Commercially available solid state pH electrodes are mainly built around Ion Selective Field Effect Transistors (ISFET).
The basic principle of the ISFET working is the control of the current flowing between two semiconductor elements (drain and souce) by electrostatic field, generated by the protonated oxide gate. Protonation of the gate is in a way identical to the process taking place in glass pH electrode, just the methodology used to measure protonation degree is different. Instead of measuring potential difference on two sides of the glass, we measure the current flowing through the transistor. The lower the pH, the more protonated and charged gate is which changes its electric field - changing in turn current flowing through the transistor. This current is a signal that can be measured to check the pH value.
ISFET electrodes can be very small when compared to the bulky glass bubble of the standard glass electrode. They are also much more sturdy, so they can be easily used in places where fragile glass electrodes will not survive. However, ISFET electrode can't be used with standard pH meters (unless it is connected through special interface) and the pH measurements are generally less precise when compared to glass electrode.