Quantum transport of pseudospin-polarized Dirac fermions in gapped graphene nanostructures
JJournal of Computational Electronics manuscript No. (will be inserted by the editor)
Quantum transport of pseudospin-polarized Dirac fermions ingapped graphene nanostructures
Leyla Majidi · Malek Zareyan
Received: date / Accepted: date
Abstract
We investigate the unusual features of the quantum transport in gapped monolayergraphene, which is in a pseudospin symmetry-broken state with a net perpendicular pseu-domagnetization. Using these pseudoferromagnets (PFs), we propose a perfect pseudospinvalve effect that can be used for realizing pseudospintronics in monolayer graphene. Thepeculiarity of the associated effects of pseudo spin injection and pseudo spin accumulation are also studied. We further demonstrate the determining effect of the sublattice pseudospindegree of freedom on Andreev reflection and the associated proximity effect in hybrid struc-tures of PFs and a superconductor in S/PF and PF/S/PF geometries. In particular, we finda peculiar Andreev reflection that is associated with an inversion of the z component of thecarriers pseudospin vector. Our results show that the gapped normal graphene behaves likea ferromagnetic graphene and the effect of the pseudospin degree of freedom in gappedgraphene is as important as the spin in a ferromagnetic graphene. Keywords
Sublattice pseudospin · Gapped graphene · Pseudospin valve · Superconductingproximity effect · Andreev reflection
PACS · · · Graphene, the two dimensional layer of the carbon atoms with honeycomb lattice structure,has attracted a great deal of attention as a new promising material for nanoelectronics, sinceits experimental realization a few years ago[1,2,3]. Graphene has a zero-gap semiconduct-ing band structure in which the charge carriers behave like 2D massless Dirac fermions witha pseudo-relativistic chiral property. The carrier type, [electron-like ( n ) or hole-like ( p )] andits density can be tuned by means of electrical gate or doping of underlying substrate. Mostof the peculiar properties of graphene is the result of its massless Dirac spectrum of thelow-lying electron-hole excitations, which in addition to the regular spin appear to come en-dowed with the two quantum degrees of freedom, the so called pseudospin and valley. The Leyla Majidi · Malek ZareyanDepartment of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), P. O. Box 45195-1159,Zanjan 45137-66731, IranE-mail: [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Leyla Majidi, Malek Zareyan pseudospin represents the sublattice degree of freedom of the graphene’s honeycomb struc-ture, and the valley defines the corresponding degree of freedom in the reciprocal lattice[2,4,5,6,7]. The effect of these additional quantum numbers has already been proven to be dras-tically important in several quantum transport phenomena in graphene, including quantumHall effect[2,3,8,9], conductance quantization[10], Klein tunneling[11,12,13] and quan-tum shot noise[14,15,16]. Interestingly, the pseudospin and the valley degrees of freedomin graphene have been proposed separately to be used for the controlling electronic devicesin the same way as the electron spin is used in spintronic and quantum computing. Rycerz etal. [17,18,19,20] demonstrated an electrostatically controlled valley filter effect in graphenenanoribbons with zigzag edge which can be used for realizing valley valve structures invalleytronics (valley-based electronics) applications. On the other hand, a pseudospin-basedversion of a spin valve has been proposed in monolayer graphene and bilayer graphene[21,22,23], which can be used for realizing pseudospintronics in graphene. Also, the possibilityof an interaction driven spontaneous breaking of the pseudospin symmetry, which can leadto the realization of pseudomagnetic states in monolayer and bilayer graphene, has beenstudied recently[24].Recent experimental progresses in proximity-inducing superconductivity in grapheneby fabrication of transparent contacts between a graphene monolayer and a superconductor,has provided a unique possibility to study relativistic-like superconductivity and proximityeffect[25,26,27]. Peculiarity of Andreev reflection (AR)[28], conversion of the electron into the hole excitations at a normal metallic-superconducting (N/S) interface, has been studiedin graphene-based N/S junctions by Beenakker, who demonstrated that unlike the retro ARfor highly doped graphene or a N metal, the dominant process for undoped N grapheneis the specular AR[29,30]. In the case of a graphene ferromagnetic-superconducting (F/S)junction, the situation is dramatically different from common F/S junctions where the sub-gap Andreev conductance decreases with increasing the exchange energy h from its value forN/S junction and vanishes for a half metallic ferromagnet with h = µ , where all carriers havethe same spin[31]. It has been shown that for the exchange energies higher than the chemicalpotential h > µ , a peculiar spin-resolved Andreev-Klein process at graphene F/S interfacecan result in an enhancement of the subgap Andreev conductance by h , up to the point atwhich the conductance at low voltages eV (cid:28) ∆ S is larger than its value for the