Second-order phase transitions in semiconductors are characterized by continuous changes in physical properties without an associated latent heat, distinguishing them from first-order transitions where abrupt changes and latent heat are present. These transitions involve a gradual evolution of order parameters, such as polarization in ferroelectrics or magnetization in magnetic materials, as the system passes through the critical temperature. The study of these transitions provides deep insights into the underlying physics of semiconductors and their functional properties.

One of the defining features of second-order phase transitions is the behavior of thermodynamic quantities near the critical point. Properties such as magnetic susceptibility, specific heat, and conductivity exhibit power-law dependencies on temperature, described by critical exponents. For example, the magnetic susceptibility χ in a ferromagnetic material diverges as χ ∝ |T - Tc|^(-γ), where Tc is the critical temperature and γ is the critical exponent. Similarly, the order parameter η, such as spontaneous polarization in ferroelectrics, follows η ∝ (Tc - T)^β below Tc. These exponents are universal, meaning they depend only on the symmetry and dimensionality of the system rather than microscopic details.

Landau theory provides a phenomenological framework for understanding second-order phase transitions. It is based on the expansion of the free energy in terms of the order parameter, assuming analyticity and symmetry. For a ferroelectric transition, the Landau free energy F can be written as F = F0 + a(T - Tc)η² + bη⁴ + ..., where a and b are positive coefficients. Above Tc, the equilibrium order parameter is zero, while below Tc, it grows continuously as η = √[a(Tc - T)/2b]. The simplicity of Landau theory makes it a powerful tool for predicting the qualitative behavior of systems near critical points, though it neglects fluctuations that become important very close to Tc.

A classic example of a second-order phase transition in semiconductors is the ferroelectric transition in barium titanate (BaTiO3). At temperatures above approximately 393 K, BaTiO3 adopts a cubic perovskite structure with no spontaneous polarization. As the temperature drops below Tc, the crystal undergoes a transition to a tetragonal phase, accompanied by the displacement of Ti ions relative to the oxygen octahedra, resulting in a spontaneous polarization. This transition is continuous, with the polarization increasing smoothly as the temperature decreases further. The dielectric constant of BaTiO3 also exhibits a pronounced peak at Tc, following a Curie-Weiss law ε ∝ |T - Tc|^(-1) in the paraelectric phase.

In contrast to second-order transitions, first-order phase transitions involve discontinuous changes in the order parameter and the release or absorption of latent heat. For instance, the semiconductor-to-metal transition in vanadium dioxide (VO2) is first-order, with an abrupt change in conductivity and a coexistence region where both phases are present. The distinction between first and second-order transitions is crucial for applications, as second-order transitions typically offer more gradual property changes, which can be advantageous in tuning device performance.

Critical phenomena in second-order transitions are also observed in transport properties. The electrical conductivity of semiconductors near a phase transition can show anomalous behavior due to fluctuations in the order parameter. For example, in magnetic semiconductors, the resistivity may exhibit a peak near Tc due to scattering from spin fluctuations. Similarly, in ferroelectric semiconductors, the carrier mobility can be influenced by polarization fluctuations, leading to non-monotonic temperature dependence.

The study of second-order phase transitions extends beyond ferroelectrics to include magnetic and structural transitions in semiconductors. For instance, the antiferromagnetic transition in chromium-doped semiconductors like (Ga,Mn)As involves a continuous change in magnetic ordering, with critical exponents reflecting the universality class of the system. Structural transitions, such as those in shape-memory alloys or charge-density-wave systems, also exhibit second-order characteristics when the lattice distortion develops continuously.

Experimental techniques for probing second-order transitions include measurements of specific heat, which typically shows a lambda-shaped anomaly at Tc, and diffraction methods, which reveal the gradual development of order. Spectroscopic techniques like Raman scattering can track soft modes, where certain phonon frequencies approach zero as Tc is approached, signaling the instability of the high-temperature phase.

Theoretical advances beyond Landau theory, such as renormalization group theory, have provided a deeper understanding of critical behavior, accounting for fluctuations that modify mean-field predictions. These approaches have been validated by high-precision experiments on model systems, confirming the universality of critical exponents.

In summary, second-order phase transitions in semiconductors are marked by continuous changes in physical properties, governed by critical exponents and describable within Landau theory. Examples like the ferroelectric transition in BaTiO3 illustrate the rich physics of these transitions, with implications for materials design and device engineering. The absence of latent heat and hysteresis makes second-order transitions particularly suitable for applications requiring smooth property tuning, distinguishing them from their first-order counterparts.