What kind of surgery was operated on Root's ear in episode / (S03E17)

Root tells Finch:

Decima severed my link to the Machine... temporarily.

But she didn't like that, so she wants you to patch 'these' into 'this'.

• 'This', the device on the left side, is (the internal part of) a cochlear implant.

• 'These', the three things on the right side, will presumably give her an umjammable connection with the Machine.

  According to Amy Acker, who plays Root, she is

  directly implanted with the Machine

  she also calls it

  the heart of the Machine

Finch seems to comply, because when Root takes the cochlear implant back a few minutes later, we can see that 'these' have been patched into 'this', as requested.

Root leaves the earbud, which she has been using until Decima's countermeasure rendered it useless, behind.

I haven't seen the episode, but according to the Wiki:

The device surgically implanted behind Root's ear is a cochlear implant, which is typically used for post-lingual adults with sensorineural (nerve) deafness. Root's hearing loss, on the other hand, is due to the loss of one of the conductive bones in her ear, a conductive hearing loss. Regardless, Root would not regain hearing in her right ear with the implant, but rather be able to hear limited sound.

I think this explains what her initial ear problems were. In this episode, she has a device surgically fitted to her ear to allow her to remain in constant communication with The Machine. As this is a fictional surgery, I'm not aware of its name!

Finally, from a different part of the Wiki:

...Later, Root and Cyrus are again attacked by Decima, and despite even Reese's assistance, Cyrus is taken. Following the kidnapping, Root comes to the Library and gives Finch a small chip that will aid her in preventing Samaritan from being activated. Finch wonders why she is no longer worried about Cyrus, and Root finally admits that she was responsible for ordering the hit that got Cyrus's friends killed in 2009, and her guilt is stifling her desire to help Cyrus now. Root decides that Finch can't help her and takes the chip she gave him earlier...

So it appears she gave Finch a chip to help prevent the activation of Samaritan, then took it back from him.