I have the desktop version of adobe after effects CS6. Is there a way that I can upgrade to after effects CC for free?

Ned is correct upgrades are never free except for the rare occasion when the manufacturer offers it on a new purchase of something that's about to release a new version. e.g., Toshiba offers Windows 8 upgrades on some of their new computers with pre-installed Windows 7. (except the truth is pc's with Win 7 installed are priced at least $125 more than the same exact pc with Win 8 pre-installed)

I've never seen Adobe offer free programs or apps - especially upgrades. If you buy one of their older versions of Creative Suite you can find really good prices though. Like CS5 premium versions are going for as low as $300 U.S. dollars.

I will also add that you should be 100% sure that CC After Effects has something you want and KNOW how to use. As Ned pointed out things change and CC may not be what you think it is. Adobe might offer you a free trial period of 2 weeks if you purchase something else....

You always have to pay for upgrades, unless you happened to purchase a version when it is just about to be replaced by a new one.  CC is a subscription version so you end up paying for it on a regular basis, whereas your CS6 purchase was a one-time investment.

Before getting CC make sure your machine meets the After Effects system requirments.  They differ between CS6 and CC.

After Effects - http://helpx.adobe.com/after-effects/system-requirements.html