Semiclassical models of electrodynamics feature a classical electromagnetic field evolving according to Maxwell’s differential equations, interacting with an atomic system that has quantum mechanical properties (Jaynes & Cummings, 1963; Jaynes, 1973; Mandel, 1976; Grandy, 1991; Marshall & Santos, 1997; Gerry & Knight, 2005). This contrasts with the standard QM model of electrodynamics (QED), which treats the electromagnetic field in terms of discrete photon particles, instead of the classical differential equations.
The basic intuition behind these semiclassical models is that electrons are locked into bound states in the atomic system, and a minimum resonant frequency is required to wedge them out of these states. Any wave that is below this minimum frequency just doesn’t resonate properly with the wave field of the electron, and passes right through.
These bound electrons have discrete, quantized energy levels because they obey wave equations, and essentially these waves must vibrate like drums or guitar strings, with an integral number of wavelengths fitting within the overall space available in an atom. The frequency dependence and quantized nature of the atomic system would hold if it interacted with anything — it would be impossible for the EM field to behave other than in this discretized manner in its interactions with atoms.
Another clue that there may be something fundamentally misplaced in the photon model is the presence of Planck’s constant h, which arises directly from adding mass to the wave equations, where the waves travel at speeds less than the speed of light (i.e., the Klein-Gordon and Dirac equations). Because light (electromagnetic radiation) has no mass, there is no reason for there to be such a constant associated with it, and the classical EM equations have no place for this constant.
Although the photoelectric effect has a fairly compelling semiclassical explanation, there are other phenomena that are harder to explain within this framework. For example, it is possible to have a system that emits a single “photon” of EM energy at a time, and this photon can then be detected later. Advocates of the photon model argue that it is only detected in one specific location, which seems like evidence for a localized little particle, and not a more broadly distributed wave.
However, we must appreciate that the source of the EM field with sufficient energy to excite an atom is typically the spontaneous emission of photons from other atomic systems. This means that these photons were “created” by a kind of mirror image of the very same discrete process involved in detecting the photons. This should impart a temporal, spatial, and energy-level discreteness to the EM radiation in the first place.
There are other statistical properties of photon emission (e.g., anticorrelations; Grangier et al., 1986; and antibunching; Hong et al., 1987) that have been proposed to be inconsistent with the semiclassical approach. Nevertheless, semiclassical accounts of these phenomena have been provided, by leveraging an additional stochastic process associated with the hypothesized zero point field (Marshall & Santos, 2007; Marshall & Santos, 1997), but this work has failed to overturn the status quo belief in photons, perhaps in part because of various important outstanding issues associated with this zero point field construct.
Overall, this semiclassical physical model requires much more complex calculations and conceptual frameworks than the simple ideas and math associated with the photon model, so from the tools vs models perspective, there isn’t much reason for people to adopt this more complex model.