Single-electron current sources: Toward a refined definition of the ampere

The control of electrons at the level of the elementary charge $e$ was demonstrated experimentally already in the 1980s. Ever since, the production of an electrical current $ef$, or its integer multiple, at a drive frequency $f$ has been a focus of research for metrological purposes. This review discusses the generic physical phenomena and technical constraints that influence single-electron charge transport and presents a broad variety of proposed realizations. Some of them have already proven experimentally to nearly fulfill the demanding needs, in terms of transfer errors and transfer rate, of quantum metrology of electrical quantities, whereas some others are currently “just” wild ideas, still often potentially competitive if technical constraints can be lifted. The important issues of readout of single-electron events and potential error correction schemes based on them are also discussed. Finally, an account is given of the status of single-electron current sources in the bigger framework of electric quantum standards and of the future international SI system of units, and applications and uses of single-electron devices outside the metrological context are briefly discussed.

Figure 1

Schematic diagram of the basic circuit for manipulating individual electrons, the single-electron box (SEB): a conducting island carrying electric charge $en$, and an electrostatically coupled external electrode with the charge $e(N-n)$ producing the gate voltage $V_g$.

Figure 2

Practical SEB. (a) Scanning electron micrograph of a realistic box structure, (b) its equivalent electric circuit, and (c) single-electron transitions in the box illustrating the “charge quantization”: a time-dependent gate voltage $V_g(t)$ (sinusoidal curve) of an appropriate amplitude drives individual electron transitions changing the box state between the two discrete charge configurations, the electron on the left or on the right island. These two charge states are detected via the detector shown in the upper right corner of (a), whose two-level output current is synchronous with the oscillating $V_g(t)$. Adapted from 334 and 335.

Figure 3

High-resolution TEM image of a cross section of an aluminum oxide tunnel junction. From 320.

Figure 4

(a) Measured thermally activated rates of forward $\Gamma (E)$ and backward $\Gamma (-E)$ tunneling in a “hybrid” SIN single-electron box at different temperatures as a function of the gate-voltage offset from the degeneracy point related to the energy change $E$ in tunneling as $E=2E_C(n_g-1/2)$. Solid lines are the theory prediction according to Eq. (9) with fitted parameters $E_C=157\mu \mathrm{eV}$, $\Delta =218\mu \mathrm{eV}$, and $1/G_T=100\mathrm{M}\Omega$. (b) The tunneling rate at degeneracy $E=0$ as a function of temperature (squares), and best fit (solid line) to Eq. (9). Adapted from 335.

Figure 5

Equivalent electric circuit of an SET.

Figure 6

Real-time detection of Andreev tunneling in an isolated SEB shown in the scanning electron micrograph of (a) and its schematic in (b). The electrometer is used for counting the single-electron and Andreev tunneling rates. (c) The tunneling rate for AR shown as dots for forward and backward directions. The lines are theoretical calculations where the nonuniformity of the tunnel barrier is taken into account. Adapted from 251.

Figure 7

Single-electron control in a semiconductor structure consisting of two lateral quantum dots measured with a quantum-point-contact (QPC) charge detector. (a) Atomic force microscope image of the structure. (b) The diagram of the equilibrium charge states of the two dots, controlled by voltages on the gates $G1$ and $G2$: empty dots (0); left ($L$), right ($R$), or both dots (2), occupied with one electron. (c) Trace of the output signal of the QPC detector (conductance $G_{\mathrm{QPC}}$) showing random single-electron transitions between these states driven by thermal fluctuations close to the degeneracy point, when the charging energies of the states (0), ($L$), and ($R$) coincide. From 220.

Figure 8

A simple schematic showing photon absorption by a generic tunnel junction, and the inelastic electron tunneling from the left side of the barrier to the right.

