1.2 Electronic transport in doped semiconductors
1.2.4 Mott’s law and Variable-Range-Hopping conduction
The basic concept of variable-range-hopping is, as the name reveals, the electronic transport by hopping over variable distances caused by the minimization of energy when undergoing a transition between two remote sites i and j. At sufficiently low temperatures, the macroscopic conductivity is determined by the states whose ener-gies are close to the Fermi-level. Mott [65] was the first one to notice that at low temperatures, the activation energy decreased with decreasing temperature, from where he derived the celebrated Mott’s law
ρ(T) =ρ0exp [(T0/T)]1/4 (1.61) with
T0 = β
kBN(μ)ξ (1.62)
where N(μ) is the density of states at the Fermi-energy,kB the Boltzmann constant, ξ the localization radius of the relevant states, and β a numerical coefficient. Mott’s original derivation could not give an exact value of the coefficient β, since he gave only a qualitative description of the emergence of variable-range-hopping transport in an impurity band with constant density of states at the Fermi-level. Fig. 1.13 shows schematically the impurity band structure which was the starting point to describe variable-range-hopping. Recalling the dominant exponential terms in the expression (1.42) for the hopping resistance Rij it is natural to suppose that, at very low temperatures, only resistances having very small values of ij will contribute to conduction. This implies that the relevant states (i,j) have to lie energetically in a narrow region near the Fermi-level, described by the energy interval 0, as shown in Fig. 1.13. At the same time, the relevant states are far away from each other, therefore, their spatial distribution can be considered uncorrelated. Furthermore,
because of the narrow width of the band0, the density of states in the region [μ±0] can be regarded as constant, so that the total number of relevant states is given by
N(0) = 2N(μ)0 (1.63)
In order to derive the resistance corresponding to all the sites i for which |i−μ| ≤ 0, the average energy barrier and distance between hops in the narrow band can be estimated as ij=0 and rij=[N(0)]−1/3, respectively. Using these average expressions, the macroscopic resistivity can be written as
ρ=ρ0exp For large values of 0, the second term dominates in the exponential part of (1.64), which implies a decreasing ρ(0) with decreasing 0. As the width of the narrow band 0 becomes smaller, the states become very rare and their decreasing overlap plays the dominant role, resulting in an increase ofρ(0). The competition between spatial overlap and energy leads to a resistance minimum when both terms in the exponent of (1.64) become equal. The temperature-dependent energy0 which yields the minimum resistance takes then the form
0(T) = (kT)3/4
[N(μ)ξ3]1/4 (1.65)
This expression, which corresponds to the
”optimal band width“ of the relevant states, contains the physical essence of variable-range-hopping transport. In particular, it describes the scenario of minimum resistivity, analogous to the obtention of the per-colation threshold in a Miller-Abrahams random resistor network. By substituting 0(T) into the resistivity expression (1.64) one arrives at Mott’s law(1.61). The T−1/4 dependence of the hopping resistivity is characteristic of variable-range-hopping with a constant density of states at the Fermi-level, scenario which was given the name of Mott variable-range-hopping (Mott-VRH), in spirit of Mott’s rather intuitive deriva-tion.
The quantity 0(T) corresponds to the average activation energy between hops of the relevant sites, and the derivative d(lnρ)/d(kT)−1 yields the activation energy at a given temperature. From (1.65) it is evident that the activation energy decreases monotonically asT3/4with decreasing temperature. Unlike the high-temperature sce-nario of constant activation energy, where the hopping length is temperature indepen-dent and of the order of the mean separation between impurities (nearest-neighbor-hopping), the average hopping length in the VRH-regime does vary with temperature.
From Eq.(1.64) we infer
R¯hop ∝[N(μ)0(T)]−1/3 ∝ ξ(T0/T)1/4 (1.66)
µ
³
³
N ( )
µ --
³
µ +
³
0 0
Figure 1.13: Section of an impurity band near the Fermi-level, with a constant density of states.
At low temperatures, the relevant states for hopping transport are the ones within a energy interval0 around the Fermi-levelμ, which corresponds to the
”optimal“ band in Mott’s derivation.
The average hopping length thus increases as T−1/4 as the temperature is lowered, and may reach values much greater than the average distance between impurities.
Despite the qualitative character of Mott’s original derivation, the description of the physics of variable-range-hopping is very consistent. Still, for a quantitative analysis (e.g. comparison with experimental data) the knowledge of the coefficient β in the numerator of the characteristic temperature T0 is necessary. A more rigorous derivation of Mott’s law has been later done by Ambegaokar et al. [55] using the percolation method, where the coefficient β could be determined. For the sake of completeness, the solution of the macroscopic hopping conductivity by the percolation method at the limit of low temperatures will be summarized. As usual, the starting point for the percolation Ansatz is the bonding criterion
2rij ξ + ij
kT ≤x (1.67)
where both terms should be taken as relevant for the search of the percolation thresh-old xc. The largest possible values of ij and rij are defined as:
max =kT x; rmax = ξx
2 (1.68)
such that ij ≤ max, which applies for both |i−μ| and |j−μ|. Ambegaokar et al. [55] formulated the percolation problem using dimensionless variables in the form
di =ri/rmax (1.69)
Δi = (i−μ)/max (1.70) so that the bonding criterion adopts the form
dij + Δij ≤1 (1.71) Considering again the impurity band structure depicted in Fig. 1.13 with a constant density of states in the region |Δi,j|<1, the total number of states will be given by
N(x) = 2N(μ)maxrmax3 = 1
4N(μ)kBT ξ3x4 (1.74) so that determining the percolation threshold xc requires to find the critical concen-tration Nc at which percolation first occurs, taking into account randomly located sites distributed in the energy interval [Δi]< 1. The percolation threshold will be equal to and the macroscopic hopping resistivity can be written as
ρ(T) =ρ0exc =ρ0exp
4nc N(μ)kBT ξ3
1/4
(1.76) This expression corresponds to Mott’s law, with β=4nc. It is not surprising that Mott’s qualitative derivation and the solution offered by the percolation method are of the same form, since both approaches consider the scenario of sufficiently low tem-peratures and the same impurity band structure depicted in Fig. 1.13. For an accurate determination of the coefficient β, the above dimensionless percolation problem was numerically solved by the Monte Carlo method [66]. An extrapolation to an infinite array gives following values
nc = 5.3±0.3; β = 21.2±1.2 (1.77) The validity of Mott’s law can be extended to lower dimensions. The quantityN(μ) can be understood as the n-dimensional density of states, and [N(0)]1/3 can be re-placed by [N(0)]1/n. Repeating Mott’s qualitative derivation described above, one obtains a law of the formρ=ρ0exp (T0/T)1/p with p=(d+ 1)−1.
On the other hand, Mott’s law loses his validity when the density of states at the Fermi-level is no longer constant. This situation might arise when sizable correlation effects are present, leading to the formation of an energy gap at the Fermi-level. The influence of a vanishing density of states at EF on the hopping conductivity will be discussed in the next section.