1. Department of Basic Science, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153, Japan
2. CREST, Japan Science and Technology Corporation (JST), Kawaguchi, Saitama 332-0012 , Japan
3. Present address: Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK.

Correspondence to: S. Komiyama¹ Correspondence and requests for materials should be addressed to S.K. (e-mail: Email: csusumu@ASone.c.u.-tokyo.ac.jp).

We use a relatively large semiconductor quantum dot (QD), which is placed in a high magnetic field and is weakly coupled through tunnel barriers to electron reservoirs (Fig. 1a and b). The magnetic field is such that the lowest orbital Landau level, LL1, (with two opposite spin polarizations) is filled, while the first excited Landau level, LL2, is occupied with a small number of electrons. At the Fermi level, LL1 and LL2 form two compressible metallic regions, which correspond, respectively, to an "outer ring" and an "inner core" of the QD (Fig. 1a)⁸. The tunnelling probability between these two metallic regions is strongly suppressed by an incompressible insulating strip that separates them. When the electrochemical potential of the outer ring, ₁, lines up with that of the reservoirs, conductance resonance takes place because electrons can tunnel between the reservoirs and the outer ring. Although the inner core does not directly contribute to the transport, its charge strongly affects the transport by electrostatic coupling to the outer ring.

Figure 1: Single-electron transistor with a quantum dot (QD) in a magnetic field.

The opposite spin states will not be distinguished in this description, as they are nearly degenerate under these experimental conditions. a, Diagram of the QD. The grey regions indicate metallic inner core and outer ring regions formed by the lowest two Landau levels (LLs). b, Energy spectra of the LLs in the QD. When an electron–hole pair is excited within the QD by absorption of a far-infrared (FIR) photon, the excited electron (hole) rapidly falls (climbs up) to the inner core (outer ring) of the QD to polarize the QD. c, Scanning electron micrography of the QD ( 700 700 nm²), formed by laterally confining the 2DEG with negatively biased metal gates. The mobility and the sheet density of the 2DEG are, respectively, 80 m² V ^-1 s^-1 and n_s 2.4 10¹⁵ m^-2. The QD contains about 350 electrons. The metal gates and the leads, spanning a length of 100 m, serve as a dipole antenna that couples FIR radiation to the QD. The QD sample is placed within a mixing chamber of a dilution refrigerator. The magnetic field is applied normal to the plane of the QD with a superconducting solenoid. d, Diagram of Coulomb conductance peaks as a function of the control gate voltage V_g. In the single-electron transistor operation, the conductance resonance takes place when the electrochemical potential of the outer ring, ₁, lines up with that of the reservoirs. Upon excitation of the electron–hole pair in the QD, the conductance peak pattern (solid line) switches to a new pattern (dotted line) as a result of the induced polarization.

High resolution image and legend (26K)

When a far-infrared (FIR) photon is absorbed by the QD upon cyclotron resonance (Fig. 1b), an electron (hole) created in the higher empty LL2 (the lower filled LL1) will rapidly give up its excess energy to the lattice and fall (climb up) to the inner core (outer ring). As the inner core is now negatively charged by -e (the electron charge), the electrochemical potential of the outer ring has decreased by -₁ = -eC₂/(C₂C ₁ + C₁₂C ₁ + C₂C₁₂), where C_i and C_ij are the capacitances formed between respective regions (Fig. 1a). This internal polarization causes the conductance-resonance peaks to shift by -V_g -₁ (Fig. 1d). This, together with the fact that the excited electron in the inner core has an extremely long lifetime owing to suppression of tunnelling, makes single FIR photon detection feasible.

In the experiments, a QD of mechanical size 700 700 nm ² is fabricated on a GaAs/AlGaAs single heterostructure crystal (Fig. 1c). As extremely weak and well defined sources of FIR radiation, we use a GaAs two-dimensional electron gas (2DEG) Hall bar⁹ as well as an n-InSb device¹⁰. When the current I_em is passed through the devices in magnetic field B _em, they emit narrow-band cyclotron radiation centred at the cyclotron frequency w_C = eB_em/ m* (refs 9, 10 ), where m* = 0.068 m ₀ and m* = 0.0014 m ₀ are, respectively, the effective masses of electrons in GaAs and InSb with the free electron mass m₀. Two different arrangements have been applied for illumination. In one arrangement, a GaAs emitter is placed in the same mixing chamber as that of the QD sample. In the other arrangement, the emitters are placed in another superconducting solenoid (T = 2 K) located outside the mixing chamber (but still in the same cryostat), facilitating independent tuning of the radiation wavelength. In either arrangement, unwanted near- and mid-infrared radiation from the emitters, if any, is completely eliminated by filtering through silicon plates and black polyethylene films.

The black circles in Fig. 2 show the positions of the conductance peaks as a function of the control gate voltage V _g and magnetic field B taken without FIR illumination. As B is raised for a given trace, electrons are transferred sequentially from the inner core to the outer ring, while the total number of electrons is kept unchanged^{7, 8}. Each time one electron leaves the inner core, the inner core is positively charged by +e, causing ₁ to increase by +₁. This process, exactly opposite to the one expected for the cyclotron-resonance excitation, yields a sequence of small jumps in the peak position by V_g = +0.6 mV for each trace.

