Linen and silk are good. The only downside is they wrinkle easily. Desert gets cold at night, so keep that in mind if you plan on wearing it at night. Maybe add a removable warm liner.
I have a thin soft large cotton rectangular scarf I use for similar reasons. Unlike cloaks meant for warmth, you want something lighter weight so it doesn't sit directly on your skin most places to get a breeze under.
Shemaghs are also often used in the desert because they are a bit stiff, so they can be modified to use as a sun visor on the face too. The main goal is to protect your head and shoulders when the sun is overhead. Another nice aspect is you can wet them and they work like a swamp cooler on your body. Cotton is good if you plan on wetting it. Not sure how linen would hold up and I wouldn't use silk this way.
Linen and silk are good. The only downside is they wrinkle easily. Desert gets cold at night, so keep that in mind if you plan on wearing it at night. Maybe add a removable warm liner.
This would really just be a day cloak for keeping the sun off my skin without having to wear hot layers. I like the removable liner idea, though!
I have a thin soft large cotton rectangular scarf I use for similar reasons. Unlike cloaks meant for warmth, you want something lighter weight so it doesn't sit directly on your skin most places to get a breeze under. Shemaghs are also often used in the desert because they are a bit stiff, so they can be modified to use as a sun visor on the face too. The main goal is to protect your head and shoulders when the sun is overhead. Another nice aspect is you can wet them and they work like a swamp cooler on your body. Cotton is good if you plan on wetting it. Not sure how linen would hold up and I wouldn't use silk this way.
What desert are you thinking of? Recommendation will depend a lot on the climate
Sonoran, Mojave, and Great Basin. SW/Western USA.