Mesoscopic Quantum Circuit Theory to Persistent Current and Coulomb Blockade

The quantum theory for mesoscopic electric circuit is briefly described. The uncertainty relation for electric charge and current modifies the tranditional Heisenberg uncertainty relation. The mesoscopic ring is regarded as a pure L-design, and the persistent current is obtained explicitly. The Coulomb blockade phenomenon appears when applying to the pure C-design. talk at the Fifth International Wigner Symposium, Vienna 25-29 August, 1997 on leave from the absence of Zhejiang University, Hangzhou 310027 China 1 Owning to the dramatic achievement in nanotechnology, there have been many studies on mesoscopic physics [1]. In present talk I briefly demonstrate a quantum mechanical theory for mesoscopic electric circuits based on the fact that electronic charge takes discrete values [2]. As the application of this approach, the persistent current on a mesoscopic ring and the Coulomb blockade phenomena are formulated from a new point of view, about which some details are presented. Most importantly, it is a physical realization of the deformation of quantum mechanics studied considerably by other authors in mathematical physics. I. QUANTIZED CIRCUIT WITH CHARGE DISCRETENESS In order to taken into account the discreteness of electronic charge. we must impose that the eigenvalues of the self-adjoint operator q̂ (electric charge ) take discrete values [2], i.e. q̂|q >= nqe|q > (n ∈ Z , qe = 1.602 × 10 −19 coulomb). Since the spectrum of charge is discrete, the inner product in charge representation will be a sum instead of the usual integral and the electric current operator P̂ will be defined by the discrete derivatives [3] ∇qe, ∇qe. Thus for the mesoscopic quantum electric circuit one will have a finite-difference Schrödinger equation [2]. The uncertainty relation for electric charge and current modifies the tranditional Heisenberg uncertainty relation, namely, ∆q̂ ·∆P̂ ≥ h̄ 2 (1 + q e h̄ < Ĥ0 >). (1) where Ĥ0 = − h̄ 2 ∇qe∇qe = − h̄ 2qe (∇qe −∇qe). The Hamiltonian of quantum LC-design in the presence of exterior magnetic flux reads Ĥ = − h̄ 2qeL (Dqe −Dqe) + 1 2C q̂ + εq̂ (2) where L and C stand for the inductance and the capacity of the circuit respectively, ε represents the voltage of an adiabatic aource, and the covariant discrete derivatives are defined by 2 Dqe := e − qe h̄ φ Q̂− e i qe h̄ φ qe , Dqe := e qe h̄ φ e −i qe h̄ φ − Q̂ qe , (3) where Q̂ := ee is a minimum ‘shift operator’ with the property Q̂|n >= en+1 |n + 1 > (α ns are undetermined phases). The hamiltonian (2) is covariant under the gauge transformation, ĜDqeĜ −1 = D qe , ĜDqeĜ −1 = D ′ qe where Ĝ := e −iβ q̂ h̄ and the gauge field φ transforms as φ→ φ = φ−β. The φ plays the role of the exterioral magnetic flux threading the circuit. II. QUANTUM L-DESIGN AND PERSISTENT CURRENT Now we study the Schrödinger equation for a pure L-design in the presence of magnetic flux, − h̄ 2qeL (Dqe −Dqe)|ψ >= E|ψ > . (4) Because its eigenstates can be simultaneous eigenstates of p̂, eq.(4) is solved by the eigenstate |p >= ∑ n∈Z κne e|n > (κn := exp(i ∑n j=1 αj)). The energy spectrum is easily calculated as E(p, φ) = 2h̄ q e sin ( qe 2h̄ (p− φ) ) (5) which has oscillatory property with respect to φ or p. Differing from the usual classical pure L-design, the energy of a mesoscopic quantum pure L-design can not be large than 2h̄/q e . Clearly, the lowest energy states are those states with p = φ + nh/qe, then the eigenvalues of the electric current ( i.e. 1 L P̂ ) of ground state can be obtained [2]. The electric current on a mesoscopic circuit of pure L-design is not null in the presence of a magnetic flux (except φ = nh/qe). This is a pure quantum characteristic. The persistent current in a mesoscopic L-design is an observable quantity periodically depending on the flux φ. Because a mesoscopic metal ring is a natural pure L-design, the formula we obtained is valid for persistent current in a single mesoscopic ring [4]. One can easily calculate the