Transport in suspended graphene

Motivated by recent experiments on suspended graphene showing carrier mobilities as high as 200,000 cm²/V s, we theoretically calculate transport properties assuming Coulomb impurities as the dominant scattering mechanism. We argue that the substrate-free experiments done in the diffusive regime are consistent with our theory and verify many of our earlier predictions including (i) removal of the substrate will increase mobility since most of the charged impurities are in the substrate, (ii) the minimum conductivity is not universal, but depends on impurity concentration with cleaner samples having a higher minimum conductivity. We further argue that experiments on suspended graphene put strong constraints on the two parameters involved in our theory, namely, the charged impurity concentration and d, the typical distance of a charged impurity from the graphene sheet. The recent experiments on suspended graphene indicate a residual impurity density of which are presumably stuck to the graphene interface, compared to impurity densities of for graphene on SiO₂ substrate. Transport experiments can therefore be used as a spectroscopic tool to identify the properties of the remaining impurities in suspended graphene.     

Coulomb Problem for <Emphasis Type="Italic">Z</Emphasis> &gt; Z<Subscript>cr</Subscript> in Doped Graphene

The dynamics of charge carriers in doped graphene, i.e., graphene with a gap in the energy spectrum depending on the substrate, in the presence of a Coulomb impurity with charge Z is considered within the effective two-dimensional Dirac equation. The wave functions of carriers with conserved angular momentum J = M + 1/2 are determined for a Coulomb potential modified at small distances. This case, just as any two-dimensional physical system, admits both integer and half-integer quantization of the orbital angular momentum in plane, M = 0, ±1, ±2, …. For J = 0, ±1/2, ±1, critical values of the effective charge Z_cr(J, n) are calculated for which a level with angular momentum J and radial quantum numbers n = 0 and n = 1 reaches the upper boundary of the valence band. For Z < Z_cr (J, n = 0), the energy of a level is presented as a function of charge Z for the lowest values of orbital angular momentum M, the level with J = 0 being the first to descend to the band edge. For Z>Z_cr (J, n = 0), scattering phases are calculated as a function of hole energy for several values of supercriticality, as well as the positions ε₀ and widths γ of quasistationary states as a function of supercriticality. The values of ε₀* and width γ* are pointed out for which quasidiscrete levels may show up as Breit–Wigner resonances in the scattering of holes by a supercritical impurity. Since the phases are real, the partial scattering matrix is unitary, so that the radial Dirac equation is consistent even for Z > Z_cr. In this single-particle approximation, there is no spontaneous creation of electron–hole pairs, and the impurity charge cannot be screened by this mechanism.     

Vacuum polarization and screening of supercritical impurities in graphene

Screening of charge impurities in graphene is analyzed using the exact solution for vacuum polarization obtained from the massless Dirac-Kepler problem. For the impurity charge below a certain critical value, no density perturbation is found away from the impurity, in agreement with perturbation theory. For the supercritical charge, however, the polarization distribution is shown to have a power law profile, leading to screening of the excess charge at large distances. The Dirac-Kepler scattering states give rise to standing wave oscillations in the local density of states which are prominent in the supercritical regime.     

Harmonic measure and winding of conformally invariant curves

The exact joint multifractal distribution for the scaling and winding of the electrostatic potential lines near any conformally invariant scaling curve is derived in two dimensions. Its spectrum f(alpha,lambda) gives the Hausdorff dimension of the points where the potential scales with distance r as H approximately r(alpha) while the curve logarithmically spirals with a rotation angle phi=lambdalnr. It obeys the scaling law f(alpha,lambda)=(1+lambda(2))f(alpha)-blambda(2) with alpha=alpha/(1+lambda(2)) and b=(25-c)/12, and where f(alpha) identical with f(alpha,0) is the pure harmonic measure spectrum, and c the conformal central charge. The results apply to O(N) and Potts models, as well as to stochastic L?wner evolution.     

Scattering in graphene with impurities: A low energy effective theory

We analyze the scattering sector of the Hamiltonians for both gapless and gapped graphene in the presence of a charge impurity using the 2D Dirac equation, which is applicable in the long wavelength limit. We show that for certain range of the system parameters, the combined effect of the short range interactions due to the charge impurity can be modelled using a single real parameter appearing in the boundary conditions. The phase shifts and the scattering matrix depend explicitly on this parameter. We argue that this parameter for graphene can be fixed empirically, through measurements of observables that depend on the scattering data.     

Anchoring platinum on graphene using metallic adatoms: a first principles investigation

First principles calculations based on spin-polarized density functional theory were used to identify metallic adatoms that would strengthen the Pt(111)/graphene interface (with a low work of separation of 0.009 J m(-2)), when the adatom was placed between the Pt(111) and the graphene. It was shown that the strength of the Pt-adatom bond, which had a metallic character, increased with the amount of charge transferred from the adatom to the Pt. The carbon-adatom bond, on the other hand, had a mixed ionic and covalent character and was weaker than the Pt-adatom bond for each of the 25 elements considered. Consequently, the total Pt(111)/graphene interface strength and, hence, the anchoring effect of the adatom were controlled by the carbon-adatom bond strength. Metals with unfilled d orbitals increased the Pt/graphene interface strength to above 0.5 J m(-2). The carbon-adatom bond strength was proportional to the ratio between the charge transferred from the adatom to the graphene (ΔZ(C)) and the charge transferred to the Pt surface (ΔZ(Pt)); i.e., the ΔZ(C)/ΔZ(Pt) ratio defined the ability of an adatom to anchor Pt to graphene. For Ir, Os, Ru, Rh and Re, ΔZ(C)/ΔZ(Pt) > 1.0, making these elements the most effective adatoms for anchoring Pt to graphene.     

Enhancement of positron-atom annihilation near the positronium formation threshold

The behavior of the positron- 2 gamma annihilation rate on an atomic target near the positronium (Ps) formation threshold is determined. When the positron energy epsilon approaches the threshold epsilon(thr) from below, the effective number of electrons contributing to the annihilation, Z(eff), grows as Z(eff) approximately A/square root of [epsilon(thr)-epsilon], where A is related to the size of the Ps formation cross section, sigma(Ps) approximately B square root of [epsilon-epsilon(thr)], by A = B square root of [2 epsilon(thr)]/32 pi (in atomic units). Taking account of the finite Ps lifetime eliminates the singularity in Z(eff) and shows that close to threshold the positron annihilation cross section is identical to the para-Ps formation cross section.     

Full counting statistics in the self-dual interacting resonant level model

We present a general technique to obtain the zero temperature cumulant generating function of the full counting statistics of charge transfer in interacting impurity models out of equilibrium from time-dependent simulations on a lattice. We demonstrate the technique with application to the self-dual interacting resonant level model, where very good agreement between numerical simulations using the density matrix renormalization group and those obtained analytically from the thermodynamic Bethe ansatz is found. We show from the exact form of counting statistics that the quasiparticles involved in transport carry charge 2e in the low bias regime and e/2 in the high bias regime.     

Z2 topological order and the quantum spin Hall effect

The quantum spin Hall (QSH) phase is a time reversal invariant electronic state with a bulk electronic band gap that supports the transport of charge and spin in gapless edge states. We show that this phase is associated with a novel Z2 topological invariant, which distinguishes it from an ordinary insulator. The Z2 classification, which is defined for time reversal invariant Hamiltonians, is analogous to the Chern number classification of the quantum Hall effect. We establish the Z2 order of the QSH phase in the two band model of graphene and propose a generalization of the formalism applicable to multiband and interacting systems.     

Z2 topological term, the global anomaly, and the two-dimensional symplectic symmetry class of Anderson localization

We discuss, for a two-dimensional Dirac Hamiltonian with a random scalar potential, the presence of a Z2 topological term in the nonlinear sigma model encoding the physics of Anderson localization in the symplectic symmetry class. The Z2 topological term realizes the sign of the Pfaffian of a family of Dirac operators. We compute the corresponding global anomaly, i.e., the change in the sign of the Pfaffian by studying a spectral flow numerically. This Z2 topological effect can be relevant to graphene when the impurity potential is long ranged and, also, to the two-dimensional boundaries of a three-dimensional lattice model of Z2 topological insulators in the symplectic symmetry class.     

Anisotropic Short-Wavelength Friedel Oscillation with Adjustable Envelope in Graphene

We investigate theoretically the intervalley charge density oscillation and the screened ionic potential in graphene caused by the intervalley scattering. We demonstrate that the contribution from the intervalley scattering is comparable with that from the intravalley scattering, and oscillation rather than decaying dominates at a large distance away from the external impurity. We show that the intervalley oscillation is strongly anisotropic because of the inequivalency between neighboring valleys. The anisotropic oscillation consists of an anisotropic short-wavelength oscillation with an anisotropic fixed wavelength and an isotropic long-wavelength envelop with an isotropic wavelength modulated by doping, making an adjustable-widthed wave-packet propagation. One weakens the screening and gets anisotropic short-wavelength oscillation by introducing short-range mechanism in graphene and graphene-like materials.     

Anisotropic Short-Wavelength Friedel Oscillation with Adjustable Envelope in Graphene

We investigate theoretically the intervalley charge density oscillation and the screened ionic potential in graphene caused by the intervalley scattering. We demonstrate that the contribution from the intervalley scattering is comparable with that from the intravalley scattering, and oscillation rather than decaying dominates at a large distance away from the external impurity. We show that the intervalley oscillation is strongly anisotropic because of the inequivalency between neighboring valleys. The anisotropic oscillation consists of an anisotropic short-wavelength oscillation with an anisotropic fixed wavelength and an isotropic long-wavelength envelop with an isotropic wavelength modulated by doping, making an adjustable-widthed wave-packet propagation. One weakens the screening and gets anisotropic short-wavelength oscillation by introducing short-range mechanism in graphene and graphene-like materials.     

Gap inhomogeneities and the density of states in disordered d-wave superconductors

We report on a numerical study of disorder effects in 2D d-wave BCS superconductors. We compare exact numerical solutions of the Bogoliubov-de Gennes (BdG) equations for the density of states rho(E) with the standard T-matrix approximation. Local suppression of the order parameter near impurity sites, which occurs in self-consistent solutions of the BdG equations, leads to apparent power-law behavior rho(E) approximately |E|(alpha) with nonuniversal alpha over an energy scale comparable to the single-impurity resonance energy Omega(0). We show that the novel effects arise from static spatial correlations between the order parameter and the impurity distribution.     

Charge state of a transition metal impurity in II–VI semiconductors

The results of calculating the electronic structure of semiconductor compounds A^IIB^VI: 3d(A = Zn; B = S, Se, Te; 3d = Sc-Cu) at a low content of 3d impurities are discussed. The excess charge of an impurity ion with respect to the charge of the zinc ion is determined for the whole series of 3d impurities. It is found that the excess charge gradually varies from +0.6|e| for the scandium impurity to ?0.2|e| for the copper impurity. Photoionization of an impurity ion is simulated by adding a hole or an electron to the impurity center. The added charge is redistributed between the impurity ion and its nearest neighbors, thus decreasing or increasing the total excess charge of the impurity center by a magnitude of ～ 0.2|e|.     

Screening of Local Magnetic Moment by Electrons of Disordered Graphene

Based on the Anderson impurity model and self-consistent approach, we investigate the condition for the screening of a local magnetic moment by electrons in graphene and the influence of the moment on electronic properties of the system. The results of numerical calculations carried out on a finite sheet of graphene show that when the Fermi energy is above the single occupancy energy and below the double occupancy energy of the local impurity, a magnetic state is possible. A phase diagram in a parameter space spanned by the Coulomb energy U and the Fermi energy is obtained to distinguish the parameter regions for the magnetic and nonmagnetic states of the impurity. We find that the combined effect of the impurity and finite size effect results in a large charge density near the edges of the finite graphene sheet. The density of states exhibits a peak at the Dirac point which is caused by the appearance of the edge states localized at the zigzag edges of the sheet.     

Ground state of graphene in the presence of random charged impurities

We calculate the carrier-density-dependent ground-state properties of graphene in the presence of random charged impurities in the substrate taking into account disorder and interaction effects nonperturbatively on an equal footing in a self-consistent theoretical formalism. We provide detailed quantitative results on the dependence of the disorder-induced spatially inhomogeneous two-dimensional carrier density distribution on the external gate bias, the impurity density, and the impurity location. We find that the interplay between disorder and interaction is strong, particularly at lower impurity densities. We show that, for the currently available typical graphene samples, inhomogeneity dominates graphene physics at low (< or approximately 10(12) cm(-2)) carrier density with the density fluctuations becoming larger than the average density.     

Visualization of nano-plasmons in graphene

We study localized plasmons at the nanoscale (nano-plasmons) in graphene. The collective excitations of induced charge density modulations in graphene are drastically changed in the vicinity of a single impurity compared to graphene's bulk behavior. The dispersion of nano-plasmons depends on the number of electrons and the sign, strength and size of the impurity potential. Due to this rich parameter space the calculated dispersions are intrinsically multidimensional requiring an advanced visualization tool for their efficient analysis, which can be achieved with parallel rendering. To overcome the problem of analyzing thousands of very complex spatial patterns of nano-plasmonic modes, we take a combined visual and quantitative approach to investigate the excitations on the two-dimensional graphene lattice. Our visual and quantitative analysis shows that impurities trigger the formation of localized plasmonic excitations of various symmetries. We visually identify dipolar, quadrupolar and radial modes, and quantify the spatial distributions of induced charges.     

Unconventional integer quantum Hall effect in graphene

Monolayer graphite films, or graphene, have quasiparticle excitations that can be described by (2+1)-dimensional Dirac theory. We demonstrate that this produces an unconventional form of the quantized Hall conductivity sigma(xy) = -(2e2/h)(2n+1) with n = 0, 1, ..., which notably distinguishes graphene from other materials where the integer quantum Hall effect was observed. This unconventional quantization is caused by the quantum anomaly of the n=0 Landau level and was discovered in recent experiments on ultrathin graphite films.