Quantum Effect in a Diode Included Nonlinear Inductance-Capacitance Mesoscopic Circuit

Transcription

1 Commun. Theor. Phys. (Beijing, China) 52 (2009) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 52, No. 3, September 15, 2009 Quantum Effect in a Diode Included Nonlinear Inductance-Capacitance Mesoscopic Circuit YAN Zhan-Yuan, ZHANG Xiao-Hong, and MA Jin-Ying School of Mathematics and Applied Physics, North China Electric Power University, Baoding , China (Received August 8, 2008; Revised May 11, 2009) Abstract The mesoscopic nonlinear inductance-capacitance circuit is a typical anharmonic oscillator, due to diodes included in the circuit. In this paper, using the advanced quantum theory of mesoscopic circuits, which based on the fundamental fact that the electric charge takes discrete value, the diode included mesoscopic circuit is firstly studied. Schrödinger equation of the system is a four-order difference equation in ˆp representation. Using the extended perturbative method, the detail energy spectrum and wave functions are obtained and verified, as an application of the results, the current quantum fluctuation in the ground state is calculated. Diode is a basis component in a circuit, its quantization would popularize the quantum theory of mesoscopic circuits. The methods to solve the high order difference equation are helpful to the application of mesoscopic quantum theory. PACS numbers: b, Ra, Ca Key words: mesoscopic circuit, quantum effects 1 Introduction In recent years, research work of macroscopic quantum effects and mesoscopic physics has attracted great interest. [1 With the dramatic development in nanotechnology and nanoelectronics, the integrated circuits and components have been miniaturized toward atomic-scale dimensions in the manufacture of electronic devices. Undoubtedly, the quantum mechanical effect of circuits and devices must be taken into account. The quantum theory of mesoscopic circuits is studied as early as 1970 s, comparing with the quantization process of a harmonic oscillator, the L-C design circuit is quantized by Louisell. [2 In the middle of 90 s, much progress has been made in many aspects, such as quantum fluctuation, squeezing effect of some fundamental mesoscopic circuits. [3 6 However, they seldom consider the discreteness of the electronic charge. In Ref. [8, an improved quantum theory for mesoscopic electric circuits considering the discreteness of electric charge is proposed, which caused a new research hotpot in mesoscopic circuits, in the framework of a profound quantum theory for mesoscopic circuits, the quantum effects of RLC circuits, [9 11 inductance and capacitance coupling circuit [12,13 transmission lines [14 are studied. The circuits studied before are linear systems. If the diode is included in the mesoscopic circuit, the system acquires a nonlinear characteristic. In this paper, a mesoscopic circuit describing the anharmonic oscillator [15 studied in the framework of quantum theory developed by Li and Chen. [8 The anharmonic oscillator is quantized, and the energy spectrum, wave functions of the system are obtained. Diode is a basis component in a circuit, its quantization would popularize the quantum theory of mesoscopic circuits. 2 Quantization of Diode Included Nonlinear Inductance-Capacitance Mesoscopic Circuit Fig. 1 Diagram of mesoscopic nonlinear inductance-capacitance circuit. A diode included nonlinear inductance-capacitance mesoscopic circuit describing the anharmonic oscillator is shown in Fig. 1. Kirchhoffs voltage rule in the circuit is expressed as [16 L q + V C (q) = 0, (1) where q is the electric charge and V c is the voltage drop across both capacitors. In Eq. (1), the value for V C will vary according to the amount of charge on the capacitances C 1 and C 2. A plot of the voltage across both capacitances as a function of the charge on the capacitances is shown in Fig. 2. When the potential drop across C 2 is Supported by National Natural Science Foundation of China under Grant No zhanyuanyan@tom.com

