We have a few pages on this very topic in our book in the misconceptions section of devops! Devops is orthogonal to roles as it's about the collaboration and cooperation between teams and across organizations. Assigning this to a specific team can be less than ideal as it puts the responsibility for collaboration and cooperation on that specific team. Some organizations are trying to use devops as a job title with the idea that a devops engineer can do twice the amount of work than someone else. This can lead to burnout and isolation for the individuals who are tasked with this challenge if that's the underlying reason for the role title. Some organizations understand that the value in devops _is_ the collaboration and rightfully recognize that collaboration isn't a 'free skill', meaning that it costs time and energy to do the relationship work associated with working with other people in your team and across other teams.
So is devops part of a full-stack developer role? Not really. Is it better as a role unto itself? Not really that either, although in some cases it's helpful to have a disrupting greenfield team that shows the way of how to devops before being incorporated back into the larger organizations current structures. Rob Cummings is one individual who has given some great talks on this subject around successful separate devops teams http://www.slideshare.net/robc77/nordstrom-devops-rollercoaster-gartner-datacenter-conf for example.