Hi Tania,
any organic matter you have handy will do. There will always be a debate on which wood (the essence, the age, the size, etc.) is best but really use what you have close to you.
Understanding the principle behind the huggel bed might help to make those decisions.
As always, observe, experiment and keep learning
Edit : you might want to have a look at Geoff Lawton's technique in desert environment. Huggelmound might not be the best solution for your climate considering that if it is too raised it might dry off quickly and delay the rotting/digestion process. It really depends on your situation (if you have strong winds, shade or no shade, etc..). As an immediate reaction to your “high desert“ location, I would say rather use that organic matter you have as heavy mulch and try to develop shade, water collection and build up life within your soil
Here's a video to help, but you can have a look at a few others you may find on the subject :
https://www.youtube.com/watch?v=keQUqRg2qZ0&feature=youtu.be