correspondingN/S structure[32,33,34]. Also, the corresponding Andreev-Klein bound states in grapheneS/F/S structure are responsible for the long-range Josephson coupling of F graphene[35,36]. Moreover, specific nonlocal proximity effect takes place in graphene-based supercon-ducting heterostructures mediating purely by a nonlocal process known as crossed Andreevreflection (CAR) which creates a spatially entangled electron-hole pair. While in ordinarynonrelativistic systems the small value of CAR conductance is canceled by the conductanceof elastic electron cotunneling (CT) process, it can be enhanced in ballistic graphene N/S/Nand F/S/F structures[37,38].In this paper, we study the effect of the pseudospin degree of freedom on quantum trans-port in gapped monolayer graphene, that presents a pseudospin symmetry broken ferromag-net (PF), with a finite pseudospin magnetization oriented vertically to the graphene plane.The magnitude of the pseudomagnetization (PM) depends on the chemical potential and itsdirection can be switched by changing the type of doping (electron n or hole p ). Based onthis observation, we propose a nonmagnetic pseudospin valve structure (PF/N/PF) with re-markably large pseudomagnetoresistance (PMR) in analogy to the giant magnetoresistance(GMR) in magnetic multilayers[39], which can be perfect for chemical potentials close tothe energy gap ( µ (cid:39) ∆ ) and appropriate lengths of the N region. More importantly, we showthat the perfect pseudospin valve effect can be reached even in higher chemical potentials uantum transport of pseudospin-polarized Dirac fermions in gapped graphene nanostructures 3 µ (cid:29) ∆ by applying an appropriate bias voltage. We further demonstrate the unusually long-range penetration of the equilibrium and non-equilibrium pseudospin polarization into the Nregion by proximity to a PF, that is in clear contrast to the induced magnetization in ordinaryF/N junctions which decays exponentially within λ F [40].Moreover, we study the effect of the pseudospin on AR and the associated proximityeffect in hybrid structures of PFs and a superconductor in S/PF and PF/S/PF geometries. Wefind that in graphene PF, due to the possibility for a small chemical potential µ , a peculiarAR occurs at S/PF interface which is associated with an inversion of the z component ofthe carriers pseudospin vector, and that this has important consequences for the proximityeffect. For an S/PF junction, we find that the Andreev-Klein reflection can enhance the pseu-dospin inverted Andreev conductance by the energy gap ∆ N to reach a limiting maximumvalue for ∆ N (cid:29) µ , which depends on the bias voltage and can be larger than the value forthe corresponding junction with no energy gap ( ∆ N (cid:28) µ ). This is similar to the behavior ofAndreev conductance with the exchange energy h in a graphene F/S junction and approvesthat the energy gap ∆ N in the band structure of normal graphene produces an effect similar tothe exchange field in F graphene. We also demonstrate that depending on the energy ε of theincident electron, µ and ∆ N , AR can be of retro or specular types, respectively, without orwith the inversion of the z component of the pseudospin vector. Furthermore, the spatially-damped oscillatory behavior of the proximity density of states in pseudoferromagnetic sideof the S/PF contact and the pseudospin switching effect in superconducting graphene pseu- dospin valve structure (PF/S/PF) confirm the crucial rule of the pseudospin in the gappednormal graphene and its similarity to the rule of spin in an F graphene.This paper is organized as follows. We introduce pseudoferromagnets (PFs) in Sec. 2,and use them in Sec. 3 to study the pseudospin valve effect in graphene PF/N/PF junction.Sections 4 and 5 are devoted, respectively, to the investigation of the proximity effect and thepseudospin injection in graphene PF/N junctions. In Sec. 6, we investigate AR in grapheneS/PF junction and present our main findings for the Andreev conductance and the proximityDOS of the S/PF junction. Section 7 is devoted to the investigation of the CT and CARprocesses in superconducting pseudospin valve structure. Finally, we present the conclusionin Sec. 8. One of the interesting features of graphene, that makes it very applicable in semiconductortechnology, is the possibility of opening a gap in the energy band structure of graphene.There are several methods to open an energy gap in the band structure of graphene. A sce-nario is placing graphene on top of an appropriate substrate which breaks the graphenesublattice symmetry and generates a Dirac mass for charge carriers. The band gap openingis observed in epitaxially grown graphene on a SiC substrate[41,42] and a hexagonal boronnitride crystal[43]. The energy band gap engineering can be also achieved through dopingthe graphene with several molecules such as
