Nope. Most single crystal growth methods involve rather strong thermal gradient. Bridgeman and float zone are two examples.I doubt whether it is possible to establish a temperature difference between two ends of a single crystal without destroying it. Even if a bulk of material is a single crystal, it might be impossible to establish a temperature difference across it without destroying its original lattice.
Phonon is only associated with its (single) crystal lattice. It is the quantum of interaction with the lattice AS A WHOLE.