Effective medium theory for electron waves in a gate-defined quantum dot array in graphene

We develop the effective medium theory (EMT) to calculate the effective ``refractive index'' of quantum dot array for electron waves in graphene. It can be used to design electron metamaterials which will control the flow of graphene electrons by achieving any desirable effective ``refractive index.'' Here the coherent-potential approximation is adopted to derive the analytic formulation of the effective ``refractive index,'' which has been successfully used in the EMT of electromagnetic metamaterials. The EMT is valid beyond the long-wavelength limit. The region of validity is identified by the comparison of the energy band structures obtained by the EMT and the rigorous multiple scattering theory. We also provide the expressions for effective ``refractive indices'' in the frame of the long-wavelength theory and the Maxwell-Garnett theory so that a simple estimation can be done in the long-wavelength limit. We perform the simulation of the electron density distribution around a metamaterial with circular cross-section composed of smaller quantum dots. The simulation results show excellent agreement with the equal-size effective circular potential, verifying the effectiveness of our theory to design electron metamaterials. Finally, we show that even though these quantum dots that make up the metamaterial are disordered, the electron density distribution is almost the same as that around the effective one again and the effective medium theory is still applicable.