CrO , NH , H O [44,45].To study quantum transport in gapped graphene nanostructures within the scattering for-malism, we first construct the quasiparticle wave functions that participate in the scatteringprocesses. We adopt the Dirac equation of the form H ψ = ( ε + µ ) ψ , (1)where H = v F ( σ . p ) + ∆ N σ z (2) Leyla Majidi, Malek Zareyan is the two-dimensional Dirac Hamiltonian in presence of an energy gap, with p = − i ¯ h ∇ the momentum operator in the x - y plane ( v F = m/s represents the Fermi velocity) and σ = ( σ x , σ y , σ z ) the vector of the Pauli matrices operating in the space of two sublatticesof the honeycomb lattice[46,47]. The two-dimensional spinor has the form ψ = ( ψ A , ψ B ) ,where the two components give the amplitude of the wave function on the two sublatticesand ε is the quasiparticle energy.For a uniform gapped graphene region, the solutions of the Dirac equation Eq. (1) aretwo states of the form ψ e ± c = e ± ik ec x e iqy (cid:18) e ∓ i α ec / ± e − φ ec e ± i α ec / (cid:19) , (3)for conduction band electrons of n -doped graphene and ψ e ± v = e ∓ ik ev x e iqy (cid:18) e ± i α ev / ± e φ ev e ∓ i α ev / (cid:19) , (4)for valance band electrons of p -doped graphene, at a given energy ε and transverse wavevector q with the energy-momentum relation ε ec ( v ) = ± [ − µ + (cid:113) ∆ N + ( ¯ hv | k ec ( v ) | ) ] . α ec ( v ) = arcsin [ ¯ hvq / (cid:113) ( ε ± µ ) − ∆ N ] is the angle of propagation of electron which has longitudinalwave vector k ec ( v ) = ( ¯ hv F ) − (cid:113) ( ε ± µ ) − ∆ N cos α ec ( v ) and φ ec ( v ) = arcsinh [ ∆ N / (cid:113) ( ε ± µ ) − ∆ N ] . The two propagation directions of electron along the x -axis are denoted by ± in ψ e ± c ( v ) .The pseudospin of such states for conduction (valance) band electrons of n - ( p -)dopedgraphene is obtained as (cid:104) σ ( k ) (cid:105) e + c ( v ) = (cid:115) − ( ∆ N ε ± µ ) ( cos α ec ( v ) ˆ x ± sin α ec ( v ) ˆ y )+ ∆ N ε ± µ ˆ z . (5)As can be seen from the above equation, the existence of a band gap makes the pseudospinto have a component perpendicular to the plane of the graphene sheet. The in-plane andout-of-plane components of the pseudospin depend on ( ε + µ ) / ∆ N , which can be tuned tounity to make the pseudospin vector to be oriented perpendicular to the sheet. Increasing ( ε + µ ) / ∆ N leads to the decrease of the out-of-plane component such that it goes to zerowhen ε + µ (cid:29) ∆ N .The total pseudomagnetization (PM) of the gapped graphene can be calculated by sum-ming the expression (5) over all the wave vectors k = ( k , q ) , P M n ( p ) = ∑ k (cid:104) σ ( k ) (cid:105) e + c ( v ) , (6)from which we find that PM only has an out-of-plane component which depends on ( ε + µ ) / ∆ N . Fig. 1 (a) shows the behavior of the out-of-plane component of PM per electron PM z / N as a function of µ / ∆ N at zero temperature ( T = µ (cid:39) ∆ N , PM z / N takes its maximum value PM z / N =
1, while increasing µ / ∆ N leads to the decreaseof PM z such that it goes to zero for highly doped gapped graphene ( µ (cid:29) ∆ N ). Also thedirection of PM can be switched by changing the type of doping between n and p . Wenote that the pseudospin polarization of a gapped graphene corresponds to a difference inthe electronic charge densities of the two triangular sublattices, which in turn produces an uantum transport of pseudospin-polarized Dirac fermions in gapped graphene nanostructures 5 −20 −10 0 10 20−101 µ / ∆ N P M z / N (a) Fig. 1 (Color online) (a) Vertical pseudomagnetization per electron PM z / N of the gapped graphene layerversus chemical potential µ ( µ is scaled to the energy gap ∆ N ). (b) Schematic illustration of the proposedpseudospin valve in monolayer graphene: The left and right regions are pseudoferromagnets (PFs) and theintermediate region is a normal graphene (N) without a band gap. (c-d) Profile of pseudomagnetization vector P M inside the two PFs (blue) and the N region of length L = λ F (pink) for two configurations of (c) paralleland (d) antiparallel, when µ (cid:39) ∆ N . in-plane electrical polarization [23]. This correspondence between the PM vector and thein-plane electrical polarization can be used for an experimental measuring of PM.So we demonstrate that a monolayer graphene with an energy gap ∆ N in its electronicband structure behaves as a pseudospin symmetry-broken ferromagnet (PF) with a perpen-dicular to the plane of graphene PM, whose direction is switched by altering the type ofdoping between n and p . Based on the above observation, we propose a nonmagnetic pseudospin valve which consistsof two PFs ( x < x > L ) with a tunable direction of PM, that are connected through a Leyla Majidi, Malek Zareyan λ F P M R eV/ ∆ N = 1eV/ ∆ N = 0.5eV/ ∆ N = 0 λ F P M R µ / ∆ N = 1.001 µ / ∆ N = 1.1 µ / ∆ N = 2 µ / ∆ N = 10 µ / ∆ N = 100 µ / ∆ N = 2(b)(a) eV / ∆ N = 0 Fig. 2 (Color online) Pseudomagnetoresistance (PMR) of the pseudospin valve versus the length of the Nregion ( L / λ F ) for (a) different values of µ / ∆ N , when eV / ∆ N =
