Electronic Properties of α− T 3 Quantum Dots in Magnetic Fields
EEPJ manuscript No. (will be inserted by the editor)
Electronic Properties of α − T Quantum Dots in Magnetic Fields
Alexander Filusch a and Holger Fehske b Institute of Physics, University Greifswald, 17487 Greifswald, Germanythe date of receipt and acceptance should be inserted later
Abstract.
We address the electronic properties of quantum dots in the two-dimensional α −T lattice whensubjected to a perpendicular magnetic field. Implementing an infinite mass boundary condition, we firstsolve the eigenvalue problem for an isolated quantum dot in the low-energy, long-wavelength approximationwhere the system is described by an effective Dirac-like Hamiltonian that interpolates between the graphene(pseudospin 1/2) and Dice (pseudospin 1) limits. Results are compared to a full numerical (finite-mass)tight-binding lattice calculation. In a second step we analyse charge transport through a contacted α − T quantum dot in a magnetic field by calculating the local density of states and the conductance within thekernel polynomial and Landauer-B¨uttiker approaches. Thereby the influence of a disordered environmentis discussed as well. PACS.
XX.XX.XX No PACS code given
Quantum matter with Dirac-cone functionality is ex-pected to provide the building block of future electron-ics, plasmonics and photonics. Against this background,above all graphene-based nanostructures were intensivelyexamined, both experimentally and theoretically, in therecent past. This is because their striking electronic prop-erties can be modified by nanostructuring and patterning,e.g., manufacturing nanoribbons [1], nanorings [2], junc-tions [3], quantum dots [4], or even quantum dot arrays [5,6]. Thereby the transport behaviour heavily relies on thegeometry of the sample (or device) and its edge shape [7,8]. The mutability of systems with Dirac nodal points,which is especially important from a technological point ofview [9], can also be achieved by applying external electric(static or time-dependent) fields. One of the options arenanoscale top gates that modify the electronic structure ina restricted area [10]. This allows to imprint junctions andbarriers relatively easy, and therefore opens new possibil-ities to study fascinating phenomena such as Klein tun-nelling [11,12], Zitterbewegung [13,14], particle confine-ment [15,16], Veselago lensing [17], Mie scattering ana-logues [18,19,20,21,22] and resonant scattering [23,24].Clearly the energy of the charge-carrier states can be ma-nipulated by (perpendicular) magnetic fields as well. Withthis the quantum Hall effect, the Berry phase curvature,the Landau level splitting and Aharonov-Bohm oscilla-tions have been investigated [2,25,26]. a alexander.fi[email protected] b [email protected] Shortly after the field of graphene was opened, Dirac-cone physics was combined with flat-band physics in amodified lattice, the α − T lattice, which is obtainedby coupling one of the inequivalent sites of the honey-comb lattice to an additional atom located at the cen-tre of the hexagons with strength α [27,28,29]. Obviously,such a lattice interpolates between graphene ( α = 0) andthe Dice lattice ( α = 1). Most notably, the flat bandcrosses the nodal Dirac points, which has peculiar conse-quences, such as an α -dependent Berry phase [30], super-Klein tunnelling [31,32], or Weiss oscillations [33]. Inter-estingly, the magneto-optical response will be also en-hanced due to the flat bands [34]. Analysing the frequency-dependent magneto-optical and zero-field conductivity ofHg − x Cd x Te [35] at the critical cadmium concentration x c (cid:39) .
17 (marking the semimetal-semiconductor transi-tion), it has been shown that this material can be linked tothe α − T model with α = 1 / √ α − T and Dice ( α = 1) models experimen-tally are cold bosonic or fermionic atoms loaded in opticallattices [30,37].The massless Dirac equation [38] provides the basisfor numerous theoretical investigations of the low-energyexcitations in these novel, strictly two-dimensional sys-tems [39,40,41,29,32,42,43,44,45,46,47,48,49,50,51,52,53], whereby the quasiparticles carry a pseudospin 1 inthe Dice lattice rather than pseudospin 1/2 in the caseof graphene. Accordingly one usually works with a three-(Dice) and two-component (graphene) realisation of thestandard Dirac-Weyl Hamiltonian. Investigating the elec-tronic properties of α − T quantum dots in magneticfields, we also start from such a description, and thereforemust implement a boundary condition when the dot is cut a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p A. Filusch et al.: α − T quantum dot out from the plane [54,39,55]. Of course, this approachhas to be approved by comparison with lattice modelresults obtained numerically [56,55,57]. Addressing thetransport behaviour of contacted dots and the influenceof disorder on that we have to work with the full latticemodel in any case.The outline of this paper is as follows. In Section 2we introduce the α − T model, discuss the continuumapproach, derive the infinite-mass boundary condition,and solve the eigenvalue problem for an isolated quan-tum dot in a constant magnetic field in dependence on α . Section 3 contains our numerical results for the eigen-value spectrum, the (local) density of states and the con-ductance. Thereby we critically examine how the con-tinuum model results compare to the numerical exacttight-binding lattice-model data (Section 3.1). Afterwardswe study transport through a quantum dot subject to amagnetic field in the end-contacted lead-sample geometrymost relevant for experiments (Section 3.2), and analyseboundary disorder effects (Section 3.3). We conclude inSection. 4. α − T model We start from the tight-binding Hamiltonian H α = − (cid:88) (cid:104) ij (cid:105) te iΦ ij a † i b j − (cid:88) (cid:104) ij (cid:105) αte iΦ ij b † i c j + ∆ (cid:88) i (cid:16) a † i a i − b † i b i + c † i c i (cid:17) + H.c. , (1)where a ( † ) , b ( † ) and c ( † ) annihilate (create) a particle in aWannier state centred at site A , B and C of the α − T lattice, respectively. The nearest-neighbour transfer am-plitude between A and B sites is given by t , and willbe rescaled by α if hopping takes place between nearest-neighbour B and C sites, see Fig. 1 (a). In this way, thescaling parameter interpolates between the honeycomblattice ( α = 0) and the Dice lattice ( α = 1). In thepresence of a vector potential A ( r ), hopping is modifiedfurther by the Peierls phase Φ ij = 2 π/φ (cid:82) ji A ( r )d r with φ = h/e .In order to implement boundary conditions below, wehave introduced a sublattice-dependent onsite potential ∆ , which opens a gap in the band structure at the chargeneutrality point. In what follows we assume that ∆ > ∆ < E . Note that a positive ∆ will shift the flat bandto the bottom of the upper dispersive one.Next we write down the corresponding continuumDirac-Weyl Hamiltonian in momentum space in the ab-sence of a magnetic field, being valid for low energies nearthe Dirac-points K ( τ = +1) and K (cid:48) ( τ = − H ϕτ = v F S ϕτ · p + U ∆ , (2) where ϕ = arctan α and τ is the valley index. In equa-tion (2), v F = 3 at/ (cid:126) is the Fermi velocity, where a refersto the lattice constant, and p = − i (cid:126) ∇ denotes the mo-mentum operator in two spatial dimensions. The compo-nents of the pseudospin vector S ϕτ = ( τ S ϕx , S ϕy ) in (three-dimensional) spin space, S ϕx = ϕ ϕ ϕ ϕ ,S ϕy = − i cos ϕ i cos ϕ − i sin ϕ i sin ϕ , (3)represent the sublattice degrees of freedom. In equa-tion (2), the matrix U = − (4)introduces a mass term, similar to σ z in the standard(spin-1/2) massive Dirac-Weyl equation. Therefore H ϕτ comprises the limiting cases of massive pseudospin 1/2( α = 0) and pseudospin 1 ( α = 1) Dirac-Weyl quasi-particles. Rescaling the energy by cos ϕ , the eigenvalues E τ,s | ψ τ (cid:105) of H ϕτ | ψ τ (cid:105) = E τ,s | ψ τ (cid:105) become E τ, = ∆ , (5) E τ,s = s (cid:112) ( v F p ) + ∆ , (6)where s = ± Implementing the so-called infinite mass boundary condi-tion (IMBC) we take up a proposal by Berry and Mon-dragon [58]. For this, we consider the Hamiltonian H ϕτ = S ϕτ · p + ∆ ( r ) U (7)(setting v F = (cid:126) = 1 in this section), with a position-dependent mass term, ∆ ( r ) U , which is zero (finite) inside(outside) a circular region D , cf. Fig. 1. Note that Her-miticity of the Hamiltonian in D implies (cid:104) n D · j τ (cid:105) ( r ) = 0at every point r of the boundary ∂D . Here, j τ = S ϕτ is thecurrent density operator and n D = (cos ϑ ( r ) , sin ϑ ( r )) isthe normal vector of D . Then the local boundary condi-tion for a general wave function ψ τ = ( ψ τ,A , ψ τ,B , ψ τ,C )is ψ τ,B (cid:12)(cid:12)(cid:12)(cid:12) r ∈ ∂D = iΓ τ ( r ) (cid:16) cos ϕ e iτϑ ( r ) ψ τ,A + sin ϕ e − iτϑ ( r ) ψ τ,C (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) r ∈ ∂D . (8) . Filusch et al.: α − T quantum dot 3 Fig. 1. (a) α − T lattice with basis { A, B, C } and Bravais-lattice vectors a and a . Next-nearest neighbours are con-nected by δ A,i ( i = 1 , ,
3) where α gives the ratio of the trans-fer amplitudes A - B and B - C . In the numerical work we usegraphene-like parameters, i.e., a lattice constant a = 0 .
