and the
width of the leads will be fixed at 
throughout this section.
Here we shall analyse results for four types of calculation:
conductance of a single quantum wire and ensemble average quantities related
to the conductance, as functions of Fermi energy 
and wire length 
.
Firstly, we discuss the results for the conductance of a single quantum wire
sample, presented in Fig. 2.
Figure 2: 
as a function of energy of a single sample of quantum wire
with rough edges and no island disorder, for wire lengths: 
and 
, and average wire width and leads width .
Reference step functions are the conductances of the perfect wire of the
width 
, length 
, and leads with (full line), and 
(i.e. no difference between leads and wire - broken line). The
inset is the conductance of a quantum wire
sample as a function of the wire length. The top picture shows the
geometry of the lead-wire-lead system used in the calculations.
The edge roughness destroys the conductance quantization steps, firstly near
the band
center for very small disorder (see case in Fig. 2). The
deterioration of the quantization spreads towards the band edge both as the
disorder and as the length of the wire is increased. In the mesoscopic regime,
when 
, the conductance
curve shows sample specific fluctuations as a function of energy and length.
However, the
amplitude of these fluctuations is of order 
, independent of energy
(Fig. 2,
L=10, L=30) or length (see inset of Fig. 2). This is a quantum
interference effect similar to the universal conductance fluctuations observed
in the mesoscopic regime (Lee and Stone 1985)) of other systems.
The particular value for the conductance is determined by the electron
wavelength (i.e. electron energy), the length of the quantum wire and the actual
realization of the disorder in a
sample. The conditions for the destructive or constructive interference of an
electron wavefront are, generally, very sensitive to all of these parameters.
The scale of the sensitivity to energy, however, differs with the length of the
wire (which is evident from Fig. 2 for 
and 
) or with
the level of disorder. This observation supports the idea that the
fluctuations do not arise from classical scattering from the rough boundary
but from the phase modulation of the electron wavefunction due to multiple
elastic
scattering in the wire (Takagaki and Ferry 1992). We estimate that the typical
spacing between peaks and valleys in the conductance as a function of energy
depends on the wire length as: 
. This is a weaker dependence on
the
wire length than in the case of the universal conductance fluctuations in the
metallic regime, where 
(Lee and Stone 1985)). It should also be noted
that the conductance falls faster near the band center than elsewhere (this
will be clear from
the results for the average conductance). When the system is in the strong
localization regime, the conductance is reduced to a set of peaks
of different amplitudes, with maximum value of 
(see Fig. 2
). The mechanism of electron transport is resonant tunnelling through
the quantum wire (Bryant 1991). When the energy of an electron coincides with
an eigenenergy of the wire the electron can be transmitted through the wire via
this localized state. The height of the peak of depends on the overlap
between the wavefunctions of this spatially
localized state inside the wire and the propagating states in the leads. As the
length of the wire is further increased, the number of peaks and their
amplitudes reduce and their positions change. The highest such peaks are
probably due to tunnelling through multiple resonant states, so-called
`necklace' states (Pendry 1987).
Figure 3: Average conductance as a function of energy, for quantum wires with
rough boundary and no island disorder, for different wire lengths:
, 
, 
and 
. Number of samples
taken for calculating average values are: 
for wire lengths 
respectively.
Next we discuss the results for the ensemble average conductance and for the
conductance fluctuations. Fig. 3 shows results for 
and 
. Conductance quantization disappears
very quickly, although the presence of a short plateau for the first
subband can be observed for the shorter wires. The boundary roughness has less
impact for longer wavelengths (smaller energies) and hence the average
conductance tends to decrease with increasing energy. As the length of the
wire increases there is a rapid decrease of near the band center. In the
strong localization regime, a broad peak
emerges near the band edge (see Fig. 3 for ), which
corresponds to the peak in the localization length(Nikolic and MacKinnon 1993). In this
regime, the average conductance of quasi-1D systems falls off exponentially
with the length (johnstone and Kunz 1983):
Figure 4: Conductance fluctuations which correspond to the case in
Fig. 4 for the wire lengths: , and
(three bold lines). Two thin lines are the localization length and the
density of states (both rescaled: 
and
) for quantum wires with boundary
roughness.
The conductance fluctuations for the case of the quantum wire with boundary roughness only, calculated for the examples from Fig. 3 by using definition Eq. (16), are presented in Fig. 4. Three characteristic regimes are shown:
(e.g. for most of the energies and
for the level of disorder assumed in our samples);
(e.g.
);
(e.g. ).
in Fig. 4 show
interesting similarities with the density of states for the same type of
quantum wires. There is a peak near the band edge which corresponds to the peak
in the DOS, which is the last remaining feature of the inverse square root
singularities from the DOS of clean quasi-1D systems (see Nikolic and MacKinnon 1993). This behaviour can be explained in the following way.