0, and (b) different values of the bias voltage eV / ∆ N , when µ / ∆ N = normal (nonpseudomagnetized) layer of length L [shown schematically in Fig. 1(b)]. Theconfiguration of PMs in the pseudospin valve can be changed from parallel to antiparallelby fixing the type of doping of one region and changing the type of the doping in the otherregion. The size of the pseudospin valve effect is determined by the extent in which theconduction of the antiparallel configuration is suppressed (similar to the spin valve effect).The pseudomagnetoresistance of a pseudospin valve is defined as PMR = G P − G AP G P , (7)where G P ( AP ) is the conductance of the parallel (antiparallel) configuration that can be cal-culated from the Landauer formula[48], G P ( AP ) = e h (cid:90) | t P ( AP ) | cos α d α . (8)Here, t P ( AP ) is the transmission amplitude of electrons through the pseudospin valve inparallel (antiparallel) configuration, which can be calculated by matching the wave functions uantum transport of pseudospin-polarized Dirac fermions in gapped graphene nanostructures 7 of three regions at the two interfaces ( x = x = L ), ψ = ψ e + c + r ψ e − c , ψ = a ψ (cid:48) ce + + b ψ (cid:48) ce − , ψ , P ( AP ) = t P ( AP ) ψ e + c ( v ) . (9)The left PF, N region and the right PF are signed by 1,2, and 3, respectively, and ψ ( (cid:48) ) e ± c ( v ) are the wave functions of Dirac equation for incoming and outgoing electrons of n - ( p -)doped graphene sheet with (without) a gap. Figure 2(a) shows the dependence of the result-ing PMR on the length of the N region L / λ F ( λ F = ¯ hv F / µ ) for different values of µ / ∆ N atzero bias voltage eV =
0. We have taken µ = µ = | µ | = µ . We observe that the pseudospinvalve effect can be perfect ( PMR =
1) for µ (cid:39) ∆ N . For these values of µ , PMR shows anoscillatory behavior with L / λ F , with an amplitude which takes the value 1 for some rangesof the length L . We note that this perfect pseudospin valve effect of the monolayer grapheneis more robust with respect to an increase of the length of the N region, as compared to thesimilar effect in a bilayer graphene pseudospin valve structure[21]. The amplitude of PMRdecreases by increasing µ / ∆ N and tends to the constant value of PMR = / µ (cid:29) ∆ N . This residual PMR is the difference in the resistance of a n - p graphene structure with that of a uniformly ( p or n ) doped graphene with the same | µ | ,which is present even in the limit PM → have found that the perfect pseudospin valve effect can be resumed by applying an appro-priate bias voltage to the valves with higher chemical potentials µ > ∆ N [see Fig. 2(b)]. Let us now study the proximity effect in hybrid structures of PFs and a normal grapheneregion in PF/N and PF/N/PF junctions. We start with a single PF/N junction in a graphenesheet in the x - y plane, where the region x < x > P M in PF and N region using Eq. (6), and by considering the contributionof the pseudospin of all incident electrons from left ( l ) and right ( r ) regions that are scatteredfrom the junction, P M i N i = { P M li N li + P M ri N ri } , (10)where P M l ( r ) i = ∑ k (cid:104) σ ( k ) (cid:105) ψ i , l ( r ) , N l ( r ) i = ∑ k ψ ∗ i , l ( r ) ψ i , l ( r ) and i denotes the PF (N region).The resulting profile of P M across the PF/N junction is demonstrated in Fig. 3 for µ (cid:39) ∆ N . It is seen that a nonzero P M is induced in N region ( ∆ N =
0) which rotates aroundthe normal to the junction ( x axis) with x . The perpendicular component PM z oscillates asa function of x with a period of order λ F , and shows only a weak decay in the scale of λ F . While the in-plane components PM x , y vanish inside PF, they are produced at the PF/Ninterface and are penetrated into the N region. PM y shows an oscillatory behavior with x similar to PM z . Interestingly, PM x is uniform inside N, which considering the decay of theother two components, implies that P M at the points in N region far from the junction isuniform and oriented perpendicular (along x axis) to the P M in the connected PF. Thisunusual proximity effect can be explained in terms of reflectionless Klein transmission ofelectrons which incident normally to PF/N interface[11,12,13]. We note to the unusually
Leyla Majidi, Malek Zareyan −10 −5 0 5 1000.511.5 P M / N x / λ F PM x PM y PM z −10 0 100.40.60.81 x / λ F | P M | / N (b) (a) Fig. 3 (Color online) Equilibrium PM of the PF/N junction when µ (cid:39) ∆ N : (a) profile of P M in PF (blue)and N region (pink) and (b) the position dependence of the
P M components. The inset of Fig. 3(b) showsthe magnitude of
P M . long-range penetration of the proximity induced PM inside the N region, which is in contrastto the ferromagnet/normal-metal junction (F/N), in which the induced magnetization decaysover short interatomic distances.The above analysis of the proximity effect in PF/N junction can be extended to the pseu-dospin valve geometry of Fig. 1(b). The profile of P M orientation in different regions ofthe PF/N/PF junction is indicated in Fig. 1 for parallel (c) and antiparallel (d) cases when L = λ F and µ (cid:39) ∆ . P M is perpendicular to the x axis and undergoes rotation across theN contact in a way that in parallel and antiparallel cases PM y and PM z , respectively, showsa change of signs at the middle of N region ( x = L / P M inside the N region is constant with x for both of parallel and antiparallel con-figurations. So the strong pseudomagnetic coupling between the two pseudoferromagneticregions, which itself can be due to the long-range penetration of pseudospin polarizationinto the N region by proximity to PFs, leads to the strong robustness of the pseudospin valveeffect with respect to increasing the length of N contact. In this section, we study the behavior of the injected pseudospin and pseudomagnetizationinto the nonpseudomagnetized N region of the PF/N junction, by the bias voltage. In a PF/N uantum transport of pseudospin-polarized Dirac fermions in gapped graphene nanostructures 9 −10 −5 0 5 10−0.300.51 x / λ F P M / N PM x PM y PM z −10 −5 0 5 10−0.300.51 x / λ F P M / N −10 −5 0 5 10−0.300.51 x / λ F P M / N −10 −5 0 5 100.31 x / λ F | P M | / N eV/ ∆ N =0.1eV/ ∆ N =1eV/ ∆ N =10 (d) (c) eV/ ∆ N = 10 (a) (b) eV/ ∆ N = 1eV/ ∆ N = 0.1 Fig. 4 (Color online) Nonequilibrium PM of the PF/N junction when µ (cid:39) ∆ N : position dependence of (a-c)the P M components and (d) the magnitude of
P M for three values of the bias voltage eV / ∆ N = . , , PM x PM y PM z −4 −2 0 1 3 5−0.500.51 x / λ F P M / N −4 −2 0 1 3 5x / λ F −0.500.51 P M / N PM x PM y PM z eV/ ∆ N = 0.1eV/ ∆ N = 0 (a) (b)(c) (d) eV/ ∆ N = 1 eV/ ∆ N = 10 Fig. 5 (Color online) Position dependence of the
P M components of PF/N/PF junction with parallel con-figuration, for different values of the bias voltage eV / ∆ N = , . , ,
10, when L = λ F and µ (cid:39) ∆ N .0 Leyla Majidi, Malek Zareyan −3 −1.5 0 1 2.5 400.20.40.60.81 | P M | / N −3 0 2.5 5 800.20.40.60.81 x / λ F | P M | / N eV/ ∆ N =0eV/ ∆ N =0.1eV/ ∆ N =1eV/ ∆ N =10 (a)(b) L / λ F = 5L / λ F = 1 Fig. 6 (Color online) Position dependence of the magnitude of
P M for parallel configuration of the PF/N/PFjunction with (a) L = λ F and (b) L = λ F , when eV / ∆ N = , . , ,
10 and µ (cid:39) ∆ N . junction, when a charge current flows across the interface, the pseudospin polarized carriersin PF contribute to the net current of PM entering the nonpseudomagnetized region and leadto the nonequilibrium PM in N region. Figure 4 shows the behavior of the nonequilibriumPM in PF and N sides of the PF/N interface for different values of the bias voltage, when µ (cid:39) ∆ N . It is seen that similar to the equilibrium case, a uniform P M is induced inside theN region far from the interface, such that its magnitude decreases by increasing the bias volt-age. This result is in contrast to the F/N junction, where the nonequilibrium magnetizationdecays exponentially within λ F [40]. Also the magnitude of P M inside the PF decreasesby increasing the bias voltage. This is due to the reduction of the z component of the pseu-dospin vector for the charge carriers that are going away from the energy band gap, and canbe seen from Figs. 4(a-c).The results of the above analysis for a PF/N/PF structure with parallel configuration ofPMs are shown in Fig. 5 for different values of the bias voltage eV / ∆ N = , . , ,
10, when µ = ∆ N and L = λ F . It is seen that in contrast to the case of equilibrium, the x component ofthe injected P M into the N region has a nonzero constant value and the symmetry of the y and z components PM y , z are broken relative to the middle of the N region. Therefore, incontrast to the constant magnitude of the induced P M in equilibrium, the magnitude of thenonequilibrium
P M depends on x and has a different behavior for different lengths of theN region [Fig. 6]. uantum transport of pseudospin-polarized Dirac fermions in gapped graphene nanostructures 11 (a) z y x (b) Fig. 7 (Color online) (a) Schematic illustration of the graphene S/PF junction. (b) The band structure of the n -doped PF to explain the two cases of Andreev reflection at S/PF interface: Right (Left) panel shows that anincident electron from the conduction band of PF with a subgap energy 0 ≤ ε ≤ µ − ∆ N ( ε ≥ µ + ∆ N ) is retro(specular) reflected as a hole in the conduction (valance)band without (with) the inversion of the z componentof the pseudospin vector at S/PF interface. σ e − and σ h + denote the pseudospin vectors of incident electronand reflected hole. v e and v h denote the velocity vectors of the electron and the hole, moving in differentdirections. Now, we consider a wide graphene S/PF junction normal to x -axis with highly doped S re-gion for x < n -doped PF for x > ∆ N = ∆ S which is taken to be real and constant. To study AR at S/PF inter-face within the scattering formalism, we first construct the quasiparticle wave functions thatparticipate in the scattering processes. In order to describe the superconducting correlationsbetween relativistic electrons and holes of different valleys, we adopt the Dirac-Bogoliubov-de Gennes (DBdG) equation:[29] (cid:18) H − µ ∆ S ∆ ∗ S µ − H (cid:19) (cid:18) uv (cid:19) = ε (cid:18) uv (cid:19) , (11) H = H − U ( r ) , (12)where H is the two-dimensional Dirac Hamiltonian with an energy gap (Eq. (2)), ε is theexcitation energy and U ( r ) the electrostatic potential is taken to be U (cid:29) µ in S region and U = u and v , are two-component spinorsof the form ( ψ A , ψ B ) .An incident electron of the conduction band from right to S/PF interface with a subgapenergy ε ≤ ∆ S can be either normally reflected as an electron or Andreev reflected as a hole.The reflected hole can be from the conduction or the valance band, depending on the electronenergy ε , µ and the energy gap ∆ N . As is shown in Fig. 7(b), as long as 0 ≤ ε ≤ µ − ∆ N the reflected hole is an empty state in the conduction band and AR is retro (middle panel),while for ε ≥ µ + ∆ N it is an empty state in the valance band and AR is specular, if ∆ N < ∆ S (right panel). The importance of AR near the Fermi level imposes the condition of ∆ N < ∆ S on size of the energy gap ∆ N . The retro reflection dominates if µ (cid:29) ∆ S + ∆ N , while thespecular reflection dominates if µ (cid:28) ∆ S − ∆ N . Using the solutions of Dirac equation forelectrons and holes of n -doped PF, the pseudospin of the incident electron and the reflectedhole of the conduction (valance)band are obtained as, (cid:104) σ ( k ) (cid:105) e − c = (cid:115) − ( ∆ N µ + ε ) ( − cos α ec ˆ x + sin α ec ˆ y )+ ∆ N µ + ε ˆ z , (13) (cid:104) σ ( k ) (cid:105) h + c ( v ) = (cid:115) − ( ∆ N µ − ε ) ( − cos α h ˆ x ± sin α h ˆ y ) ± ∆ N | µ − ε | ˆ z . (14)Here, α h = arcsin [ ¯ hvq / (cid:113) ( µ − ε ) − ∆ N ] indicates the angle of propagation of the hole ata transverse momentum q with energy-momentum relation ε hc ( v ) = µ ∓ (cid:113) ∆ N + ( ¯ hv | k h | ) for the hole from the conduction (valance)band. As can be seen from the above equations when an electron from the conduction band is reflected as a hole in the valance band, thesign of the gap-induced z component of the pseudospin vector (cid:104) σ z (cid:105) is changed, while inthe case of the conduction band hole, it retains its sign. This is shown schematically in themiddle (right) panel of Fig. 7(b) for the case of Andreev reflected hole from the conduction(valance)band without (with) the inversion of (cid:104) σ z (cid:105) upon AR at S/PF interface. Thus, forthe incident electron and the reflected hole being from different types of bands, we havean inversion of the z component of the pseudospin vector upon AR at S/PF interface. Inthe following we will show how the pseudospin (cid:104) σ z (cid:105) inversion by AR leads to peculiarproperties of S/PF and PF/S/PF systems.To evaluate the Andreev conductance of an S/PF junction, we use the Blonder-Tinkham-Klapwijk (BTK) formula[49]: G c ( v ) = e h ˜ N ( eV ) (cid:90) α c ( − | r c ( v ) | + | r A , c ( v ) | ) cos α e d α e , (15)where r c ( v ) and r A , c ( v ) denote the amplitudes of normal and Andreev reflections, respectively.˜ N ( ε ) = W ( µ + ε ) / π ¯ hv F (cid:113) ( µ + ε ) − ∆ N is the number of transverse modes in a sheet ofgapped graphene of width W and α c = arcsin [ (cid:113) ( µ − ε ) − ∆ N / (cid:113) ( µ + ε ) − ∆ N ] is thecritical angle of incidence above which the Andreev reflected waves become evanescent anddo not contribute to any transport of charge.We calculate the amplitudes of normal and Andreev reflections by matching the wavefunctions of PF and S region at the interface x =
0. The wave functions inside PF and Sregion are as follows: ψ c ( v ) PF = ψ e − c + r c ( v ) ψ e + c + r A , c ( v ) ψ h + c ( v ) , (16) ψ S = a ψ S + + b ψ S − , (17)where ψ e ( h ) ± c ( v ) and ψ S ± are the solutions of DBdG equation for the quasiparticles inside the n -doped PF and S region, respectively, and the two cases of the Andreev reflected holes uantum transport of pseudospin-polarized Dirac fermions in gapped graphene nanostructures 13 ~ Fig. 8 (Color online) Dependence of the Andreev conductance of graphene S/PF contact on the gap ∆ N / µ (in units of the chemical potential) at three bias voltages eV / ∆ S = , . ,
1. Left (Right) inset shows that anincident electron from the conduction band of PF with ∆ N / µ ≤ / ( + eV / ∆ N ) ( ∆ N / µ ≥ / ( eV / ∆ N − ) )is reflected as a hole in the conduction (valance)band without (with) the inversion of the z component of thepseudospin vector at S/PF interface. from the conduction (valance)band without (with) the inversion of (cid:104) σ z (cid:105) are denoted by c ( v ) in ψ c ( v ) PF .Figure 8 shows the Andreev conductance of the graphene S/PF junction as a functionof ∆ N / µ for ∆ N / ∆ S = . ∆ N < µ / ( + eV / ∆ N ) , the conductance decreases monotonically with ∆ N / µ . In this interval, theincident electron and the reflected hole are from the conduction band and therefore AR iswithout the inversion of (cid:104) σ z (cid:105) [see left inset of Fig. 8]. The density of states of the con-duction band hole decreases by increasing ∆ N / µ . Thus, the amplitude of AR and hencethe Andreev conductance decreases with ∆ N / µ and goes to zero at ∆ N = µ / ( + eV / ∆ N ) ,where the density of states of the conduction band hole vanishes. The absence of hole statesfor µ / ( + eV / ∆ N ) < ∆ N < µ / ( eV / ∆ N − ) causes a gap in conductance, which decreaseswith eV / ∆ S and goes towards smaller ∆ N / µ . For ∆ N ≥ µ / ( eV / ∆ N − ) , the pseudospin (cid:104) σ z (cid:105) inverted Andreev conductance increases monotonically with ∆ N / µ . In this regime, thetransport is between the conduction and the valance band and the incident electron of theconduction band is reflected as a hole in the valance band. So the pseudospin (cid:104) σ z (cid:105) of thereflected hole changes sign [see right inset of Fig. 8] and the density of states of the holeincreases with ∆ N / µ , resulting in an enhancing Andreev conductance. Such a peculiar ARis associated with a Klein tunneling of the n -type carriers to the p -type carriers. The en-hancing conductance reaches a limiting maximum value for ∆ N (cid:29) µ , which depends on thebias voltage and can be larger than the value for the corresponding S/N structure ( ∆ N (cid:28) µ ).The limiting value of the Andreev