2 No. 3 Quantum Effect in a Diode Included Nonlinear Inductance-Capacitance Mesoscopic Circuit 535 lower than a certain voltage, the diodes do not conduct, the circuit equivalent capacitance C e is that of the two capacitances 1/C e = 1/C 1 + 1/C 2. As the charge on the capacitances increases, the potential drop across C 2 will reach a certain voltage and the diodes begin to conduct. The circuit equivalent capacitance increases and the slope of the line shown in Fig. 2 decreases to the value of 1/C 1. In this instance, the diodes act as closed switches and effectively remove the capacitance C 2 from the circuit. Fig. 2 Voltage versus charge across both capacitors. In Eq. (1), the form of V c (q) is given by the following piecewise function { 1/Ce q(t) < q b, V c (q) = q (2) 1/C 1 otherwise, where q b is the critical charge required to conduct the diodes. The three straight line segments can be approximated by a continuous of the form V c (q) = q µ νq3, (3) where µ and ν are the fitting values corresponding to experimental values. [7 As a matter of fact, the electric charges are discrete. So in the process of quantization of the mesoscopic circuits, the eigenvalues of the self-adjoint operator ˆq should take discrete values, following the quantization method in [8, ˆq q = nq e q, (4) where n Z, q stand for eigenstates of electric charge for the circuits and q e = Coulomb, is the elementary charge. A minimum shift operators is introduced, ˆQ = exp(iq eˆp/ h), (5) the important self-adjoint momentum operator is defined as ˆP = h 2i ( q e + qe ) = h 2iq e ( ˆQ ˆQ ), (6) in which, the right and left discrete derivative operators are qe = ( ˆQ 1)/q e, qe = (1 ˆQ )/q e, (7) and the free Hamilton operator is Ĥ 0 = h2 2L ( q e qe ) = h2 2 ( ˆQ + ˆQ + 2) Quantized Hamilton of the system is = h2 2Lq e ( qe qe ). (8) Ĥ = h2 2Lq e ( qe qe ) + ˆq2 2µ νˆq4 4. (9) Now, we consider a representation in which the operator ˆp is diagonal and called ˆp presentation. using the following relations p ˆQ 2 p = 2π h3 2 q e p 2 δ(p p ), p ( qe qe ) p = 4π h q 2 e [ cos ( qe h p ) 1 δ(p p ). (10) The Schrödinger equation of nonlinear inductancecapacitance mesoscopic circuit in ˆp presentation is ν h h2 p4ψ + 2µ p 2 ψ { h 2 } + qe 2L[cos(q ep/ h) 1 + E ψ = 0. (11) In above, the nonlinear inductance-capacitance mesoscopic circuit is quantized in the framework of the quantum theory for mesoscopic circuits based on discrete charge space, it is a four order difference equation. 3 Wave Function and Quantum Fluctuations of System Ordinarily, a linear inductance-capacitance-resistancepower source systems Schrödinger equations are two order difference equation, they could be solved through transforming them into a formation of Mathieu equation. [8,11,13 To solve the Schrödinger equation of the nonlinear inductance-capacitance mesoscopic circuit system, we extended the perturbative method [18 of two order difference equation to four order difference equation, this method is applicable for other high order difference equations. By introducing π q e h p = 2z, λ = 64 h2 νlq 6 e + 64E νq 4 e ξ = 32 h2 νlqe 6,, (12) the Schrödinger equation of nonlinear inductancecapacitance mesoscopic circuit is convert to: z 4ψ z2ψ + [λ 2ξ cos(2z)ψ = 0. (13)