142 nmand a transfer integral t = 3 .
033 eV which sets the energyscale. (b) Continuum model energy dispersion near K or K (cid:48) when ∆ = 0 with two linear dispersive bands and a flat bandat E = 0. (c) α − T dot setup with a constant magnetic field,perpendicular to the ( x, y ) plane. The quantum dot D (blueregion) with radius R and zero gap ( ∆ = 0) is surroundedby a ring of width W (grey, dashed border) having a gapfulband structure ( ∆ > n D is perpendicular to theboundary. The variable Γ τ ( r ) can be obtained from the solutionof the scattering problem at a planar mass step, H ϕτ = S ϕτ · p + ∆Θ ( x ), where the height of the barrier is assumedto be larger than the energy ( ∆ > | E | ) and the Heavisidestep function divides the ( x, y )-plane in regions I for x < x >
0. In doing so, we will consider only thedispersive states, since (cid:104) j τ (cid:105) = 0 for the flat band states.In region I, the wave function with wave vector k =( k x , k y ) and propagation direction θ k = arctan k y /k x is ψ I τ,s = 1 √ τ cos ϕ e − iτθ k sτ sin ϕ e iτθ k e i kr + r τ √ τ cos ϕ e iτθ k − sτ sin ϕ e − iτθ k e i k (cid:48) r . (9)Here, k (cid:48) = ( − k y , k x ) denotes the wave vector of the re-flected wave having a valley-dependent reflection coeffi-cient r τ .In region II, the wave function takes the form ψ II τ,s = t τ √ τ a τ,s b τ,s τ b τ,s e − qx + ik y y d τ,s , (10) where t τ denotes the valley-dependent transmission coef-ficient, ( k x , k y ) = ( iq, k y ), and a τ,s = − i cos ϕ (cid:113) ( q − τ k y ) ( ∆ + E ) , (11) b τ,s = (cid:113) ( q − k y )( ∆ − E ) , (12) c τ,s = − i sin ϕ (cid:113) ( q + τ k y ) ( ∆ + E ) , (13) d τ,s = (cid:113) ∆q + Ek y − τ k y q cos 2 ϕ ( ∆ + E ) . (14)Obviously, ψ II τ,s is an evanescent wave perpendicular to theboundary but oscillatory along ∂D .Enforcing the continuity of the wave function at x = 0, ψ I τ,s,B = ψ II τ,s,B , (15)cos ϕ ψ I τ,s,A + sin ϕ ψ I τ,s,C = cos ϕ ψ II τ,s,A + sin ϕ ψ II τ,s,C , (16)and performing the limit ∆ → ∞ ( q → ∞ ), we obtain r τ,s = is + cos ϕ e − iτθ k + sin ϕ e + iτθ k is − cos ϕ e iτθ k + sin ϕ e − iτθ k . (17)Since | r τ,s | = 1 ∀ E , the incoming wave is perfectly re-flected at the boundary, regardless of τ and s . Insertingthe full wave function (9) with (17) and n D ( x = 0) ≡ e x into equation (8), we find Γ τ = τ .Clearly the whole scattering problem can be rotatedby any angle ϑ , i.e., for the α − T lattice the IMBC at ∂D becomes: ψ τ,B = iτ (cid:0) cos ϕψ τ,A e iτϑ + sin ϕψ τ,C e − iτϑ (cid:1) . (18)At α = 0 we reproduce the IMBC of graphene [58]. α − T quantum dot ina perpendicular magnetic field We now consider a circular quantum dot of radius R in aconstant magnetic field, B = B e z , related to the vectorpotential A = B/ − y, x, x, y ) → ( r, φ ), the (minimal-coupling) Hamiltonianis H ϕτ = v F S ϕτ · ( p + e A ) + ∆U Θ ( r − R ) . (19)In the quantum dot region D ( r < R ) we have ∆ = 0 and H ϕτ = τ (cid:126) ω c ϕL τ, − ϕL τ, + ϕL τ, − ϕL τ, + . (20)Here, L τ, ∓ = − ie ∓ iτφ (cid:110) ∂ ρ ± τL z (cid:126) ρ ± τ ρ (cid:111) , L z = − i (cid:126) ∂ φ , (cid:126) ω c = √ (cid:126) v F /l B , l B = (cid:113) (cid:126) eB , and ρ = r/ √ l B . Rota-tional symmetry ([ H ϕ , J z ] = 0 ) suggests the ansatz: ψ Dτ = χ τ,A e i ( m − τ ) φ χ τ,B e imφ χ τ,C e i ( m + τ ) φ . (21) A. Filusch et al.: α − T quantum dot With this, for the dispersive band states , we obtain thefollowing differential equation for the χ τ,B component:0 = (cid:26) ∂ ρ + 1 ρ ∂ ρ − m + 4 ε τ +2 τ cos 2 ϕ − (cid:18) m ρ + ρ (cid:19)(cid:27) χ τ,B , (22)yielding ψ Dτ,s = N τ cos ϕρ − m + τ f τ,A e i ( m − τ ) φ iε τ ρ − m L − mn τ ( ρ ) e imφ τ sin ϕρ − m − τ f τ,C e i ( m + τ ) φ e − ρ . (23)The L ba ( x ) are the generalized Laguerre polynomials, f τ,A = (cid:40) − L − m +1 n + − ( ρ ) , if τ = +1( n − + 1) L − m − n − +1 ( ρ ) , if τ = − f τ,C = f − τ,A , (25) n τ = ε τ + ( τ cos 2 ϕ − / ε τ = ( E τ,s / (cid:126) ω c ) , m is the total angular quantumnumber, and N is a normalization constant. Note that χ τ,C (cid:54) = χ − τ,A ∀ ϕ , implying a valley asymmetry for α < r = R , where n D = n r = (cos φ, sin φ ), we obtain0 = cos ϕρ τ f τ,A ( ρ )+ sin ϕf τ,C ( ρ ) − ε τ ρ τ L − mn τ ( ρ ) (cid:12)(cid:12)(cid:12)(cid:12) ρ = R/ √ l B . (26)As a result, the energy eigenvalues E τ,sn ρ ,m, are deter-mined by the (positive and negative) zeros of this equa-tion, where n ρ = 1 , , . . . is the radial quantum num-ber. At α = 0, these eigenvalues are related to thosederived previously for graphene [47,39,55] by replacing m → ( m − R (or large- B ) limit, we can exploit the re-lation between Laguerre polynomials L ba ( x ) and confluenthypergeometric functions of the first kind M ( a, b, x ): L ba ( x ) = (cid:32) a + bb (cid:33) M ( − a, b + 1 , x ) . (27)In leading order, M ( − a, b +1 , x → ∞ ) takes the form [59]: M ( − a, b + 1 , x ) = Γ ( b + 1) Γ ( − a ) e x x − a − b − (cid:2) O ( | x | − ) (cid:3) . (28)Substituting this into equation (26), we obtain sin( πn τ ) =0. Consequently n τ = 0, 1, 2,. . . and the energy eigenvalues(Landau levels) become [30] E τ,n τ ,s = s (cid:126) ω c (cid:114) n τ + 12 (1 − τ cos 2 ϕ ) . (29) For the flat band states , a similiar calculation gives ψ Dτ, = sin ϕρ − m + τ f τ,A e i ( m − τ ) φ ϕρ − m − τ f τ,C e i ( m + τ ) φ e − ρ . (30)Since ψ Dτ,B ≡
0, this is always compatible with the IMBC.Clearly, E τ, = 0 [cf. equation (5)]. Figure 2 presents the analytical results for the magneticfield dependence of the energy spectra of (isolated) α − T quantum dots with IMBC. For all α , we observe flat bandsat E = 0 (red lines) and a merging of the quantumdot states to the Landau levels characterised by quan-tum number n τ (dotted curves) when the magnetic fieldincreases. Note that n τ = n τ ( n ρ , m ) (the data show theresults for n ρ ≤ | m | ≤ B [cf. equation (29)], i.e., they are notequidistant.In the graphene-lattice model ( α = 0, top panels), wearrive at the same conclusions as previous work [55,47],also for larger total angular and radial quantum num-bers. According to the IMBC, the spectra show a bro-ken particle-hole symmetry and E m (cid:54) = E − m , even for B = 0 [where the eigenvalues are twofold degenerate( E τ = E − τ )]. For B > E τ = − E − τ . Combining the spectra of bothvalleys K and K (cid:48) , the symmetry is restored.In the α − T -lattice model with 0 < α < B = 0, i.e., we find E m (cid:54) = E − m and valley degeneracy E τ = E − τ . Clearlytime-reversal symmetry is broken at B > E τ (cid:54) = − E − τ . As a consequence, the eigenvalues vary dif-ferently when B is increased. Such valley-anisotropy hasbeen found in the magneto-optical properties of (zigzag) α − T nanoribbons [34].For the Dice-lattice model ( α = 1, bottom panels),we have a specific situation. Here, E m = E − m at B =0, i.e., the state is now fourfold degenerate. When B > K and K (cid:48) points.Let us now discuss the convergence of the eigenvaluesagainst the Landau levels in some more detail. The firstLandau level comprises all eigenvalues with m <
