## Persistent Currents in Normal Metal Rings: Old Questions, New Answers

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##### Date

2010-08-02

##### Authors

Ulrich Eckerna

Peter Schwabb

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##### Abstract

<p>The Aharonov-Bohm effect, first described in 1959, is among the most spectacular effects of quantum mechanics, emphasizing the role the electromagnetic potentials – and not the electromagnetic fields – play for the wave-like motion of quantum particles. Considering a ringlike geometry in a constant perpendicular magnetic field, a direct consequence is that all properties of a charged system are periodic functions of the magnetic flux, Φ, the flux periodicity given by the fundamental flux quantum, Φ0 = h/e. This result is based on the particular combination, p + eA, which appears in the Hamiltonian of the system, where p is the momentum, and A the vector potential; here we consider electronic systems, and the charge of an electron is –e.<br />
In equilibrium, the system’s properties can be calculated from the partition function, which involves a trace over all states of the systems: hence in the classical limit any flux dependence disappears (Bohr-van-Leeuwen-Theorem), and the persistent current, I(Φ) = –∂F(Φ)/∂Φ, vanishes; F(Φ) denotes the thermodynamic potential. Thus very small systems and very low temperatures are required for a finite (non-zero) I(Φ) to exist.<br />
In fact “normal” persistent currents, of the order of a few nA, have been seen in several experiments, for temperatures below 1 K [1–4]. In contrast to the experiments [1–3] which used a SQUID technique in order to detect the magnetic moment induced by the current, the most recent study [4] employed a nano-electromechanical technique: the rings were placed on a cantilever, whose oscillation frequency can be measured with extremely high accuracy. The perimeter of the studied rings varied between 0.6 and 1.6 μm.<br />
Assuming time reversal invariance, the Fourier expansion of the persistent current is given by I(Φ) = I1 sin (2πΦ/Φ0) + I2 sin (4πΦ/Φ0) + … .<br />
Because of the disorder always present in small rings, due to the fabrication process, the amplitudes I1 and I2 are random quantities. For example, I1 fluctuates from ring to ring; in particular, it changes its sign, hence the average is expected (and was found) to be zero. The size of ‹(I1)2›1/2, on the other hand, has been debated for many years; the theoretical prediction ‹(I1)2›1/2 ≈ Ec/Φ0 [5] was now convincingly confirmed [4]; here Ec = hD/L2 is called Thouless energy, D is the diffusion constant, and L the perimeter of the ring. Concerning the second harmonic, I2, it was pointed out rather early [6] that the effective electron-electron interaction gives an important contribution, which nevertheless is too small compared to the experimental result [1]. A recent paper discusses the question whether a small amount of paramagnetic impurities can resolve this discrepancy [7]. For an introduction into persistent currents, see [8].</p>

