmissions h T n i ; for clarity only the data for the first ten conduct-ing channels are presented. In figure 4 we show a typical output of our comparativestudy for two Au wires with the same number of the conduct-ing channels ( N c = 34 ), so that one can compare directly theindividual channel transmission. It can be seen that the indi-vidual transmissions for both roughness models are in a goodagreement. This illustrates the above mentioned universal-ity; note that the individual transmissions for both roughnessmodels coincide albeit the values of the roughness RMS androughness correlation length in considered roughness modelsare (intentionally) not the same.The universality exists also within the chosen roughnessmodel. Specifically, all results of this paper and paper [14],obtained for the step-shaped roughness, are the same for anychoice of δ and ∆ x , if they are plotted in dependence on L/ξ .Finally, the main result of this paper (Fig. 10b in sectionIV) is that the ring with rough edges supports the ballistic per-sistent current I typ ≃ ev F /L in spite of L ≫ l . This results isuniversal simply due to its insensitivity to the edge roughness. FIG. 5: The 2D ring with the inner radius R and outer radius R .The mean radius is R = ( R + R ) / , the ring width W = R − R .The data in the next two figures are calculated for R = 6 . nmand R = 15 . nm (the ring width W = 9 nm, the ring length L = 2 πR = 70 nm), and for m ∗ equal to the free electron mass. III. SINGLE-ELECTRON STATES IN CLEAN RINGS:EFFECT OF HARTREE-FOCK INTERACTION
In this section we study the single-electron states in cleanmetal rings. In section III.A we calculate the exact non-interacting-electron states. We point out that the ring ge-ometry produces the centrifugal force which pushes the non-interacting states towards the outer ring edge and makes themfundamentally different from the states in the stripe geometry.In sections III.B, III.C and III.D we consider the Hartree-Fockinteraction and we find that the non-interacting-electron ringmodel fails. Namely, the Hartree-Fock interaction eliminatesthe centrifugal force and causes that the true single-electronstates in the ring are in fact similar to those ones in the stripe.This similarity has a serious implication. We have seen insection II that the stripe with rough edges possesses a ballisticchannel (channel n = 1 ) even if L ≫ l . The same has to holdfor the corresponding ring. The ring with rough edges shouldtherefore support ballistic persistent current I typ ≃ ev F /L for L ≫ l . This effect will be studied in section IV. A. Clean ring with non-interacting electrons
Consider the 2D ring (figure 5) in the form of the annu-lus with the inner radius R and outer radius R . The non-interacting electrons in the ring without magnetic flux are de-scribed by the Schrodinger equation H ψ ( r, ϕ ) = Eψ ( r, ϕ ) , (13)where ψ is the wave function, E is the energy, and H = − ~ m ∗ (cid:18) ∂ ∂r + 1 r ∂∂r + 1 r ∂ ∂ϕ (cid:19) + V ( r ) . (14)Here r and ϕ are the polar electron coordinates (figure 5), and V ( r ) is the confining potential V ( r ) = (cid:26) , R < r < R ∞ , elsewhere , (15) r r r R R R R R R m = 0 m = _+1 m = _+2m = _+3 m = _+4 m = _+5m = _+6 m = _+7 m = _+8m = _+9 m = _+10 m = _+11 ξ n = , m ( r ) [ n m - / ] FIG. 6: The full lines show the exact wave functions ξ n =1 ,m ( r ) ofthe non-interacting electrons in the ring geometry (Fig. 5). Thesewave functions are normalized as (2 π/L ) R R R drr | ξ n,m ( r ) | = 1 .The dotted lines show the electron wave function in the 2D stripe, χ n =1 ( r ) = p /W sin (cid:2) πW ( r − R ) (cid:3) . If one sets into the equation (13) the wave function in the form ψ ( r, ϕ ) = 1 √ L e imϕ ξ ( r ) , m = 0 , ± , ± , . . . (16)where ξ ( r ) is the radial wave function and m is the angularquantum number, one obtains the radial Schrodinger equation (cid:20) − ~ m ∗ (cid:18) ∂ ∂r + 1 r ∂∂r − m r (cid:19) + V ( r ) (cid:21) ξ ( r ) = Eξ ( r ) (17)This equation determines the spectrum of energies E n,m andwave functions ξ n,m ( r ) , where n = 1 , , . . . . We obtain E n,m and ξ n,m ( r ) exactly by solving the equation (17) numerically.Magnetic flux Φ can be introduced by applying the sub-stitution m → ( m + Φ / Φ ) in the Hamiltonian of equation(17) and in the factor e imϕ of equation (16). If we do so, thewave functions ξ n,m ( r ) and ξ n, − m ( r ) are no longer degener-ate. However, we find that the difference between them smalland we therefore discuss only ξ n,m ( r ) calculated for Φ = 0 .Figure 6 shows the wave functions ξ n =1 ,m ( r ) calculatedfor the ring in figure 5. They are compared with wave func-tion χ n =1 ( r ) = p /W sin (cid:2) πW ( r − R ) (cid:3) which holds forthe stripe geometry (equation (17) describes the stripe geome-try if the term r ∂∂r is skipped and the term − m r is replaced by − m R ). The difference between the electron states in the ringand electron states in the stripe is clearly visible: in the ringthe function ξ n =1 ,m ( r ) is shifted towards the outer ring edgeby the centrifugal potential ∝ m r and towards the inner ringedge by term r ∂∂r . Evidently, the shift towards the outer edgedominates for large | m | . This shift means that the electrons inchannel n = 1 increasingly hit the outer ring edge.A similar finding has been reported in works [30, 31] wherethe non-interacting electron states in metallic rings were ana-lyzed in terms of the semiclassical trajectories. In the non-interacting model [30, 31] the electron wave functions aregoverned exclusively by the straight-line trajectories. In theannular geometry with L ≫ W it is clear on the first glancethat any straight-line trajectory has to hit the outer ring edgemany times in order to make one trip around the ring. Inparticular, the so-called whispering gallery modes [30, 31]hit solely the outer edge, in accord with our observation that ξ n =1 ,m ( r ) tends to be localized at r = R . As a result, onefinds [31] in channel n = 1 the mean free path l ∼ W whenthe ring edges are rough. We will see that these findings, in-cluding l ∼ W for n = 1 , are artefacts of the non-interactingmodel: they fail in the presence of the Hartree-Fock interac-tion. B. Hartree-Fock equation for clean ring
