Yes, there probably is. The Harry Potter fandom is vast and diverse, and often crossovers with other popular franchises occur in fanfiction. The idea of Harry having a sonic screwdriver, which is from Doctor Who, could lead to some really interesting and creative fan - written stories.
You can start by looking on major fanfiction platforms like Archive of Our Own (AO3) or FanFiction.net. Just use the search terms 'Harry Potter' and'sonic screwdriver' and see what comes up.
There's a chance. Fanfiction.net is a huge platform. But finding that specific combination might not be straightforward. You'll need to use the search function effectively and browse through relevant categories to increase your odds.
It's possible. FanFiction.net is a large platform with diverse content. However, finding specific fanfics like this might take some searching and filtering through the available stories.
One possible plot could be that during the regeneration process of the 9th Doctor, something goes awry with the Time Lord's normal regeneration cycle. Maybe there's interference from an outside force, like a powerful alien entity that wants to disrupt the natural order. This interference causes the regeneration to skip a few incarnations and directly jump to the 11th Doctor. The new 11th Doctor then has to deal with the after - effects of this abnormal regeneration. He might have some strange memories or abilities that are a mix of what the 9th Doctor was supposed to pass on and some new, unexpected elements.
They could also meet in a time - locked situation. The 12th Doctor is trying to solve a mystery from his past, and the 13th Doctor stumbles upon the same situation in her present. At first, they don't understand how they are both involved, but as they start to piece things together, they discover that it's all part of a much larger, more sinister plot. Along the way, they might meet some old friends or foes from their respective tenures as the Doctor. And in the end, they work together to outwit the mastermind behind the plot, all while having some really interesting conversations about their experiences as the Doctor.
In some fanfictions, the 10th and 11th Doctor are brought together by a time anomaly. They are forced to cooperate to fix it, and along the way, they encounter old friends and enemies. The contrast between the 10th's more traditional style and the 11th's more zany approach creates interesting dynamics. They might also have to deal with the consequences of their previous actions in different timelines, which adds an extra layer of complexity to the story.
In the fanfiction, the 11th doctor, with his eccentric and energetic personality, could initially be a bit confused by the 13th doctor's more modern approach. But then they could start to combine their wits. For example, the 11th doctor's knowledge of ancient alien races could be paired with the 13th doctor's understanding of current galactic politics. They could work together to save a planet that is on the brink of destruction due to a complex web of time - related issues. The encounter could also lead to some humorous moments, like when they argue about who has the better sonic screwdriver or who has had the more exciting adventures.