<p>The Aharonov-Bohm effect, first described in 1959, is among the most spectacular effects of quantum mechanics, emphasizing the role the electromagnetic potentials – and not the electromagnetic fields – play for the wave-like motion of quantum particles. Considering a ringlike geometry in a constant perpendicular magnetic field, a direct consequence is that all properties of a charged system are periodic functions of the magnetic flux, Φ, the flux periodicity given by the fundamental flux quantum, Φ0 = h/e. This result is based on the particular combination, p + eA, which appears in the Hamiltonian of the system, where p is the momentum, and A the vector potential; here we consider electronic systems, and the charge of an electron is –e.<br /> In equilibrium, the system’s properties can be calculated from the partition function, which involves a trace over all states of the systems: hence in the classical limit any flux dependence disappears (Bohr-van-Leeuwen-Theorem), and the persistent current, I(Φ) = –∂F(Φ)/∂Φ, vanishes; F(Φ) denotes the thermodynamic potential. Thus very small systems and very low temperatures are required for a finite (non-zero) I(Φ) to exist.<br /> In fact “normal” persistent currents, of the order of a few nA, have been seen in several experiments, for temperatures below 1 K [1–4]. In contrast to the experiments [1–3] which used a SQUID technique in order to detect the magnetic moment induced by the current, the most recent study [4] employed a nano-electromechanical technique: the rings were placed on a cantilever, whose oscillation frequency can be measured with extremely high accuracy. The perimeter of the studied rings varied between 0.6 and 1.6 μm.<br /> Assuming time reversal invariance, the Fourier expansion of the persistent current is given by I(Φ) = I1 sin (2πΦ/Φ0) + I2 sin (4πΦ/Φ0) + … .<br /> Because of the disorder always present in small rings, due to the fabrication process, the amplitudes I1 and I2 are random quantities. For example, I1 fluctuates from ring to ring; in particular, it changes its sign, hence the average is expected (and was found) to be zero. The size of ‹(I1)2›1/2, on the other hand, has been debated for many years; the theoretical prediction ‹(I1)2›1/2 ≈ Ec/Φ0 [5] was now convincingly confirmed [4]; here Ec = hD/L2 is called Thouless energy, D is the diffusion constant, and L the perimeter of the ring. Concerning the second harmonic, I2, it was pointed out rather early [6] that the effective electron-electron interaction gives an important contribution, which nevertheless is too small compared to the experimental result [1]. A recent paper discusses the question whether a small amount of paramagnetic impurities can resolve this discrepancy [7]. For an introduction into persistent currents, see [8].</p>

<p>The Aharonov-Bohm effect, first described in 1959, is among the most spectacular effects of quantum mechanics, emphasizing the role the electromagnetic potentials – and not the electromagnetic fields – play for the wave-like motion of quantum particles. Considering a ringlike geometry in a constant perpendicular magnetic field, a direct consequence is that all properties of a charged system are periodic functions of the magnetic flux, Φ, the flux periodicity given by the fundamental flux quantum, Φ0 = h/e. This result is based on the particular combination, p + eA, which appears in the Hamiltonian of the system, where p is the momentum, and A the vector potential; here we consider electronic systems, and the charge of an electron is –e.<br /> In equilibrium, the system’s properties can be calculated from the partition function, which involves a trace over all states of the systems: hence in the classical limit any flux dependence disappears (Bohr-van-Leeuwen-Theorem), and the persistent current, I(Φ) = –∂F(Φ)/∂Φ, vanishes; F(Φ) denotes the thermodynamic potential. Thus very small systems and very low temperatures are required for a finite (non-zero) I(Φ) to exist.<br /> In fact “normal” persistent currents, of the order of a few nA, have been seen in several experiments, for temperatures below 1 K [1–4]. In contrast to the experiments [1–3] which used a SQUID technique in order to detect the magnetic moment induced by the current, the most recent study [4] employed a nano-electromechanical technique: the rings were placed on a cantilever, whose oscillation frequency can be measured with extremely high accuracy. The perimeter of the studied rings varied between 0.6 and 1.6 μm.<br /> Assuming time reversal invariance, the Fourier expansion of the persistent current is given by I(Φ) = I1 sin (2πΦ/Φ0) + I2 sin (4πΦ/Φ0) + … .<br /> Because of the disorder always present in small rings, due to the fabrication process, the amplitudes I1 and I2 are random quantities. For example, I1 fluctuates from ring to ring; in particular, it changes its sign, hence the average is expected (and was found) to be zero. The size of ‹(I1)2›1/2, on the other hand, has been debated for many years; the theoretical prediction ‹(I1)2›1/2 ≈ Ec/Φ0 [5] was now convincingly confirmed [4]; here Ec = hD/L2 is called Thouless energy, D is the diffusion constant, and L the perimeter of the ring. Concerning the second harmonic, I2, it was pointed out rather early [6] that the effective electron-electron interaction gives an important contribution, which nevertheless is too small compared to the experimental result [1]. A recent paper discusses the question whether a small amount of paramagnetic impurities can resolve this discrepancy [7]. For an introduction into persistent currents, see [8].</p>