We still consider the single-electron states in the form ψ n,m ( r, ϕ ) = 1 √ L e imϕ ξ n,m ( r ) , (18)however, they are now described by the Hartree-Fock equation [ H + H ( r )] ψ nm ( r, ϕ ) + F nm ( r, ϕ ) = E nm ψ nm ( r, ϕ ) , (19)where H is the Hamiltonian of the non-interacting electrons(Eq.14), H ( r ) is the Hartree interaction and F nm ( r, ϕ ) is theFock interaction. The Hartree interaction reads [29] H ( r ) = − e πǫ Z π dϕ Z R R dr ′ r ′ ρ ( r ′ ) p r + r ′ − rr ′ cos ϕ , (20)where ǫ is the permittivity of the metal and [29] ρ ( r ) = − eL X n X m (cid:2) | ξ n,m ( r ) | − | χ n ( r ) | (cid:3) , (21)is the space charge density. Here we sum over all occupiedstates ( n, m ) , the factor of incorporates two spin orienta-tions, and χ n ( r ) = p /W sin [ nπ ( r − R ) /W ] (22)is the wave function in the stripe. The charge density (21) isdue to the ring geometry: if we skip in equation (17) the term r ∂∂r and replace the term − m r by − m R , we obtain the stripegeometry with solution ξ n,m ( r ) ≡ χ n ( r ) and ρ ( r ) = 0 . Ingolden rings there are many occupied channels and the term eL P n P m | χ n ( r ) | in (21) is equal to the charge density ofthe positive ion background. Finally, the Fock interaction isoperative between the electrons of like spin. It reads F nm ( ϕ, r ) = − e πǫ Z R R dr ′ r ′ Z π dϕ ′ (1 / √ L ) e imϕ ′ ξ nm ( r ′ ) p r + r ′ − rr ′ cos( ϕ − ϕ ′ ) × L X n ′ X m ′ e im ′ ( ϕ − ϕ ′ ) ξ n ′ m ′ ( r ) ξ n ′ m ′ ( r ′ ) , (23) where we sum over all occupied states ( n ′ , m ′ ) . Evidently, F nm ( ϕ, r ) = 1 √ L e imϕ F nm ( r ) , (24)where F nm ( r ) is the radial part of F nm ( ϕ, r ) : F nm ( r ) = − e πǫ L Z R R dr ′ r ′ Z π dθ e − imθ ξ nm ( r ′ ) p r + r ′ − rr ′ cos( θ ) × X n ′ X m ′ e im ′ θ ξ n ′ m ′ ( r ) ξ n ′ m ′ ( r ′ ) . (25)If we set into Eq. (19) equations (18) and (24), we obtain (cid:20) − ~ m ∗ (cid:18) ∂ ∂r + 1 r ∂∂r − m r (cid:19) + V ( r ) + H ( r ) (cid:21) ξ n,m ( r )+ F nm ( r ) = E n,m ξ n,m ( r ) . (26)The last equation is the radial Hartree-Fock equation.Equation (26) can be solved numerically by means of theHartree-Fock iterations. In the first iteration step, the Hartreeterm (eqs. 20 and 21) and Fock term (25) are calculated bysetting for ξ nm ( r ) the exact non-interacting ring states andequation (26) is solved numerically. This gives a new set ofstates ξ nm ( r ) . In the second iteration step, the Hartree termand Fock term are calculated for ξ nm ( r ) obtained in the firstiteration step and equation (26) is solved again. After manyiterations a self-consistent solution is achieved; the wave func-tions ξ nm ( r ) obtained in two successive steps show a negligi-ble difference. Since the self-consistent calculation is com-putationally cost, in this paper we obtain the self-consistentHartree-Fock results only for rings with a single occupiedchannel. For the multi-channel rings we perform either onlythe first Hartree-Fock iteration step or a so-called restrictedself-consistent Hartree-Fock calculation. In spite of these re-strictions, we are able to draw a few key conclusions. C. Hartree-Fock results: failure of the non-interacting model
Figure 7 shows again the wave functions ξ n =1 ,m ( r ) for thering in figure 5. The full lines show the exact non-interactingring states (taken from the preceding figure), the dotted linesshow the self-consistent Hartree-Fock results. For simplicity,in this Hartree-Fock calculation the electron number in thering is restricted to in order to occupy only channel n = 1 .The results clearly illustrate why the non-interacting modelfails. As | m | increases, the non-interacting electron states (fulllines) are pushed by the centrifugal force towards the outerring edge. However, the Hartree-Fock interaction repels theelectrons back. The Hartree-Fock wave functions are almostsymmetric around the center of the ring cross section even forlarge | m | . Furthermore, this symmetric shape is so narrowthat the wave-function tails do not reach the ring edges. Thisimplies that the electrons in channel n = 1 move around the ξ n = , m ( r ) [ n m - / ] r r r R R R R R R m = 0 m = _+1 m = _+2m = _+3 m = _+4 m = _+5m = _+6 m = _+7 m = _+8 FIG. 7: Electron wave functions ξ n =1 ,m ( r ) for the ring in Fig.5. The full lines show the results for the non-interacting electrons(taken from the preceding figure) and the dotted lines show the self-consistent results for the electrons that interact via the Hartree-Fockinteraction. These Hartree-Fock calculations were performed for electrons in the ground-state n = 1 , m = 0 , ± , ± , . . . , ± ; each( n, m ) is occupied by two electrons with opposite spins. The permit-tivity ǫ is assumed to be equal to the permittivity of vacuum. ring ballistically - without collisions with the ring edges. Thishowever also means that channel n = 1 will be ballistic evenif the ring edges are rough, similarly as we have seen for thestripe geometry (the bottom right panel of figure 2).Figure 7 also suggests that the true single-electron states ofthe clean ring, the Hartree-Fock states ξ n =1 ,m ( r ) , can be wellapproximated by the non-interacting-electron wave-functionof the clean stripe, χ n =1 ( r ) = p /W sin (cid:2) πW ( r − R ) (cid:3) .Clearly, the function χ n =1 ( r ) captures the fact that the ef-fect of the ring curvature is compensated by the Hartree-Fockfield. Additionally, it is not as narrow as the Hartree-Fockstates ξ n =1 ,m ( r ) and thus suppresses the collisions with thering edges less effectively (one does not need to worry thatthe suppression is overestimated). Unlike χ n =1 ( r ) , the exactnon-interacting ring states ξ n =1 ,m ( r ) evidently fail to mimicthe true single-electron ring states. Now we show that thesefindings hold also for the multi-channel rings. D. Hartree-Fock results continued: multi-channel rings
