By using highly sensitive millikelvin superconducting quantum interference device magnetometry, the magnitude of the Curie temperature as a function of the Mn concentration x is determined for thoroughly characterized Ga1$-${}xMnxN. The interpretation of the results in the frame of tight-binding theory and of Monte Carlo simulations allows us to assign the spin interaction to ferromagnetic superexchange and to point out the limited accuracy of state-of-the-art ab initio methods in predicting the magnetic characteristics of dilute magnetic insulators.

Molecular beam epitaxy has been employed to obtain Ga${}_{1$-${}x}$Mn${}_{x}$N films with $x$ up to 10% and Curie temperatures ${T}_{\text{C}}$ up to 13 K. The magnitudes of ${T}_{\text{C}}$ and their dependence on $x$, ${T}_{\text{C}}(x)$\propto${}{x}^{m}$, where $m=2.2\ifmmode\pm\else\textpm\fi{}0.2$, are quantitatively described by a tight-binding model of superexchange interactions and Monte Carlo simulations of ${T}_{\text{C}}$. The critical behavior of this dilute magnetic insulator shows strong deviations from the magnetically clean case ($x=1$), in particular, (i) an apparent breakdown of the Harris criterion, (ii) a nonmonotonic crossover in the values of the susceptibility critical exponent ${$\gamma${}}_{\text{eff}}$ between the high temperature and critical regimes, and (iii) a smearing of the critical region, which can be explained either by the Griffiths effects or by macroscopic inhomogeneities in the spin distribution with a variance $$\Delta${}x=(0.2\ifmmode\pm\else\textpm\fi{}0.1)$%.

We employ two tight-binding (TB) approaches to systematically study the electronic structure and hole or electron transfer in B-DNA monomer polymers and dimer polymers made up of $N$ monomers (base pairs): (I) at the base-pair level, using the onsite energies of base pairs and the hopping integrals between successive base pairs, i.e., a wire model and (II) at the single-base level, using the onsite energies of the bases and the hopping integrals between neighboring bases, i.e., an extended ladder model since we also include diagonal hoppings. We solve a system of $M$ (matrix dimension) coupled equations [(I) $M=N$, (II) $M=2N]$ for the time-independent problem, and a system of $M$ coupled first order differential equations for the time-dependent problem. We perform a comparative study of stationary and time-dependent aspects of the two TB variants, using realistic sets of parameters. The studied properties include HOMO and LUMO eigenspectra, occupation probabilities, density of states and HOMO-LUMO gaps as well as mean over time probabilities to find the carrier at each site [(I) base pair or (II) base], Fourier spectra, which reflect the frequency content of charge transfer, and pure mean transfer rates from a certain site to another. The two TB approaches give coherent, complementary aspects of electronic properties and charge transfer in B-DNA monomer polymers and dimer polymers.