Probing Johnson noise and ballistic transport in normal metals with a single-spin qubit

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Science  06 Mar 2015:
Vol. 347, Issue 6226, pp. 1129-1132
DOI: 10.1126/science.aaa4298

Listen to the quantum art of noise

Electrons in metals are subject to thermally induced noise that can generate tiny magnetic fields. For quantum electronic applications, the noise and magnetic fields can be damaging. Kolkowitz et al. show that the spin properties of single defects in diamond can be used to probe the noise. The findings provide insight into how the noise is generated, which could help to mitigate its damaging effects in sensitive quantum electronic circuits.

Science, this issue p. 1129


Thermally induced electrical currents, known as Johnson noise, cause fluctuating electric and magnetic fields in proximity to a conductor. These fluctuations are intrinsically related to the conductivity of the metal. We use single-spin qubits associated with nitrogen-vacancy centers in diamond to probe Johnson noise in the vicinity of conductive silver films. Measurements of polycrystalline silver films over a range of distances (20 to 200 nanometers) and temperatures (10 to 300 kelvin) are consistent with the classically expected behavior of the magnetic fluctuations. However, we find that Johnson noise is markedly suppressed next to single-crystal films, indicative of a substantial deviation from Ohm’s law at length scales below the electron mean free path. Our results are consistent with a generalized model that accounts for the ballistic motion of electrons in the metal, indicating that under the appropriate conditions, nearby electrodes may be used for controlling nanoscale optoelectronic, atomic, and solid-state quantum systems.

Understanding electron transport, dissipation, and fluctuations at submicrometer length scales is critical for the continued miniaturization of electronic (1, 2) and optical devices (35), as well as atom and ion traps (610), and for the electrical control of solid-state quantum circuits (11). Although it is well known that electronic transport in small samples defies the conventional wisdom associated with macroscopic devices, resistance-free transport is difficult to observe directly. Most of the measurements demonstrating these effects make use of ohmic contacts attached to submicrometer-scale samples and observe quantized but finite resistance corresponding to the voltage drop at the contact of such a system with a macroscopic conductor (12, 13). Techniques for noninvasive probing of electron transport are being actively explored (14, 15), because they can provide insights into electronic dynamics at small length scales. Our approach makes use of the electromagnetic fluctuations associated with Johnson noise close to a conducting surface, which can be directly linked to the dielectric function at similar length scales, providing a noninvasive probe of electronic transport inside the metal. Measurements of these fluctuations at micrometer length scales with cold, trapped atoms showed excellent agreement with predictions based on diffusive electron motion (79), whereas millimeter–length scale measurements with superconducting quantum interference devices (SQUIDs) have been demonstrated for use as an accurate, contact-free thermometer (16).

Our approach makes use of the electronic spin associated with nitrogen-vacancy (NV) defect centers in diamond to study the spectral, spatial, and temperature dependence of Johnson noise emanating from conductors. The magnetic Johnson noise results in a reduction of the spin lifetime of individual NV electronic spins, thereby allowing us to probe the intrinsic properties of the conductor noninvasively over a wide range of parameters. Individual, optically resolvable NV centers are implanted ∼15 nm below the surface of a ∼30-μm-thick diamond sample. A silver film is then deposited or positioned on the diamond surface (Fig. 1A). The spin sublevels Embedded Image and Embedded Image of the NV electronic ground state exhibit a zero-field splitting of Embedded ImageGHz (1720). The relaxation rates between the Embedded Image and Embedded Image states provide a sensitive probe of the magnetic field noise at the transition frequencies Embedded Image, where Embedded Image is the magnetic field along the NV axis, g ≈ 2 is the electron g-factor, and μB is the Bohr magneton (21, 22) (Fig. 1B).

Fig. 1 Probing Johnson noise with single-spin qubits.

(A) The thermally induced motion of electrons in silver generates fluctuating magnetic fields (Embedded Image), which are detected with the spin of a single NV. The NV is polarized and read out through the back side of the diamond. (B) The NV spin is polarized into the Embedded Imagestate using a green laser pulse. Spin relaxation into the Embedded Image states is induced by magnetic field noise at ∼2.88 GHz. After wait time τ, the population left in Embedded Image is read out by spin-dependent fluorescence. All measurements shown were performed at low magnetic fields (Embedded Image). (C) Spin relaxation data for the same single shallow-implant NV before silver deposition (open blue squares), with silver deposited (red circles) and after the silver has been removed (open blue triangles). (D) Spin relaxation for a single NV close to a silver film prepared in the Embedded Image state (red circles) and in the Embedded Image state (open orange circles). (Inset) Spin relaxation for a single native NV in bulk diamond in the Embedded Image state (blue circles) and in the Embedded Image state (open light blue circles).

