160
A. M. Korol et al.
At the same time, it is known that the characteristics of structures based on Dirac
materials are significantly influenced by the difference in the values of the Fermi
velocity in different parts of the structure [10–20]. A lot of various structures with
non-equal Fermi velocities in different regions of the given structure were studied in
last years. They comprise the graphene-based single- and double-barrier structures,
various types of superlattices including the quasiperiodic ones, superconducting
junctions, structures based on the topological insulators, etc. [10–20].
Motivated by the above considerations, in this paper, we study the ballistic transmission of quasielectrons through a rectangular potential barrier in the α-T 3 model
and show that it depends strongly on the relation between the parameters α and β,
where β is equal to the ratio of the Fermi velocities in the barrier and out-of-barrier
areas. By changing the values of the parameters α and β, one can flexibly control the
transmission properties of the structure under consideration within a wide range.
2 Model and Formulae
The Dirac-like equation for the considered model can be represented as follows
[1–8]:
⎛
⎝
0
f cos ϕ
0
f
∗ cos ϕ
0
f sin ϕ
0
f
∗ sin ϕ
0
⎞
⎠ ψ + U I 0 ψ = Eψ,
(1)
where U is the external potential which corresponds to the rectangular barrier and is
equal to
U (x) =
⎧
⎨
⎩
0, x ≤ 0
U, 0 < x < D
0, D << x
(2)
in different regions of the given structure; I 0 is the identity matrix.
The quantity f in (1) is equal to
f = v F
k x − ik y
.
(3)
For our purpose, it suffices to take into consideration only one K valley in the
hexagonal Brillouin zone. The quantities v F and k x acquire different values in the
barrier and out-of-barrier regions. The parameter ϕ is introduced for convenience:
ϕ = arctgα, α is a parameter showing the coupling strength of the central atom with
the atom at the hexagon vertices; for the dice lattice α = 1, for graphene α = 0.
The eigenfunctions in the (1) can be represented as follows:
A. M. Korol et al.
At the same time, it is known that the characteristics of structures based on Dirac
materials are significantly influenced by the difference in the values of the Fermi
velocity in different parts of the structure [10–20]. A lot of various structures with
non-equal Fermi velocities in different regions of the given structure were studied in
last years. They comprise the graphene-based single- and double-barrier structures,
various types of superlattices including the quasiperiodic ones, superconducting
junctions, structures based on the topological insulators, etc. [10–20].
Motivated by the above considerations, in this paper, we study the ballistic transmission of quasielectrons through a rectangular potential barrier in the α-T 3 model
and show that it depends strongly on the relation between the parameters α and β,
where β is equal to the ratio of the Fermi velocities in the barrier and out-of-barrier
areas. By changing the values of the parameters α and β, one can flexibly control the
transmission properties of the structure under consideration within a wide range.
2 Model and Formulae
The Dirac-like equation for the considered model can be represented as follows
[1–8]:
⎛
⎝
0
f cos ϕ
0
f
∗ cos ϕ
0
f sin ϕ
0
f
∗ sin ϕ
0
⎞
⎠ ψ + U I 0 ψ = Eψ,
(1)
where U is the external potential which corresponds to the rectangular barrier and is
equal to
U (x) =
⎧
⎨
⎩
0, x ≤ 0
U, 0 < x < D
0, D << x
(2)
in different regions of the given structure; I 0 is the identity matrix.
The quantity f in (1) is equal to
f = v F
k x − ik y
.
(3)
For our purpose, it suffices to take into consideration only one K valley in the
hexagonal Brillouin zone. The quantities v F and k x acquire different values in the
barrier and out-of-barrier regions. The parameter ϕ is introduced for convenience:
ϕ = arctgα, α is a parameter showing the coupling strength of the central atom with
the atom at the hexagon vertices; for the dice lattice α = 1, for graphene α = 0.
The eigenfunctions in the (1) can be represented as follows: