Persistent Currents in Normal Metal Rings

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Science  09 Oct 2009:
Vol. 326, Issue 5950, pp. 272-275
DOI: 10.1126/science.1178139


Quantum mechanics predicts that the equilibrium state of a resistive metal ring will contain a dissipationless current. This persistent current has been the focus of considerable theoretical and experimental work, but its basic properties remain a topic of controversy. The main experimental challenges in studying persistent currents have been the small signals they produce and their exceptional sensitivity to their environment. We have developed a technique for detecting persistent currents that allows us to measure the persistent current in metal rings over a wide range of temperatures, ring sizes, and magnetic fields. Measurements of both a single ring and arrays of rings agree well with calculations based on a model of non-interacting electrons.

An electrical current induced in a resistive circuit will rapidly decay in the absence of an applied voltage. This decay reflects the tendency of the circuit’s electrons to dissipate energy and relax to their ground state. However, quantum mechanics predicts that the electrons’ many-body ground state (and, at finite temperature, their thermal equilibrium state) may contain a persistent current (PC), which flows through the resistive circuit without dissipating energy or decaying. A dissipationless equilibrium current flowing through a resistive circuit is counterintuitive, but it has a familiar analog in atomic physics: Some atomic species’ electronic ground states possess nonzero orbital angular momentum, which is equivalent to a current circulating around the atom.

One of the major insights of mesoscopic condensed-matter physics is that this analogy remains valid even when the electrons experience a static disorder potential, as in a resistive metal (1). Theoretical treatments of PCs in resistive metal rings have been developed over a number of decades [see (1, 2) and references therein]. Calculations that take into account the electrons’ inevitable coupling to the static disorder potential and a fluctuating thermal bath predict several general features. A micrometer-diameter ring will support a PC of I ~ 1 nA at temperatures T ≲ 1 K. A magnetic flux Φ threading the ring will break time-reversal symmetry, allowing the PC to flow in a particular direction around the ring. Furthermore, the Aharonov-Bohm effect will require I to be periodic in Φ with period Φ0 = h/e, thereby providing a clear-cut experimental signature of the PC.

These predictions have attracted considerable interest, but measuring the PC is challenging for a number of reasons. For example, the PC flows only within the ring and so cannot be measured with a conventional ammeter. Experiments to date (2, 3) have mostly used superconducting quantum interference devices (SQUIDs) to infer the PC from the magnetic field it produces. Interpretation of these measurements has been complicated by the SQUID’s low signal-to-noise ratio (SNR) and the uncontrolled back action of the SQUID’s ac Josephson oscillations, which may drive nonequilibrium currents in the rings. In addition, SQUIDs perform optimally in low magnetic fields; this limits the maximum Φ that can be applied to the rings, allowing observation of only a few oscillations of I(Φ) and complicating the subtraction of background signals unrelated to the PC.

Experiments to date have produced a number of confusing results in apparent contradiction with theory and even among the experiments themselves (2, 3). These conflicts have remained without a clear resolution for nearly 20 years, suggesting that our understanding of how to measure and/or calculate the ground-state properties of as simple a system as an isolated metal ring may be incomplete.

More recent theoretical work has predicted that the PC is highly sensitive to a variety of subtle effects, including electron-electron interactions (47), the ring’s coupling to its electromagnetic environment (8), and trace magnetic impurities within the ring (9). These theories have not explained all of the experimental results to date, but they do indicate that accurate measurements of the PC would be able to address a number of interesting questions in many-body condensed-matter physics (in addition to resolving the long-standing controversy described above).

We measured the PC in resistive metal rings using a micromechanical detector with orders of magnitude greater sensitivity and lower back-action than SQUID-based detectors. Our approach allows us to measure the PC in a single ring and arrays of rings as a function of ring size, temperature, and the magnitude and orientation of the magnetic field over a much broader range than has been possible previously. Quantitative agreement is found between these measurements and calculations based on a model of diffusive, non-interacting electrons. This agreement is supported by independent measurements of the rings’ electrical properties.

Figure 1, A to C, shows single-crystal Si cantilevers with integrated Al rings [their fabrication is described elsewhere (10)]. All the PC measurements were made in magnetic fields well above the critical field of Al, ensuring that the rings were in their normal (rather than superconducting) state. The parameters of the four ring samples measured are given in Table 1.

Fig. 1

(A) Cantilever torque magnetometry schematic. An array of metal rings is integrated onto the end of a cantilever. The cantilever is mounted in a 3He refrigerator. A magnetic field B is applied at an angle θ from the plane of the rings. The out-of-plane component of B provides magnetic flux Φ through the ring. The in-plane component of B exerts a torque on the rings’ magnetic moment and causes a shift in the cantilever’s resonant frequency δν. Laser interferometry is used to monitor the cantilever’s motion and to determine δν. (B) A scanning electron micrograph of several Si cantilevers similar to those used in the experiment. The light regions at the end of some of the cantilevers are arrays of Al rings. The individual rings are visible in (C), which shows a magnified view of the region in (B) outlined in red. (D) Raw data showing δν as a function of B for an array of N = 1680 rings with r = 308 nm at T = 365 mK and θ = 45°. (E) PC inferred from the frequency shift data in (D) after subtracting a smooth background from the raw data. The lefthand axis shows the total current IΣ in the array, and the righthand axis shows the estimated typical per-ring current Ityp = IΣ/Embedded Image. Oscillations with a characteristic period of ~20 mT (corresponding to Φ = h/e) are visible in (D) and (E).

Table 1

Sample parameters. For each of the four ring samples, the rings’ mean radius r, linewidth w, and thickness d are listed, along with the number N of rings in the sample. The electrons’ diffusion constant D, extracted from the fits in Fig. 3, is given. The stated errors are statistical errors in the fits. An additional 6% error in D is estimated for uncertainties in the overall calibration, as discussed in the SOM text. The fifth sample is the codeposited wire used in the transport measurements described in the SOM text. For this sample, D was determined from the wire’s resistivity.

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In the presence of a magnetic field B, each ring’s current I produces a torque on the cantilever τ=μ×B as well as a shift δν in the cantilever’s resonant frequency ν. Here μ=πr2In^ is the magnetic moment of the PC, r is the ring radius, and n^ is the unit vector normal to the ring. We infer I(B) from measurements of δν(B); the conversion between δν(B) and I(B) is described in the supporting online material (SOM) text.

To monitor ν, we drive the cantilever in a phase-locked loop. The cantilever is driven via a piezoelectric element, and the cantilever’s displacement is monitored by a fiber-optic interferometer (11). The cantilever’s thermally limited force sensitivity is ~2.9 aN/Hz1/2 at T = 300 mK, corresponding to a magnetic moment sensitivity of ~11 μB/Hz1/2 and a current sensitivity of ~20 pA/Hz1/2 for a ring with r = 400 nm at B = 8 T. By comparison, SQUID magnetometers achieve a current sensitivity ≳5 nA/Hz1/2 for a similar ring (1214). The noise temperature of the cantilever and the electron temperature of a metal sample at the end of a cantilever both equilibrate with the fridge temperature for the conditions we used (11).

The frequency shift of a cantilever containing an array of N = 1680 lithographically identical rings with r = 308 nm at T = 323 mK is shown (Fig. 1D) as a function of B. Oscillations with a period 20 mT, corresponding to a flux h/e through each ring, are visible in the raw data. Depending on r and θ (the angle between B and the plane of the ring), we observe as many as 450 oscillations over a 5.5-T range of B (figs. S12 to S17).

