Electronic properties of strained graphene

Abstract

Graphene is an atomically thick single layer of carbon atoms arranged according to a honeycomb lattice. Its quite recent discovery, due to Geim and Novoselov, and the realization of sufficiently large graphene flakes in the laboratory have stimulated an enormous outburst of both experimental and theoretical investigation (Science 306, 666 (2004)). Indeed in 2010, Geim and Novoselov were awarded the Nobel Prize in Physics for groundbreaking experiments regarding the two-dimensional material graphene (Rev. Mod. Phys. 83, 837 (2011), Rev. Mod. Phys. 83, 851 (2011)). The electronic band structure of graphene consists of two bands. In pristine graphene, they touch each other at the Fermi level in a linear, cone-like fashion at the so-called Dirac points K and K0. These two special points coincide with the corners of the hexagonal first Brillouin zone (1BZ). Transport measurements show that graphene has a remarkably high electron mobility at room temperature. Specifically, the electron mobility for graphene on a silicon dioxide (SiO2) substrate is 105 cm2 V−1 s−1, whereas for suspended graphene (i.e. without substrate) it is 106 cm2 V−1 s−1 (Phys. Rev. Lett. 101, 096802 (2008)), which are values comparable with those obtained in more complex systems, as a modulation-doped field transistor (MODFET). Besides a high mobility, graphene presents a relatively high optical transparency, in addition to a remarkable flexibility, robustness and environmen- tal stability. These properties make graphene an attractive material for applications also in photonics, optoelectronics and plasmonics (Nature Photonics 4, 611 (2010)). Graphene is also notable for its remarkable mechanical properties. In particular, recent ab initio calculations (Phys. Rev. B 76, 064120 (2007)) as well as experiments (Nature 457, 706 (2009)) have demonstrated that graphene single layers can reversibly sustain elastic deformations as large as 20%. In microelectronics, the effect of strain is often used to modify the electronic and transport properties of materials in order to improve the performance of the devices. In graphene, the application of strain (e.g. by stretching or bending) allows to tune its electronic properties. Recently, there has been a great interest towards the study and the realization of graphene-based electronic devices designed by a suitable tailoring of the electronic struc- ture exploiting not only electric field effect but also applied strain. Both these techniques allow to tune the electronic properties of graphene in a reversible and clean way, i.e. without adding any source of disorder. Therefore, an in-depth knowledge of the effects of the strain on graphene could be exploited to improve graphene-based devices. In this Thesis, we study theoretically the influence that applied strain can have on several properties related the electronic structure, such as the optical properties, the plasmonic properties, and the transport properties. First of all, we have discussed the strain dependence of the electronic band struc- ture, and derived the strain and electric field dependence of the optical conductivity of graphene under uniaxial strain. Within a tight-binding model, including strain- dependent nearest neighbour hoppings and orbital overlaps, we have interpreted the evolution of the band dispersion relations with strain modulus and direction in terms of the proximity to several electronic topological transitions (ETT). These correspond to the change of topology of the Fermi line as a function of strain. In the case of graphene, one may distinguish among three distinct ETTs. We also recover the evolution of the location of the Dirac points, which move away from the corners of the 1BZ as a function of strain. For sufficiently small strain modulus, however, one may still linearly expand the band dispersion relations around the new Dirac points, thereby recovering a cone approximation, but now with elliptical sections at constant energy, as a result of the strain-induced deformation. For increasing strain, two inequivalent Dirac points may merge into one, which usually occurs at either midpoint M` (` = 1, 2, 3) of the 1BZ boundary, depending on the strain direction. This corresponds to the breaking down of linearity of the band dispersions along a given direction through the Dirac points, the emergence of low-energy quasipar- ticles with an anisotropic massive low-energy spectrum, and the opening of a gap in the energy spectrum. Besides, we confirm that such an event depends not only on the strain modulus, but characteristically also on the strain direction. In particular, no gap opens when strain is applied along the armchair direction. We derived the energy dependence of the density of states (DOS), and recovered a linear dependence at low energy within the cone approximation, albeit modified by a renormalized strain-dependent slope. In particular, such a slope has been shown to increase with increasing strain modulus, re- gardless of the strain direction, thus suggesting that applied strain may obtain a steeper DOS in the linear regime. We have also calculated the DOS beyond the Dirac cone approximation. The proxim- ity to ETTs gives rise to (possibly degenerate) Van Hove singularities in the density of states, appearing as logarithmic peaks in the DOS. Finally, we generalized our previous results for the optical conductivity to the case of strained graphene. We studied the frequency dependence of the longitudinal optical conductivity as a function of strain modulus and direction, as well as of field orientation. Our main results are that (a) logarithmic peaks appear in the optical conductivity at sufficiently high frequency, and can be related to the ETTs in the electronic spectrum under strain, and depending on the strain direction; (b) the relative weight of the peaks in general depends on the strain direction and field orientation, and contributes to the generally anisotropic pattern of the optical conductivity as a function of field orientation; (c) the opening of a band gap, where allowed, is signalled by a vanishing optical conductivity. The optical conductivity is directly related to measurable quantities, such as the transmittance and reflectance. Thus, an experimental study of the optical conductivity in the visible range of frequen- cies as a function of strain modulus and direction, as well as of field orientation, should enable one to identify the occurrence of the three distinct ETTs predicted for graphene. In addition, according to our results, the asymmetry