An adhesive membrane comprises (a) a conductor; (b) a composite material comprising conductive particles at least partially embedded in an electrically insulating layer disposed on the conductor; and (c) a pressure sensitive adhesive layer disposed on the composite material, the conductive particles being capable of electrically connecting the conductor to a second conductor under application of sufficient pressure therebetween, the conductive particles having no relative orientation and being disposed so that substantially all electrical connections made between the conductor and a second conductor will be in the z direction, and the combined thickness of the electrically insulating layer and the pressure sensitive adhesive layer being greater than the size of the largest conductive particle when the largest conductive particle is measured in the z direction.