We investigate the emergence of extra Dirac points in the electronic structure of a periodically spaced barrier system, i.e., a superlattice, on single-layer graphene, using a Dirac-type Hamiltonian. Using square barriers allows us to find analytic expressions for the occurrence and location of these new Dirac points in k space and for the renormalization of the electron velocity near them in the low-energy range. In the general case of unequal barrier and well widths the new Dirac points move away from the Fermi level and for given heights of the potential barriers there is a minimum and maximum barrier width outside of which the new Dirac points disappear. The effect of these extra Dirac points on the density of states and on the conductivity is investigated.

Two-dimensional (2D) cell cultures growing on plastic do not recapitulate the three dimensional (3D) architecture and complexity of human tumors. More representative models are required for drug discovery and validation. Here, 2D culture and 3D mono- and stromal co-culture models of increasing complexity have been established and cross-comparisons made using three standard cell carcinoma lines: MCF7, LNCaP, NCI-H1437. Fluorescence-based growth curves, 3D image analysis, immunohistochemistry and treatment responses showed that end points differed according to cell type, stromal co-culture and culture format. The adaptable methodologies described here should guide the choice of appropriate simple and complex in vitro models.

We evaluate the dispersion relation for massless fermions, described by the Dirac equation, and for zero-spin bosons, described by the Klein-Gordon equation, moving in two dimensions and in the presence of a one-dimensional periodic potential. For massless fermions the dispersion relation shows a zero gap for carriers with zero momentum in the direction parallel to the barriers in agreement with the well-known "Klein paradox". Numerical results for the energy spectrum and the density of states are presented. Those for fermions are appropriate to graphene in which carriers behave relativistically with the "light speed" replaced by the Fermi velocity. In addition, we evaluate the transmission through a finite number of barriers for fermions and zero-spin bosons and relate it with that through a superlattice.

We evaluate the electronic transmission and conductance in bilayer graphene through a finite number of potential barriers. Further, we evaluate the dispersion relation in a bilayer graphene superlattice with a periodic potential applied to both layers. As model we use the massless Dirac-Weyl equation in the continuum model. For zero bias the dispersion relation shows a finite gap for carriers with zero momentum in the direction parallel to the barriers. This is in contrast to single-layer graphene where no such gap was found. A gap also appears for a finite bias. Numerical results for the energy spectrum, conductance, and the density of states are presented and contrasted with those pertaining to single-layer graphene.

We review the energy spectrum and transport properties of several types of onedimensional superlattices (SLs) on single-layer and bilayer graphene. In single-layer graphene, for certain SL parameters an electron beam incident on an SL is highly collimated. On the other hand, there are extra Dirac points generated for other SL parameters. Using rectangular barriers allows us to find analytical expressions for the location of new Dirac points in the spectrum and for the renormalization of the electron velocities. The influence of these extra Dirac points on the conductivity is investigated. In the limit of d-function barriers, the transmission T through and conductance G of a finite number of barriers as well as the energy spectra of SLs are periodic functions of the dimensionless strength P of the barriers, Pd(x) = V (x)/hv F , with v F the Fermi velocity. For a Kronig-Penney SL with alternating sign of the height of the barriers, the Dirac point becomes a Dirac line for P = p/2 + np with n an integer. In bilayer graphene, with an appropriate bias applied to the barriers and wells, we show that several new types of SLs are produced and two of them are similar to type I and type II semiconductor SLs. Similar to single-layer graphene SLs, extra 'Dirac' points are found in bilayer graphene SLs. Non-ballistic transport is also considered.

