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Pseudo-diffusive magnetotransport in graphene

Transport properties through wide and short ballistic graphene junctions are studied in the presence of arbitrary dopings and magnetic fields. No dependence on the magnetic field is observed at the Dirac point for any current cumulant, just as in a classical diffusive system, both in normal-graphene-normal and normal-graphene-superconductor junctions. This pseudo-diffusive regime is however extremely fragile respect to doping at finite fields. We identify the crossovers to a field-suppressed and a normal ballistic transport regime in the magnetic field - doping parameter space, and provide a physical interpretation of the phase diagram. Remarkably, pseudo-diffusive transport is recovered away from the Dirac point in resonance with Landau levels at high magnetic fields.

Low energy excitations in a monolayer of carbon atoms arranged in a honeycomb lattice, known as a graphene sheet, have the remarkable peculiarity of being governed by the 2D massless Dirac equation, which is responsible for a variety of exotic transport properties as compared to ordinary metals. Particularly striking is that for clean, undoped graphene the density of states is zero, but not so the conductivity, which remains of the order of the quantum unit e 2 /h [1,2]. Another intriguing fact is that a wide and short strip of undoped graphene exhibits "pseudo-diffusive" transport properties in the absence of electron-electron interactions and impurity scattering [3]. By pseudo-diffusive it is meant that transport properties are indistinguishable from those of a classical diffusive system. These include the full transport statistics (in particular the Fano factor F = 1/3 and the conductance G ∝ W/L [3], where W is the width and L is the length of the graphene strip), the critical current [4] and I-V characteristics [5] in Josephson structures, as well as the relation of the normal metal-superconductor conductance to the normal transmissions [6]. The same behaviour can be expected in bilayer graphene [7]. In fact, all of the above similarities can be explained by noting that at the Dirac point (i.e., for undoped graphene) transport occurs entirely via evanescent modes with a transmission that is equal to the diffusive transport theory result (evaluated at k F l = 1, l ≡ mean free path [24]) without quantum corrections [3,6,8], Here k y is the transverse momentum of the channel. In diffusive systems the above relation holds independently of an externally applied magnetic field in the limit of many channels (classical limit), for which any quantum weak localization correction is negligible [8].
The question we raise here is: Does the diffusive behaviour of ballistic graphene persist in the presence of a magnetic field? We will show that for zero doping the equivalence is preserved for any magnetic field. Remarkably for a ballistic system, the applied magnetic field does not affect the transport statistics (for any current cumulant) at the Dirac point. For graphene with disorder this has recently also been demonstrated at the conductivity level [9,10,11]. At sufficiently strong magnetic fields an exponentially small chemical potential is enough to enter a field-suppressed transport regime. However, at resonance with the Landau levels (LLs) the pseudo-diffusive behaviour is recovered for all current cumulants. For even higher dopings one observes a final crossover to the ballistic magnetotransport regime, since clean graphene then resembles a ballistic normal metal.
The magnetic field introduces a fundamental quantummechanical length scale known as the magnetic length l B = ( /|e B|) 1/2 . In complete absence of scattering, localized LLs are well formed and ballistic transport is suppressed. The only contribution to transport in this regime comes from resonant tunneling exactly at Landau energies. In usual metals, this happens when the cyclotron diameter 2r c = 2l 2 B k F is smaller than the relevant scattering length (set by system size, disorder or temperature). If 2r c is of the order of or larger than the scattering length, then delocalized states contribute to transport leading to Shubnikov-de Haas oscillations and the Quantum-Hall effect. For wide and short ballistic strips the relevant scattering scale is the strip length L, and scattering on lateral boundaries can be ignored. In graphene, the lowest LL lies precisely at the Dirac point. Besides, at this point k F = 0 and thus r c = 0 independently of the magnetic field. Therefore, in contrast to the normal metal strip and to the high doping limit, at the Dirac point no delocalized bulk transport should take place for any magnetic field and resonant tunneling should be field independent.
