Surface-electrode (SE) rf traps are a promising approach to manufacturing complex ion-trap networks suitable for large-scale quantum information processing. In this paper we present analytical methods for modeling SE traps in the gapless plane approximation, and apply these methods to two particular classes of SE traps. For the SE ring trap we derive analytical expressions for the trap geometry and strength, and also calculate the depth in the absence of control fields. For translationally symmetric multipole configurations (analogs of the linear Paul trap), we derive analytical expressions for electrode geometry and strength. Further, we provide arbitrarily good approximations of the trap depth in the absence of static fields and identify the requirements for obtaining maximal depth. Lastly, we show that the depth of SE multipoles can be greatly influenced by control fields.
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