conductance for ∆ N (cid:29) µ decreases by increasing ∆ N / ∆ S from its value for a specular AR in corresponding S/N structure ∆ N (cid:28) ∆ S ( G / G = / ) and vanishes for ∆ N > ∆ S , while for ∆ N (cid:28) µ it increases by increasing ∆ N / ∆ S and tends tothe corresponding value of a retro type AR ( G / G = ) as ∆ N → ∆ S [50]. So the behavior λ F N ( x , ε ) / N ( ε ) ∆ N / µ = 0.5 ∆ N / µ = 1 ∆ N / µ = 2 ∆ N / µ = 5 Fig. 9 (Color online) The behavior of the proximity density of states (DOS) inside the pseudoferromagneticregion versus x / λ F for different values of ∆ N / µ , when ε / ∆ S = . ∆ N / ∆ S = . of the Andreev conductance with ∆ N / µ is similar to that of a graphene F/S junction with h / µ , where AR of n - n type carriers for h < µ changes to the Andreev-Klein reflection ofthe n - p type carriers for h > µ [32]. This shows that the energy gap ∆ N in the band structureof normal graphene behaves like an exchange energy in F graphene and enhances the sub-gap Andreev conductance of S/PF junction, which is accompanied by the inversion of the z component of the pseudospin vector for the reflected hole relative to the incident electron.To complete the analysis of the present section, we evaluate the proximity density ofstates (DOS) inside the pseudoferromagnetic region by using the formula [51] N ( ε , r ) = ∑ k | ψ k ( r ) | δ ( ε ( k ) − ε ) , (18)where ψ k ( r ) corresponds to the eigenfunction of energy ε ( k ) and the sum is over all stateswith the wave vectors k . Replacing Eq. (16) in the above equation, we find the total subgapDOS inside the pseudoferromagnetic region for the case of AR with (cid:104) σ z (cid:105) inversion as, N ( ε , x ) N ( ε ) = (cid:115) − ( ∆ N µ + ε ) (cid:90) π / − π / | ψ ( r ) vPF | cos α e d α e , (19)where N ( ε ) = ( µ + ε ) / ( π ¯ hv F ) (cid:113) ( µ + ε ) − ∆ N is the DOS of a pseudoferromagneticlayer. Figure 9 shows the behavior of the proximity DOS inside the pseudoferromagneticregion in terms of the dimensionless distance x / λ F for different values of ∆ N / µ , when ε / ∆ S = . ∆ N / ∆ S = .
1. We see that there are two phenomena to consider in describ-ing the spatial variations of N ( x ) , when an energy gap is present in the band structure ofnormal graphene. The first phenomenon is the short distance decay at the interface with aslope which increases by increasing ∆ N / µ . The other important phenomenon is the damped uantum transport of pseudospin-polarized Dirac fermions in gapped graphene nanostructures 15 oscillation of N ( x ) , caused by the momentum shift between Andreev correlated electron-hole pair with opposite (cid:104) σ z (cid:105) directions. The period of oscillations is determined by ¯ hv F / ∆ N ,which is similar to an S/F structure where the period of DOS oscillations in the ballisticlimit is given ¯ hv F / h [52,53,54]. This shows the similarity of the effect of an spin-splittingexchange field h with the energy gap ∆ N , which behaves as a pseudospin-splitting field [seeEq. (2)]. We note that the appropriate method to probe the DOS oscillations in S/PF junctionis the local scanning of the surface of the pseudoferromagnetic region by scanning tunnel-ing microscopy (STM). So the spatially damped oscillatory behavior of the DOS inside thepseudoferromagnetic region confirms that the energy gap ∆ N in the band structure of normalgraphene produces an effect similar to the exchange field in F graphene. Finally, we study the nonlocal quantum transport in PF/S/PF junction that constitutes a su-perconducting pseudospin valve structure. We calculate the normal and Andreev reflectionamplitudes ( r and r A , respectively) in the left PF and the transmission amplitudes of the elec-tron ( t ) and the hole ( t A ) into the right PF of both parallel and antiparallel configurations, bymatching the wave functions of the two PFs and S region at the two interfaces ( x = x = L ), ψ (cid:48) = ψ e + c + r ψ e − c + r A ψ h − c , ψ (cid:48) = a ψ S + + b ψ S − + a (cid:48) ψ S (cid:48) + + b (cid:48) ψ S (cid:48) − , ψ (cid:48) , P ( AP ) = t ψ e + c ( v ) + t A ψ ( (cid:48) ) h + c ( v ) . (20)Here, the left PF, S region, and the right PF are signed by 1,2, and 3, respectively, and ψ ( (cid:48) ) h + c ( v ) is the solution of the Dirac equation for conduction (valance)band hole of the n - ( p -)dopedPF). Replacing the reflection and transmission amplitudes in BTK formula, we obtain theconductance of AR, CT, and CAR processes for parallel and antiparallel alignments of PMs.In CAR process an electron excitation and a hole excitation at two separate pseudoferromag-netic leads are coupled by means of Andreev scattering processes at two spatially distinctinterfaces. We find that for all incoming waves with two bias voltages eV = ± ( µ − ∆ N ) , ARprocess is suppressed and the cross-conductance in the right PF depends crucially on theconfiguration of PMs in the two PFs. We find that the transport is mediated purely by CT inparallel configuration and changes to the pure CAR in the low energy regime, by reversingthe direction of PM in the right PF. This suggests a pseudospin switching effect between thepure CT and pure CAR in PF/S/PF structure, which can be seen from Eq. (21), for the rightgoing conduction (valance)band electron (hole) of n - ( p -)doped PF, (cid:104) σ ( k ) (cid:105) e ( h )+ c ( v ) = ± (cid:115) − ( ∆ N µ + ε ) ( cos α ec ˆ x + sin α ec ˆ y ) ± ∆ N µ + ε ˆ z . (21)Figure 10 shows the behavior of the conductance of CT and CAR processes, respec-tively, in parallel and antiparallel alignments of PMs versus the length of the S region fortwo values of ∆ N / ∆ S , when µ / ∆ N = . eV = µ − ∆ N . It is seen that the CT processis favored for short junctions L (cid:28) ξ s , while the CAR process is suppressed in this regime. ξ S G / G ξ S G / G CTCT CAR ∆ N / ∆ S = 10 ∆ N / ∆ S = 1 CAR
