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Photophoretic forces could levitate thin 10 centimeter-scale structures in Earth's stratosphere indefinitely. We develop analytical models of the thermal transpiration force on a bilayer sandwich structure in the stratosphere. Lofting is maximized when the layers are separated by an air gap equal to the mean free path (MFP), when about half of the layers' surfaces consist of holes with radii MFP, and when the top layer is solar-transmissive and infrared-emissive while the bottom layer is solar-absorptive and infrared-transmissive. We use the models to design a 10 cm diameter device with sufficient strength to withstand forces that might be encountered in transport, deployment, and flight. The device has a payload of about 285 mg at an altitude of 25 km; enough to support bidirectional radio communication at over 10 Mb/s and limited navigation. Such devices could be useful for atmospheric science or telecommunications on Earth and Mars. Structures a few times larger might have payloads of a few grams.