Evanescent wave transport and shot noise in graphene: ballistic regime and effect of disorder
R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi, J. Salmilehto, A. F. Morpurgo, P. J. Hakonen
备注：
Extended version (19 pages, 10 figures) of Phys. Rev. Lett. 100, 196802 (2008). Additional data on the effect of disorder and non-parallel leads. Submitted for publication in Journal of Low Temperature Physics for the Proceedings of the International Symposium on Quantum Phenomena and Devices at Low Temperatures (ULTI 2008), Espoo, Finland
We have investigated electrical transport and shot noise in graphene field effect devices. In large width over length ratio $W/L$ graphene strips, we have measured shot noise at low frequency ($f$ = 600--850 MHz) in the temperature range of 4.2--30 K. We observe a minimum conductivity of $\frac{4e^{2}}{πh}$ and a finite and gate dependent Fano factor reaching the universal value of 1/3 at the Dirac point, i.e. where the density of states vanishes. These findings are in good agreement with the theory describing that transport at the Dirac point should occur via evanescent waves in perfect graphene samples with large $W/L$. Moreover, we show and discuss how disorder and non-parallel leads affect both conductivity and shot noise.
We study the transport through evanescent waves in graphene quantum dots of different geometries. The transmission is suppressed when the leads are attached to edges of the same majority sublattice. Otherwise, the transmission depends exponentially on the distance between leads in rectangular dots, and as a power law in circular dots. The transmission through junctions where the transmitted and reflected currents belong to the opposite valley as the incoming one depends on details of the lattice structure at distances comparable to the atomic spacing.
In Dirac materials, the low energy excitations behave like ultra-relativistic massless particles with linear energy dispersion. A particularly intriguing phenomenon arises with the intrinsic charge transport behavior at the Dirac point where the charge density approaches zero. In graphene, a 2-D Dirac fermion system, it was predicted that charge transport near the Dirac point is carried by evanescent modes, resulting in unconventional pseudo-diffusive charge transport even in the absence of disorder. In the past decade, experimental observation of this phenomenon remained challenging due to th…
We study the the transport properties of multiterminal ballistic graphene samples, concentrating on the conductance matrix, fluctuations and cross-correlations. Far away from Dirac point, the current is carried mostly by propagating modes and the results can be explained with the conventional semiclassical picture familiar from ray optics, where electrons propagate along a single direction before scattering or reaching the terminals. However, close to the Dirac point the transport is due to evanescent modes which do not have to follow a rectilinear path. As we show in this Letter, this propert…
Universal properties of spin-Hall effect in ballistic 2D electron systems are addressed. The net spin polarization across the edge of the conductor is second order, ~λ^2, in spin-orbit coupling constant independent of the form of the boundary potential, with the contributions of normal and evanescent modes each being ~ \sqrtλ but of opposite signs. This general result is confirmed by the analytical solution for a hard-wall boundary, which also yields the detailed distribution of the local spin polarization. The latter shows fast (Friedel) oscillations with the spin-orbit coupling entering via …
We consider a wave propagating through a thin disordered slab inside a wire or waveguide of finite width. In the dense weak scattering limit, the statistics for the complex reflection and transmission coefficients (the coherent field) is found to depend dramatically on the contribution of evanescent modes or closed channels, leading to an effective refractive index whose real part is quite sensitive to the closed channels inclusion. In contrast, evanescent modes play no role in the statistical average of transmittances and reflectances. The theoretical predictions, based on the perturbative Bo…
The dynamical approach is applied to ballistic transport in mesoscopic graphene samples of length L and contact potential U. At times shorter than both relevant time scales, the flight time and \hslash/U, the major effect of the electric field is to create electron - hole pairs, i.e. causing interband transitions. In linear response this leads (for width W>>L) to conductivity pi/2 e^{2}/h. On the other hand, at times lager than the two scales the mechanism and value are different. It is shown that the conductivity approaches its intraband value, equal to the one obtained within the Landauer-Bu…
We show that graphene-based quantum pumps can tap into evanescent modes, which penetrate deeply into the device as a consequence of Klein tunneling. The evanescent modes dominate pumping at the Dirac point, and give rise to a universal response under weak driving for short and wide pumps, in close analogy to their role for the minimal conductivity in ballistic transport. In contrast, evanescent modes contribute negligibly to normal pumps. Our findings add a new incentive for the exploration of graphene-based nanoelectronic devices.
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An analytic theory of electron transport in disordered graphene in a ballistic geometry is developed. We consider a sample of a large width W and analyze the evolution of the conductance, the shot noise, and the full statistics of the charge transfer with increasing length L, both at the Dirac point and at a finite gate voltage. The transfer matrix approach combined with the disorder perturbation theory and the renormalization group is used. We also discuss the crossover to the diffusive regime and construct a ``phase diagram'' of various transport regimes in graphene.