0; thehigher Landau levels have contributions with m < n τ .This holds for K and K (cid:48) , independent of α . Obviously,the eigenvalues with positive (negative) energies cross theLandau levels first, before they converge towards these val-ues from below (above) at the K ( K (cid:48) ) point when the mag-netic field increases. The greater α , the more pronounced . Filusch et al.: α − T quantum dot 5 − . − . . . . E [ e V ] K , α =0.0 K , α =0.0 − . − . . . . E [ e V ] K , α =0.25 K , α =0.25 B [T] − . − . . . . E [ e V ] K , α =1.0 B [T] K , α =1.0 Fig. 2.
Eigenvalue spectra of an α − T dot with radius R =20 nm. Solid lines give the solutions of (26) as a function ofthe perpendicular magnetic field B in valleys K (left) and K (cid:48) (right) when α = 0, 0.25, and 1 (top to bottom). Only resultswith n ρ = 1 (blue), 2 (violet) and 3 (orange) with − ≤ m ≤
10 are shown. Flat bands are marked in red. Dashed black linesgive the Landau levels (29). this kind of “overshooting” appears to be. This effect (be-ing largest at α = 1) is not observed for negative (pos-itive) energies at K ( K (cid:48) ). We note that in certain casesthe eigenvalue levels form a wide band of states and canbe hardly resolved after bending up. In addition, lookingfor instance at the blue curves for α = 0 .
25 ( K (cid:48) point, E > K (cid:48) -eigenvalue setsis reached only for larger values of the magnetic field. Theinset demonstrates an avoided crossing for m = −
3: While B [T] − . − . . . . E [ e V ] K , α = 1 / √ B [T] K , α = 1 / √ m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − m = − Fig. 3.
Eigenvalue spectra of a quantum dot with α = 1 / √ B = 200 T (other model parameters and notation as in Fig.2). Inset: Magnification of solutions with m = −
3, and n ρ = 1(blue) respectively n ρ = 2 (violet), at an avoiding crossing. the eigenvalue belonging to n ρ = 1 (blue curve) convergesto the first Landau level, the eigenvalue with n ρ = 2 (vio-let curve) tends to the second one. The same happens forthe curves with other values of m . We now analyse the validity range of the continuum modelderived in the low-energy charge carrier regime close tothe Dirac points K and K (cid:48) . For this we consider thecase of a circular dot imprinted on the α − T lattice,whereby the dot region is not surrounded by an infinitemass medium but by a ring (of width W with finite masspotential ∆ , cf. Fig. 1), which has the same lattice struc-ture as D . In this way particularly good result can beachieved if ∆/t > a/W . The eigenvalue problem of such afinite (non-interacting) system can be solved numerically,e.g., in a very efficient way by using the kernel polynomialmethod [60]. By the kernel polynomial method we havealso direct access to the local (L) density of states (DOS),LDOS( E ) i = (cid:88) l |(cid:104) i | l (cid:105)| δ ( E − E l ) (31)( i is a singled out lattice site and n numbers the single-particle eigenvalues), the DOSDOS( E ) = (cid:88) n δ ( E − E n ) , (32)and the integrated (I) DOSIDOS( E ) = E (cid:90) −∞ DOS( E (cid:48) )d E (cid:48) . (33)Figure 4 contrasts the DOS of our quantum dot lat-tice model with the eigenvalues of the continuum model, A. Filusch et al.: α − T quantum dot − . − . . . . E [ e V ] α =0.0 − . − . . . . E [ e V ] α =0.25 − . − . . . . E [ e V ] α = 1 / √
25 50 75 100 B [T] − . − . . . . E [ e V ] α =1.0 − − − − log(DOS) Fig. 4.