The impact of Johnson noise emanating from a polycrystalline silver film deposited on the diamond surface (Fig. 1C) is evident when comparing the relaxation of a single NV spin below the silver (red circles) to the relaxation of the same NV before film deposition and after removal of the silver (open blue squares and triangles, respectively). At room temperature and in the absence of external noise, the spin lifetime is limited by phonon-induced relaxation to Embedded Image ms. With the silver nearby, the lifetime of the Embedded Image state is reduced to Embedded Image μs, which we attribute to magnetic Johnson noise emanating from the film. To verify that the enhanced relaxation is due to magnetic noise, we compare the lifetime of the Embedded Image state, which has magnetic dipole allowed transitions to both of the Embedded Image states, to that of the Embedded Image state, which can only decay directly to the Embedded Image state (Fig. 1D). As expected for relaxation induced by magnetic noise, the Embedded Image state has approximately twice the lifetime of the Embedded Image state (23). This is in contrast to the observed lifetimes when limited by phonon-induced relaxation (Fig. 1D, inset), where the Embedded Image and Embedded Image states have almost identical lifetimes (24). In what follows, we define Embedded Image as the lifetime of the Embedded Image state.

To test the scaling of Johnson noise with distance (d) to the metal, we deposit a layer of SiO2 on the diamond surface with a gradually increasing thickness (Fig. 2A). We characterize the thickness of the SiO2 layer as a function of position on the sample (Fig. 2B, inset) and deposit a 60-nm polycrystalline silver film on top of the SiO2. The conductivity of the silver film is measured to be 2.9 × 107 S/m at room temperature. By measuring the relaxation rates Embedded Image of individual NVs at different positions along the SiO2 ramp, we extract the distance dependence of the noise (Fig. 2B), with the uncertainty in the distance dominated by the variation in the implanted depth of the NVs (taken to be Embedded Image nm). To ensure that the measured rates are Johnson noise limited, we measure the spin relaxation of 5 to 10 randomly selected NVs per location along the ramp and plot the minimum observed rate at each location (23). As expected (79), the magnitude of the noise increases as the NVs approach the silver surface.

Fig. 2 Distance dependence of NV relaxation close to silver.

(A) A gradual SiO2 ramp (slope of ∼0.2 nm/μm) is grown on the diamond surface, followed by a 60-nm silver film. (B) The NV relaxation rate is measured as a function of position along the ramp, which is then converted to distance to the film. At each point, 5 to 10 NV centers are measured, and the minimum rate measured is plotted (red circles). The horizontal error bars reflect 1 SD in the estimated distance to the film including the uncertainty in NV depth, while the vertical error bars reflect 1 SD in the fitted relaxation rate. The red dashed line shows the expected relaxation rate with no free parameters after accounting for the finite silver film thickness. (Inset) Thickness of the ramp as a function of lateral position along the diamond sample (blue curve). The red crosses correspond to the positions along the sample where the measurements were taken.

To investigate the dependence of the noise on temperature and conductivity, we deposit a 100-nm polycrystalline silver film on a diamond sample and measure the Embedded Image of a single NV beneath the silver over a range of temperatures (∼10 to 295 K). The measured relaxation rate for a single NV near the silver increases with temperature (red circles in Fig. 3A), as expected for thermal noise, but the scaling is clearly nonlinear. This can be understood by recognizing that the conductivity of the silver film is also a function of temperature and that the magnitude of the thermal currents in the silver depends on the conductivity. To account for this effect, a four-point resistance measurement of the silver film is performed to determine the temperature dependence of the bulk conductivity of the silver film (Fig. 3B).

Fig. 3 Temperature dependence of NV relaxation close to polycrystalline silver.

(A) The measured relaxation rate of a single NV spin under a polycrystalline silver film as a function of temperature (red data points). The error bars reflect 1 SD in the fitted relaxation rate. The conductivity of the silver film as a function of temperature shown in (B) is included in a fit to Eq. 2, with the distance to the film as the single free parameter (red dashed line). The extracted distance is Embedded Image nm. (B) The conductivity of the 100-nm-thick polycrystalline silver film deposited on the diamond surface is measured as a function of temperature. (Inset) Grain boundaries within the polycrystalline silver film, imaged using electron backscatter diffraction. The average grain diameter is 140 nm, with a SD of 80 nm.

To analyze the dependence of the NV spin relaxation rate on distance, temperature, and conductivity, we use the model of (6), in which an electronic spin-1/2 qubit with Larmor frequency Embedded Image is positioned at a distance d from the surface of a metal. For silver at room temperature, the skin depth at Embedded Image is Embedded Image μm; consequently, when Embedded Image nm, we are in the “quasi-static” limit Embedded Image. The thermal limit Embedded Image is valid for all temperatures in this work. In this regime, the magnetic noise spectral density perpendicular to the silver surface is given byEmbedded Image(1)where Embedded Image is the temperature-dependent conductivity of the metal as defined by the Drude model. This scaling can be intuitively understood by considering the magnetic field generated by a single thermal electron in the metal at the NV position, Embedded Image, where the thermal velocity Embedded Image, Embedded Image is the effective mass of electrons in silver, and Embedded Image is the electron charge. In the limit Embedded Image, screening can be safely ignored, and the NV experiences the magnetic field spectrum arising from N independent electrons in a volume V; Embedded Image, where Embedded Image is the electron density and Embedded Image is the correlation time of the noise, given by the average time between electron scattering events; and Embedded Image, where Embedded Image is the electron mean free path and Embedded Image is the Fermi velocity. Recognizing that the NV is sensitive to the motion of electrons within a sensing volume Embedded Image, we arrive at the scaling given by Eq. 1, with Embedded Image. Applying Fermi’s golden rule and accounting for the orientation and spin-1 of the NV yields the relaxation rate for the Embedded Image stateEmbedded Image (2)where Embedded Image is the electron g-factor, Embedded Image is the Bohr magneton, and Embedded Image is the angle of the NV dipole relative to the surface normal vector (23). In Fig. 2B, the inverse scaling with distance d predicted by Eq. 1 is clearly evident for NVs very close to the silver. At distances comparable to the silver film thickness, Eq. 1 is no longer valid, but we recover excellent agreement with the no-free-parameters prediction of Eq. 2 by including a correction for the thickness of the silver film (red dashed line in Fig. 2B), which is measured independently. The measured relaxation rates as a function of temperature are also in excellent agreement with the predictions of Eq. 2 (red dashed line in Fig. 3A), while the extracted distance of Embedded Image nm is consistent with the expected depth (23).

Notably, very different results are obtained when we replace the polycrystalline film with single-crystal silver. For this experiment, a 1.5-μm-thick single-crystal silver film grown by sputtering onto silicon (23, 25, 26) is placed in contact with the diamond surface. The measured conductivity of the single-crystal silver exhibits a much stronger temperature dependence (blue line in Fig. 4A) as compared to that of the 100-nm-thick polycrystalline film. Figure 4B presents the measured relaxation rate as a function of temperature for an NV in a region in direct contact with the single-crystal silver (blue squares). The dashed blue line corresponds to the temperature-dependent rate predicted by Eq. 2, which strongly disagrees with the experimental results. Specifically, because the measured silver conductivity increases faster than the temperature decreases in the range from room temperature down to 40 K, Eq. 2 predicts that the relaxation rate should increase as the temperature drops, peaking at 40 K and then dropping linearly with temperature once the conductivity saturates. Instead, the Embedded Image of the NV consistently increases as the temperature drops, implying that at lower temperatures, the silver produces considerably less noise than expected from Eq. 2. We observe similar deviation from the prediction of Eq. 2 for all 23 NVs measured in the vicinity of the single-crystal silver (23).

Fig. 4 Temperature dependence of NV relaxation close to single-crystal silver.

(A) Measured conductivity of single-crystal (blue curve) and polycrystalline (red curve, same as Fig. 3B) silver as a function of temperature. (Inset) Electron backscatter diffraction image of the single-crystal silver film showing no grain boundaries, and the observed diffraction pattern. (B) Relaxation of a single NV spin under single-crystal silver as a function of temperature (blue squares). The error bars reflect 1 SD in the fitted relaxation rate. Equation 2 is fit to the data from 200 to 295 K (blue dashed line). A nonlocal model (23) is fit to the data (blue solid line); the extracted distance between the NV and the silver surface is Embedded Image nm. (C) Cartoon illustrating the relevant limits, where the noise is dominated by diffusive electron motion (left, Embedded Image) and ballistic motion (right, Embedded Image). (D) The same data as in (B) were taken for 23 NVs at varying distances from the film. The Embedded Image of each NV at 103 K (top) and 27 K (bottom) is plotted against the extracted depth (blue triangles). The horizontal error bars reflect 1 SD in the fitted distance to the film, while the vertical error bars reflect 1 SD in the fitted relaxation time. The nonlocal model (solid colored lines) saturates at a finite lifetime determined by Eq. 3 (bottom, dashed black line), whereas the local model does not (dashed colored lines).