Figure 1E shows the data from Fig. 1D after subtracting the smooth background and converting the data from δν(B) to I(B) using the expressions in the SOM text. The lefthand axis in Fig. 1E shows IΣ, the total PC inferred from the measurement, which is the sum of the PC from each ring in the array. The righthand axis shows the estimated typical single-ring PC: Ityp=IΣ/N. This relationship between Ityp and IΣ arises because the PC in each ring is predicted to oscillate as a function of B with a phase that depends on the ring’s microscopic disorder, and thus is assumed to be random from ring to ring. This assumption is verified below.

To establish that δν provides a reliable measure of the PC, Ityp(B) was measured as a function of several experimental conditions: the laser power incident on the cantilever, the amplitude and frequency of the cantilever’s motion, the polarity and orientation of the magnetic field, and the presence or absence of room-temperature electronics connected to the cryostat. These data are shown in the SOM text and indicate that the measurements of Ityp(B) are independent of these parameters (for the conditions of our experiment) and reflect the equilibrium PC in the rings.

Figure 2, A to C, shows Ityp(B) for arrays of rings with three different radii: r = 308, 418, and 793 nm. We have also measured a single ring with r = 418 nm (Fig. 2D). Figure 2, E to H, shows |I˜typ(fΦ)|, the absolute value of the Fourier transform of the data in Fig. 2, A to D [fΦ is the flux frequency in units of (h/e)−1]. Figure 2, I to L, shows Gtyp(δB), the autocorrelation of Ityp(B) for each of these samples. Gtyp(δB) is calculated from measurements of Ityp(B) taken over a much broader range of B than is shown in Fig. 2, A to D; the complete data are shown in the SOM text.

Fig. 2

PC versus magnetic field in (A) the 308-nm array for T = 365 mK, θ = 45°; (B) the 418-nm array for T = 365 mK, θ = 45°; (C) the 793-nm array for T = 323 mK, θ = 6°; and (D) the 418-nm ring for 365 mK, θ = 45°. In each case, a smooth background has been removed. (E to H) Fourier transforms of the data in (A) to (D). The expected h/e and h/2e periodicities are indicated by the horizontal blue bars. The bars’ widths reflect the rings’ linewidth w. A small h/2e peak is present in (E) (visible in the log-scale graph, inset). (I to L) The autocorrelation functions of the data shown in (A) to (D), but computed over a field range ΔB larger than shown in (A) to (D): ΔB = (I) 5.4 T, (J) 5.3 T, (K) 0.6 T, and (L) 1.1 T (full data are shown in the SOM text).

A number of conclusions can be drawn from a qualitative examination of these data. First, Ityp(B) oscillates with a period h/e but also contains an aperiodic modulation that broadens the peaks in I˜typ(fΦ) and causes Gtyp(δB) to decay at large δB. This modulation is due to the fact that we apply a uniform B to the sample, leading to magnetic flux inside the metal of each ring given by ΦM = BAM where AM is the area of the metal projected along B. This leads to a new effective disorder potential [and hence a randomization of the phase of the I(B) oscillations] each time ΦM changes by ~Φ0 (15). As a result, the peaks in I˜typ(fΦ) span a band of fΦ roughly bounded by the rings’ inner and outer radii (the blue bars in Fig. 2, E to H), and the decay of Gtyp(δB) is found to occur on a field scale [defined as the half width at half maximum of Gtyp(δB)(15)] Bc = κΦ0/AM. Here κ is a constant that is predicted (16) to be 1; we find 1 < κ < 3 in these samples. For the array samples, ring-to-ring variations in r (estimated to be ~1%) should contribute negligibly to Bc and the peak widths in I˜typ(fΦ). The fact that the r = 418 nm array and the r = 418 nm single ring show similar peak width and Bc indicates that ring-to-ring variations in r do not affect the signal appreciably.

It is clear from Fig. 2 that the PC is smaller in larger rings. This is consistent with the prediction (17) that the typical amplitude Ih/e(T = 0) of the h/e-periodic Fourier component of I(Φ) at T = 0 corresponds roughly to the current produced by a single electron diffusing around the ring at the Fermi energy, and hence should scale as 1/r2. In addition, Ih/e(T) is predicted (17) to decrease on a temperature scale (known as the Thouless temperature) TT1/r2, corresponding to the scale of disorder-induced correlations in the ring’s spectrum of single-electron states.

In Fig. 2E, a small peak at fΦ = 2 can be seen, corresponding to the second harmonic of I(Φ). This harmonic has attracted particular attention because under some conditions, it has a component that is not random from ring to ring (4, 12, 18, 19). The signal from such a nonrandom “average” current would scale as IΣ(avg)N rather than N. Furthermore, the amplitude of IΣ(avg)can be strongly enhanced by electron-electron interactions (46) and other effects (8, 9). However, IΣ(avg) arises because of time-reversal symmetry within the metal, which in our experiments is broken by ΦM. We calculate that ΦM suppresses IΣ(avg) by a factor ~e2πr/1.3B (where the magnetic length B=h/eB), which for this experiment should render IΣ(avg) unobservably small. As a result, the peak in Fig. 2E at fΦ = 2 presumably reflects the random component of the second harmonic of I(Φ), which is predicted (17) to have a zero-temperature amplitude Ih/2e(0) = Ih/e(0)/23/2, to be suppressed on a temperature scale = TT/4, and to produce a signal with the same N scaling as Ih/e.

We now turn to a more quantitative analysis of the data. Theory predicts (17) that, for each independent realization of the disorder potential, Ih/pe [the pth harmonic of I(Φ)] is drawn randomly from a distribution with a mean Ih/pe=0 and a root mean square (rms) value Ih/pe21/2, which in general is nonzero. Here represents an average over disorder potentials. The quantity Ih/pe21/2 can be calculated explicitly as a function of r, T, p, and the electrons’ diffusion constant D for a variety of models.

To compare our data against these calculations, we make use of the fact that Ih/pe21/2 can be extracted from a measurement of IΣ(B) when the measurement record spans many Bc. When this condition is satisfied, averages performed with respect to B are equivalent to averages performed with respect to disorder realizations, and it is straightforward to show that the area under a peak in |I˜typ(fΦ)|2 (Fig. 2, E to H) at fΦ = p is simply related to Ih/pe21/2[(fΦ+fΦ)1fΦ_fΦ+(|I˜typ(fΦ)|2b(fΦ))dfΦ]1/2=Ih/pe21/2(1)Here b is the noise floor in |I˜typ(fΦ)|2 and is estimated from the portions of the data away from the peaks. We take the limits of integration fΦ+ and fΦ to be roughly the values of fΦ corresponding to h/pe flux periodicity through the outer and inner radii of the ring, respectively. In previous experiments, Ih/pe2 could only be determined by measuring several rings, one ring at a time (3, 20). This approach was limited by the low SNR achieved in single-ring measurements and practical limits on the number of nominally identical rings (15) that could be measured.

Measurements of Ih/e21/2 for each sample and Ih/2e21/2 for the smallest rings are shown as a function of T for θ = 45° (open symbols) and θ = 6° (solid symbols) (Fig. 3). It can be seen that the PC in larger rings decays more quickly with T than in smaller rings, and that Ih/2e21/2 decays more quickly than Ih/e21/2, which is consistent with the discussion above. In addition, the agreement between the data for the r = 418 nm array and the r = 418 nm single ring indicates that the PC signal scales as N and hence that the PC is random from ring to ring.

Fig. 3

Temperature dependence of the h/e and h/2e Fourier components of the current per ring. The vertical axis indicates Embedded Image and Embedded Image, the rms values of the Fourier amplitudes of the persistent current. In each data set, the open points were taken with θ = 45°, whereas for the solid points, θ = 6°. The arrows indicate the data points derived from I(B) measurements taken over a magnetic field range much greater than Bc; other data points are derived from the scaling of I(B) measured over a smaller range of B, as described in the SOM text. The lines (solid for array samples, dotted for the single ring) are fits to the prediction for non-interacting diffusive electrons. The electron diffusion constant D is the only fitting parameter and is listed in Table 1.