Logarithmic density of states, log(DOS) (grey curves),of an α − T quantum dot ( R = 20 nm) embedded in a circularring-barrier potential ∆/t = 0 . W = 5 nm). For com-parison, the continuum model eigenvalues of Figs. 2 and 3 areincorporated (yellow curves). Again the flat band is marked inred. − . − . . . . D O S [ ] B = 2 T − . − . . . . E [eV] D O S [ ] B = 140 T . . . I D O S [ ] . . . . I D O S [ ] Fig. 5.
DOS (black lines, left axis) and integrated DOS(dashed lines, right axis; included for E ≤ α − T quantum dot (where α = 1 / √
3) in a perpendicular magneticfields: B = 2 T (upper panel) and B = 140 T (lower panel).Yellow vertical lines mark the energy eigenvalues of the con-tinuum model with IMBC. For B = 140 T, the Landau levelsare included (red lines). Other model parameters are as in Fig.4. in dependence on the strength of the applied magneticfield B , for different values of α . In general we can saythat the continuum model provides an excellent approxi-mation to the exact data for negative energies, regardlessof B and α . At this point let us emphasise once again thatif we had used a negative ∆ , positive and negative energyresults would change roles. Comparing the data, one hasto remember that the numerical exact tight-binding ap-proach takes into account larger angular momenta ( m )than our continuum model calculation; therefore addi-tional eigenvalues will appear also for E <
0. In the caseof graphene ( α = 0), we obtain a very good agreementalso for positive energies, even though some features, suchas the anti-crossing of energy levels, are not reproducedin the continuum model [55]. At finite α (and E > α is growing. These states are mainly localised atthe quantum dot’s boundary (see below), and can be re-lated to the sublattice-dependent potential ∆ along ∂D .Similar “anomalous” in-gap states were also found in two-dimensional pseudospin-1 Dirac insulators and have beenattributed to the boundary between two regions with dif-ferent flat-band positions in a gapped Dice-lattice sys- . Filusch et al.: α − T quantum dot 7 tem [61]. The edge states in our system have the sameorigin: The position of the flat band is shifted by ∆ whenchanging from region I to II.Figure 5 compares the DOS of the tight-bindingquantum-dot model and the distribution of the eigen-values in the continuum IMBC model (with n ρ ≤ | m | ≤
20) for weak and strong magnetic fields. Theheights of the steps in the integrated DOS can be takenas measure of the spectral weight of the correspondingeigenstates, particularly with regard to the degeneracy ofthe levels (note that the IDOS is not drawn for
E >
E <
0. The sector
E > B = 2 T,upper panel), the Landau levels are more difficult to iden-tify. For high magnetic fields ( B = 140 T, lower panel),states with large angular quantum numbers m contributeto each Landau level. Note that we have included in thefigure series of states which are not yet converged for the n ρ - and m -values used (vertical dashed yellow lines). We now consider a more realistic situation, where the α − T quantum dot is contacted by leads. The bound-ary of this “device” is realised covering the whole setupby a sheath of width W with a gapful band structure dueto a (finite) mass term ∆ , see Fig. 6. To determine the con-ductance between the left (L) and right (R) leads in thelimit of vanishing bias voltage, we employ the Landauer-B¨uttiker approach [62]: G = G (cid:88) m ∈ L ,n ∈ R | S n,m | (34)with G = 2 e /h . G is the maximum conductance perchannel. The scattering matrix between all open (i.e., ac-tive) lead channels, S n,m , can be easily calculated withthe help of the Phyton -based toolbox
Kwant [63].Figure 7 shows the conductance of the contacted α −T quantum dot as a function of energy at weak (upper panel)and strong (lower panel) magnetic fields. The conductanceessentially probes the extended (current-carrying) statesof the dot. Again, we choose α = 1 / √
3, in order to allowfor a direct comparison with the DOS data of the iso-lated dot depicted in Fig 5. Let us first consider the case ∆ lead = 0 (black dashed lines). For B = 2 T, we see thatthe first five peaks at E < m is not a goodquantum number anymore. For positive energies we recog-nise larger deviations from the continuum eigenvalues asis the case for the DOS (cf. Fig. 5); overall much more Fig. 6.