A golden 2D ring of size considered in figure 5 containsabout thousand electrons which occupy about thirty channels n . A self-consistent Hartree-Fock analysis of such many-electron ring is beyond our computational possibilities. How-ever, useful information can be obtained already when onlythe first Hartree-Fock iteration is performed for the ring with afew occupied channels. Results of such calculation are shownin figure 8 for the ring with four occupied channels. The fulllines show the exact non-interacting ring states and the dotted ξ n = , m ( r ) -0.500.5 ξ n = , m ( r ) r -0.500.5 ξ n = , m ( r ) r r m = 0 m = _+3 m = _+7m = _+10 m = _+17m = _+14 m = _+16m = 0 m = _+8 m = _+9m = _+4 m = _+13m = 0m = 0 m = _+6 ξ n = , m ( r ) [ n m - / ] R R R R R R FIG. 8: Electron wave functions ξ n =1 ,m ( r ) , ξ n =2 ,m ( r ) , ξ n =3 ,m ( r ) ,and ξ n =4 ,m ( r ) for the ring in figure 5. The ring is filled by electrons, these electrons occupy four channels, each occupiedstate ( n, m ) contains two electrons with opposite spins. Specif-ically, in channel n = 1 there are electrons in states m =0 , ± , . . . , ± , in channel n = 2 there are electrons in states m = 0 , ± , . . . , ± , channel n = 3 contains electrons in states m = 0 , ± , . . . , ± , and channel n = 4 contains electrons instates m = 0 , ± , . . . , ± . The figure shows the results for selectedvalues of m . The full lines are the results for the non-interactingelectrons (obtained by solving equation 17). The dotted lines are theHartree-Fock results due the first Hartree-Fock iteration step. lines show the Hartree-Fock states due to the first iteration.The following features are worth noticing.As before, the exact non-interacting states are pushed to-wards the outer ring edge by centrifugal force, while theHartree-Fock interaction repels the electrons in the oppositedirection. In particular, most of the Hartree-Fock wave func-tions in channel n = 1 is now shifted towards the inner edgerather than towards the outer edge. Thus, the key feature ofthe exact non-interacting states (the strong shift towards theouter edge by the centrifugal force) tends to diminish whenthe states are subjected to their own Hartree-Fock field. Thisis a clear sign that the exact non-interacting states fail to de-scribe the true single-electron states in metallic rings. Thisalso means that the modeling of the single-electron states inmetallic rings by means of the straight-line paths [30] fails forreal metal rings: the electron paths in presence of the Hartree-Fock field cannot be the straight lines.Figure 8 also shows that the Hartree-Fock states approachthe non-interacting states as the channel number n increases.Indeed, with the increase of n the centrifugal term ∝ m be-comes less important because the larger the number n thesmaller the occupied angular numbers m in channel n . Asa result, the exact non-interacting states approach the stripe-geometry limit χ n ( r ) = p /W sin (cid:2) n πW ( r − R ) (cid:3) , and be-come robust against the Hartree-Fock field.In summary, the first Hartree-Fock iteration step in figure 8shows that the exact non-interacting electron model fails to de-scribe the true single-electron states in a clean multi-channelmetal ring. We have performed a similar first-iteration-stepcalculation also for three other rings with the same size butwith a larger electron number: , , and occupied chan-nels. We have seen a clear trend: the larger the electron num-ber, the stronger the shift of the non-interacting states towardsthe outer ring edge and the larger the opposite-oriented shiftof the Hartree-Fock states. In other words, with increasingFermi energy the difference between the non-interacting statesand Hartree-Fock states increases and the failure of the non-interacting-ring model is more pronounced.What are the true self-consistent Hartree-Fock states inmulti-channel rings? The multi-channel self-consistent cal-culation is for us too cost; a feasible task is the restrictedself-consistent Hartree-Fock calculation. This means that wecalculate the wave functions ξ n,m ( r ) self-consistently for oneselected channel (say channel n = 1 ) by assuming that theelectrons in channel n = 1 interact with the self-consistentHartree-Fock potential due to the electrons in channel n = 1 and with the non-self-consistent Hartree-Fock potential dueto the electrons in channels n = 2 , . . . . The non-self-consistent means that the Hartree-Fock potential due to chan-nels n = 2 , , . . . is calculated by setting for ξ n =2 ,m ( r ) , ξ n =3 ,m ( r ) , . . . the exact non-interacting states rather than theself-consistent Hartree-Fock states.In figure 9 we show again the non-interacting electronstates (panel a) and Hartree-Fock states from the first iter-ation step (panel b), and we compare them with results ofthe restricted self-consistent Hartree-Fock calculation (panelc). Unlike the non-interacting states in panel a, the Hartree-Fock wave functions in panel c are repelled back to the cen-ter. Moreover, when compared with the wave functions inpanels a and b, the wave functions in panel c show a ten-dency to be compressed to the same symmetric form. Thistendency suggests that the fully self-consistent Hartree-Fockprocedure would make the wave functions in panel c evenmore symmetric and even closer to each other. Another sup-port for this suggestion is provided by the single-channel-ringstudy in figure 7, where the (fully self-consistent) Hartree-Fock states ξ n =1 ,m ( r ) are indeed almost the same and almostperfectly symmetric. So we believe that the stripe-geometrylimit χ n =1 ( r ) = p /W sin (cid:2) πW ( r − R ) (cid:3) , proposed abovefor the single-channel rings, approximates well also the truesingle-electron states ξ n =1 ,m ( r ) in multi-channel rings.Note that χ n =1 ( r ) approximates quite well already the (notfully self-consistent) results in panel c. First, it captures thetendency of the Hartree-Fock interaction to compensate the ef-fect of the ring curvature. Second, one sees in panel c that the r χ n = ( r ) ξ n = , m ( r ) [ n m - / ] R R (a) (b)(c)(d) m = _+17m = 0 m = _+17m = 0 m = _+17m = 0 FIG. 9: Electron wave functions ξ n =1 ,m ( r ) with m = 0 , ± , · · · ± in the ring with four occupied channels, considered in Fig. 8.Results from figure 8 are shown again in panels (a) and (b), wherepanel (a) shows the exact non-interacting wave functions and panel(b) shows the Hartree-Fock wave functions due to the first iterationstep. Panel (c) shows the results of the restricted self-consistent Hartree-Fock calculation (see the main text). Panel (d) shows thestripe-geometry solution χ n =1 ( r ) = p /W sin (cid:2) πW ( r − R ) (cid:3) . wave function tails near the edge points R and R are mostlysuppressed much more than the tails of χ n =1 ( r ) . Therefore,the Hartree-Fock states in panel c have to feel the edge rough-ness (if any) less efficiently than it is felt by state χ n =1 ( r ) .Thus, approximation ξ n =1 ,m ( r ) ≃ χ n =1 ( r ) certainly doesnot underestimate the edge roughness scattering in the ring.Finally, approximation ξ n =1 ,m ( r ) ≃ χ n =1 ( r ) can be ex-tended to all n as ξ n,m ( r ) ≃ χ n ( r ) , because the effect of thecentrifugal force diminishes with increasing n (figure 8). Inconclusion, the true single-electron states of the clean metallicring, the self-consistent Hartree-Fock states, can be approxi-mated by the non-interacting states of the clean metallic stripe, ψ n,m ( r, ϕ ) ≃ √ L e imϕ r W sin h n πW ( r − R ) i , (27)of course, with eigen-energies E n,m ≃ ~ π m ∗ W n + ~ m ∗ R m . (28)To add magnetic flux Φ , substitution m → ( m +Φ / Φ ) has tobe used on the right-hand side of equations (27) and (28). Ap-proximation ξ n,m ( r ) ≃ χ n ( r ) captures the fact that the effectof Φ on the numerically obtained ξ n,m ( r ) is hardly visible.We have seen in section II that in the stripe with roughedges the channel n = 1 is ballistic for L ≫ l . Since ξ n =1 ,m ( r ) ≃ χ n =1 ( r ) , channel n = 1 has to be ballisticalso in the ring with rough edges, and such ring should there-fore support ballistic persistent current I typ ≃ ev F /L evenif L ≫ l . This ballistic current is studied in the next sec-tion. In contrast to our result, the non-interacting model pre-dicts [30, 31] for channel n = 1 the diffusive mean free path l ∼ W , whenever the ring with rough edges is of size L ≫ W .This prediction is an artefact of the non-interacting model. IV. PERSISTENT CURRENTS IN RINGS WITH GRAINBOUNDARIES AND ROUGH EDGES
Assume that the wires in figures 1a and 1b are circularlyshaped in the plane of the 2D gas and the wire ends are con-nected. So we have a 2D ring with grain boundaries and a2D ring with rough edges.
What are the persistent currentsin such rings?
In this section we answer the question bymeans of simple intuitive arguments (section IV.A) and bymeans of the first-principle simulation (section IV.B). Simu-lation results for typical persistent currents are presented insection IV.C, section IV.D presents the sample-specific cur-rents. In section IV.E we simulate typical persistent currentsin rings with combined disorder due to the rough edges andgrain boundaries, compare them with experiment [5], and ex-plain the anomalous experimental data.
A. Intuitive arguments
For rings with random grain boundaries one can safely ex-pect the standard diffusive result I typ ≃ ( ev F /L )( l/L ) , be-cause the corresponding metallic stripe shows the standarddiffusive resistance (left panels of figure 2). This expectationagrees with our microscopic results shown later. We note that our grain-boundary model (figure 1a) is universal in the sensethat any other grain-boundary model with random boundarieswould give again the diffusive conductance and diffusive per-sistent current. Indeed, diffusive transport is caused by therandom orientation and random positions of grain boundaries,not by microscopic details of the individual boundary.For the rings with rough edges the situation is different.We have seen in section III that the electron states in cleanrings and clean stripes are similar, in particular ξ n =1 ,m ( r ) ≃ χ n =1 ( r ) . In addition, in section II we have seen that channel n = 1 in the stripe with rough edges possesses at L ≫ l thetransmission h T i ≃ (the right panel of figure 2b). Since ξ n =1 ,m ( r ) ≃ χ n =1 ( r ) , channel n = 1 has to be ballisticalso in the ring with rough edges and the persistent currentin such ring can be estimated as follows. Assume roughlythat h T n i = 1 for n = 1 and h T n i ∼ l/L for all other n .In this model, channel n = 1 contributes by ballistic current I typ = ev F /L while the total contribution from other chan-nels is diffusive, I typ ≃ ( ev F /L )( l/L ) , and negligible for L ≫ l . Thus, multichannel rings with rough edges shouldsupport at L ≫ l the typical currents I typ ≃ ev F /L , expectedto exist only in ballistic single-channel rings. In terms of classical paths, the rough edges scatter all elec-trons except for a small part of those that move (almost) in par-allel with the edges. This small part, mainly the electrons thatoccupy channel n = 1 , hits the edges rarely and thus movesalmost ballistically. We recall (see section III) that the mo-tion parallel with the edges exists in the ring geometry owingto the Hartree-Fock interaction. It eliminates the effect of thering geometry and establishes relation ξ n =1 ,m ( r ) ≃ χ n =1 ( r ) . B. Microscopic model