For short wires (i.e. quasi-ballistic transport) the important length is the
elastic mean free path 
. If increases (as a function of energy) in the
quasi-ballistic regime, then the conductance fluctuations decrease - as the
scattering in the wire is reduced, and vice versa. Since is roughly
inversely proportional to the DOS (when the DOS is increased the scattering
rate increases and therefore decreases), then the conductance fluctuations
might be expected to mirror the DOS. As is increased with respect to
(and still 
) then the fluctuations will increase towards their
maximal value, which is reached in the mesoscopic regime. The quantum wires of
length in Fig. 4 show a relatively wide region of energy in
which the conductance fluctuations are independent of energy, with a value
which is close to the universal value for quasi-1D systems (
(Lee and Stone 1985)).
For long wires () we have the strong localization regime and 
follows the curve for the localization length 
(see
Fig. 4), i.e. the average conductance, since these are related
(Eq. 17). That 
and are proportional can be
understood by using the picture of `open' and `shut' channels, or `maximal
fluctuations', in a quantum wire in the strong localization regime
(Pendry et al. 1992).
This terminology is associated with the distribution function 
for the
eigenvalues 
(
) of 
, where 
is the transmission matrix: it has a peak at 
, and a tail which
extends to 
(Pendry et al. 1992, MacKinnon 1992). Each eigenvector of 
defines a conducting `micro-channel' with the
corresponding conductance in units of 
(Pendry et al. 1992). As
the size of the system is increased, the peak at strengthens at the
expense of the tail, but the shape of the tail remains the same.
For long lengths, therefore, most of the micro-channels will have conductance
of order 0 since the bulk of the distribution is around 0. Only a small
fraction of the channels, with values for of order 1 (roughly
), will contribute to the conductance. Such a distribution of
, implies that the fluctuations tend to the maximum possible value
consistent with their mean. However, the moments of the conductance,
, for 2D and 3D cases do not show
the same size dependence as the moments of 
, within the metallic regime. Universal conductance
fluctuations are affected by the correlations between the s rather than
by the distribution of the s themselves. UCFs (i.e. with a universal value
ca. ) are restricted to disordered systems in the metallic regime where
perturbation theory is applicable. When localization effects are important,
however, the correlations between the micro-channels change and a different
sort of fluctuations is observed (Pendry et al. 1992). The statistics of a single
micro-channel were found to be crucial in the strong localization regime.
The average conductance of the disordered quasi-1D system shows an
asymptotically
exponential decrease with the length (johnstone and Kunz 1983). This single parameter
dependence suggests that the conductance statistics of quasi-1D samples, with
a length much longer than the localization length, is dominated by a single
channel. Hence a similar size dependence for the conductance moments should be
expected as for , since
the distribution of s is important, rather than the correlation between
them. This picture of the conductance statistics in the strong localization
regime shows that the conductance fluctuations are proportional to the average
conductance.
Figure 5: Average conductance as a function of wire length for real wire
without islands for energies 
and 
- top figure.
The corresponding conductance fluctuations are
shown in the lower figure as well as fluctuations for energies 
and 
. Number of samples is 
.
In Fig. 5 results are given for the conductance as a function
of wire length, for a fixed energy. The results for the average conductance
are presented for two values of the energy and two more examples of conductance
fluctuations are added. The decrease of conductance is more marked in the
quasi-ballistic (near ballistic) regime (short wires), where the difference
between and 
is negligible.
The difference between the two averages increases as the disorder or length of
the wire is increased (Figures 3 and 5). This
shows that the average is dominated by a small number of
samples with conductance of order . The divergence of the two curves
indicates that the conductance distribution is transforming to the form which
has a peak for small conductances and a long tail towards larger conductances,
i.e. it is transforming to a log-normal distribution (johnstone and Kunz 1983).
For wires of length the decrease of the average conductance
with the length of the wire is mainly determined by
the localization length. The value for the localization length can be obtained
from Fig. 5, where 
is fitted with a
straight line and 
is determined from its slope,
for wires of length 
.
The results for the conductance fluctuations are also shown in
Fig. 5. The fluctuations first increase for very short wire
lengths (
). This is the nearly ballistic regime where the fluctuations
grow as the disorder (or wire length) increases. On the other side, for long
wires, i.e. the strong localization regime (), the fluctuations
decrease as the wire
becomes longer, due to the overall decrease of the conductance. In this regime
the fluctuations between wires of the same length depend only on the
localization length (see Fig. 5). The fluctuations take
maximal values between these two regimes.
Fluctuations for wires with many subbands ( and in
Fig. 5) have a sharp peak for short lengths and then start to
decrease well before the wire has become longer than . The
fluctuations in the wires with a few subbands ( and
Fig. 5) even appear to be independent of length for several
ranges of in the intermediate region. This could be called the universal
region (Ando and Tamura 1992, Tamura and Ando 1991), although the constant value of the conductance
fluctuations is not equal to the universal value
obtained for quasi-1D systems in the perturbation theory (Lee and Stone 1985).
