Double Quantum Wire Magnetic Response
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Double Quantum Wire Magnetic Response Anatoly Yu. Smirnov1, Lev G. Mourokh2 and Norman J.M. Horing2 1 D-WaveSystems Inc., 320-1985 W.Broadway Vancouver, British Columbia, Canada V6J 4Y3 2 Department of Physics and Engineering Physics Stevens Institute of Technology, Hoboken, NJ 07030 ABSTRACT The induced magnetic moment of a biased semiconductor tunnel-coupled parallel double quantum wire system is examined here. The wires are in a series arrangement with tunnel coupling to each other and to leads. Their parallel lengths and associated continuous spectrum are taken in the direction perpendicular to the lead-to-lead current. The equations of motion for the double-wire electron Green’s function are formulated and analyzed using the transfertunneling Hamiltonian formalism. We determine the average magnetic moment of the doublewire system induced by a magnetic field applied perpendicular to the plane of the structure and we show that there are crossovers between diamagnetic and paramagnetic behavior, depending on the bias voltage, equilibrium chemical potential of the leads and temperature. INTRODUCTION
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Left Lead
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There has recently been intensified interest in extended tunnel-coupled structures, in particular, parallel quantum wires [1-3], which have symmetric and antisymmetric states analogous to those of quantum dots [4,5]. Such double-quantum-wire systems have been fabricated by split-gate methods [6] or by cleaved edge overgrowth [7]. The double-wire system that we examine in regard to magnetic moment is illustrated in Fig.1. It consists of two tunnel-coupled parallel quantum wires of finite length L with wire separation 2d placed between two metallic leads. There is electron confinement in two dimensions (y and z) in both wires, and the energy spectrum in the third direction (x) is taken to be continuous. Subject to the bias voltage, electrons tunnel sequentially from the left lead to the left wire, tunnel from the left wire to the right wire, and, finally, tunnel from the right wire to the right lead. We assume that tunneling from wire to wire is much faster than tunneling from the leads to wires, because the potential barriers between leads and wires are higher than the barrier between the wires themselves. We neglect the Coulomb interaction between electrons. A magnetic field is
Right Lead
2d Figure 1. Schematic of the double-wire system.
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taken to be applied perpendicular to the plane of this structure, facilitating the analysis of magnetic properties of the system. With the use of the nonequilibrium Green’s functions, we analyze the magnetic moment of the double-wire system. GREEN’S FUNCTION FORMULATION FOR DOUBLE-WIRE SYSTEM The electron field operators in the left and right wires are described as ˆ ( r , t ) = ψ ( y )ψ ( z )ψˆ ( x, t ), Ψ L L 0 L ˆ ( r , t ) = ψ ( y )ψ ( z )ψˆ ( x, t ), Ψ R
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0
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(1) (2)
where ψL(y)=ψ0(y+d), ψR(y)=ψ0(y-d), ψ0 is the ground state wave function for the y and z directions, and ψˆ L ( x, t ) , ψˆ R ( x, t ) are the electron
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