y83 ff3 fs4 fc0 sc0 ls0 wsb">These values are close to the reported 1.1 × 106/m ob-
tained by fitting the results for a variety of multilayer
graphene films synthesized by CVD [15].
Regarding the linear temperature dependence of the
conductivity, theoretically this dependence has been pre-
dicted for a monolayer in the ballistic regime [16] and for
bilayer graphene for diffusive transport mediated by
(a)
100 150 200 250 300 350
0.10
0.15
0.20
0.25
C o n d u c t a n c e (m S)
Temperature (K)
W=203 m
W=92 m
Stra ig h t lin e fit
(b)
(b)
Figure 3. (a) Schematic of the geometry of the multilayer
graphene stripe along with electrical contacts. (b) Measured
electrical conductance as a function of temperature for the
two kinds of films. The corresponding sheet resistance at
room temperature is: ~707 /sq for W = 92 μm and ~812
/sq for W = 203 μm.
disorder [17], in both cases at high temperatures. But it
should be noted that even in the early theoretical work on
graphite, a linear dependence of the electrical conductiveity
along the graphene layers was also predicted [18]. Alth-
ough bilayer graphene was grown on the copper wire in
our case, in the collapsed situation and owing to ts lateral
dimension, the film can be considered as a four layer
graphene. But some care should be taken since the lateral
edges along the dimension L (see schematic in Figure
3(a)) may contribute to the electrical conduction because
ideally the edges are curved and closed borders.
3.3. A Cross-Stripe Junction Device
On the other hand, taking advantage of the shape of the
obtained multilayer graphene films, we built devices in
the typical cross-stripe junction used in tunneling spec-
troscopy [19]. Firstly, two copper wires (~5 mm long, 63
μm diameter) covered with multilayer graphene were
crossed and fixed on a glass slide substrate using an epoxy
glue in their extremities, then copper was etched and the
sample washed with deionized water. Current (I) against
voltage (V) characterization was made by injecting cur-
rent into two adjacent arms and measuring voltage across
the opposite arms of the device [19]. In Figure 4(a)
conductance near zero bias as a function of temperature
of the junction is shown, and in Figure 4(b) I(V) curves
at three different temperatures in a log-log scale are pre-
sented. Due to the geometrical configuration of the device
one should expect that the primary contribution to the
electrical transport across the junction is perpendicular to
the graphene layers, at least in the layers in close contact
between the two stripes. An estimation of the electrical
conductivity of the junction can be done as follows. The
area corresponds to the intersection of the two stripes (92
μm × 92 μm in this case), for the length the graphite in-
terlayer distance is taken as a first approximation, the
conductance at room temperature is (see Figure 4(b)) 15
× 10–3/; all these finally yield to a value of ~6 × 10–4/m
for the conductivity. If this value is taken as the conduc-
tivity perpendicular to the layers, and a value of ~1 × 106/
m (see Section 3.2) for the conductivity along the gra-
phene layers; then an anisotropy factor of the parallel to
the perpendicular conductivity of ~1.7 × 109 is obtained.
Clearly, this value does not represent a physical charac-
teristic of the graphite structure because a value of ~3 ×
103 for the anisotropy factor for crystalline graphite has
been reported [20]. In other words: the conductivity of
the junction between the two stripes is at least of the order
of 105 less than the conductivity of the contact between
two graphene layers in the ideal structure of graphite.
Due to this fact, and that the conductance near zero bias
decreases when temperature decreases (Figure 4(a)) and
also because the differential conductance increases as the
Copyright © 2012 SciRes. WJNSE
D. MENDOZA
56
(a)
(b)
Figure 4. Electrical characteristics of the cross-stripe junc-
tion device. (a) Conductance against temperature measured
at a fixed current of 10 μA; (b) Current as a function of voltage
for positive and negative polarities at different tempera-
tures: green triangles (89 K), red circles (296 K) and blue
squares (380 K).
voltage bias increases (Figure 4(b)), it is very likely that
the cross-stripe device is a tunnel junction [19].
As a last observation, it should be noted that superlin-
ear behavior on the current dependence I~Vα, specifically
with α = 3/2, has been predicted for graphene within the
framework of Schwinger´s pair production and Klein
tunneling [21-24]. Under some specific conditions, a
linear behavior for small voltages is also found [23,24].
As a guide for the eye, in Figure 4(b) the linear and su-
perlinear ~V3/2 behaviors are plotted. Note that the linear
behavior is reasonably well reproduced for low voltages,
being a small vertical shift the difference for the three
temperatures, and a ~V3/2 tendency appears to be a good
option for higher voltages. We believe that our device
has the appropriate geometry to observe tunneling phe-
nomena, possibly Klein tunneling, because particles tun-
nel from one electrode to the other through a barrier, that
in this case could be vacuum. Probably, tunneling takes
place between the two adjacent parallel graphene layers
of the multilayer graphene stripes that form the junction.
As the voltage bias across the junction is changed, there
is a relative shift of the Fermi level on the two sides of
the barrier, and therefore scanning a range of energies
around the Fermi energy [19] of graphene. In any case, it
would be interesting to built hybrid structures, using
multilayer graphene stripe as one electrode and super-
conducting or magnetic films as the counter electrode in
the cross-stripe geometry. Experiments in this direction
are currently in process in our laboratory.
4. Conclusion
Bilayer graphene was grown on copper wires by means
of CVD method with methane as the carbon source. Af-
ter etching the copper wire, the bilayer tube collapses
forming stripes of four-layer graphene. A linear depend-
ence of the electrical conductance as a function of tem-
perature is found for this kind of films. Using the mate-
rial obtained by this method, a cross-stripe junction is
built, its electrical conductance behavior as a function of
voltage bias and temperature indicates that the device is a
kind of tunnel junction. It is proposed that this kind of
device might be appropriate to observe Klein tunneling.
5. Acknowledgements
I thank Carlos Flores IIM-UNAM for the transmission
electron microscopy observations.
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