Molecular electronics offers new possibilities for continually miniaturizing the electronic devices beyond the limits of standard silicon-based technologies. There have been significant experimental and theoretical efforts to build thiophene molecular junctions and study their quantum transport properties. However, in most of these studies Au is used as lead material. It is well known that the fabrication of the traditional molecular device is now hindered by technological difficulties such as the oxidation of metallic contacts, and the interface instability between the organic molecule and the inorganic metallic electrodes. In this paper, we use the graphene electrodes to construct a series of thiophene-based devices. The graphene electrodes proposed in this paper are able to avoid such problems. Moreover, the stability of graphene electrodes at room temperature paves the way to studying the electron transport through a single molecule under the ambient conditions. Firstly, we design a series of molecular rectifying devices based on thiophene dimer and its derivatives, in which the hydrogen atom on the thiophene monomer is substituted with a representative electron-donating group (NH2) and electron-withdrawing group (NO2). Secondly, we investigate systematically the electronic transport properties through these molecular junctions by performing the first principles calculations based on density functional theory and nonequilibrium Green's function. The calculated results show that these thiophene molecular devices substituted with NH2 and NO2 possess the rectifying behavior and negative differential resistance properties. Furthermore, we also find that the position of substituent group NH2 or NO2 has a major influence on the electronic transport properties. In order to explore the physical mechanism behind these transport properties, the electronic structures of the molecules, the transmission spectrum, and the molecular projected eigenstates are analyzed. The results reveal that the position of NH2 can adjust the intensity of the negative differential resistance. When the NH2 group is close to the molecular end, the negative differential resistance behavior in this molecular device is more prominent than in other molecules. In addition, the position of NO2 can change the direction of the rectification. When the NO2 group is close to the molecular end, the current in negative bias is larger than in positive bias, resulting in a negative rectification. In contrast, when the NO2 group is close to the molecular centre, a positive rectification occurs. Our results can provide a worthy complement to thiophene molecular experiment, and also has a guiding significance for designing other molecular electronic devices.