Minimal conductivity and signatures of quantum criticality in ballistic graphene bilayer

We study the ballistic conductivity of graphene bilayer in the presence of next-nearest neighbor hoppings between the layers. An undoped and unbiased system was found in Ref. [1] to show a nonuniversal (length-dependent) conductivity $\sigma(L)$, approaching the value of $\sigma_\star=3/\pi\simeq{}0.95$ for large $L$. Here we demonstrate one-parameter scaling and determine the scaling function $\beta(\sigma)=d\ln{}\!\sigma/d\ln{}\!L$. The scaling flow has an attractive fixed point [$\,\beta(\sigma_\star)=0$, $\beta'(\sigma_\star)<0\,$] reproducing the scenario predicted for random impurity scattering of Dirac fermions with Coulomb repulsion, albeit the system considered is perfectly ballistic and interactions are not taken into account. The role of electrostatic bias between the layers is also briefly discussed.Comment: RevTeX, 5 pages, 4 figure

Magnetoconductance of the Corbino disk in graphene: Chiral tunneling and quantum interference in the bilayer case

Quantum transport through an impurity-free Corbino disk in bilayer graphene is investigated analytically, by the mode-matching method for effective Dirac equation, in the presence of uniform magnetic fields. Similarly as in the monolayer case (see Refs. [1,2]), conductance at the Dirac point shows oscillations with the flux piercing the disk area $\Phi_D$ characterized by the period $\Phi_0=2\,(h/e)\ln(R_{\rm o}/R_{\rm i})$, where $R_{\rm o}$ ($R_{\rm i}$) is the outer (inner) disk radius. The oscillations magnitude depends either on the radii ratio or on the physical disk size, with the condition for maximal oscillations reading $R_{\rm o}/R_{\rm i}\simeq\left[\,R_{\rm i}t_{\perp}/(2\hbar{}v_{F})\,\right]^{4/p}$ (for $R_{\rm o}/R_{\rm i}\gg{}1$), where $t_\perp$ is the interlayer hopping integral, $v_F$ is the Fermi velocity in graphene, and $p$ is an {\em even} integer. {\em Odd}-integer values of $p$ correspond to vanishing oscillations for the normal Corbino setup, or to oscillations frequency doubling for the Andreev-Corbino setup. At higher Landau levels (LLs) magnetoconductance behaves almost identically in the monolayer and bilayer cases. A brief comparison with the Corbino disk in 2DEG is also provided in order to illustrate the role of chiral tunneling in graphene.Comment: Typos corrected; acknowledgment added. RevTeX, 13 pages, 7 figure

Pseudodiffusive conductance, quantum-limited shot noise, and Landau-level hierarchy in biased graphene bilayer

We discuss, by means of mode-matching analysis for the Dirac equation, how splittings of the Landau-level (LL) degeneracies associated with spin, valley, and layer degrees of freedom affect the ballistic conductance of graphene bilayer. The results show that for wide samples ($W\gg{}L$) the Landauer-B\"{u}ttiker conductance reaches the maximum $G\simeq{}se^2/(\pi{h})\times{}W/L$ at the resonance via each LL, with the prefactor varying from $s=8$ if all three degeneracies are preserved, to $s=1$ if all the degeneracies are split. In the absence of bias between the layers, the degeneracies associated with spin and layer degrees of freedom may be split by manipulating the doping and magnetic field; the conductance at the zeroth LL is twice as large, while the conductance at any other LL equals to the corresponding conductance of graphene monolayer. The presence of bias potential allows one also to split the valley degeneracy. Our results show that the charge transfer at each LL has pseudodiffusive character, with the second and third cumulant quantified by ${\cal F}=1/3$ and ${\cal R}=1/15$ (respectively). In case the electrochemical potential is allowed to slowly fluctuate in a finite vicinity of LL, the resulting charge-transfer characteristics are still quantum-limited, with ${\cal F}\simeq{}0.7$ and ${\cal R}\simeq{}0.5$ in the limit of large fluctuations. The above values of ${\cal F}$ and ${\cal R}$ are also predicted to be approached in the limit of high source-drain voltage difference applied. The possible effects of indirect interlayer hopping integrals are also briefly discussed.Comment: Minor revisions, refs. added; new Section V describing the possible effects of indirect hoppings between the layers. Figure files optimized for the faster download. RevTeX, 13 pages, 10 figure

