A dematerialisation circuit was an essential part of a TARDIS which enabled it to dematerialise from normal space into the Time Vortex and rematerialise back from it. The Doctor's Type 40 TARDIS used a Mark I model dematerialisation circuit, while the Master's TARDIS used a Mark II. (TV: Terror of the Autons)
The First Doctor removed the dematerialisation circuit from his TARDIS and took it with him when he and Susan went out exploring so Stoyn couldn't steal the ship and return to Gallifrey. (AUDIO: The Beginning)
During his exile on Earth, the Time Lords removed the dematerialisation codes from the Third Doctor's mind and deactivated the dematerialisation circuit of his TARDIS. He coveted the Master's working dematerialisation circuit, and when the opportunity struck he swapped it for his own defective circuit. Although he learned he could not use the Mark II circuit with his TARDIS, the Doctor was cheered to realise the Master was now as trapped on Earth as he. (TV: Terror of the Autons, PROSE: The Eight Doctors)
However, the Doctor lost the Master's dematerialisation circuit following the destruction of the Keller Machine. Shortly afterwards, the Doctor received a telephone call from the Master, who had his dematerialisation circuit back and was now free to leave Earth once more, while the Doctor had to remain in exile. (TV: The Mind of Evil).
Once the Time Lords had rescinded the Doctor's exile as thanks for his defeat of the renegade Time Lord Omega, they also restored his knowledge of the dematerialisation codes and provided him with a new circuit. (TV: The Three Doctors)
The Third Doctor took a part of his dematerialisation circuit with him when he took the Brigadier's godson Albert to a London zoo. The young boy stole it from the Doctor's pocket and used it to transport himself into the animal pens, but when he entered the tiger's cage, it caught his hand in its mouth. The Doctor entered the cage, then instructed Albert how to use the circuit to free himself. (AUDIO: Walls of Confinement)
When the Doctor's first TARDIS escaped Gallifrey and crash-landed on Valderon, its dematerialisation circuit was one of many circuits destroyed in the attempt. The Type 50 stole the Doctor's Type 40's dematerialisation circuit as a temporary replacement, instantly condemning it for being unreliable and almost burnt out. The dematerialisation circuit was later returned when the Type 50 was dying. (AUDIO: Prisoners of Fate)
The dematerialisation circuit of the Master's TARDIS was destroyed as a result of the Master's attempt to escape a Mondasian colony ship which was being pulled into a black hole. Ultimately, the Master would come across his future incarnation, Missy, who provided him with a spare dematerialisation circuit which she had remembered to keep on hand as a result of their encounter. (TV: The Doctor Falls)