Quantum interference rectifier

Physica E 9 (2001) 631–634

www.elsevier.nl/locate/physe

Quantum interference recti er V.A. Geylera; ∗;1 , I.Yu. Popovb a Institut

fur Mathematik, Humboldt-Universitat zu Berlin, Rudower Chausee 25, 12489, Berlin, Germany of Higher Mathematics, Leningrad Institute of Fine Mechanics and Optics, Sablinskaya 14, St.-Petersburg, 197101, Russia

b Department

Received 28 August 2000; received in revised form 16 October 2000; accepted 17 October 2000

Abstract Electron transport in bent quantum wire in the presence of a magnetic eld which is orthogonal to the system plane is considered. Possible constructions of “quantum interference switch” and “quantum interference recti er” are suggested. ? 2001 Elsevier Science B.V. All rights reserved. PACS: 72.10.−d Keywords: Electronic transport; Quantum interference

The development of nanoelectronics gives one a possibility to create a class of nanosized arti cial objects: quantum dots and antidots, quantum wires, etc. Such systems are often referred as mesoscopic. The goal is to stress that the systems involved are large enough to be shaped by the experimentalist and at the same time so small that quantum eects like interference are manifested on them. Therefore, a way is opened to absolutely new technologies, for example, quantum computing. But, simultaneously, it poses a problem of development of new electronic devices based on quantum principles. There are a number of works in which possible constructions of dierent ∗

Corresponding author. Tel.: +49-30-2093-2358; fax: +49-302093-2727. E-mail address: [email protected] (V.A. Geyler). 1 On leave of absence from Department of Mathematics, Mordovian State University, Saransk, 430000, Russia.

quantum devices are suggested: quantum interference transistor, quantum capacitor, quantum multiplexer, etc. (see, e.g., [1–8]). This problem is closely related with the problem of description of the behaviour of low-dimensional quantum systems, which is especially dicult in the case when the system is in a magnetic eld. The goal of the present paper is to suggest constructions of quantum recti er and quantum switch governed by external magnetic eld and to describe the electron transport in the systems. First, we solve basic scattering problem for a bent quantum wire in a magnetic eld which is orthogonal to the system plane. Then, we apply the result to the description of new quantum devices. We use 1D approximation for the quantum wire. Consider bent quantum wire (Fig. 1) consisting of three parts: two semi-in nite straight wires L1 ; L2 and

1386-9477/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 0 ) 0 0 2 8 5 - X

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The opposite direction on L1 to that on Larc leads to the dierence in signs in Eqs. (1) and (2). The solution of the scattering problem has the following form: 1 (x) = exp(ikx=˜) on L1 ; arc (’) = a1 exp(ip1 r’=˜) + a2 exp(ip2 r’=˜); on Larc ; 2 (x) = b1 exp(ikx=˜) + b2 exp(−ikx=˜) on L2 ; Fig. 1. Bent quantum wire. r is the radius, and ’ is the sectorial angle of the arc.

an arc Larc of radius r and sectorial angle ’0 . We assume that the system is in a magnetic eld B which is uniform in a neighbourhood of the arc and vanishes outside it. The momentum operator kˆ on L1 and L2 is ˜ d : kˆ = i dx In turn, the momentum operator pˆ on Larc has the form   ˜ 1 d + ; pˆ = r i d’ 2

where  = r B=0 (0 is the magnetic ux quantum, 0 = 2˜c=|e|). We stress that in the onedimensional arc, the momentum operator pˆ is independent of a gauge choice for the magnetic eld because this operator depends on the magnetic ux  only. As for the parameter ’ on Larc , one has 06’6’0 : The Hamiltonian of the system has the following form: 2 kˆ ; H1 = H2 = 2m for L1 and L2 ,

˜ ˜ ; p2 = −k − ; r r are determined by the energy conservation law:  2 ˜ k2 1 p+ : = 2m r 2m p1 = k −