induced by uniaxial strain in the op- tical conductivity causes an observable degree of dichroism. Indeed, the optical response of uniaxially strained graphene to linearly polarized light depends on the direction of the polarization. Moreover, the optical response of graphene can give information about the magnitude and the direction of strain in a graphene sample. Finally, these results about the effect of uniaxial strain on the electronic structure and optical conductivity are in agreement with recent ab initio calculations (EPL 92, 67001 (2010)). After an in-depth study of the changes of electronic structure due to uniaxial strain, we dealt with the strain-induced modifications of the plasmons. By studying the elec- tronic polarization, we have derived the dispersion relation of the plasmon modes in graphene. Besides including electron-electron correlation at the random phase approx- imation (RPA) level, we have considered local field effects (LFE), that are specific to the peculiar lattice structure under study, and we have also taken into account the z-extension of the electron wave functions. Both terms are sizable in electron-electron scattering processes with large exchange momentum (q /a). As a consequence of the two-band character of the electronic band structure of graphene, we have found in general two plasmonic branches: (1) a low-energy branch, with a square- root behavior at small wavevectors, and (2) a high-energy branch, weakly dispersing at small wavevectors. In particular, we have found that the high-energy plasmon mode disappears neglecting LFE. While in the absence of LFE only scattering processes with momenta within the 1BZ are considered, LFE allow to include all scattering processes with arbitrarily low wavelengths, thereby taking into account the discrete nature of the crystalline lattice. Hence, the Umklapp electron-electron scattering processes have fun- damental role in order that the system sustains the high-energy plasmon mode. More- over, we have found an intermediate energy pseudo-plasmon mode, associated with a logarithmic divergence of the polarization, which can be related to an interband transi- tion between the Van Hove singularities in the valence and conduction bands of graphene, and it can be identified with a ! transition. In graphene, to date there are measure- ments about the low energy plasmon (Nature 487, 77 (2012), Nature Photonics 6, 749 (2012)) and the pseudo-plasmon excitation (Phys. Rev. B 77, 233406 (2008)), whereas there is no clear experimental evidence about the high energy plasmon. Usually, experi- mental methodologies to detect plasmon dispersion relation, such as electron energy loss spectroscopy (EELS), measure the collective excitation at small wavevectors (q ! 0). The detection of the high energy branch at small wavevector could be difficult, first of all, because of the reduced spectral weight associated with the high energy branch, but also because these plasmons could be damped by the promotion of electrons from the valence band into the higher ( ) energy band. Due to the robustness of the Dirac cones with respect to the application of uniax- ial strain, for sufficiently small strain modulus, it is possible to use the massless Dirac approximation in order to describe the low energy electronic properties. In particular, exploiting the massless Dirac approximation, we have studied the dependence on applied uniaxial strain of density-density and current-current linear-response electronic correla- tion functions of graphene. Starting from these linear correlation functions, it is possible to obtain analytical results about several measurable quantities of strained graphene, such as the plasmonic dispersion relation, the optical conductivity, as well as the static magnetic and electric susceptibilities. After deriving a general correspondence between strained and unstrained correlation functions, we derived the strain dependence of the low-energy plasmon dispersion relation and of the optical conductivity. Specifically, we found that the prefactor in the pq-dependence of the plasmon frequency develops an anisotropic character, with maximum (minimum) occurring when the wavevector is orthogonal (longitudinal) to the direction of applied strain. We have obtained that uni- axial strain induces an anisotropy on both the plasmonic dispersion relation and the electronic dispersion relation. Hence, we presume that the application of uniaxial strain on graphene could induce a modification of the plasmaronic resonance. We remind that plasmaron is an excitation which arises from the coupling of charge carriers and plas- mons. Indeed, by means of a heuristic argument we found that uniaxial strain applied on graphene should induce a shifting and broadening of the plasmaron resonance en- ergy, proportionally to the strain modulus. Therefore, by suitably applying uniaxial strain, one gains further control on the energy of the plasmaronic excitation, besides the possibility of tuning the relative dielectric constant r. In addition, we have derived a strain-induced anisotropic enhancement of the deviations from the photonic behavior of the theoretically predicted transverse collective excitation, which should facilitate its experimental detection in suitably strained graphene samples. Finally, we have studied the effect of a strain-induced one-dimensional profile on sev- eral ballistic transport properties of graphene. This study may be useful for the realiza- tion of a new class of ballistic devices designed by a suitable tailoring of the electronic structure exploiting not only the electric field effect but also applied strain. In particular, we have studied the cases of a single strain-induced sharp barrier, and of a superstructure of several, periodically repeated, such sharp barriers. In both cases, we have dealt with the analysis of the angular dependence of the tunneling transmission, the conductivity, and the Fano factor. In particular, we have found that a strain-induced superlattice in graphene can accommodate additional resonant quasiparticle states, besides the ones usually found across a single barrier. We thus surmise that a strain-induced superlat- tice in graphene could be used as a filter for well-defined electronic resonant modes. After considering the cases of a single sharp tunneling barrier, and of a superstructure of several, periodically repeated, such sharp barriers, we have specifically studied the more realistic case in which both the modulus of applied uniaxial strain, and possibly an applied gate potential, depend continuously on position.