We start with the clean ring. According to section III, thetrue single-electron states of the clean ring (the Hartree-Fockstates) can be approximated by the non-interacting electronstates of the clean stripe as show equations (27) and (28). Onecan define variables x and y by transformation Rϕ → x and ( r − R ) → y , and rewrite equations (27) and (28) as ψ n,m ( x, y ) = 1 √ L e ik m x r W sin h n πW y i (29)and E n,m = ~ π m ∗ W n + ~ m ∗ L k m , (30)where k m = πL ( m + Φ / Φ ) . If we take the Hamiltonianof the clean stripe (Hamiltonian (2) without disorder) andwrite Schrodinger equation Hψ ( x, y ) = Eψ ( x, y ) , the wavefunctions (29) and eigen-energies (30) are evidently its solu-tions. It is customary to view this approach as a quasi-1Dapproximation in which the non-interacting-electron states ofthe 2D ring are naively mapped on the non-interacting elec-tron states of the straight stripe via transformation Rϕ → x , ( r − R ) → y . In fact, this mapping is not a quasi-1D approx-imation for the non-interacting 2D states. The states mappedon the non-interacting states of the stripe are the Hartre-Fockstates of the ring, and this mapping is due to the fact thatthe Hartree-Fock interaction acts against the centrifugal forceand eliminates the effect of the ring geometry. If one uses thismapping, one in fact captures the key effect of the Hartree-Fock interaction without any Hartree-Fock calculation.In case of disordered rings, the Hartree-Fock interaction isexpected to play a key role in the rings with rough edges (seethe discussion in section III). In this case the Hartree-Fockanalysis would be even more tedious than for the clean rings.Fortunately, the mapping approach is a reasonable and viablealternative which can easy be extended to disordered rings.We bend the disordered 2D stripe in figure 1 to form a 2Dring similar to that one in figure 5, but disordered. We describethe electron states in the ring by Hamiltonian of the constitut-ing stripe, by Hamiltonian (2). The ring is mapped on thestripe by assuming that the x coordinate in Hamiltonian (2) isthe electron position along the ring circumference and y is theposition along the ring radius. We can thus apply directly thescattering matrix calculation for the disordered stripe (section0 I t y p / I
9 34 0.5 0.87 2120 77 10.0 1.9 7130 115 5.0 2.6 10570 270 25 6.1 27590 347 5.0 8.7 35190 347 40 8.7 322 I t y p / I
9 34 15 0.2 21 9 34 10 0.1 3415 57 30 0.4 1530 115 50 0.2 6870 270 350 0.2 43090 347 500 0.2 569 l I t y p / I t y p t heo r l I t y p / I t y p t heo r W N C d G R G l W N C ∆ x δ l [nm] [nm] [nm] [nm] [nm] [nm] [nm] Ring with grain boundaries Ring with rough edges (b)(a)
FIG. 10: Typical persistent current I typ versus L/l in disordered Auring. The ring parameters are shown,
Φ = − . h/e , l has beenobtained from the wire resistivity (figure 2). The arrows point theparameters studied further in figure 12. Symbols are our data, fulllines show formula I theortyp = 1 . ev F /L )( l/L ) . II). Of course, now this calculation has to be supplemented bycyclic boundary conditions [12] ψ (0 , y ) = exp( − i π Φ / Φ ) ψ ( L, y ) , ∂ψ∂x (0 , y ) = exp( − i π Φ / Φ ) ∂ψ∂x ( L, y ) , (31)where the exponential factor is the Peierls phase. We set intoequations (31) the expansion (8) and rewrite them as (cid:18) A − (0) B + ( L ) (cid:19) = (cid:20) Q − ( φ ) Q ( φ ) 0 (cid:21) (cid:18) A + (0) B − ( L ) (cid:19) , (32)where Q is the N × N matrix with terms Q αβ = e i π Φ / Φ δ αβ .The scattering matrix equation (9) has to be fulfilled togetherwith cyclic conditions (32). This happens for discrete energies E = E j (Φ) which we find for a given ring numerically [17].Again, it is tempting to consider the above mapping ap-proach as a quasi-1D approximation [12] and to think abouta truly-2D calculation for non-interacting-electrons [with dis-order introduced in the 2D-ring Hamiltonian (14)]. We recallthat the truly-2D calculation without the Hartree-Fock inter-action fails to describe the true single-electron states in cleanrings and rings with rough edges. The mapping approach cap-tures the key effect of the Hartree-Fock interaction.Once we know the ring spectrum E j (Φ) , we calculate thesample-specific current I = − P ∀ E j ≤ E F dE j /d Φ and even-tually the typical current I typ ≡ h I i / , where h I i is aver-aged over a small energy window at E F . Technical details ofaveraging are explained in [17] and also in section IV.D. N C = 347, ∆ x = 40 nm, L/ l = 120N C = 34, L/ l = 18N C = 347, ∆ x = 5 nm, L/ l = 120 N C = 270, L/ l = 60N C = 77, L/ l = 37N C = 115, L/ l = 80 I t y p / I FIG. 11: Typical persistent current I typ in the ring with rough edgesas a function of the total number of channels ( N ) considered in thesimulation. The same parameters and symbols are used as in figure10(b), the considered ring lengths are shown as L/l . C. Results for typical currents: a comparison for random grainboundaries and rough edges
Figure 10 shows our main results. For the rings withgrain boundaries one can see that our data for I typ agree (atlarge L ) with the diffusive result I theortyp = 1 . ev F /L )( l/L ) .This agrees with experiments [7, 8], illustrates the univer-sality (the white-noise-like properties) of our random-grain-boundary model, and confirms the intuitive expectations ofsection IV.A.For the rings with rough edges, however, our data for I typ are systematically (not regarding the data fluctuations) closeto the ballistic one-channel value I = ev F /L , albeit L ≫ l , N c ≫ , and h ρ i ∝ L . All this agrees with experiment [5, 13]and this agreement is discussed in detail in section IV.E. Inthe preceding text we have arrived at result I typ ∼ ev F /L intuitively by assuming, that the electrons in channel n = 1 almost entirely avoid the scattering with rough edges and thuscarry the ballistic current ∼ ev F /L . Now we make this intu-itive argument more precise.In figure 11 we show how the typical current in the ringwith rough edges depends on the number of channels ( N )considered in the simulation. It is (roughly) N -independentfor N & , no matter how large N c is. In other words, thecurrents ∼ I in rings with rough edges exist owing to theopen channels n = 1 , , . . . , N effc , where N effc ∼ for anyvalue of N c (see also figure 3). Since h T i ∼ , our intuitiveargument invokes that the value I typ ∼ I will survive also ifone chooses N as small as N = 1 . Figure 11 shows that thisis not the case. For instance, in the ring with N c = 347 and L/l = 120 the current for N → is quite close to zero. Thisis easy to understand: Once the channel n = 1 cannot commu-nicate with other channels, the transmission h T i ∼ tends tobe suppressed to zero by Anderson localization, present in anysufficiently long 1D disordered system. Communication witha few other channels is needed to restore h T i ∼ and toobtain I typ ∼ I . FIG. 12: Persistent currents in a ring with grain boundaries, a ringwith rough edges, and