10.1 Fundamentals of Electric Conductivity; Diffusive versus Ballistic Conductivity 231
Under the influence of the voltage V the electrons are accelerated, they obtain
the energy E = eV. Applying Planck’s equation E
hv
=
e
λ
to express the energy
of the electrons and inserting into Eq. (10.2), one finally obtains
G
Nev
Vl
Ne v
El
Ne
h l
Ne
h n
=
=
=
=
e
e
2
2
2 1
λ
.
(10.3)
The term n
l
= λ
stands for the electron wave mode number. Each electron wave
mode may have two modes (spin up and spin down) leading to N = 2n, therefore, one finally obtains for the conductance of a short thin wire with one mode:
G
e
h
=
2
2
.
(10.4a)
Assuming m active modes in a wire, the conductance is
G m
e
h
mG
=
=
2
2
0 .
(10.4b)
Equation (10.4b) is fundamental; it says that the electric conductivity of a small
thin wire increases stepwise with the increment G
e
h
0
2
5
2
7 72 10
=
=
×
−
.
S, the
“conductivity quantum”. In Eq. (19.4b), the length and the cross section of the
wire no longer appear; therefore, in the ballistic case, the electric conductance
is independent of material and geometry. The inverse value R G
0
0
1 26
=
= kΩ is
called the “resistance quantum”, also known as von Klitzing constant.
Analyzing Eq. (10.2), one learns that the conductance decreases with increasing voltage; however, because of the existence of the resistance quantum, this
happens in steps. Except for graphene, the quantized phenomena described
above are observed at low temperatures, only. Otherwise, thermal energy is
larger than or in the range of the energy difference between two neighboring
electron wave modes; hence the steps are smeared out, as the different modes
may be activated thermally and not by the electric field. In such a case, the I–V
diagram looks like one following Ohm’s law. A rigorous treatment of quantized
electric conductance is given in a review paper of Datta [1].
Ballistic conductivity is independent of the material and the geometry of the
wire. (This statement has to be taken cum grano salis (with a pinch of salt), as the
free path length of the electrons depends on the material.) However, ballistic
conductivity depends on the applied voltage, as the number of modes excited by
the electrical field depends on the applied voltage. Each one of the excited modes
contributes to the electrical conductivity in an equal step, the conductivity quantum.
At low temperatures, the electrical conductivity increases in steps with the applied
voltage. This is depicted in Figure 10.3.
Under the influence of the voltage V the electrons are accelerated, they obtain
the energy E = eV. Applying Planck’s equation E
hv
=
e
λ
to express the energy
of the electrons and inserting into Eq. (10.2), one finally obtains
G
Nev
Vl
Ne v
El
Ne
h l
Ne
h n
=
=
=
=
e
e
2
2
2 1
λ
.
(10.3)
The term n
l
= λ
stands for the electron wave mode number. Each electron wave
mode may have two modes (spin up and spin down) leading to N = 2n, therefore, one finally obtains for the conductance of a short thin wire with one mode:
G
e
h
=
2
2
.
(10.4a)
Assuming m active modes in a wire, the conductance is
G m
e
h
mG
=
=
2
2
0 .
(10.4b)
Equation (10.4b) is fundamental; it says that the electric conductivity of a small
thin wire increases stepwise with the increment G
e
h
0
2
5
2
7 72 10
=
=
×
−
.
S, the
“conductivity quantum”. In Eq. (19.4b), the length and the cross section of the
wire no longer appear; therefore, in the ballistic case, the electric conductance
is independent of material and geometry. The inverse value R G
0
0
1 26
=
= kΩ is
called the “resistance quantum”, also known as von Klitzing constant.
Analyzing Eq. (10.2), one learns that the conductance decreases with increasing voltage; however, because of the existence of the resistance quantum, this
happens in steps. Except for graphene, the quantized phenomena described
above are observed at low temperatures, only. Otherwise, thermal energy is
larger than or in the range of the energy difference between two neighboring
electron wave modes; hence the steps are smeared out, as the different modes
may be activated thermally and not by the electric field. In such a case, the I–V
diagram looks like one following Ohm’s law. A rigorous treatment of quantized
electric conductance is given in a review paper of Datta [1].
Ballistic conductivity is independent of the material and the geometry of the
wire. (This statement has to be taken cum grano salis (with a pinch of salt), as the
free path length of the electrons depends on the material.) However, ballistic
conductivity depends on the applied voltage, as the number of modes excited by
the electrical field depends on the applied voltage. Each one of the excited modes
contributes to the electrical conductivity in an equal step, the conductivity quantum.
At low temperatures, the electrical conductivity increases in steps with the applied
voltage. This is depicted in Figure 10.3.