Electrochemical Methods: Fundamentals and Applications of Allen J. Bard presents biosensors in one of its chapters. It is a milestone of electrochemistry, therefore I suppose you can found what you need.
Ronald Pethig and Stewart Smith have written a rather comprehensive introduction into the field: "Introductory Bioelectronics", which was first published in 2013.
In general, such textbooks face the question of which audience they address, i.e. whether the reader comes from the life sciences or from engineering. In case of their book, the content of the first two chapters is well know to life scientists or students in that field, but a good introduction for engineers. The reamining eight chapters give a very good overview of practically all R&D questions in bioelectronics, i.e. electrochemistry, biosensors and their instrumentation, microfluidics, dielectrophoresis and the integration of a lab-on-chip systems. I can very much recommend it.
Recently a new book about label-free biosensing is published: "Label-free biosensing: Advances materials, devices and applications", Springer 2018.
The book addresses state-of-the-art technologies and emerging developments in the field of label-free biosensing and the underlying transducer principles, materials, device fabrication, and applications.