Figure 9

NIS junctions influenced by a hot environment. (a) Geometry of a NIS junction made of aluminum (low contrast) as the superconductor and copper (high contrast) as the normal metal. The tapered ends lead to large pads. (b) Typical $I\mathrm{\text{−}}V$ characteristics, measured at 50 mK for a junction with $R_T=30\mathrm{k}\Omega$. Linear leakage, i.e., nonvanishing subgap current due to coupling to the environment, can be observed. The dotted line is the corresponding theoretical line from the $P(E)$ theory and $RC$ environment with dissipation $R$ at $T_{\mathrm{env}}=4.2\mathrm{K}$. (c) Measured $I\mathrm{\text{−}}V$ curves of an NIS junction with $R_T=761\mathrm{k}\Omega$ on a ground plane providing a large protecting capacitance against thermal fluctuations (solid symbols) and of a similar junction with $R_T=627\mathrm{k}\Omega$ without the ground plane (open symbols). Solid lines present the theoretical results for capacitance $C=10$ and 0.3 pF. The resistance and the temperature of the environment are set to $R=2\Omega$ and $T_{\mathrm{env}}=4.2\mathrm{K}$, respectively. The inset shows $I\mathrm{\text{−}}V$ curves based on the full $P(E)$ calculation as functions of the shunt capacitance $C$. The colored lines are reproduced on this graph from the main figure. Adapted from 300.

Figure 10

(a) Radiative heat flow is caused by the photons which carry energy between resistors $R_1$ and $R_2$ at temperatures $T_1$ and $T_2$, respectively. The heat transport can be modeled by having voltage fluctuations $\delta V_i$ as shown in (b). Here we have assumed total transmission. The assumption can be relaxed by adding a nonzero impedance to the loop.

Figure 11

A three-junction pump. Schematics shown in (a), where the pump is biased by voltage $V$ and with gate voltages $U_1$ and $U_2$. (b) The stability diagram of the three-junction pump on the plane of the gate voltages at zero bias voltage. For operation of the pump, see text. From 315.

Figure 12

Predicted relative cotunneling-induced error vs inverse temperature for multijunction pumps, with $N=4$ (circles) and $N=5$ (squares). The computer simulations (points) and the predictions of analytic results (lines) are shown. Parameters are $R_T=20R_K$, $f=4\times {10}^{-4}/R_{T}C$, and $CV/e=-0.15$. Adapted from 176.

Figure 13

The seven-junction pump. (Left) The schematic of the pump, with six islands, each with a gate. The electrons are pumped to and from the external island on the top, and the charge on the island is detected by a single-electron electrometer. (Middle) The voltage $V_p$ on the external island vs time when pumping $\pm e$ with a wait time of 4.5 s in between. (Right) The pumping error vs temperature of the measurement, demonstrating the 15 ppb accuracy at temperatures below 100 mK. Adapted from 200.

Figure 14

The hybrid NIS turnstile. Top left: A scanning electron micrograph of a SINIS turnstile, which is a hybrid single-electron transistor with superconducting leads and a normal-metal island. Top right and bottom: Current of a turnstile under rf drive on the gate at different operation points with respect to the dc gate position and the rf amplitude of the gate. Adapted from 302 and 203.

Figure 15

Schematic picture of pumping (a) with a normal SET, (b) with a hybrid SET with $E_C=\Delta$, and (c) with a hybrid SET with $E_C=2\Delta$. The shaded areas are the stability regions of the charge states $n=0$ and 1. The edges of the normal SET stability regions are drawn in all figures with dashed black lines. The long shaded lines represent the transition thresholds from states $n=0$ and 1 by tunneling through the left ($L$) or the right ($R$) junction in the wanted forward ($F$, solid line) or unwanted backward ($B$, dashed line) direction. The thick black line corresponds to pumping with constant bias voltage $e{V}_b/\Delta =1$ and a varying gate voltage. From 203.

Figure 16

(a) Stability diamonds for single-electron tunneling (solid lines) and Andreev tunneling (dotted lines) for a sample with $E_C>\Delta$. (b) Stability diamonds for $E_C<\Delta$. (c) The first pumping plateau of the high-$E_C$ device as a function of the gate-voltage amplitude $A_g$. The solid symbols show pumped current with $f=10\mathrm{MHz}$ and three different bias voltages. Dotted lines are the simulated traces with the corresponding biasing. (d) The same data as in (c) but now for the low-$E_C$ device showing excess current due to Andreev tunneling. Adapted from 9.

