Quantization Of Mesoscopic Quartz Piezoelectric Crystal Equivalent Circuit

Along with the dramatic achievement of mesoscopic physics and nanoelectronics, it has been a strong and definite trend in the miniaturization of integrated circuits, the quantum mechanical effect of circuits and devices themselves must be taken into account, the basic theory and method which used to research classical circuit will not suitable any more. So, a quantum theory to mesoscopic circuits is demanded. Based on the fundamental fact that the electric charge takes discrete value, a finite difference Schr?dinger equation is deduced, and mathematic solution of finite-difference Schr?dinger equation are complicated, it is now the main difficulty in the application of the theory, the solution always calculated with WKB method in the condition of a small parameter.Perturbative method based on power series is another method to solve Schr?dinger equation. The energy and wave function of mesoscopic RLC circuit are calculated with this method. Perturbative method could be extended to solve four order difference equation, so, the detail energy spectrum and wave functions of the diode included mesoscopic circuit are obtained.In present paper, by using the advanced quantum theory of mesoscopic circuits,a mesoscopic quartz piezoelectric crystal equivalent circuit is quantized in the framework of advanced quantum theory of mesoscopic circuits. In momentum presentation, the Schr?dinger equation of the circuit is a Mathieu function, To solve the finite difference Schr?dinger equation, an improved parameter perturbative method is proposed when WKB and perturbative method are inapplicable. With this method, the detail energy spectrum and wave functions are obtained. As an application, the current quantum fluctuation in the ground state is calculated.The results of paper would be helpful to the design of integrate circuit, the improved parameter perturbative method would popularize the quantum theory of mesoscopic circuits.