7 research outputs found
Single-layer and bilayer graphene superlattices: collimation, additional Dirac points and Dirac lines
We review the energy spectrum and transport properties of several types of
one- dimensional superlattices (SLs) on single-layer and bilayer graphene. In
single-layer graphene, for certain SL parameters an electron beam incident on a
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
analytic 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
{\delta}-function barriers, the transmission T through, 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, P{\delta}(x) ~ V
(x). For a Kronig-Penney SL with alternating sign of the height of the barriers
the Dirac point becomes a Dirac line for P = {\pi}/2 + n{\pi} 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 as in single-layer
graphene extra "Dirac" points are found. Non-ballistic transport is also
considered.Comment: 26 pages, 17 figure
Extra Dirac points in the energy spectrum for superlattices on single-layer graphene
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.Comment: 7 pages, 8 figures, accepted for publication in Phys. Rev.
Dirac electrons in a Kronig-Penney potential: dispersion relation and transmission periodic in the strength of the barriers
The transmission T and conductance G through one or multiple one-dimensional,
delta-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
delta-function barrier V(x,y) / hbar v_F = P delta(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 = 2 pi n 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) pi.Comment: 5 pages, 6 figure
Kronig-Penney model on bilayer graphene: spectrum and transmission periodic in the strength of the barriers
We show that the transmission through single and double {\delta}-function
potential barriers of strength P in bilayer graphene is periodic in P with
period {\pi}. 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 {\delta}-function barriers (Kronig-Penney model) in bilayer
graphene is periodic in P with period 2{\pi}. For P smaller than a critical
value, the spectrum exhibits two Dirac points while for P larger than this
value an energy gap opens. These results are extended to the case of a
superlattice of {\delta}-function barriers with P alternating in sign between
successive barriers; the corresponding spectrum is periodic in P with period
{\pi}.Comment: 11 pages, 12 figure
Dirac and Klein-Gordon particles in one-dimensional periodic potentials
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.Comment: 9 pages, 12 figure
Dirac Spectrum in Piecewise Constant One-Dimensional Potentials
We study the electronic states of graphene in piecewise constant potentials
using the continuum Dirac equation appropriate at low energies, and a transfer
matrix method. For superlattice potentials, we identify patterns of induced
Dirac points which are present throughout the band structure, and verify for
the special case of a particle-hole symmetric potential their presence at zero
energy. We also consider the cases of a single trench and a p-n junction
embedded in neutral graphene, which are shown to support confined states. An
analysis of conductance across these structures demonstrates that these
confined states create quantum interference effects which evidence their
presence.Comment: 10 pages, 12 figures, additional references adde