The solid lines are fits to theoretical predictions in which Ih/pe21/2 is calculated for diffusive non-interacting electrons. This calculation closely follows that of (17) but takes into account the presence of the large magnetic field B inside the metal (which lifts the spin degeneracy and breaks time-reversal symmetry) as well as spin-orbit scattering (the rings’ circumference exceeds the spin-orbit scattering length, as discussed in the SOM text). We findIh/pe2(T)=g(p2TTT)Ih/pe2(0)(2)where g(x)=π63x2n=1nexp[(2π3nx)1/2], Ih/pe2(0)1/2=0.37p3/23eD(2πr)2, and TT=π2DkB(2.πr)2

The data from each sample were fit separately, in each case using D as the only fitting parameter. The best-fit values of D are listed in Table 1. These values are typical for high-purity evaporated Al wires of the dimensions used here (21, 22); however, to further constrain the comparison between our data and theory, we also independently determined D from the resistivity of a co-deposited wire (the wire’s properties are listed in Table 1). This measurement is described in detail in the SOM text and provides a value of D in good agreement with the values extracted from the PC measurements. The values of D in Table 1 show a correlation with the samples’ linewidths, which may reflect the increased contribution of surface scattering in the narrower samples.

The calculation leading to Eq. 2 assumes the phase-coherent motion of free electrons around the ring. Measurements of the phase coherence length LΦ(T) in the co-deposited wire are described in the SOM text and show that LΦ r for nearly all the temperatures at which the PC is observable. The closest approach between LΦ and 2πr at a temperature where the PC can still be observed occurs in the 308-nm array at T = 3 K, where we find LΦ(3 K) = 1.86 × (2πr). It is conceivable that the more rapid decrease in Ih/e21/2 observed in this sample above T = 2 K (Fig. 3) is due to dephasing; however, it is not possible to test this hypothesis in the other samples, because the larger rings’ PC is well below the noise floor when LΦ(T) = 1.86 × (2πr). To the best of our knowledge the effect of dephasing on the PC has not been calculated.

Our measurement of the PC in normal metal rings over a wide range of temperatures, ring sizes, array sizes, magnetic field magnitudes, and magnetic field orientations with high SNR, excellent background rejection, and low measurement back-action indicates that the rings’ equilibrium state is well described by the diffusive non-interacting electron model. In addition to providing a clear experimental picture of PCs in simple metallic rings, these results open the possibility of using measurements of the PC to search for ultra–low temperature phase transitions (6) or to study a variety of many-body and environmental effects relevant to quantum phase transitions and quantum coherence in solid-state qubits (23, 24). Furthermore, the micromechanical detectors used here are well suited to studying the PC in circuits driven out of equilibrium (for example, by the controlled introduction of microwave radiation) (8). The properties of PCs in these regimes have received relatively little attention to date but could offer new insights into the behavior of isolated nanoelectronic systems.

Supporting Online Material

SOM Text

Figs. S1 to S18

References and Notes

References and Notes

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  25. We thank M. Devoret, R. Ilic, T. Ojanen, and J. C. Sankey for their assistance. A.C.B.-J., W.E.S., and J.G.E.H. are supported by NSF grants 0706380 and 0653377. F.v.O. is supported in part by the Deutsche-Israelische Projektkooperation. J.G.E.H. acknowledges support from the Sloan Foundation. A.C.B.-J. acknowledges support from UNESCO-L’Oreal. L.G. is supported in part by U. S. Department of Energy grant DE-FG02-08ER46482. F.v.O and L.G. acknowledge the hospitality of the Kavli Institute for Theoretical Physics in the final stages of this work.
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