To analyze these observations, we note that the conventional theoretical approach (6) resulting in Eq. 2 treats the motion of the electrons in the metal as entirely diffusive, using Ohm’s law, Embedded Image, to associate the bulk conductivity of the metal with the magnitude of the thermal currents. While accurately describing the observed relaxation rates next to the polycrystalline material, where the resistivity of the film is dominated by electron scattering off grain boundaries (Fig. 3B, inset), this assumption is invalid in the single-crystal silver film experiments, particularly at low temperatures. Here, the measured conductivity of the single-crystal film indicates that the mean free path Embedded Image is greater than 1 μm, considerably exceeding the sensing region determined by the NV-metal separation, and thus the ballistic motion of the electrons must be accounted for. Qualitatively, the correlation time of the magnetic noise in this regime is determined by the ballistic time of flight of electrons through the relevant interaction region Embedded Image (Fig. 4C). This results in a saturation of the noise spectral density and the spin relaxation rate Embedded Image as either the NV approaches the silver surface or the mean free path becomes longer at lower temperatures (23), with the ultimate limit to the noise spectrum given by:

Embedded Image(3)

This regime of magnetic Johnson noise was recently analyzed theoretically (11) using the Lindhard form nonlocal dielectric function for the metal modified for finite electron scattering times (23, 27, 28). Comparison of this model (solid line in Fig. 4B) to the data, with distance again as the only free parameter, yields excellent agreement for all 23 measured NVs (23). Figure 4D shows the measured Embedded Image times at 103 and 27 K for each NV as a function of extracted distance (blue triangles). Of the 23 NVs measured, 15 are in a region of the diamond sample in direct contact with the silver (23). Excellent agreement between the nonlocal model (solid lines) and the data is observed for all 23 NVs at all 12 measured temperatures. Apparent in Fig. 4D is the saturation of the relaxation rate as the NV approaches the silver surface and as the mean free path becomes longer at lower temperatures (dashed black line), as predicted by Eq. 3.

Although ballistic electron motion in nanoscale structures has previously been studied and utilized (12, 13), our approach allows for noninvasive probing of this and related phenomena and provides the possibility for studying mesoscopic physics in macroscopic samples. The combination of sensitivity and spatial resolution demonstrated here enables direct probing of current fluctuations in the proximity of individual impurities, with potential applications such as imaging of Kondo states and probing of novel two-dimensional materials (29), where our technique may allow for the spatially resolved probing of edge states (12). Likewise, it could enable investigation of the origin of Embedded Image flux noise by probing magnetic fluctuations near superconducting Josephson circuits (30, 31). Finally, as Johnson noise presents an important limitation to the control of classical and quantum mechanical devices at small length scales (610), the present results demonstrate that this limitation can be circumvented by operating below the length scale determined by the electron mean free path.

Supplementary Materials

Materials and Methods

Figs. S1 to S7

Tables S1 to S3

References (3234)

References and Notes

  1. Materials and methods are available as supporting material on Science Online.
  2. Acknowledgments: We thank E. Demler, A. Bleszynski Jayich, B. Myers, A. Yacoby, M. Vavilov, R. Joynt, A. Poudel, and L. Langsjoen for helpful discussions and insightful comments. Financial support was provided by the Center for Ultracold Atoms, the National Science Foundation (NSF), the Defense Advanced Research Projects Agency Quantum-Assisted Sensing and Readout program, the Air Force Office of Scientific Research Multidisciplinary University Research Initiative, and the Gordon and Betty Moore Foundation. S.K. and A.S. acknowledge financial support from the National Defense Science and Engineering Graduate fellowship, V.E.M. from the Society of Fellows of Harvard University, and S.K. from the NSF Graduate Research Fellowship. All fabrication and metrology were performed at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Infrastructure Network, which is supported by the NSF under award no. ECS-0335765. The CNS is part of Harvard University.
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