Figure 17

Two typical geometries for a superconducting bias lead: (a) A lead having a constant cross section determined by the thickness $d$ and width $w$. The length of the line is $l$. (b) A sector-shaped lead characterized by an opening angle $\theta$, initial radius $r_0$, and final radius $r_t$. For the picture $\theta$ is set to 180°. The colored parts on top denote a quasiparticle trap connected via an oxide barrier. (c) Quasiparticle density $n_{\mathrm{qp}}$ along a constant-cross-section line with various oxide trap transparencies $k$, and (d) along an opening line. In the plots, $n_{\mathrm{qp}}$ is scaled by $n_i=Dl{P}_{\mathrm{inj}}/A_i$, where the injection area $A_i$ equals $wd$ for (c) and $\theta r_{0}d$ for (d). For the leads in (b) and (d), we also use the notation $x=(r-r_0)/l$ with $l=r_t-r_0$ and have used values $r_0=20\mathrm{nm}$ and $r_t=5\mu \mathrm{m}$.

Figure 18

(a) Lateral and (b) vertical quantum-dot arrangements. All quantum-dot pumps and turnstiles discussed are in the lateral arrangement. The electrons tunnel between the dot and the source and drain reservoirs. The tunnel barriers between the dot and the reservoirs are created either by the electrostatic potentials of nearby gate electrodes or by different materials such as AlGaAs. The gate arrangement for (c) the accumulation and (d) depletion mode quantum dots in the lateral arrangement.

Figure 19

The first single-electron current source based on quantum dots by 218, 217. (a) SEM image of the device from the top; (b), (c) operation principle; and (d) measured $I\mathrm{\text{−}}V$ curves reported. The gate configuration corresponds to the case in Fig. 18d. The different $I\mathrm{\text{−}}V$ curves are measured while driving the turnstile with different center-gate [gate C in (a)] voltages, rf amplitudes, and phase differences. The curves are not offset and the dashed lines show the current levels $nef$ with $n=-5,\dots ,5$. Adapted from 218, 217.

Figure 20

(a) Schematic illustration of the device and (b) observed current plateaus during the turnstile operation. From 280.

Figure 21

(a) SEM image of the device and (b) pumped current through it in the experiments by 2. Different values of the current correspond to different dc voltages $V_{G1}$ [see (a)]. Adapted from 2.

Figure 22

(a) SEM image of the device and (b) observed current plateaus up to 1 MHz pumping frequency on a silicon quantum dot. From 295.

Figure 23

(a) Measured current plateaus for different frequencies of the turnstile operation with the device shown in Fig. 22a. From 296. (b) SEM image of the silicon quantum-dot device and a schematic measurement setup employed in the experiments by 64. (c) Measured current plateaus (solid line) and the corresponding theoretical curve (dashed line) by 64. The insets show zooms at the $n=0$ (bottom) and $n=1$ (top) plateaus. The dashed lines show $\pm {10}^{-3}$ relative deviation from the ideal $ef$ level. From 64.

Figure 24

Current plateau in the electron pumping experiments as a function of the middle-gate voltage at 547 MHz operation frequency. The dashed lines show $\sigma =\pm 10\mathrm{fA}$ uncertainty in the electrometer calibration. The top left inset shows the device used as the electron pump. The top right inset shows current plateaus at 1 GHz pumping frequency and the bottom inset shows the pumped current as a function of the operation frequency. From 48.

Figure 25

(a) SEM image of the device with a schematic measurement setup. (b) Current plateaus obtained by using a sine wave drive at different frequencies and magnetic fields. (c) Relative difference of the pumped current from $ef$ using a sine wave form and a tailored arbitrary wave form at different frequencies. The rightmost data point denoted by an asterisk shows the result with the potential of the entrance gate shifted by 10 meV from the optimal operation point. From 131.

