  Using a one-dimensional tight-binding Anderson model, we study a disordered
nanowire in the presence of an external gate which can be used for depleting
its carrier density (field effect transistor device configuration). In this
first paper, we consider the low temperature coherent regime where the electron
transmission through the nanowire remains elastic. In the limit where the
nanowire length exceeds the electron localization length, we derive three
analytical expressions for the typical value of the thermopower as a function
of the gate potential, in the cases where the electron transport takes place
(i) inside the impurity band of the nanowire, (ii) around its band edges and
eventually (iii) outside its band. We obtain a very large enhancement of the
typical thermopower at the band edges, while the sample to sample fluctuations
around the typical value exhibit a sharp crossover from a Lorentzian
distribution inside the impurity band towards a Gaussian distribution as the
band edges are approached.
