634
A. P. Pospelov et al.
an atom, is very much the same as the time of complete interruption of the direct
conduction.
In our work, the control parameter of the kinetics of local dissolution of the
conduction channel is the electric resistance of the point contact. This is due to the
following reasons. It is known that resistance of a conductor with a nonuniform
cross section is largely determined by the resistance of the section with the smallest
diameter. It is more so for a higher degree of nonuniformity. A quantitative measure
of the influence of a monotonous nonuniformity of cross section can be the length
of a conductor with the resistance being half the total resistance. If we approximate
a conductor with a nonuniform cross section by a truncated cone with a height L and
diameters of the lower and upper bases d 1 and d 2 , respectively (d 1 > d 2 ), we can
easily find that the length of the section with a halved resistance adjacent to the basis
with the smaller diameter from this equation:
L 1 / 2 =
L
3
d 1
d 2
− 1
.
(6)
Apparently, if d 1 = d 2 , that is for a uniform conductor L 1/2 = L/2. But if d 2 =
0.1d 1 , then the total conductor resistance is halved over its section with a length
L/29. A growing conicity, which simulates the anode dissolution of the conduction
channel in a GES, results in the point-contact resistance tending to the resistance of
the area of contact between the dendrite tip and the counterelectrode. This means
that with a GES we tend to describe the Yanson point contact within the model of an
aperture rather than the model of a long channel [14].
Earlier, it was shown that point contacts formed in the area of contact of the
growing copper dendrite and the copper counterelectrode are characterized by a
ballistic regime of flight of electrons through the conduction channel with large
electron mean free paths at room temperature and thus are the Yanson point contacts
[3, 4]. To such contacts, we can apply Sharvin’s formula [15, 16]:
R 0 =
16
3π
ρl
d 2 ,
(7)
where R 0 is the point-contact resistance for a zero-voltage bias; ρl is a product of
the electric resistivity ρ and the electron mean free path l, which is a characteristic
constant of each metal.
The product ρl can be estimated using the following reasoning. It is known that
the mean free path of a conduction electron can be found from this relation:
l = v F τ r ,
(8)
where v F is the electron velocity at the Fermi surface; τ r is the relaxation time. The
velocity of electron at the Fermi surface can be estimated from this equation [17]:
A. P. Pospelov et al.
an atom, is very much the same as the time of complete interruption of the direct
conduction.
In our work, the control parameter of the kinetics of local dissolution of the
conduction channel is the electric resistance of the point contact. This is due to the
following reasons. It is known that resistance of a conductor with a nonuniform
cross section is largely determined by the resistance of the section with the smallest
diameter. It is more so for a higher degree of nonuniformity. A quantitative measure
of the influence of a monotonous nonuniformity of cross section can be the length
of a conductor with the resistance being half the total resistance. If we approximate
a conductor with a nonuniform cross section by a truncated cone with a height L and
diameters of the lower and upper bases d 1 and d 2 , respectively (d 1 > d 2 ), we can
easily find that the length of the section with a halved resistance adjacent to the basis
with the smaller diameter from this equation:
L 1 / 2 =
L
3
d 1
d 2
− 1
.
(6)
Apparently, if d 1 = d 2 , that is for a uniform conductor L 1/2 = L/2. But if d 2 =
0.1d 1 , then the total conductor resistance is halved over its section with a length
L/29. A growing conicity, which simulates the anode dissolution of the conduction
channel in a GES, results in the point-contact resistance tending to the resistance of
the area of contact between the dendrite tip and the counterelectrode. This means
that with a GES we tend to describe the Yanson point contact within the model of an
aperture rather than the model of a long channel [14].
Earlier, it was shown that point contacts formed in the area of contact of the
growing copper dendrite and the copper counterelectrode are characterized by a
ballistic regime of flight of electrons through the conduction channel with large
electron mean free paths at room temperature and thus are the Yanson point contacts
[3, 4]. To such contacts, we can apply Sharvin’s formula [15, 16]:
R 0 =
16
3π
ρl
d 2 ,
(7)
where R 0 is the point-contact resistance for a zero-voltage bias; ρl is a product of
the electric resistivity ρ and the electron mean free path l, which is a characteristic
constant of each metal.
The product ρl can be estimated using the following reasoning. It is known that
the mean free path of a conduction electron can be found from this relation:
l = v F τ r ,
(8)
where v F is the electron velocity at the Fermi surface; τ r is the relaxation time. The
velocity of electron at the Fermi surface can be estimated from this equation [17]: