Photon-Induced Kondo Satellites in a Single-Electron Transistor

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Science  28 May 2004:
Vol. 304, Issue 5675, pp. 1293-1295
DOI: 10.1126/science.1096377


We measure the differential conductance of a single-electron transistor (SET) irradiated with microwaves. The spin-entangled many-electron Kondo state produces a zero-bias peakin the dc differential conductance if the quantum dot in the SET contains an unpaired electron. When the photon energy hf is comparable to the energy width of the Kondo peak and to e (the charge on the electron) times the microwave voltage across the dot, satellites appear in the differential conductance shifted in voltage by ±hf/e from the zero-bias resonance. We also observe an overall suppression of the Kondo features with increasing microwave voltage.

The observation of the Kondo effect in nanometer-size semiconductor structures (1, 2) has caused a renaissance in the study of this quantum mechanical many-body phenomenon. It has been known since the 1930s [see, e.g., (3)] that a small concentration of magnetic impurities in a metal radically changes the conductance at low temperatures. However, only with the development of scaling and renormalization group theory techniques could the equilibrium properties of this strongly interacting system be predicted. The Kondo problem involves the coupling between an unpaired electron localized on an impurity and the surrounding delocalized electrons in a metal. The coupling leads to screening of the localized electron's spin by the delocalized electrons with opposite spin, so that a spin-zero singlet is formed below the Kondo temperature TK. In 1988 it was proposed on the basis of theory (4, 5) that a quantum dot containing an unpaired electron coupled to conducting leads would be analogous to a magnetic impurity coupled to its host metal. Such a quantum dot coupled to two leads (the drain and the source), with a gate electrode nearby, is a SET. In the absence of the Kondo coupling, the conductance of a SET at zero drain-source bias (Vds = 0) is very small, except for values of the gate voltage at which two charge states of the quantum dot are degenerate. Thus, the zero-bias conductance as a function of gate voltage consists of a series of Coulomb charging peaks, one for each electron added to the dot. The Kondo effect enhances the conductance between these peaks when the dot contains a spin, because the screening of the spin creates a new spin-entangled many-electron quantum state that extends from one lead through the dot into the other lead.

SETs provide new ways of studying the Kondo effect that are not possible with magnetic impurities in metals (6). In particular, the capability of applying a voltage between the two leads of a SET makes it possible to study the Kondo effect out of equilibrium. It has been predicted that one such nonequilibrium phenomenon, photon-assisted Kondo conductance, should provide a new spectroscopy of the Kondo singlet (715).

Figure 1 shows the density of states in each lead of the SET. At equilibrium, the Kondo coupling results in a peak in the density of states at the Fermi level of width ∼kBTK (Fig. 1A). The differential conductance Gd = dI/dVds is determined by the tunneling of electrons from one lead into the density of states of the other lead, and is enhanced near zero bias because of the peak in the density of states. Applying a bias Vds (Fig. 1B) causes Gd to decrease. Thus, the signature of the Kondo effect is a sharp peak in Gd at Vds = 0.

Fig. 1.

Density of states in a SET. The vertical axis is electron energy. The horizontal direction is a schematic cross section of the device, showing the drain lead, the SET island (central region), and the source lead. (A) At zero drain-source bias (Vds = 0), when the Fermi levels in the drain and source (Ed and Es, respectively) are equal, the singularities in the two leads overlap, giving rise to Kondo-enhanced differential conductance. (B) At finite drain-source bias, the Kondo singularities are not at the same energy, and the excess differential conductance at zero bias is eliminated. (C) Restoration of the Kondo effect at finite bias with high-frequency radiation. The radiation-induced density of states in each lead contains satellites of the main Kondo singularity, offset by integer multiples of the photon energy hf. When eVds = hf, a peak in the differential conductance is restored. (D) Scanning electron micrograph of a device similar to that used in the experiment (g, gate electrode).

The Kondo singlet is expected to have an intrinsic time scale, related to its width through the uncertainty principle. Application of a microwave ac voltage between the source and drain at a frequency f higher than kBTK/h is expected to create new peaks in the density of states displaced by energies nhf, where n is an integer (Fig. 1C). Thus, peaks in the differential conductance are predicted in the presence of microwaves not only at Vds = 0 but also at satellite voltages Vds = ±nhf/e. An intuitive way to describe this is that electrons are excited by energy hf in one lead and then tunnel to the Fermi energy in the other lead. For many years the search for this effect has been unsuccessful. It has been reported (16) that the zero-bias peak in conductance is reduced by microwaves, but no satellites have been observed. Indeed, theoretical work has shown that the decoherence of the Kondo coupling by the microwave excitation might be so strong that it destroys the Kondo effect before the satellites can emerge (13, 14).

We report on the observation of photon-induced satellites in the Kondo conductance of a SET. The effect appears to be quite delicate, which may be why it has not been previously seen. We only observe it when Embedded Image, where Embedded Image is the amplitude of the microwave voltage between the source and drain. We must also have hf not much larger than kBTK because the decoherence eliminates all Kondo features at large Embedded Image, but at the same time, hf > kBTK is required for quantum effects. Because we use a cavity (17) to excite our SET with a narrow frequency band, radio-frequency noise that would cause decoherence but no sidebands is excluded, and heating is minimized. In addition, the small size of our quantum dot makes non-Kondo excitations of the dot less likely.

The SETs used in this work are similar to those studied in recent experiments (1, 18, 19) and are formed by using electron beam lithography to pattern metallic gates on a GaAs/AlGaAs hetero-structure with Si modulation doping. Figure 1D is a micrograph of a device similar to the one used in the measurements. The voltage Vg on the gate electrode is used to vary the electrochemical potential of the dot and is tuned so that the dot contains about 50 confined electrons.

The SET is located outside of a small (diameter ∼1 mm) aperture in a thin (100 μm) wall of a square cavity resonator (17). The resonator is driven via a coaxial transmission line by a generator operating at room temperature. The dc leads connected to the SET pick up the oscillating magnetic flux from the aperture generating Embedded Image. This setup is different from the more traditional scheme for coupling the high-frequency signal to the dot, which involves a connection between the coaxial line and the sample wiring that uses a dc-blocking capacitor (16, 20). Further details of our method are given in (17).

The microwave signals are probably induced in all the leads simultaneously. However, theory (14) suggests that the Kondo effect is less sensitive to microwave gate voltage than to Embedded Image, especially midway between the Coulomb charging peaks. Moreover, a given voltage on a gate has less influence on the dot than does the same voltage on the source or drain because of the respective capacitances. We therefore assume that microwave excitation by the gate is negligible.

For fkBTK/h, the response of the device to an ac voltage is expected to be adiabatic. One can then predict the response from the dc characteristics, and this also allows us to calibrate Embedded Image versus microwave power P (17). Figure 2A shows the evolution of the differential conductance Gd with Embedded Image for a peak with a relatively large kBTK. For comparison, we show the calculated adiabatic response (Fig. 2B). In the adiabatic regime, Gd is the dc differential conductance at a given voltage, multiplied by the probability of having that voltage. Therefore (17), the Gd peak at Vds = 0 splits in two with increasing Embedded Image (Fig. 2C). At large Embedded Image, both the data (Fig. 2A) and the calculation (Fig. 2B) show the expected double-peak structure with peak separation approximately linear in Embedded Image. The corresponding values of Embedded Image are shown on the lower horizontal axis of Fig. 2A.

Fig. 2.

(A) Evolution of Gd of a high-TK Kondo peak with increasing microwave power delivered to the cavity with the SET at T = 100 mK. (B) Calculated adiabatic response. The amplitude of the ac drain-source voltage Embedded Image in (A) is calibrated by matching the microwave-induced Kondo peak splitting at large power to the splitting in the calculated response shown in (B). (C) Diagram illustrating the splitting of the peak under adiabatic oscillatory bias. Dashed line: Kondo peak with no oscillations present. Solid line: Expected (time-averaged) Kondo data when an oscillatory term of amplitude Embedded Image is added to Vds.

Although the measured peak splittings behave as expected, the overall response shown in Fig. 2A is clearly not adiabatic: The maximum conductance decreases much more rapidly for microwave excitation than for the calculation of the low-frequency response. This may be related to the universal suppression of the Kondo peak by microwave radiation predicted theoretically (13, 14). However, power-dependent heating may also be present at microwave frequencies, even though we have shown (17) that at very low frequencies, heating is negligible up to the highest ac voltage levels we use in the microwave measurements.

To explore the nonadiabatic regime, we have found a Kondo peak with lower TK than that in Fig. 2. This can be done by adjusting the voltages on the electrodes to decrease the coupling between the leads and the dot. Figure 3A shows Gd as a function of Vds at several values of the microwave amplitude Embedded Image. For comparison, the zero-power trace as well as the traces calculated in the adiabatic limit for the same values of Embedded Image are shown (Fig. 3B).

Fig. 3.

(A) Gd at increasing values of Embedded Image with microwave excitation at f = 13.47 GHz. The conductance scale corresponds to the bottom curve; moving up from the bottom, each successive trace is offset by 0.04 e2/h for clarity. The values of Embedded Image, starting with the top trace, are 29, 45, 60, 67, and 144 μV. The expected positions of the two satellites ± hf/e are indicated by dashed lines. (B) Gd at values of Embedded Image matching those in the left plot, calculated in the adiabatic limit using the measured zero-power trace (top curve).

The striking feature of the data in Fig. 3A is that before the Kondo peak splits in two with increasing Embedded Image, it develops two satellites, one on each side, in contrast with the adiabatic predictions in Fig. 3B. As Embedded Image increases, the central peak height decreases while the satellite peaks grow. Eventually, at Embedded Image, the central peak is lost and only the satellites remain. Note that there is a region in Embedded Image where the separation between the satellites is very close to 2hf/e. At still higher values of Embedded Image, the satellite positions become dependent on Embedded Image.

The evolution of the peak shape with microwave voltage is shown in Fig. 4A. As for the data in Fig. 2, the overall weight of the Kondo features decreases rapidly. However, we now clearly see the satellites at voltages independent of the microwave excitation.

Fig. 4.

(A) Evolution of Gd with increasing microwave voltage applied to the SET. The horizontal axis shows the ac amplitude of the drain-source voltage Embedded Image, calibrated as explained in the text. The data are taken for T = 100 mK at the sample location. (B) VS, half the separation in voltage between the satellites, as a function of Embedded Image for the data in (A). The horizontal line corresponds to Vs = hf/e.

In the range of Embedded Image where the central peak is present together with the satellites, the satellite maxima are shifted toward zero by the rapidly changing background of the central peak. To correct for this systematic shift, which amounts to about 5 to 15% depending on Embedded Image, we fit the data to a triple Lorentzian when the central peak is present, and use the outer Lorentzians to determine the voltage separation VS of each satellite from the central peak (Fig. 4B). Clearly, the satellite peaks remain pinned at a fixed bias as Embedded Image changes by about a factor of 2 before they start shifting with Embedded Image. The energy of the satellites determined in this way matches the photon energy to within experimental error, which indicates that the satellites are photon-induced. The satellites gain their maximum visibility around Embedded Image, as expected from theory (13, 14).

Our studies of the Kondo conductance of a SET in the presence of high-frequency radiation show that when the photon energy is somewhat greater than the Kondo scale and the microwave voltage is carefully chosen, single-photon satellite peaks in the differential conductance can be observed. We also find that the Kondo features are suppressed by microwave radiation, making it difficult to observe even the first sideband, let alone higher order ones.

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