Graphene nanoribbons (GNRs) are natural waveguides for electrons in graphene. Nevertheless, unlike micron-sized samples, conductance is nearly suppressed in these narrow graphene stripes, mainly due to scattering with edge disorder generated during synthesis or cut. A possible way to circumvent this effect is to define an internal waveguide that isolates specific modes from the edge disorder and allows ballistic conductance. There are several proposals for defining waveguides in graphene; in this manuscript, we consider strain folds and scalar potentials and numerically evaluate these proposal…
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We have investigated shot noise in graphene field effect devices in the temperature range of 4.2--30 K at low frequency ($f$ = 600--850 MHz). We find that for our graphene samples with large width over length ratio $W/L$, the Fano factor $\mathfrak{F}$ reaches a maximum $\mathfrak{F} \sim$ 1/3 at the Dirac point and that it decreases strongly with increasing charge density. For smaller $W/L$, the Fano factor at Dirac point is significantly lower. Our results are in good agreement with the theory describing that transport at the Dirac point in clean graphene arises from evanescent electronic st…
Typical persistent current ($I_{typ}$) in a normal metal ring with disorder due to random grain boundaries and rough edges is calculated microscopically. If disorder is due to the rough edges, a ballistic current $I_{typ}\simeq e v_F/L$ is found in spite of the diffusive resistance ($ \propto L/l$), where $v_F$ is the Fermi velocity, $l$ is the mean free path, and $L \gg l$ is the ring length. This ballistic current has a simple interpretation: It is due to a single coherent electron that moves in parallel with the edges and thus does not feel the roughness. Our calculations explain a puzzling…
It is known that Josephson junction (JJ) hosting scattering centers with energy-dependent scattering amplitudes which breaks the $ω\rightarrow-ω$ symmetry (where $ω$ is the excitation energy of electron about the Fermi level) exhibits finite thermoelectric response. In contrast, here we show that even in a ballistic JJ this symmetry is broken and it leads to a non-zero thermoelectric response when the junction length is of the order of coherence length of the superconductor and the corresponding thermoelectric coefficient confirms to the universal sinusoidal dependence on $ϕ_{12}$, where $ϕ_{1…
We calculate the shot noise generated by evanescent modes in graphene for several experimental setups. For two impurity-free graphene strips kept at the Dirac point by gate potentials, separated by a long highly doped region, we find that the Fano factor takes the universal value F = 1/4. For a large superlattice consisting of many strips gated to the Dirac point interspersed among doped regions, we find F = 1/(8 ln 2). These results differ from the value F = 1/3 predicted for a disordered metal, providing an unambiguous experimental signature of evanescent mode transport in graphene.
In previous work we have found a regime in ballistic quantum dots where interelectron interactions can be treated asymptotically exactly as the Thouless number $g$ of the dot becomes very large. However, this work depends on some assumptions concerning the renormalization group and various properties of the dot obeying Random Matrix Theory predictions at scales of the order of the Thouless energy. In this work we test the validity of those assumptions by considering a particular ballistic dot, the Robnik-Berry billiard, numerically. We find that almost all of our predictions based on the earli…
Typical persistent current ($I_{typ}$) in a mesoscopic normal metal ring with disorder due to rough edges and random grain boundaries is calculated by a scattering matrix method. In addition, resistance of a corresponding metallic wire is obtained from the Landauer formula and the electron mean free path ($l$) is determined. If disorder is due to the rough edges, a ballistic persistent current $I_{typ} \simeq e v_F/L$ is found to coexist with the diffusive resistance ($ \propto L/l$), where $v_F$ is the Fermi velocity and $L \gg l$ is the ring length. This ballistic current is due to a single …
We demonstrate that branching of the electron flow in semiconductor nanostructures can strongly affect macroscopic transport quantities and can significantly change their dependence on external parameters compared to the ideal ballistic case even when the system size is much smaller than the mean free path. In a corner-shaped ballistic device based on a GaAs/AlGaAs two-dimensional electron gas we observe a splitting of the commensurability peaks in the magnetoresistance curve. We show that a model which includes a random disorder potential of the two-dimensional electron gas can account for th…
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Two dimensional electron gases based on SrTiO$_3$ are an intriguing platform for exploring mesoscopic superconductivity combined with spin-orbit coupling, offering electrostatic tunability from insulator to metal to superconductor within a single material. So far, however, quantum effects in SrTiO$_3$ nanostructures have been complicated by disorder. Here we introduce a facile approach to achieving high mobility and patterning gate-tunable structures in SrTiO$_3$, and use it to demonstrate ballistic constrictions with clean normal state conductance quantization. Conductance plateaus show two-f…
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We present low-temperature magnetotransport measurements on graphene rings encapsulated in hexagonal boron nitride. We investigate phase-coherent transport and show Aharonov-Bohm (AB) oscillations in quasi-ballistic graphene rings with hard confinement. In particular, we report on the observation of $h/e$, $h/2e$ and $h/3e$ conductance oscillations. Moreover we show signatures of magnetic focusing effects at small magnetic fields confirming ballistic transport. We perform tight binding calculations which allow to reproduce all significant features of our experimental findings and enable a deep…
