Hopping conduction and the Coulomb gap; applications to Fe₃O₄, Ti₄O₇ and impurity conduction in Si:P

Abstract

Proceses in which electrons move by hopping from a full to an empty localized site in a degenerate electron gas are considered. One example is impurity conduction in doped and compensated silicon (e.g. Si:P). For the conductivity a variation as A exp (−B/T^1/4) was predicted some years ago if the “interatomic” Coulomb interaction between electrons on different sites is neglected. If it is not neglected, a “Coulomb gap” E_C of order e²/κa or in some cases less is introduced, where a is the distance between centres, as first pointed out in 1970 by Pollak. The activation energy by single-electron hops cannot be less than E_C, so only if kT≫E_C should they lead to the T^1/4 law. Following earlier work (Mott 1976), it is shown that many-electron hops may lead to this law at low temperatures, with the same value of B but a smaller and T-dependent value of A. Somewhat different conclusions presented by Knotek and Pollak and others due to Efros and Shklovskii are discussed. Particular attention is given to the thermopower S, predicted to behave as T^1/2 dlnN(E)/dE for hopping conduction both of single electron or (probably) of many-electron types; a new suggestion is an intermediate range of T where kT≅E_C in which S=(k/e) (E_C/2kT).

These concepts are then applied to Fe₃O₄ and Ti₄O₇, in which the number of carriers is half that of the available sites, and in which the carriers take up some ordered array at low temperatures, with a sharp increase in conductivity and a breakdown of long-range order as T rises through the Verwey temperatures T_V. The relative advantages of treating the system as an array of heavy particles (perhaps polarons) and as a charge-density wave, disappearing at the Verwey temperature, are discussed. The thermopower above T_V is not consistent with the former hypothesis, if it is assumed that the Hubbard ∪ (the mean intraatomic energy of a pair of electrons) is large, and the Heikes formula used; the fault in our view is the neglect of Coulomb interaction in the latter formula. Long-range order is known to be suppressed in Fe₃O₄ by replacing about 2% of the oxygen by fluorine, and in Ti₄O₇ by less than 1% of V₄O₇. The electron gas in these alloys at all temperatures is described as a “Fermi glass”, or “Wigner glass”, namely a degenerate gas of electrons with degeneracy temperature ∼ 1000K, all electrons being in Anderson localized states; the random field is due to the impurities and, in the sense of Hartree-Fock, to the other localized electrons. Conduction is by variable-range hopping; the “Coulomb gap” E_C is of the same order as the activation energy for conduction in the ordered state. The intermediate range of T where S=(k/e) (E_C/2kT) is clearly shown in Fe₃O_4−xF_x. In the pure state (x=0) below T_V it is pointed out that there is no clear distinction between a charge density wave and a crystallization of polarons; a band treatment is applicable in either case. The electrical properties are determined mainly by exces oxygen. Above T_V examination of the thermopower leads to the conclusion that in the pure material the electron gas is a Wigner glass, the electrons being in Anderson states solely induced by the other localized electrons. The entropy which drives the transition is that arising from the linear specific heat K²TN (E_F) of the “Wigner glass,” not that resulting from the disordered arrangement of electrons on the sites. In this point our model is similar to heavy particle models such as that of Ihle and Lorenz, but the calculation of the entropy is different. A problem which appears unsolved is why so small a concentration of impurity can stabilise the disorder.