a clean ring for the parameters marked by thearrows in Fig. 10, for L = 375 nm, and for Φ = − . h/e . For bothdisordered rings, the considered parameters ensure l ( E F ) = 21 nmat the Au Fermi level ( E F = 5 . eV). Figure (a) shows the single-electron current I j versus the eigen-energy E j . Figure (b) shows thetotal current I = P ∀ E j ≤ E F I j obtained by summing the currents inthe figure (a) for E F varied from to . eV. Figures (c) and (d) showthe same data as the figures (a) and (b), but for a small energy windowbelow the Au Fermi level. The data are scaled by I = ev F /L , thedata points are connected by full lines which serve as a guide for theeye, the bars depict the energy increment ∆ E = 2 π ~ v F /L . Figure(e) shows the typical current I typ ≡ h I i / . Averaging over theenergy window in figure (d) gives the values shown by dashed lines: I typ /I ≃ . l/L ) for the ring with grain boundaries, I typ /I ≃ . for the ring with rough edges, and I typ /I ≃ √ N c for the cleanring [32]. The circles show the data obtained by varying the numberof channels, N , from N = 1 to N > N c (here N c = 34 ). D. Sample-specific currents
To provide further insight, figure 12 shows the sample-specific currents in two selected rings from figure 10 (boldarrows) and in a clean ring. Figure 12(a) shows the de- pendence I j versus E j , figure 12(b) shows the total current I = P ∀ E j ≤ E F I j versus E F . Evidently, the ring with roughedges exhibits remarkably larger currents than the ring withgrain boundaries, albeit both rings are of the same size andposses the same value of l .Figures 12(c) and 12(d) focus on a small energy windowbelow the Au Fermi level. One can see that I j in the ring withrough edges exhibits sharp peaks with the sign alternating andoscillating with period ∆ E = 2 π ~ v F /L . This period is twicethe inter-level distance in the ballistic single-channel ring,which suggests that the peaks are due to the quasi-ballisticchannel n = 1 . [We recall that h T i ∼ also for L/l ≫ ,as is shown in the right panel of figure 2(b).] However, theheight of the peaks is affected also by other channels, because,as discussed above, channel cannot keep h T i ∼ withoutcommunicating with a few other channels.In figure 12(d) one can see that in the ring with rough edgesalso the total current I ( E F ) oscillates with period ∆ E . Theamplitudes of the total current are close to I , and thereforethe typical currents of size ∼ I appear in figure 10(b).In fact, already the data for the clean ring show I ( E F ) os-cillating with period ∆ E . However, the amplitude of I is ∼ √ N c I [32] and the amplitude of I n is I , where thefactor of is due to the spin. Evidently, the rough edges re-duce I from ∼ √ N c I to ∼ I , but they do not change theoscillation period set by the clean ring. Note that also the ringwith grain boundaries exhibits the oscillating persistent cur-rent. These oscillations are chaotic and correlated with cor-relation length ∼ ( l/L )∆ E , predicted [12, 13] for the white-noise-like disorder.Figure 12(e) shows the typical current. The dashed linesshow the values of I typ obtained from the data in figure 12(d),the circles show I typ in dependence on N . For all threerings one sees, that the circles approach with raising N the N -independent value (the large N limit) represented by thedashed line. It can be seen that a reliable estimate of I typ inthe ring with grain boundaries requires N & N c , while for thering with rough edges one only needs N ∼ no matter howlarge N c is. This is due to the effective number N effc ∼ ,as has already been explained in the beginning of this section. E. Combined effect of rough edges and random grains:Comparison with experiment
In experiment [5] the persistent current ∼ I was observedin the Au ring with L ≃ l and W = 90 nm. Indeed, fig-ure 10(b) demonstrates I typ ∼ I also for L/l ≃ and W = 90 nm. The difference is that the work [5] has re-ported l ≃ W ( l = 70 nm for W = 90 nm) while our val-ues of l in Fig. 10(b) [see also figure 3(a)] are at least twotimes larger than W ; the edge roughness alone cannot pro-duce l ≃ W . In reality the edge roughness coexists withother types of disorder. Reference [5] did not specify disor-der in measured samples, but Webb mentioned in [35] thatthe grains in the rings of work [5] were much larger than W I t y p / I l < ρ > - / N C
90 347 5.0 8.7 700 0.2 0 85.2 0.9590 347 5.0 8.7 700 0.2 π /16 97.0 0.9090 347 5.0 8.7 700 0.2 π /8 100 0.895 6 7 8 9 10L [ µ m]00.510 20 40 60 80 100L / l -5 -4 -3 -2 -1 < T n >
40 60 80 100L / l I t y p / I t y p t heo r W[nm] N C ∆ x[nm] δ [nm] d G [nm] R G α l [nm] ξ / N C l Wire/ring with rough edges and bamboo-like grains (d G = 7.7W)d G α n = 1n = 347 α = 0 α = 0 α = π /8 α = π /16 (a)(b) (c)(d) FIG. 13: Transport in Au wires and Au rings with rough edgesand bamboo-like grains. The angle α of the grain boundary is cho-sen at random from the interval ( − α , α ) , where α is the param-eter: α = 0 means the ideal bamboo shape with the boundaryperpendicular to the wire [37–39]. The table shows all parametersand the resulting l and ξ . Figure (a) shows the mean resistance h ρ i versus L/l , figure (b) show the transmission h T n i versus L/l for α = 0 . Open symbols in figure (c) show the typical current I typ /I versus L for various α , the full symbols show the maximum cur-rents. Figure (d) shows the I typ data from figure (c) normalized by I theortyp = 1 . ev F /L )( l/L ) and plotted in dependence on L/l . (say in Ref. [36] d G ≃ W ). The grains with d G ≫ W areknown as bamboo-like grains [37–39]. Of course, d G ≫ W and l ≃ W [5] means l ≪ d G , which suggests that the grainboundaries were not the main source of scattering in work [5].If the random grain boundaries (or impurities) were the mainsource of scattering, the measured persistent current [5] wouldbe ∼ ( l/L ) I rather than ∼ I (c.f. Fig. 10 and Ref. [17]).What remains is the edge roughness and it indeed explains themysterious coexistence of results I typ ≃ I , L/l ≫ , and h ρ i ∝ L . What happens if one adds the bamboo-like grains?
Since d G ≫ W , we fit R G to obtain l ≃ W . Figure 13shows such a study for the same W and similar L as in Ref.[5]. In figure 13(a) we see again the diffusive law h ρ i ∝ L/l ,but now l ≃ W , like in Ref. [5]. Figure 13(b) shows thatthe transmission through channels , , and a few more is stilllarge (between and . ), though not as large as in the wirewith rough edges only [c.f. the right panel of Fig. 2(b)]. Asuppression of the transmission, caused by a combined effectof the rough edges and bamboo-like grains, is visible for all channels. Consequently, l ≃ W . Similarly, the typicalcurrents in figures 13(c) and 13(d) are suppressed in com-parison with the pure edge-roughness case [Fig. 10(b)], but they still grossly exceed the value I theortyp = 1 . ev F /L )( l/L ) .Figure 13(c) presents the maximum currents, because Ref. [5]in fact reported the current amplitudes rather than I typ . Theseamplitudes were between ∼ . I and ∼ I and essentiallythe same show our data (the full symbols). V. SUMMARY AND CONCLUDING REMARKSA. Summary
In our paper, persistent currents in mesoscopic normal-metal rings with disorder due to the rough edges and randomgrain boundaries have been calculated by means of the single-particle scattering-matrix method. In addition, the diffusiveresistance of corresponding metallic wires has been obtainedfrom the Landauer formula and the diffusive electron meanfree path has been determined. Our calculations capture twocrucial points. First, disorder is described microscopically; wedo not rely on the approximation of the spatially homogeneouswhite noise. Second, our description of the single-electronstates in the ring captures the key effect of the Hartree-Fockinteraction, the cancelation of the centrifugal force by an op-posite oriented Hartree-Fock field.Our main results (Fig. 10) are the following. If disorder isdue to the random grain boundaries, our results for the typ-ical persistent current agree with the white-noise-related for-mula I typ ≃ ( ev F /L )( l/L ) and recent experiments [7, 8].However, if the disorder is due to the rough edges, we findthe ballistic-like current I typ ≃ ev F /L albeit the resistance isdiffusive ( ∝ L/l ) and L ≫ l . In other words, the multichan-nel disordered metal ring of length L ≫ l supports the cur-rent I typ ≃ ev F /L , expected to exist only in a single-channeldisorder-free ring. This finding agrees with experiment [5].Thus, figure 10 naturally explains the difference betweenthe experiment [5] and experiments [7, 8]. It simply sug-gests that disorder in samples of works [7, 8] was white-noise-like (most likely mainly due to the random grain boundaries),while disorder in samples of work [5] was likely mainly dueto the rough edges. Ideally, the ballistic persistent current isinherent to metallic rings with rough edges. However, accord-ing to our data in figure 13 , it survives (slightly suppressed)also when the bamboo-like polycrystalline grains are added inorder to emulate the polycrystallinity of the real rings [5].The microscopic origin of the ballistic persistent current inmetallic rings with rough edges has been explained. The bal-listic current is mainly due to the electrons that occupy chan-nel n = 1 . Classically speaking, these electrons move (al-most) in parallel with the ring edges and therefore avoid theedge roughness scattering. The reason why they move in par-allel with the ring edges in spite of the ring geometry is theHartree-Fock interaction; it acts against the centrifugal forceand eliminates the effect of the ring geometry. In terms ofclassical paths, the electron paths in presence of the Hartree-Fock field are not the straight lines; the field deflects them3from the outer ring edge and the resulting electron wave func-tion is centered between the edges almost symmetrically.Finally, we recall that all our results are universal. Thetransport results obtained for the grain boundary model in fig-ure 1(a) hold for any other grain boundary model in whichthe orientation and positions of the boundaries are random.The transport results obtained for the step-shaped-roughnessmodel in figure 1(b) hold also for models with a smoothlyvarying roughness (section II.C). The universality exists alsowithin our specific roughness model; all our results are robustagainst the change of parameters δ , ∆ x , N c , l , and L , if theyare plotted in dependence on L/ξ . Therefore, a missing infor-mation on the nature of disorder in measured samples [5, 7, 8]is not crucial for our conclusions. Anyway, our values of δ and ∆ x are close to the real ones [40]. In principle, we couldattempt to reproduce the measured values of I typ and l ex-actly by fitting the parameters of disorder. This should makesense if new experiments determine I typ and l together withthe parameters of disorder in measured samples. B. Remark on robustness of the 2D results against 3D effects
Our results were obtained for the 2D model of fig-ure 1, while the experimental samples [5, 7, 8] are three-dimensional. We want to point out that the extension of our2D study to 3D would not change our results remarkably. Theeffect of 3D can be estimated without an explicit calculation.In our 2D wire (Fig. 1.b) the roughness scattering is due tothe wire edges. In real 3D wires the roughness scattering isin general due to the wire edges (side walls) as well as due tothe top and bottom surfaces. In spite of this difference the 3Dsample preserves the key feature of our 2D model. Namely,the electrons in the ground 1D channel (now the channel withquantum numbers n y = 1 and n z = 1 , where z is the verticaldirection) still move almost in parallel with the sample edgesand sample surfaces, and therefore avoid the roughness scat-tering. Thus, the transmission through the ground 1D chan-nel has to be ballistic, similarly as we have seen