Using Eq. (1), one comes to the relations a1 + a2 = 1; a1 − a2 = −1: Hence, a1 = 0; a2 = 1: Consequently, one has arc (’) = exp(−i(rk + ˜)’=˜): Then, using Eq. (2), one gets for the coecients b1 ; b2 : b1 + b2 = exp(−i(rk + ˜)’0 =˜); b1 − b2 = −exp(−i(rk + ˜)’0 =˜); so

pˆ 2 ; 2m for Larc . At the points A1 ; A2 (see Fig. 1) the following gauge invariant boundary conditions are used. At the point A1 one has ˆ 1 (0) = −p k ˆ arc (0) (1) 1 (0) = arc (0); Harc =

and at the point A2 : arc (’0 ) = 2 (0);

where p1 ; p2 ,

ˆ 2 (0): p ˆ arc (’0 ) = k

(2)

b1 = 0; b2 = exp(−i(rk + ˜)’0 =˜): Thus, 2 (x) = exp(−i(rk + ˜)’0 =˜)exp(−ikx=˜): As a result, the transmission coecient from L2 to L1 is t2→1 = exp(i(rk + ˜)’0 =˜):

(3)

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The transmission coecient from L1 to L2 is obtained in an analogous way. We start from the following form of the solution 2 (x) = exp(ikx=˜); arc (’) = a1 exp(ip1 r’=˜) + a2 exp(ip2 r’=˜); 1 (x) = b1 exp(ikx=˜) + b2 exp(−ikx=˜):

Fig. 2. Quantum switch. Dark area is the domain where there is a magnetic eld.

Condition (2) gives us a1 exp(ip1 r’0 =˜) + a2 exp(ip2 r’0 =˜) = 1; a1 exp(ip1 r’0 =˜) − a2 exp(ip2 r’0 =˜) = 1: Hence, arc (’) = exp(ip1 r(’ − ’0 )): Relations (1) lead to the following system for the coecients b1 ; b2 : b1 + b2 = exp(−ip1 r’0 =˜);

Fig. 3. Quantum recti er. Dark area is the domain where there is a magnetic eld.

b1 − b2 = −exp(−ip1 r’0 =˜): Consequently, 1 (x) = exp(−i(rk − ˜)’0 =˜)exp(ikx=˜): As a result, the transmission coecient from L1 to L2 is as follows: t1→2 = exp(i(rk − ˜)’0 =˜):

(4)

Let us discuss possible applications of the results. First, note that the absolute values of both the transmission coecients are equal to 1, i.e. the corresponding transmission probabilities are equal to 1 too. The phase of the wave function only changes after the transmission of the charged particle through the system. Hence, the results can be observed experimentally in interference eects only. We suggest two quantum interference devices based on the eects. The rst one is a “quantum switch” (Fig. 2). The incoming electron wave in Lin is split into two identical waves at point C1 . The waves pass through the arcs L1 and L2 in opposite directions. They come to point C2 having, respectively, supplementary phase factors exp(i(kr − ˜)’0 =˜) and exp(i(kr + ˜)’0 =˜). In the wire Lout , one has a superposition of this waves. If the dierence between the mentioned phases is  + 2n, i.e. if ’0  = =2 + n; n integer, then the outgoing wave vanishes. Thus, we have a switch controlled by the magnetic eld B.

The second device is a “quantum recti er” (Fig. 3). In this case, the incoming electron wave in Lleft is split into two identical waves at point C1 as well. But here the wave in L2 only acquires a supplementary phase factor exp(2i(kr − ˜)=˜). Let 2(kr=˜ − ) = 2n, in other words, kr=˜ −  = n; n is an integer. Then, the superposition of the waves in Lright gives us the transmission coecient from left to right for the whole device being equal to 1. Consider the transmission in the opposite direction. Here the incoming electron wave is in Lright . An analogous procedure leads to the supplementary phase factor exp(2i(kr + ˜)=˜). Let 2(kr˜−1 + ) =  + 2n0 , i.e., kr˜−1 +  = 12 + n0 ; n0 is an integer. Then the superposition of the waves in Lleft vanishes and gives us the transmission coecient from right to left for the whole device being equal to 0. Therefore, if both the conditions kr˜−1 −  = n and kr˜−1 +  = 12 + n0 are valid, the device is a quantum recti er. Taking into account that k 2 is xed by the Fermi level, we have two parameters (r and B) to ensure both the mentioned relations. Acknowledgements This work was supported by the DFG, the Program “Universities of Russia” and the RFBR (Grant

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98-01-03308). I.Y.P. thanks Mordovian State University for the hospitality during the preparation of the article. We are grateful to the referee for useful comments. References [1] F. Sols, F. Macucci, U. Ravaioli, K. Hess, Appl. Phys. Lett. 54 (1989) 350. [2] P. Exner, P. Seba, Phys. Lett. A 129 (1988) 477.

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