The transmission T and conductance G through one or multiple one-dimensional, δ-function barriers of two-dimensional fermions with a linear energy spectrum are studied. T and G are periodic functions of the strength P of the δ-function barrier V (x, y)/ vF = P δ(x). The dispersion relation of a Kronig-Penney (KP) model of a superlattice is also a periodic function of P and causes collimation of an incident electron beam for P = 2πn and n integer. For a KP superlattice with alternating sign of the height of the barriers the Dirac point becomes a Dirac line for P = (n+1/2)π.PACS numbers: 71.10. Pm, 81.05.Uw The study of particle motion in periodic potentials is at the heart of condensed matter physics and it is usually assumed that the energy spectrum is parabolic. One of the earliest examples is the well-known, one-dimensional (1D) Kronig-Penney (KP) model, 1 that consists of an infinite succession of very thin (W → 0) and very high (V 0 → ∞) barriers, referred to as δ-function barriers, but such that their product P ∝ W V 0 remains constant. This results in minibands in the electron spectrum.One may wonder though how such results are modified if the energy is linear in wave vector. Such a spectrum occurs for relativistic electrons with energy E = cp >> E 0 = m 0 c 2 , where c is the speed of light and m 0 the bare electron rest mass. Even without neglecting E 0 a strict 1D Dirac KP model was considered for relativistic quarks 2 . It is also known that electrons can transmit perfectly, upon normal incidence, through arbitrarily wide and high barriers, referred to as Klein paradox or Klein tunneling 3 . With the discovery of graphene 4 , a one-atom thick layer of carbon atoms, another system became available in which particles (electrons) moving in two dimensions, have a linear spectrum, E = v F k, with k = (k x , k y ) the wave vector. Importantly, carriers in graphene behave as chiral, massless fermions described by Dirac's equation without the mass term, and move with the Fermi velocity v F ≈ c/300. There is a wealth of exceptional properties of graphene, see e.g., Refs 5 .Because the carriers in graphene move in two dimensions, tunneling through barriers is inherently twodimensional (2D) and depends on the direction of the incident electron beam even in the absence of a magnetic field. Many authors, including ourselves, have studied this tunneling, through single, multiple barriers, and superlattices 6,7 . Surprisingly, tunneling through δfunction barriers has received very little attention 8 and we are not aware of any Dirac KP model for a superlattice in graphene. An interesting development was the application of periodic potentials to graphene that turned it into a self-collimating material despite the rather unusually high potentials used 9 .Motivated by all these results and the absence of a systematic treatment of KP barriers or superlattices, we study in this work the transmission through such structures as well as the dispersion relation of a KP superlattice. Although the model may appear a bit unrea...

We show that the transmission through single and double δ-function potential barriers of strength P = V W b / vF in bilayer graphene is periodic in P with period π. For a certain range of P values we find states that are bound to the potential barrier and that run along the potential barrier. Similar periodic behaviour is found for the conductance. The spectrum of a periodic succession of δ-function barriers (Kronig-Penney model) in bilayer graphene is periodic in P with period 2π. For P smaller than a critical value Pc, the spectrum exhibits two Dirac points while for P larger than Pc an energy gap opens. These results are extended to the case of a superlattice of δ-function barriers with P alternating in sign between successive barriers; the corresponding spectrum is periodic in P with period π.

Two-dimensional (2D) culture of cancer cells in vitro does not recapitulate the three-dimensional (3D) architecture, heterogeneity and complexity of human tumors. More representative models are required that better reflect key aspects of tumor biology. These are essential studies of cancer biology and immunology as well as for target validation and drug discovery. The Innovative Medicines Initiative (IMI) consortium PREDECT (www.predect.eu) characterized in vitro models of three solid tumor types with the goal to capture elements of tumor complexity and heterogeneity. 2D culture and 3D mono- and stromal co-cultures of increasing complexity, and precision-cut tumor slice models were established. Robust protocols for the generation of these platforms are described. Tissue microarrays were prepared from all the models, permitting immunohistochemical analysis of individual cells, capturing heterogeneity. 3D cultures were also characterized using image analysis. Detailed step-by-step protocols, exemplary datasets from the 2D, 3D, and slice models, and refined analytical methods were established and are presented.