Now we have a look at the charging time of the capacitance which isRCΣ. If this charging time is
much faster than a quantum fluctuation
RCΣ<~ 2 CΣ
e^2(204)
the system has time to move an electron on the island. The capacitance cancels out and we can
calculate the critical resistance:
R <2 ~
e^2
≈ 8 kΩ (205)If the resistance is smaller than 8 kΩthe Coulomb blockade can be suppressed by quantum fluctuation
otherwise it can not be suppressed. But all the calculations were not very precise and we used the
uncertainty relation so one generally says that if the resistance between the crystals in a material is
below the resistance quantum
h
e^2
≈ 25. 5 kΩ, (206)quantum fluctuations can suppress the Coulomb blockade. If it is above, you get an insulator at low
temperatures.
The critical resistance can also be related to the dimensions of the crystals:
Rcrit≈
h
e^2=
ρl
wt, (207)
withwas width,las length,tas thickness andρas the resistivity of the crystal. If we assume that
the crystals are squared which means thatw=land that they have about the thickness of a regular
crystalt≈ 0. 2 nm, the resistivity then has to beρ= 500μΩcm. In general you can say that materials
with resistivities> 1 mΩcmtend to be a Mott insulator.
Another way to check if you have a Mott insulator is to look at the formula of the conductivity that
just depends on the mean free path and the electron density:
σ=
ne^2 l
~(3π^2 n)^1 /^3. (208)
By measuring the conductivity and knowing the electron density you can calculate the mean free path
and if it is smaller than the size of an atom the free electron model is wrong and electron-electron
interactions need to be considered.
Disorder plays also a role in the Mott transition because it favours the insulating state. As you
can see in fig. 93 a linear chain of islands with uniform tunnel barriers below the resistance quantum
characterizes a metal. But if there are random tunnel barriers and some of them have resistances above
the resistance quantum, the whole section between them form an island. Although these islands are
metallic in between those barriers, the Coulomb blockade effect at the barrier forces the material to
go into the insulating state.
The Mott transition is also relevant for high-temperature superconductors. They have a conductivity
which is 100 times worse than the conductivity of a metal like copper or aluminium but as you cool them
down they become superconductors. But if you change the charge doping (oxygen concentration) they