To confirm this hand-waving picture we analyze theoretically magnetotransport effects through normal-graphene-normal (N) and normal-graphenesuperconductor (NS) wide ballistic junctions at arbitrary dopings and magnetic fields. From an experimental point of view, transport properties of lightly doped graphene in contact with superconductors is currently being investigated [12]. Moreover, the properties of graphene in strong magnetic fields are also a subject of great interest [13,14,15], in particular in relation to weak (anti)localization [16,17,18,19,20]. Here we consider a clean graphene sheet of width W (assumed to be the largest lengthscale in the system) in the y-direction through which transport occurs in the x-direction. For x < −L/2 it is covered by a normal contact, and for x > L/2 it is covered either by a superconducting contact or a normal one. The central region is lightly doped, leading to a finite Fermi energy µ measured relative to the Dirac point, which can be varied by an external gate voltage. The contact regions are modeled as heavily doped graphene, with Fermi energy µ c conveniently fixed to infinity with respect to both µ and the superconducting gap ∆. The boundary conditions in the y-direction are irrelevant [3] for large aspect ratios W/L ≫ 1. We choose periodic boundaries for simplicity. A constant external magnetic field B is applied perpendicular to the graphene sheet. We assume the electrodes to be magnetically shielded, e.g. by covering them with materials with high-magnetic permeability. In the Landau gauge we can write the vector potential as A = (0, Bx, 0) for |x| < L/2, and constant in the contact regions. This gauge is convenient since the motions in x-and y-direction are uncoupled and k y remains a good quantum number. We neglect Zeeman splitting, so that the electron spin only enters as a degeneracy factor of 2 in the following calculation. Finally, we note that edge currents generally give a negligible contribution to transport in the W ≫ L limit.
We will compute the N and the NS (Andreev) inverse longitudinal resistivity, ρ −1 xx = G(L/W ), expressed in terms of the conductances [3,6] and the shot noise using corresponding expressions in terms of the transmission for normal conducting contacts T ky [21]. Note that the above expressions for the NS case are valid only if T ky is left-right symmetric, which is not in general true in the presence of a magnetic field. In our particular setup it does indeed turn out to be symmetric. The transmission through the central region is obtained by imposing current conservation at the interfaces, which translates into continuity of the wavefunction. In the chosen gauge the scattering problem is effectively one-dimensional, the transverse mode profile e ikyy being the same in all regions, so we will only discuss the x-dependence of the wavefunctions from now on. The contact region eigenstates at energy ǫ = v F k 2 x + k 2 y − µ c + µ with respect to the central strip Dirac point are given by The spinor lives in the space of the two triangular sublattices that conform the graphene hexagonal lattice, s = ±1 is the 'valley' quantum number (for the degenerate K and K ′ points), and z k ≡ exp[i arg(k x + ik y )], which tends to Imposing continuity for each {k y , ǫ} at the interfaces results in the following s−independent transmission probability ± L/2 lB + k y l B . This gives the general transmissions T ky for arbitrary doping and magnetic field. It reproduces previously known results at B = 0 and non-zero doping [4] as well as Eq. (1) for µ = 0.
In Fig. 1(a) we plot ρ −1 xx as a function of the Fermi energy for increasing values of the ratio L/l B ∝ √ B. We recover the results obtained without magnetic field, namely that σ N and σ NS [where σ = ρ −1 xx (B = 0)] tend to the known quantum-limited minimal conductivity value 4e 2 /πh at zero doping, whereas for |µ|L/ v F ≫ 1 the slope of the asymptotes tends to 0.38π and 0.25π for the NS and N junction respectively [6]. Remarkably, as we increase B, all the ρ −1 xx curves remain unchanged at the Dirac point µ = 0. The Dirac-point Fano factor in Fig.  1(b) is also unaffected by magnetic fields, and takes the classical diffusive value (1/3 for the N and 2/3 for the NS junction). This happens for any current cumulant, since at µ = 0 the transmission given in Eq. (5) reduces to Eq.
However, for µ > 0 the resistivities and the Fano factors do depend on the magnetic field. In particular, for 2r c < L (and above a certain critical value of L/l B ) transport can take place only at resonance with the LLs (µL/ v F = √ 2nL/l B ), while for other dopings ρ −1 xx is suppressed like e −(L/lB) 2 /2 for the N junction and as e −(L/lB) 2 for the NS one. The width of the resonances at the LL energies vanishes for 2r c ≪ L, as we consider no disorder [15]. Remarkably, ρ −1 xx at these resonances for large fields coincides with the one at the Dirac point 4e 2 /πh, a theoretical value that is usually associated strictly with zero doping and that is interestingly at odds with some experimental findings [1]. In fact it can be analytically demonstrated that not only the conductance but the whole pseudo-diffusive transport statistics is recovered at the resonances for high magnetic fields. Under this perspective, the field-independent resistivity at µ = 0 can be understood as due to resonant transport through the zero-th LL that remains pinned at the Dirac point. The field suppressed regime is apparent for small but finite doping in Fig. 1(a), where ρ −1 xx strongly decreases with increasing value of L/l B . Correspondingly, the bulk Fano factor reaches the tunneling limit value (1 for the N and 2 for the NS junction) as transport gets suppressed [see Fig. 1(b)], in which limit the noise of the edge currents not considered here could be visible. Increasing the Fermi energy further one enters the regime 2r c > L, where ρ −1 xx is composed of two parts. The first part is linear in µL/ v F , in agreement with the scaling with L behaviour of a ballistic conductor subject to a magnetic field (L independent conductance). In particular, for sufficiently high dopings, all curves in Fig. 1(a) become parallel and tend to the same (average) slope as the zero-field conductivity. The second contribution to ρ −1 xx is an oscillating part, which for 2r c > L can no longer be explained by the resonance with LLs, since in that regime the effect of the boundaries is dominating the level structure in the central region. In fact, for 2r c ≫ L the oscillations become equally spaced and are explained rather by a Fabry-Perot type effect, connected to resonant tunneling through the structure.