Figure 26

(a) Scanning electron micrograph of the sluice used in the experiments by 384 with a simplified measurement setup. (b) Magnified view of the island of the device shown in (a) with four Josephson junctions. (c) Measured pumped current with the sluice (solid lines) as a function of the magnitude of the gate-voltage ramp such that $n$ corresponds to the ideal number of elementary charges $e$ pumped per cycle. The inset shows the steplike behavior observed in the pumped current. From 384.

Figure 27

Pumped average charge by 124 for a single pumping cycle of a sluice pump near vanishing voltage bias as a function of the gate charge offset $n_g$ and span during the pumping cycle $\Delta n_g$. From 124.

Figure 28

Schematic of the single-charge injector and its operation principle. Starting from step 1 where the Fermi energy level of the conductor lies in between two energy levels of the dot, its potential is increased by $\Delta$ by moving one occupied dot level above the Fermi energy (step 2). One electron then escapes from the dot. After that the potential is brought back to the initial value (step 3), where one electron can enter the dot, leaving a hole in the conductor. One edge channel of the quantum $RC$ circuit is transmitted into the dot, with transmission $D$ tuned by the QPC gate voltage $V_{\mathrm{G}}$. From 102.

Figure 29

ac quantization. $|I_w|$ as a function of $2e{V}_{\mathrm{exc}}/\Delta$ for different dot potentials at $D\approx 0.2$ (left) and $D\approx 0.9$ (right). Dots are measured values and lines are theoretical predictions. Insets schematically show the dot density of states $N(\epsilon )$. The vertical lines indicate the dot potential for the corresponding experimental data. From 102.

Figure 30

(a) Schematic layout of the self-assembled quantum-dot electron pump, (b) transmission electron microscopy image of the quantum dots, (c) 3D sketch of the conduction band profile of the structure under zero bias, and (d) saturation current for two different pump wavelengths ($\lambda =6.7\mu \mathrm{m}$: curve A and $\lambda =10\mu \mathrm{m}$: curve B). The difference provides a current plateau that should be $2ef$ (thick horizontal line). Inset: Variations of the measured current with respect to the average value. From 283.

Figure 31

(a) Model of the shuttle device proposed by 138). (Top) Dynamic instabilities occur since in the presence of a sufficiently large bias voltage $V$ the grain is accelerated by the electrostatic force toward the first electrode, then toward the other one. A cyclic change in direction is caused by the repeated loading of electrons near the negatively biased electrode and the subsequent unloading of the same charge at the positively biased electrode. As a result, the sign of the net grain charge alternates, leading to an oscillatory grain motion and charge transport. (Bottom) Charge variations on a cyclically moving metallic island. The dashed lines in the middle describe a simplified trajectory in the charge-position plane, when the island motion by $\delta x$ and discharge by $2\delta q$ occur instantaneously. The solid trajectory describes the island motion at large oscillation amplitudes. Periodic exchange of the charge $2q=2CV+1$ between the island and the leads results in the net shuttle current $I=2\delta qf$, where $f$ is the shuttle frequency Adapted from 138. (b) Scanning electron micrograph of a nanopillar between two electrodes. From 208. (c) Electron micrograph of the quantum bell: The Si beam (clapper) is clamped on the upper side of the structure. ac gates $G1$ and $G2$ are used for the actuation of the clapper $C$. Electron transport is measured from source $S$ to drain $D$ through the island on top of the clapper. From 94. (d) A false-color SEM image of the nanomechanical SET. A gold island is located at the center of a doubly clamped freely suspended silicon nitride string. The gold island can shuttle electrons between the source and drain electrodes when excited by ultrasonic waves. From 210.

Figure 32

(a) Scanning electron micrograph of parallel turnstiles. The turnstiles are biased with a common bias $V_b$ and driven with a common rf gate voltage $V_{\mathrm{rf}}$. Gate offset charges are compensated by individual gate voltages $V_{g,i}$. (b) Output current $I$ for ten parallel devices tuned to the same operating point producing current plateaus at $I=10Nef$. The curves are taken at different $V_b$ shown in the top left part of the panel. From 250.

