On Unix, to execute a command $cmd in the background, the one and only allowed standard output redirection syntax is "> /path/to/file &". No other valid standard output redirection syntax will allow a command to be ran in the background.
<?php
// The following will be ran in the background
exec($cmd . " > /path/to/file &");
// All the following will NOT be ran in the background
exec($cmd . " >> /path/to/file &");
exec($cmd . " &> /path/to/file &");
exec($cmd . " &>> /path/to/file &");
exec($cmd . " 2>&1 > /path/to/file &");
exec($cmd . " 2>&1 >> /path/to/file &");
?>