In the inset of Fig. 1(a) the ratio G bulk NS /G bulk N is plotted as a function of µ for the same values of L/l B as in the main panel. At the Dirac point the ratio goes to one. At µ > 0 the suppressed magnetotransport manifests itself as a decaying G bulk NS /G bulk N ∝ e −(L/lB) 2 /2 , until doping reaches the ballistic threshold and the ratio starts growing again, finally reaching its asymptotic value 0.38/0.25 = 1.52. As explained in Ref. [6], this value is expected in normal ballistic systems with Fermi wavelength mismatch. Note again here that, for sufficiently suppressed G bulk , the edge contribution [23] neglected here will dominate transport.
All the previous behaviours can be condensed in a quantitative way in the phase diagram shown in Fig. 2. It contains three regions corresponding to the three different transport regimes, namely: pseudo-diffusive (red), field-suppressed (blue) and ballistic (green), in the L/l B and µL/ v F = k F L parameter space. The corresponding crossover lines between regions are solid (dashed) for the N (NS) junction (note that the background colours correspond to the boundaries of the N case). The boundaries for the pseudo-diffusive region have been calculated assuming a maximum deviation of ±10% with respect to the Dirac point conductivity 4e 2 /πh. At low fields the width of the pseudo-diffusive window that brackets the Dirac point is roughly field independent, whereas for L/l B > 1.8 for N (1.35 for NS) the window closes down as exp[−(L/l B ) 2 /4]. Physically this means that at these higher fields the quasi-diffusive transport regime is extremely fragile respect to doping, and an exponentially fast crossover to the field-suppressed (localized) regime takes place. The boundaries of the latter (blue region) were set by a crossover criterion ρ −1 xx < 0.1(4e 2 /πh). Its spiked shape is due to the peaked contributions to the field-suppressed ρ −1 xx discussed in the analysis of Fig. 1, which are produced by resonant tunneling through LLs. When the magnetic field is increased, the positions of these peaks shift to higher dopings, converging on radial lines with slope 1/ √ 2n, while their width decreases exponentially. Above a certain value of L/l B the pseudodiffusive regime is recovered and the resonances are thus coloured in red. The third region (green) is characterized by ρ −1 xx ∝ L at fixed field and doping (which would correspond to radial lines in the phase diagram), and is therefore a ballistic transport regime. As expected from the arguments in the introduction, the boundary of the field-suppressed region closely follows the ballistic threshold 2r c = L. Finally, intermediate regions (white) are characterized by strongly oscillating conductivities.
In conclusion, by computing the general transmission probabilities through short and wide graphene junctions, we have found that the transport properties at the Dirac point exactly match those of a classical diffusive system even in the presence of a magnetic field, which actually does not affect transport at all at zero doping. This behaviour, which is associated to the existence of a zero-th LL pinned at the Dirac point, is however found to be exponentially fragile respect to doping for high fields. By analyzing inverse longitudinal resistivity and higher current cumulants we have identified and interpreted the three distinct regimes that appear at finite magnetic fields and dopings, corresponding to pseudo-diffusive, field-suppressed and ballistic transport, and computed the phase diagram for the N and NS junctions in the relevant field-doping parameter space. Transport resonances at the LL energies are found in the field suppressed regime, with ρ −1 xx and all higher bulk current cumulants saturating to the pseudo-diffusive Dirac point values at high fields. The width of these resonances decreases exponentially with magnetic field, although broadening due to disorder in real samples is expected, thus facilitating experimental observation. The reappearance of pseudodiffusive transport at finite doping could shed light on the 1/π discrepancy between experiments and theoretical results for the conductivity at the Dirac point.