The body of research on (III,Mn)V diluted magnetic semiconductors (DMSs) initiated during the 1990s has concentrated on three major fronts: (i) the microscopic origins and fundamental physics of the ferromagnetism that occurs in these systems, (ii) the materials science of growth and defects, and (iii) the development of spintronic devices with new functionalities. This article reviews the current status of the field, concentrating on the first two, more mature research directions. From the fundamental point of view, (Ga,Mn)As and several other (III,Mn)V DMSs are now regarded as textbook examples of a rare class of robust ferromagnets with dilute magnetic moments coupled by delocalized charge carriers. Both local moments and itinerant holes are provided by Mn, which makes the systems particularly favorable for realizing this unusual ordered state. Advances in growth and postgrowth-treatment techniques have played a central role in the field, often pushing the limits of dilute Mn-moment densities and the uniformity and purity of materials far beyond those allowed by equilibrium thermodynamics. In (III,Mn)V compounds, material quality and magnetic properties are intimately connected. This review focuses on the theoretical understanding of the origins of ferromagnetism and basic structural, magnetic, magnetotransport, and magneto-optical characteristics of simple (III,Mn)V epilayers, with the main emphasis on (Ga,Mn)As. Conclusions are arrived at based on an extensive literature covering results of complementary ab initio and effective Hamiltonian computational techniques, and on comparisons between theory and experiment. The applicability of ferromagnetic semiconductors in microelectronic technologies requires increasing Curie temperatures from the current record of $173\mathrm{K}$ in (Ga,Mn)As epilayers to above room temperature. The issue of whether or not this is a realistic expectation for (III,Mn)V DMSs is a central question in the field and motivates many of the analyses presented in this review.

We consider the Landauer formula, relating conductances to transmission matrices, for a two-dimensional system in a magnetic field. We argue that the magnetoresistance, R, and the Hall resistance, ${\mathrm{R}}_{\mathrm{H}}$, satisfy the sum rule (R+${\mathit{R}}_{\mathrm{H}}$ ${)}^{\mathrm{-}1}$=(${\mathit{e}}^2$/h)Tr(${\mathit{t}}^{\mathrm{°}}$t) where t is the transmission matrix. For zero field our expressions reduce to the usual multichannel Landauer formulas. In the absence of dissipation, R approaches zero, t approaches a unit matrix, and quantized values are obtained for the Hall resistance.

We present a theoretical survey of ferromagnetic transition temperatures in cubic (III,Mn)V semiconductors based on a model with $S=5/2\mathrm{}$ local moments exchange coupled to itinerant holes in the host semiconductor valence band. Starting from the simplest mean-field theory of this model, we estimate the $T_c$ enhancement due to exchange and correlation in the itinerant-hole system and the $T_c$ suppression due to collective fluctuations of the ordered moments. We show that high critical temperatures in these ferromagnetic semiconductors require both the large magnetic susceptibility contribution from the valence-band heavy holes and the large spin stiffness resulting from a complex valence-band structure that includes the more dispersive light holes. Our calculations demonstrate that the model of carrier-induced ferromagnetism of these systems is consistent with high critical temperatures observed experimentally in (III,Mn)V semiconductors.

We demonstrate voltage control of the magnetic anisotropy of a (Ga,Mn)As device bonded to a piezoelectric transducer. The application of a uniaxial strain leads to a large reorientation of the magnetic easy axis, which is detected by anisotropic magnetoresistance measurements. Calculations based on the mean-field kinetic-exchange model of (Ga,Mn)As provide a microscopic understanding of the measured effect. The reported smooth voltage control of the uniaxial in-plane anisotropy, electrically induced magnetization switching, and detection of unconventional crystalline components of the anisotropic magnetoresistance illustrate the generic utility of our multiferroic system in providing device functionalities and in the research of micromagnetic and magnetotransport phenomena in diluted magnetic semiconductors.

We present a theoretical survey of magnetocrystalline anisotropies in (Ga,Mn)As epilayers and compare the calculations to available experimental data. Our model is based on an envelope function description of the valence-band holes and a spin representation for their kinetic-exchange interaction with localized electrons on ${\text{Mn}}^{2+}$ ions, treated in the mean-field approximation. For epilayers with growth-induced lattice-matching strains we study in-plane to out-of-plane easy-axis reorientations as a function of Mn local-moment concentration, hole concentration, and temperature. Next we focus on the competition of in-plane cubic and uniaxial anisotropies. We add an in-plane shear strain to the effective Hamiltonian in order to capture measured data in bare unpatterned epilayers, and we provide microscopic justification for this approach. The model is then extended by an in-plane uniaxial strain and used to directly describe experiments with strains controlled by post-growth lithography or attaching a piezostressor. The calculated easy-axis directions and anisotropy fields are in semiquantitative agreement with experiment in a wide parameter range.

We report on a theoretical study of dc transport coefficients in (Ga,Mn)As diluted magnetic semiconductor ferromagnets that accounts for quasiparticle scattering from ionized Mn$^{2+}$ acceptors with a local moment $S=5/2$ and from non-magnetic compensating defects. In metallic samples Boltzmann transport theory with Golden rule scattering rates accounts for the principle trends of the measured difference between resistances for magnetizations parallel and perpendicular to the current. We predict that the sign and magnitude of the anisotropic magnetoresistance can be changed by strain engineering or by altering chemical composition.