for the 2Dwire (Fig.2). Consequently, the 3D rings have to carry for L/l >> the ballistic current I typ ≃ ev F /L , similarly as the2D rings in figure 10(b).Further, the roughness scattering in 3D does not modifiesthe mean free path l remarkably in comparison with 2D. In-deed, in real 3D wires the roughness amplitude (RMS) of thetop and bottom surfaces is usually of the order of one latticeconstant ( ∼ . nm; see e.g. the paper [41]), which is far lessthan the roughness amplitude at the edges (RMS ∼ nm - nm; see the experiment [40] and our present paper). Sincethe roughness-limited mean free path is proportional to thesquare of the RMS [14], the effect of the top and bottom sur-faces on the mean free path has to be two orders of magnitudeweaker than the effect of the edges. It is thus very likely thatthe roughness scattering in the 3D wires of reference [5] ismainly due to the wire edges.Finally, unlike the 2D wire in figure 1, the edges of the 3D wire are the side walls and the edge roughness at such sidewalls in general scatters the electrons also in the vertical ( z )direction. In comparison with our purely 2D scattering, thismay decrease the roughness-limited mean free path say by afew tens of percent. However, this cannot affect the ballistic-like motion in the ground 1D channel, responsible for the bal-listic current I typ ≃ ev F /L at L/l >> . C. Remark on an angle dependence of roughness scattering
In the non-interacting-electron model of the 2D ring thewave functions are dominated by the straight-line electronpaths [30, 31]. To incorporate the edge roughness scatter-ing in that model, it was assumed [31] that any straight-linepath which hits the ring edge is reflected diffusively no matterwhat is the incidence angle (the angle between the path and theedge). However, a realistic probability of diffusive reflection,derived by Ziman and Soffer [33, 34] for a free wave imping-ing the surface with uncorrelated roughness, strongly dependson the incidence angle. It is equal to unity for perpendicu-lar incidence but approaches zero for small incidence angles.Note that the realistic angle dependence of the edge roughnessscattering is inherent to our scattering matrix method.Indeed, the tendency to a specular reflection at small anglesis manifested by the channel transmission T n . Let us look atthe right panel of figure 2b in detail. Classically, the chan-nel number n corresponds to the angle between the classicaltrajectory and edge, and n = 1 corresponds to the smallestnonzero classical angle allowed by the quantum confinement.Consider say L ≃ . ξ ≃ l . In case of the diffusive re-flection assumed by work [31], for L/l = 120 we should ob-serve T n ∼ l/L ∼ / for all n including n = 1 . However,this is not the case; the right panel of figure 2b shows that T n isbetween 1 and 0.1 for n = 1 , , . . . , . Evidently, the electronmotion in these channels is much more ballistic than diffusive.When this realistic angle dependence is combined with theHartree-Fock interaction, the metallic rings with rough edgessupport for L ≫ l ballistic current I typ ≃ ev F /L . D. T = 1 as a general feature of any diffusive wire and T n =1 ≃ in the wire with rough edges: Two different things We note that the transmission T n =1 ≃ in the wirewith rough edges (right panel of figure 2(b)) has noth-ing in common with the well-known bimodal distribution / p (1 − T ) T , which exist in any diffusive conductor [24]and diverges for T = 1 . Transmissions T in the bimodaldistribution are the eigen-values of the t + t matrix [24], our T n = P N C m =1 | t n,m | are the diagonal elements of the t + t ma-trix. In other words, the channels corresponding to the eigen-values T in the distribution / p (1 − T ) T are the eigen-states of the t + t matrix and the channels corresponding to ourdiagonal elements T n are the plane-wave states. This differ-ence deserves a few remarks.4The bimodal distribution / p (1 − T ) T as a general prop-erty of any diffusive conductor with white-noise-like disor-der [24] coexists with the diffusive persistent current I typ ≃ ( ev F /L )( l/L ) in the corresponding disordered ring [12]. Thismeans that the eigenvalues T = 1 in the bimodal distributiondo not cause any ballistic persistent current. The reason whythe current is diffusive in spite of T = 1 , is most likely thatthe eigenvalue T = 1 does not necessarily mean the ballistictransmission (a well known example is the perfect transmis-sion in case of resonant tunneling).For disorder due to rough edges the situation is fundamen-tally different. In this case the eigen-values T still follow thebimodal distribution / p (1 − T ) T , however, this has noth-ing in common with the ballistic-like persistent current foundby us. The ballistic-like current is due to the appearance ofthe diagonal element T n =1 ≃ . Specifically, any wire in thestatistical ensemble of wires with rough edges exhibits the di-agonal element T n =1 ≃ independently on the choice of theFermi energy and wire length. It is easy to check for any ofour simulated wires, that the electron plane wave which en-ters the wire in channel n = 1 remains (almost) unscatteredbetween any two successive scatterers inside the disorderedregion. As a result, the ring made of such wire supports thepersistent current dominated by the ballistic channel n = 1 ,that is, I typ ≃ ev F /L . In summary, the reason for appear-ance of I typ ≃ ev F /L is the ballistic behavior of the diagonalelement T n =1 ; the fact that the bimodal distribution showseigenvalues T = 1 is irrelevant. Acknowledgement
We thank the Texas Advanced Computing Center (TACC)at the University of Texas at Austin for providing grid re-sources. We thank for grant VEGA 2/0206/11. ∗ Electronic address: [email protected][1] Y. Imry,
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