Drawing of the α − T lattice quantum dot (radius R ) contacted by leads (width l W ). The boundary condition isrealised by a W -wide stripe with mass term ∆ that covers thewhole element. The leads are docked by an additional massterm ∆ lead (blue region); the homogenous magnetic field B points out of the plane. In the calculations we use R = 20 nm, ∆/t = 0 . W = 5 nm, and l W = 80 √ a − W . − . − . . . . . . . . G [ G ] B = 2 T(1) (2) − . − . . . . E [eV] G [ G ] B = 140 T(3) Fig. 7.
Conductance of the contacted α − T quantum dotwith α = 1 / √ B = 2 T (top)and B = 140 T (bottom). The other dot parameters are asindicated in Fig. 6. Results for ∆ lead = 0 ( ∆ lead = 0 . conductive channels appear. At B = 140 T, we observethe expected Landau level quantisation of the conduc-tance. Obviously, the steps respectively plateaus are lesspronounced at positive and larger absolute values of theenergy once again. The conductance quantisation basically A. Filusch et al.: α − T quantum dot −
20 0 20 − y [ n m ] (1) −
20 0 20 − y [ n m ] (2) −
20 0 20 x [nm] − y [ n m ] (3) . . . L D O S . . . L D O S . . . L D O S [ − ] Fig. 8.
LDOS for the contacted α − T quantum dot at theresonances indicated in Fig. 7 by (1), (2) [ B = 2 T; two upperpanels] and (3) [ B = 140 T; lowest panel] for ∆ lead = 0 . breaks down if the cyclotron diameter d c = 2 | E | /v F eB ex-ceeds the lead width l W ; in this case the charge carriers,moving on a cyclotron trajectory along the quantum dotcircumference, will miss the way out at the right lead.Working with additional barriers at the lead contacts( ∆ lead = 0 . E = − .
055 eV −
20 0 20 − y [ n m ] (1) −
20 0 20 − y [ n m ] (2) −
20 0 20 x [nm] − y [ n m ] (3) . . . L D O S . . . L D O S . . . L D O S [ − ] Fig. 9.
LDOS of the contacted α −T quantum dot surroundedby a disordered circular ring. The LDOS is shown for a single(but typical) realisation of the random mass term, where the ∆ i are drawn out of the intervall [0,1.6], i.e., ˆ ∆ = 0 .
8. Again weconsider the resonances (1), (2) [ B = 2 T; two upper panels]and (3) [ B = 140 T; lowest panel] with system parameters asin Figs. 6, 7 and 8. (1), E = 0 .
059 eV (2) and E = 0 . B = 2 Tand B = 140 T, respectively. For (1), the LDOS is almostrotationally symmetric (owing to the leads there is someweak asymmetry) and has a maximum at the centre ofthe quantum dot. This is in accord with the correspondingcontinuum solution ( m = 0, sn ρ = − τ = − m ,will lead to more complicated LDOS pattern (not shown).For (2), the LDOS is more or less localised at the bound-ary of the quantum dot, i.e., this resonance will not corre-spond to a bulk state as (1). Note that we find almost thesame conductances, G/G (cid:39) .
98 (1) and
G/G (cid:39) . Filusch et al.: α − T quantum dot 9 (bulk or edge) state. In both cases, we observe some scat-tering and ‘localisation’ effects at the edges of the (lead)mass barrier. At resonance (3), the LDOS at the quan-tum dot boundary is also much larger than those in thebulk (although by a factor of ten smaller compared to tocases (1) and (2); note the different scale of the color bar).Regardless of this, G/G (cid:39) .
9, i.e., we have almost threeperfect transport channels. In this case we already enteredthe quantum Hall regime, where quantum Hall edge statesevolve which differ in nature from the edge state (2).
As a matter of course, imperfections will strongly influencethe transport through contacted Dirac-cone systems [64,65,57]. This holds true even up to the point of completesuppression, e.g., by Anderson localisation [66]. Neverthe-less most of these nanostructures appear to be conduct-ing [67,68], simply because the (Anderson) localisationlength exceeds the device dimensions for weak disorderin one or two dimensions [67,68]. In our case, the disordercaused by the boundary of the quantum dot is of particu-lar importance. To model these disorder effects, we let themass term fluctuate in the circular ring of width W . Moreprecisely, we assume ∆ → ∆ i in equation (1), where ∆ i is evenly distributed in the interval [ ∆ − ˆ ∆, ∆ + ˆ ∆ ] withˆ ∆ < ∆ , i.e., ˆ ∆ > α = 0 (graphene)and within the Dirac approximation. Long-range disor-der, on the other hand, gives rise to intravalley scatteringwhich is not sufficient to localise the charge carriers [70].Figure 9 illustrates how the LDOS shown in Fig. 8for three characteristic resonances will change if we ran-domise the mass potential ∆ i with strength ˆ ∆ = 0 . ∆ from zero to its final value0.8. Thereby the positions of the resonances (1) and (2)are slightly shifted compared to the ordered case: We find E = − .