Quantum-limited shot noise and quantum interference in graphene based Corbino disk

This is a theoretical study of finite voltage effects on the conductance, the shot noise power, and the third charge-transfer cumulant for graphene-based Corbino disk in the presence of external magnetic fields. Periodic magnetoconductance oscillations, predicted in Refs. [1,2], become invisible for relatively small source-drain voltages, as the current decays rapidly with magnetic field. Quantum interference still governs the behavior of higher charge-transfer cumulants.Comment: Minor revisions, PACS Nos. added. RevTeX, 5 pages, 3 figure

Conditions for Conductance Quantization in Mesoscopic Dirac Systems on the Examples of Graphene Nanoconstrictions

Ballistic transport through an impurity-free section of the Corbino disk in graphene is investigated by means of the Landauer-B\"{u}ttiker formalism in the mesoscopic limit. In the linear-responce regime the conductance is quantized in steps close to integer multiples of $4e^{2}/h$, yet Fabry-Perot oscillations are strongly suppressed. The quantization arises for small opening angles $\theta\lesssim\pi/3$ and large radii ratios $R_2/R_1\gtrsim{}10$. We find that the condition for emergence of the $n$-th conductance step can be written as $\sqrt{n}\theta/\pi\ll{}1$. A brief comparison with the conductance spectra of graphene nanoribbons with parallel edges is also provided.Comment: Typos corrected. RevTeX, 5 pages, 4 figures. Presented on "XVI National Conference of Superconductivity", October 7-12, 2013, Zakopane, Polan

Thermoelectric properties of gapped bilayer graphene

Unlike in conventional semiconductors, both the chemical potential and the band gap in bilayer graphene (BLG) can be tuned via application of external electric field. Among numerous device implications, this property also designates BLG as a candidate for high-performance thermoelectric material. In this theoretical study we have calculated the Seebeck coefficients for abrupt interface separating weakly- and heavily-doped areas in BLG, and for a more realistic rectangular sample of mesoscopic size, contacted by two electrodes. For a given band gap ($\Delta$) and temperature ($T$) the maximal Seebeck coefficient is close to the Goldsmid-Sharp value $|S|_{\rm max}^{\rm GS}=\Delta/(2eT)$, the deviations can be approximated by the asymptotic expression $|S|_{\rm max}^{\rm GS}-|S|_{\rm max}=(k_B/e)\times\left[\frac{1}{2}\ln{u}+\ln{}2-\frac{1}{2}+{\cal O}(u^{-1})\right]$, with the electron charge $-e$, the Boltzmann constant $k_B$, and $u = \Delta/(2k_BT)\gg{}1$. Surprisingly, the effects of trigonal warping term in the BLG low-energy Hamiltonian are clearly visible at few-Kelvin temperatures, for all accessible values of $\Delta\leqslant{}300\,$meV. We also show that thermoelectric figure of merit is noticeably enhanced ($ZT>3$) when a rigid substrate suppresses out-of-plane vibrations, reducing the contribution from $ZA$ phonons to the thermal conductivity.Comment: Minor revisions; notation clarified, a comment effective doping added. RevTeX; 9 pages; 5 figure

Lifshitz transition and thermoelectric properties of bilayer graphene

This is a numerical study of thermoelectric properties of ballistic bilayer graphene in the presence of trigonal warping term in the effective Hamiltonian. We find, in the mesoscopic samples of the length $L>10\,\mu{}$m at sub-Kelvin temperatures, that both the Seebeck coefficient and the Lorentz number show anomalies (the additional maximum and minimum, respectively) when the electrochemical potential is close to the Lifshitz energy, which can be attributed to the presence of the van Hove singularity in a bulk density of states. At higher temperatures the anomalies vanish, but measurable quantities characterizing remaining maximum of the Seebeck coefficient still unveil the presence of massless Dirac fermions and make it possible to determine the trigonal warping strength. Behavior of the thermoelectric figure of merit ($ZT$) is also discussed.Comment: Typos corrected. RevTeX, 11 pages, 8 figure