Fig. 10 (Color online) Plots of the conductance for CT and CAR processes, respectively, in parallel ( G CAR →
0) and antiparallel ( G CT →
0) alignments of PMs versus the length of the S region for two values of ∆ N / ∆ S = ,
10, when µ / ∆ N = . eV = µ − ∆ N . The CT conductance drops by increasing the length L , while the CAR conductance peaksat L < ξ s . We see that the conductance of CT and CAR processes have oscillatory behav-ior with L / ξ s and increase by increasing ∆ N / ∆ S from their values for the correspondinggraphene N/S/N structure. Also we can see that in contrast to the graphene N/S/N structure,CT and CAR processes are present for long lengths of the S region, respectively, in paralleland antiparallel PM configurations. This effect is similar to graphene F/S/F structure [38]and shows that the gapped normal graphene behaves like an F graphene. In conclusion, we have demonstrated the unusual features of the pseudospin polarized quan-tum transport in graphene-based hybrid structures of normal (N) regions, superconductors(S) and gapped regions as pseudoferromagnets (PFs). A gapped graphene is in a sublatticepseudospin symmetry-broken state with a net pseudomagnetization (PM) oriented perpen-dicularly to the plane of graphene. The magnitude of PM depends on the ratio of the chem- uantum transport of pseudospin-polarized Dirac fermions in gapped graphene nanostructures 17 ical potential to the energy gap µ / ∆ N and its direction is switched by changing the type ofdoping between n and p . Based on this observation, we have proposed a perfect pseudospinvalve (PF/N/PF junction) with pseudomagnetoresistance PMR =
1, for µ (cid:39) ∆ N and appro-priate contact length L , whose magnetization alignments can be controlled by altering thetype of their doping. We have shown that this perfect pseudomagnetic valve effect is pre-served even for very large lengths L (cid:29) λ F . Also, it can be resumed at large chemical poten-tials by applying an appropriate bias voltage. We have explained this strong robustness of theperfect pseudomagnetic switching with respect to increasing of the contact length, in termsof an unusually long-range penetration of an equilibrium and nonequilibrium pseudospinpolarization into the normal region by proximity to a PF. The induced pseudomagnetizationvector P M undergoes a damped spatial precession around the normal to the PF/N junctionand tends to be uniform along the normal at the large distances x (cid:29) λ F from the junction.Furthermore, we have found that upon a certain condition, Andreev reflection (AR) ofan electron from an S/PF interface is associated with an inversion of the perpendicular com-ponent of its pseudospin, and that this has important consequences for the proximity ef-fect in S/PF and PF/S/PF geometries. For an S/PF junction system, we have found thatthe Andreev-Klein reflection can enhance the amplitude of AR and the resulting Andreevconductance by ∆ N . In particular, we have shown that depending on the bias voltage theAndreev conductance of weekly doped PF ( µ (cid:28) ∆ N ) can be larger than its value for thecorresponding graphene S/N junction. This is similar to the behavior of Andreev conduc- tance with the exchange energy h in a graphene ferromagnet-superconductor junction. Wehave further studied the proximity density of states (DOS) in pseudoferromagnetic side ofthe S/PF contact, which exhibit a damped-oscillatory behavior as a function of the distancefrom the interface. The period of DOS oscillations is found to be inversely proportionalto the energy gap ∆ N . The proximity DOS in ferromagnetic graphene shows similar spa-tial oscillations with a period determined by 1 / h . For a superconducting pseudospin valve(PF/S/PF) structure, we have found that the transport is mediated purely by elastic electroncotunneling process in parallel alignment of PMs and crossed Andreev reflection process inantiparallel configuration, that is accompanied by pseudospin switching effect. This is againsimilar to the behavior of the corresponding superconducting structure with ferromagneticgraphene and confirms that, in this respect, the effect of the sublattice pseudospin degree offreedom in gapped graphene is as important as the spin in a ferromagnetic graphene. Acknowledgements
We gratefully acknowledge support by the Institute for Advanced Studies in Basic Sci-ences (IASBS) Research Council under grants No. G2009IASBS110 and No. G2010IASBS110. We thankA. G. Moghaddam for fruitful discussions. L. M. acknowledges the financial support of Marco Polini and theorganizers of the school NSPM2011 held in Erice, Italy.
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