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shape of the wells varying in wide ranges. It can be seen
(Figure 7) that narrowing of the interelectrode vacuum
space leads to a significant change in the CVC parame-
ters—in the height of the “effective” barrier φ and in
“field amplification coefficient” β. The calculated pa-
rameters are:
1: for 0.40 mm—φ = 0.4 eV, β = 20;
2: for 0.08 mm—φ =0.8 eV, β = 68;
3: for 0.04 mm—φ =1.6 eV, β = 153.
This effect can be related to a magnetic field. Mag-
netic properties of the metal anode can influence Landau
quantization in a magnetic field. But as the pseudo-field
is not classical, it is not absolutely clear how quantization
in a pseudo-magnetic field depends on the magnetic
properties of the environment. Perhaps the reason is that
the electrons’ motion on cyclotron orbits creates a clas-
sical magnetic field which is responsible for the effect of
the environment. The cyclotron orbits of larger radius
create more extended in normal direction magnetic fields.
Thus, while metal anode is getting closer, the orbits of
larger diameter are getting blocked firstly, as we can see
in Figure 7. At such a high sensitivity of superinjection
to the effect of the environment it is interesting to note
the independence of the “effective” barrier height on
dielectric type, contact material (Al, Cu and TiN are used)
and carbyne film thickness. The following dielectrics
were tested: SiO2, ZnO and organic dielectrics with a
bandgap of 2 - 3 eV. In all cases when the area of current
was equal (the same contact structure), almost identical
“effective” barriers were observed regardless of the car-
byne film thickness: 0.32 eV and 0.37 eV for holes and
for electrons accordingly. The calculation was made here
approximately on the basis of some evaluation of the
“effective” barrier’s cross-section in Richardson-Dush-
man model.
Extreme system sensitivity to a very minor variation of
some parameters with a strong resistance to the others is
the feature of quantum systems like topological insula-
tors. Summarizing the results we can construct the ener-
getic diagram of the relativistic Landau levels and of the
superinjection process as consequent activations on them
(see Figure 8).
The cyclotron orbits of electrons correspond to the
Landau levels n. Carriers are sequentially activated on
them and go to the orbit of bigger radius. The first barrier
(which can be seen at CVC as “an effective barrier”)
appears to be the highest. Dirac point ED in carbyne is
approximately 0.02 eV above Fermi level EF; the “effec-
tive” barrier height for holes injection is less than for
electrons for 0.04 eV. The essence of the traversal elec-
tric field effect is in orbits distortion and in change of
orbital velocity of carriers. Under the action of an exter-
nal “pulling of electrons” field the orbital velocity at the
carriers’ entry point to carbyne decreases in proportion to
Figure 8. The scheme of the superinjection process. (a) The
energetic diagram of superinjection as consequent thermal
activations on the relativistic Landau levels; (b) A top view
on injecting structure—metal contact surrounded by car-
byne. Carriers are sequentially activated on the cyclotron
orbits of bigger radius and finally injected downwards to a
dielectric layer. The bottom image shows the effect of an
external transversal electric field—circular orbits extend
into elliptic, and the orbital speed becomes variable.
the field intensity, correspondingly the effective barrier
reduces (solving exact Schrödinger equation for 2d-
Diracs particles in magnetic field is beyond the frames of
this article). A “retarding” field acts similarly: an orbital
velocity at the entry point increases and the effective
barrier occurs to be higher. For charge carriers to be
passed through the high barriers many consecutive acti-
vations are required, so final cyclotron orbit could have
quite big diameter. It may not even fit on the area of
carbyne—this may explain the observed hole injection
blocking while diameter of the metal well narrowing [5].
Meanwhile electron injection remains constant: less num-
ber of activations is required and the radius of the final
orbit becomes smaller. It is easy to note that classical
charge carriers cannot be sequentially activated: the pro-
bability of return downwards motion on an energy ladder
plenty times higher than the probability of upwards mo-
tion. It is only possible when carriers move without scat-
tering. While their upwards motion on an energy ladder,
charge carriers absorb phonons energy on the area of
cyclotron orbits, but the inverse process is impossible
because of the effect of “topological protection”.
The main oriented carbyne’s puzzle—the vertical sta-
bility of the chains—can be explained by the existence of
a giant pseudo-magnetic field as well. Conducting chains
align along the field lines due to the interaction of the
electrons moving along the chain with the pseudo-mag-
netic field (this can occur via the secondary classic mag-
netic field as we proposed above or we should assume
the pseudo-magnetic field extent in normal direction as
far as chains long). A slight deviation in their motion
along the field is forbidden by quantization of the trans-
verse motion energy because of the energy of the first
Landau level (~0.3 eV) several times more than the en-
ergy of thermal motion at 300 K. Thus, the oriented car-
byne looks like chains, which are “frozen” in the pseudo-
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