3 536 YAN Zhan-Yuan, ZHANG Xiao-Hong, and MA Jin-Ying Vol. 52 For the concrete values of the parameter ν, L, Plank constant and the elementary electric charge, ξ 10 1, perturbative method is valid. ψ = ψ (0) + ξψ (1) + ξ 2 ψ (2) + ξ 3 ψ (3) +, λ = λ (0) + ξλ (1) + ξ 2 λ (2) + ξ 3 λ (3) +, (14) where ψ (0) and λ (0) = k 4 + (8/µq 2 el)k 2 are zero order approximate of wave function and eigenvalue, ψ (1,2,3,...) and λ (1,2,3,...) are high order corrections of wave function and eigenvalue. When k = 0, 1, 2,..., ψ have separate solutions. Wave function ψ (0,1,2) and eigenvalue λ (0,1,2) are determined by the following equations: ψ (0) z ψ (0) 2 z nψ (0) = 0, z ψ (1) 2 z nψ (1) + λ (1) ψ (0) 2 cos(2z)ψ (0) = 0, z ψ (2) 2 z nψ (2) + λ (1) ψ (1) + λ (2) ψ (0) 2 cos(2z)ψ (1) = 0. The detail calculation process of the wave function and energy lever are in the appendix. Calculated to second order approximate, the separate periodic wave function are: 2ψ0 = h2 cos2z 2 ) 2 2 cos4z 6 1 2/ 6 (2 1/ 2)(1 4/µνq2 e ) + ψ h2 cos3z 2 ) 2 [ 4 cos3z = cosz / 6 (5 4/ 4 cos5z )2 (39 12/ 2)(5 4/µνq2 e ) + ψ1 2 h2 sin3z 2 ) 2 [ 4 sin 3z = sin z / 6 (5 4/) 2 4 sin5z (39 12/)(5 2 4/) [ ψ h2 2 = cos2z + 6 2/ cos4z 2 ) 2 cos6z 15 6/ 6 (20 6/ 2)(15 6/µνq2 e ) + ψ2 2 h2 sin 4z 2 ) 2 sin 6z = sin 2z /µqe 2L + 6 (20 6/µqe 2L)(15 6/µq2 e L) + (16) while the superscripts + and specify the even and odd parity solution respectively, the corresponding energy levels are E 0 = h2 Lqe 4 h / E 1 + = 3 h2 q2 e 8µ νq4 e 64 + h 4 L 2 νqe(5 8 4/) E1 = h2 q2 e 8µ νq4 e 64 h 4 L 2 νqe 8(5 4/µνq2 e ) + E + 2 = h2 Lq 2 e + q2 e 2µ νq4 e h 4 L 2 νq 8 e (2/µνq2 e 1) + h 4 L 2 νq 8 e (15 6/µνq2 e ) + E2 = h2 Lqe q2 e 2µ νq4 e 4 + h 4 L 2 νqe(15 8 6/) In calculation, we found that the energy spectrum E + k and E k would have the same expression when k 3. This does not mean the degenerate of energy level because we only calculated to ξ 2 phase, however we can conclude that the energy level will become more and more close with the increasing of k. To verify our results, let ν = 0 and µ = C 1 in Eqs. (17) and (18), the system becomes to a LC oscillator, our results are the same as the results in Ref. [19. As an application of the results obtained above, we calculated the average values of currents P and P 2 in the ground state. P 0 = 0, (15) (17)

4 No. 3 Quantum Effect in a Diode Included Nonlinear Inductance-Capacitance Mesoscopic Circuit 537 P 2 h 2 0 = ( 2 2)2 (2 2 1)2 ( 2 [ 4) µν + 346qeµ 2 2 ν 2 302qeµ 4 3 ν q6 eµ 4 ν 4 26qeµ 8 5 ν 5 + 2qe 10 µ 6 ν 6 q 2 e + 16 h4 µ 2 92 h 4 µ 3 ν L2 qe h4 µ 4 ν 2 L2 qe 6 55 h4 µ 5 ν 3 L2 qe h4 µ 6 ν 4 L2 qe 2L2 qe h8 µ 4 4 h 8 µ 5 ν h 8 µ 6 ν 2 + L4 L4 qe 18 qe 16 qe 14L4. (18) From Eq. (18), we can see that the average values of the currents are zero, whereas the average values of the square of the currents are not. This indicated the existence of the current quantum fluctuations, it is the shot noise in the system. In the above, using an advanced quantum theory to mesoscopic circuits developed recently by Li and Chen, which based on the fundamental fact that the electric charge takes discrete value, a diode included nonlinear inductancecapacitance mesoscopic circuit is quantized. The Schrödinger equation in ˆp representation is a four order difference equation. Using the power series and perturbative method, the detail energy spectrum and wave functions are obtained and verified, as an application of the results, the current quantum fluctuation in the ground state is calculated. This paper is an application of the profound quantum mesoscopic circuits theory to the diode included circuit, the methods of quantization and solution process would be helpful to the popularization of the mesoscopic quantum theory. Appendix A: Solution of Four Order Difference Equation In Sec. 3 Eq. (14), the Schrödinger equation in ˆp representation is a four order difference equation. To make the equation simple, let a = 8/ 2, Eq. (14) becomes 2 z4ψ + a z2ψ + [λ 2ξ cos(2z)ψ = 0. (A1) ψ(z) could be periodic function of π or 2π. We show the calculation process of ψ(z) with the periodic π, let ψ(z) = [c k sin(kz) + d k cos(kz), (A2) k=0,2,4,... insert Eq. (A2) into Eq. (A1), calculate to the order k = 8, λd 0 ξd (16c 2 4ac 2 ξc 4 )sin 2z + ( 2ξd d 2 4ad λd 2 ξd 4 )cos 2z + ( ξc 256c 4 16ac 4 + λc 4 ξc 6 )sin 4z + ( ξd 256d 4 16ad 4 + λd 4 ξd 6 )cos4z + ( ξc c 6 36ac 6 + λc 6 ξc 8 )sin 6z + ( ξd d 6 36ad 6 + λd 6 ξd 8 )cos6z + ( ξc c 8 64ac 8 + λc 8 )sin 8z + ( ξd d 8 64ad 8 + λd 8 )cos8z = 0. (A3) Obviously, the coefficients satisfy the following matrix equation 16 4a + λ ξ 0 0 ξ a + λ ξ 0 0 ξ a + λ ξ 0 0 ξ a + λ ξ λ ξ ξ 16 4a + λ ξ ξ a + λ ξ ξ a + λ ξ ξ a + λ ξ c 2 c 4 c 6 c 8 = 0, d 0 d 2 d 4 d 6 d 8 (A4) = 0. (A5) The same matrix equations of 2π period ψ could be obtained with the same method. We could solve λ and c k, d k through this two equations theoretically, however, it is too difficult to calculate the roots for an equation of 5-th Order. if ξ is small, perturbative theory is valid. The zero order approximate of λ is easy obtained from the diagonal elements of coefficients matrix, λ = k 4 + ak, k = 0, 1, 2, 3, (A6)

5 538 YAN Zhan-Yuan, ZHANG Xiao-Hong, and MA Jin-Ying Vol. 52 According to Eqs. (15) and (16), higher order approximates of λ k and ψ k could be calculated. For example, we calculate λ + 1 and ψ+ 1. When k = 1, λ (0) 1 = a 1. The zero approximates odd solution of ψ (0) 1 could obtained from the first formula of Eq. (16) the solution is ψ (0) 1 = cosz. Insert ψ (0) 1 into the second formula of Eq. (16), ψ (0) z 4 + a 2 ψ (0) z (a 1)ψ (0) = 0, z 4 + a 2 ψ (1) z (n 1)ψ (1) + (λ (1) 1)cosz cos3z = 0, in the equation above, let λ (1) = 1, ψ (1) has a period solution, satisfy: the solution of ψ (1) is Insert ψ (0), ψ (1) into the third formula of Eq. (16), z 4 + a 2 ψ (1) z (n 1)ψ (1) = cos3z, ψ (1) = cos3z 80 8a. (A7) z 4 + a 2 ψ (2) z (a 1)ψ (2) + cos3z 80 8a + λ(2) cosz cos5z 80 8a cosz 80 8a = 0. Let λ (2) = 1/(80 8a), ψ (2) has a period solution, satisfy: the solution of ψ (2) is z 4 + a 2 ψ (2) z (a 1)ψ (2) = cos3z 80 8a + cos5z 80 8a, ψ (2) = cos3z (80 8a) cos5z (624 24a)(80 8a). (A8) So, the final results of ψ 1 + and λ+ 1 are ψ 1 + cos3z = cosz + ξ 80 8a + ξ2[ cos3z (80 8a) cos5z, (A9) (624 24a)(80 8a) λ + 1 = a 1 + ξ + 1 ξ2 80 8a. (A10) References [1 W.H. Zurek, Phys. Today 44 (1991) 36. [2 W.H. Louisell, in Quantum Statistical Properties of Radiation 1, John Wiley, New York (1973). [3 C.P. Sun and L.H. Yu, Phys. Rev. A 49 (1994) 592. [4 L.H. Yu and C.P. Sun, Phys. Rev. A 51 (1995) [5 H.Y. Fan and X.T. Liang, Chin. Phys. Lett. 17(3) (2000) 174. [6 J.S. Wang and C.Y. Sun, Int. J. Theor. Phys. 37 (1998) [7 E.L.M. Flerackers, H.J. Janssen, and L. Beerden, Am. J. Phys. 53 (1985) 574. [8 Y.Q. Li and B. Chen, Phys. Rev. B 53 (1996) [9 Y.Q. Li, in Proc. 5th Winger Symposium, eds. P. Kasperkovitz and D. Grau, World Scientific, Singpore (1998) 307. [10 T. Lu and Y.Q. Li, Mod. Phys. Lett. B 16 (2002) 975. [11 J.X. Liu and Z.Y. Yan, Commun. Theor. Phys. 45 (2006) [12 J.S. Wang, T.K. Liu, and M.S. Zhan, Phys. Lett. A 276 (2000) 155. [13 J.X. Liu and Z.Y. YAN, Commun. Theor. Phys. 44 (2005) [14 J.C. Flores, Phys. Rev. B 64 (2001) [15 R.H. Enns and G. McGuire, in Nonlinear Physics with Maple for Scientists and Engineers, Birkhauser Boston, Cambridge (1997) Chap. 2, p. 12. [16 S. Zhang, C.I. Um, and K.H. Yeon, Chin. Phys. Lett. 19 (2002) 985. [17 J.X.Liu and Z.Y. Yan, Commun. Theor. Phys. 44 (2005) [18 T.Z. Liu and L.Q. Chen, in Nonlinear Vibrations, Higher Education Press, Beijing (2001). [19 Y.Q. Li and B. Chen, Commun. Theor. Phys. 29 (1998) 137.