  Radiation detection in applications ranging from high energy physics to
medical imaging rely on solid state detectors, often hybrid pixel detectors
with (1) reverse biased p-n junction pixel sensors and (2) readout ASICs,
attached by flip-chip-bonding. Transient signals characteristics are important
in, e.g., matching ASIC and sensor design, modeling and optimizing detector
parameters and describing timing and charge sharing properties. Currently
analytical forms of transient signals are available for only a few limited
cases (e.g., drift or diffusion) or for the steady state (which is not relevant
for high energy radiation detection). Tools are available for (relatively slow)
numerical evaluation of the transient charge transport. We present here the
first analytical solutions of partial differential equations describing
drift-diffusion-recombination charge transport in planar p-n junction sensors
in a variety of conditions: (1) undepleted, (2) fully depleted, (3) taking into
account the gradual velocity saturation, and (4) overdepleted. We deduce the
Green's functions which can be applied to any detection problem through simple
convolution with the initial conditions. We compare the analytical solutions
with Monte Carlo simulations and industry standard simulations (Synopsys
Sentaurus), demonstrating good agreement. Using the analytical equations
enables fast modeling of the influence of various detector parameters on
tracking, imaging and timing performance, describing performance and enabling
optimizations for different applications. Finally, we illustrate this model
with applications in 3D+T (x,y,z,time) photon tracking and 4D+T
(x,y,theta,phi,time) relativistic charged particle tracking.