Figure 33

(a) Circuit diagram of a charge-counting device. Electric charge $Q$ on the island on the left is monitored. The island is coupled to an electrometer island via capacitor $C_x$ and also tunnel coupled to an external conductor. The single-electron box configuration illustrated here requires only one tunnel junction with capacitance $C_j$. In addition, there is capacitance $C_0$ to ground, which accounts also for gate electrodes and any parasitic capacitances. The probing current $I_{\det }$ through the detector is sensitive to the charge on the coupling capacitor, which is a fraction $C_x/C_{\Sigma }$ of the total charge $Q$, where $C_{\Sigma }=C_x+C_0$. The detector is a single-electron tunneling transistor based on Coulomb blockade, and hence the total capacitance of the detector island $C_{\det }$ is of the order of 1 fF or less. (b) Circuit diagram of a general noisy electrical amplifier that can also be adapted to describe the electrometers of single-electron experiments. From 82. For the configuration shown in (a), one has for input impedance $Z_{\mathrm{in}}(\omega )=1/j\omega C_{\mathrm{in}}$, where $C_{\mathrm{in}}^{-1}=C_x^{-1}+C_{\det }^{-1}$. The input voltage is related to the island charge $Q$ through $V_{\mathrm{in}}=Q/C_{\Sigma }$. The noise source $I_N$ represents backaction and $V_N$ the noise added by the electrometer at the output referred to the input. The gain of the amplifier is given by $G$. The output impedance $Z_{\mathrm{out}}$ equals the differential resistance at the amplifier operation point.

Figure 34

Detector backaction mechanisms. The backaction can originate by direct electromagnetic (EM) coupling either by variations in pump biasing or by high-frequency photon-assisted tunneling (PAT). Another source of backaction is via heat conduction. The detector located in proximity of the device typically heats up. The heat can then be conducted to the device by either phononic or photonic coupling.

Figure 35

Circuit diagram of a practical implementation of the electron-counting capacitance standard. Switches NS1 and NS2 are cryogenic needle switches. From 61.

Figure 36

Fabrication of metallic devices. (a) Buildup of a trilayer resist structure and exposure in the electron-beam writer; (b) development of the top PMMA layer; (c) transfer of the pattern formed in the resist into the Ge layer by reactive ion etching; (d) creation of the undercut in the bottom resist and removal of the top resist by oxygen plasma; (e) angle deposition of metals with an oxidation in between; (f) the resulting structure after the lift-off process.

Figure 37

Simplified sketch of the most accurate routes to information on ${ϵ}_{J,K}$. Direct measurement of $R_K$ together with an independent measurement of the fine structure constant ($\alpha$) yields a value for ${ϵ}_K$. Values for the sum of ${ϵ}_J$ and ${ϵ}_K$ can be obtained from the combination of the so-called watt balance experiment ($K_J^2{R}_K$) and a measurement of the Avogadro constant ($N_A$), or from the combination of $\alpha$ and measurements of low-field gyromagnetic ratios (${\gamma }_{\mathrm{lo}}$). Less accurate information is provided by measurements of high-field gyromagnetic ratios $K_J$ and the Faraday constant $F=e{N}_A$, and by the QMT.

Figure 38

(a)–(c) Variants of Ohm’s law triangles where the quantized current ($I_S$) is compared to resistance ($R$) calibrated against QHR and to JVS. (a) The quantized current is magnified by a CCC, which allows room-temperature null detection of the voltage difference ($\Delta V$). (b) Triangle with a high-value cryogenic resistor. The current balance $\Delta I$ can be determined, e.g., with the help of a CCC. (c) QMT experiment where the null detection is performed by a room-temperature transimpedance amplifier. (d) ECCS experiment. In the first phase (A), the electron pump charges the cryocapacitor $C\approx 2\mathrm{pF}$. An SET electrometer ($E$) is used to generate a feedback voltage ($V$) that maintains the potential of the island at zero. Hence all the charge is accumulated to the cryocapacitor and not to the stray capacitance. The feedback voltage constitutes the third part of the $Q=CV$ type triangle. In the second phase (B), the cryocapacitor is calibrated against the reference $C_{\mathrm{ref}}$ which is traceable to a calculable capacitor. From 199.