057 eV (1) and E = 0 .
058 eV (2) for the sampleused in Fig. 9. Since the plateau structure is completelydestroyed for the (disordered) high-field case B = 140 T,we will leave E = 0 . G/G ( ˆ ∆ = 0 .
8) = 0 . (cid:39) G/G (0). Acompletely different behaviour is observed for the “edge-state” resonance (2). Here the LDOS is not homogeneouslydistributed along the periphery region anymore. Insteadwe find an imbalance between energy states (and asso-ciated transport channels) in the upper and lower halfof the quantum dot, which depends on the specific sam-ple of course. For other realisations the LDOS will belarger in the lower half of the quantum dot. In any casethe conductance is substantially reduced, however, for example, we have G/G ( ˆ ∆ = 0 .
8) = 0 .
54 for the de-picted realisation. The effect of the disorder is similarlystrong for the quantum Hall edge-state resonance (3),
G/G ( ˆ ∆ = 0 .
8) = 1 . ∆ , for resonances (1), (2)and (3) and three different disorder realisations each. De-spite the strong fluctuations at larger values of ˆ ∆ , whichclearly result from large local differences of the onsite ener-gies and a varying overlap of energetically adjacent states,one observes a noticeable reduction of the conductance forthe states (2) and (3) located primarily near the quantumdot boundary whereas the conductance of the bulk state(1) is only little affected. Since the spatial dimensions ofthe device are in the nanoscale regime, the conductanceof our setup is not self-averaging. Determining the prob-ability distribution for the LDOS and conductances froma large assembly of disorder realisations [71] could be apromising approach to deal with this problem, but this isbeyond the scope of the present work. To summarise, we considered a generalisation of bothgraphene and Dice lattices, the so-called α − T lattice,and studied the electronic properties of a quantum dot,imprinted on this material, in a perpendicular static mag-netic field. The quantum dot boundary condition was im-plemented in a consistent manner by an infinite mass term . . . . . . . . . ˆ∆ [ t ] . . . . . . . G [ G ] (1)(2)(3) Fig. 10.
Conductance
G/G for the resonances (1) [blackcurves], (2) [blue curves] and (3) [red curves] (cf., Fig. 7) cal-culated at different (discrete) disorder strengths ˆ ∆ . Resultsobtained for the disorder realisation used in Fig. 9 (two otherdisorder realisations) are marked by solid (dashed) lines, whichshould guide the viewer’s eye only. All other parameters are asin the previous figures.0 A. Filusch et al.: α − T quantum dot (circular ring having a finite band gap) in the contin-uum (tight-binding model) description. For an isolatedquantum dot we analysed the magnetic-field dependenceof the eigenvalue spectra at the K and K (cid:48) Dirac nodalpoints and demonstrated significant differences betweenthe graphene, Dice and α − T continuum model results,particularly with respect to the degeneracy and the con-vergence towards the Landau levels at high fields. Thecomparison of our analytical results with exact numeri-cal data for the α − T tight-binding lattice shows thatthe states with negative band energies were generally sat-isfactory reproduced (if not too far away from the neu-tral point), whereas the lattice effects play a more promi-nent role at positive energies. For an contacted quantumdot, our transport calculations confirm the existence oftransport channels, i.e., current carrying states, at weakmagnetic fields, and Landau level quantisation of the con-ductance (related to quantum Hall edge states) at largerfields. The local density of states reveals the different phys-ical nature of these states. The LDOS not only indicateshow the boundary and the contacts affect the electronicstructure, but also how disorder in the quantum dot’ssurrounding will influence its transport behaviour. Whiletransport channels related to bulk resonances were lessimpacted, edge channel resonance and quantum hall edgestates are strongly affected, giving rise to a significant re-duction of the conductance.All in all, we are optimistic that the (strong) magneto-response of valley-contrasting quasiparticles in α − T model materials provides a good basis for promising val-leytronics applications in near future. The authors are grateful to R. L. Heinisch and C. Wurl forvaluable discussions.
Authors contribution statement
Both authors contributed equally to this work.
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