Figure 2: Positions of the Coulomb conductance peaks in the plane of V _g versus B.

Data shown for observations without FIR illumination (black circles) and with FIR illumination (white circles; I_em = 3.5 A). Without illumination, each trace maps out the boundary conditions at which the total number of electrons on the QD increases by one as V_g is increased. For a given trace, electrons are transferred sequentially from the inner core (the first excited Landau level, LL2) to the outer ring (the lowest orbital Landau level, LL1) as B is raised, while the total number of electrons is kept unchanged⁸. Each time one electron leaves the inner core, the inner core is charged by + e and causes a stepwise discontinuous shift of the peak position by V_g 0.6 mV. This intra-QD charge transfer repeats approximately each time one more flux quantum is added to LL1. The average period of the steps, B = 20 mT , suggests an effective dot area of 500 nm², a size which is consistent with the dimensions for the QD. Under illumination, the conductance resonance occurs at shifted V_g positions (white circles), which form the lines that are separated from the non-excited traces in the negative direction by an amplitude approximately equal to the step height (V_g -0.6 mV) of the non-excited traces. For certain ranges of B, conductance resonance is seen to occur also at the doubly shifted positions ( V_g -1.2 mV). Such B ranges correspond to the regime where the lifetime of excited states are especially long and the doubly excited states are visible. All the data points in this figure are taken with a time constant much larger than the average time interval of the conductance switching described in Fig. 3. The arrangement for FIR illumination is the same as that for Fig. 3a.

High resolution image and legend (13K)

Single FIR photons have been detected in the B range of 3.4–4.15 T, where the inner LL2 contains from twenty electrons down to one electron. Figure 3a shows a typical example obtained on one conductance peak at T = 0.05 K and B = 3.67 T. When very weak FIR illumination is turned on ( I_em = 2 A), the original conductance resonance is occasionally switched off, while the finite conductance shows up as a few spikes distributed over a V_g range shifted towards the more negative direction of V_g. When the intensity of illumination increases (I_em = 3.5 A ), the original conductance resonance nearly vanishes, while conductance in the shifted V_g range grows to form a dense array of spikes defining a distinct envelope of another resonance line centred at the shifted position by V_g = - 0.6 mV. Detailed studies reveal that the peak position of the new conductance resonance (V_g = - 0.6 mV) is independent of the FIR intensity and that the disappearance of the original conductance resonance (integrated over time) is exactly compensated by the appearance of the new conductance resonance (integrated over time) at any level of FIR intensity. These findings indicate that this new conductance resonance is due to the state of the QD in which one extra electron is excited by the cyclotron resonance in the inner core. Finite conductance takes the form of spikes, also in a V_g range shifted by V _g = -1.2 mV at I_em = 3.5 A , which corresponds to a doubly excited state when two electron–hole pairs are excited within the QD.

Figure 3: Conductance resonance and state switching.

a, Conductance-resonance curve with and without far-infrared illumination. A GaAs-emitter is placed within the mixing chamber, where B_em is larger than B for the QD, by about 3%, in order to meet the plasma-shifted cyclotron resonance of the QD (ref. 11). The incident FIR power on the effective antenna area, 100 m ², at the position of the QD sample, is less than 0.003 fW (10⁴ photons per second, in terms of photon flux) at I_em = 3.5 A. The time constant of measurements is 1 ms and the gate voltage is scanned over 3 min for each curve. Without FIR illumination, the conductance resonance yields a smooth curve (shown at the top). With FIR illumination (I _em = 2.5 and 3.5 A), the smooth curve is replaced by a sequence of frequently switching spikes. The envelope of the spikes reveals a new conductance resonance at the position shifted by V_g = -0.6 mV from the original peak position. Signals due to the doubly excited state with two extra electrons in the inner core can be also recognized at the position shifted by V_g = -1.2 mV. b, Time-trace representation of the data of a with V_g fixed at the original peak position and I_em = 3.5 A. The arrangement of FIR illumination is the same as that for a. c, The number of switching-off events counted over ten seconds is plotted as a function of the electrical input fed to the GaAs emitter. The emitter is outside the mixing chamber, where B_em is set to be at the resonance position. The input power of 0.1 mW roughly corresponds to the FIR intensity of I_em = 3.5 A for a and b. (See the legend of Fig. 4.) The control gate voltage V_g for the QD is fixed at the original conductance-peak position.

High resolution image and legend (32K)

The shifted conductance resonance, recognized as the envelope of conductance spikes under the FIR illumination, shows neither width broadening nor amplitude reduction compared to the original resonance line. This indicates that the effective electron temperature is not appreciably elevated (not by more than 10 mK) by the FIR illumination. We have studied the influence of elevated lattice temperature up to 0.4 K in additional experiments, and confirmed that no conductance switches as observed here are induced. Thermal activation is thus irrelevant to the phenomena.

Similar effects of FIR illumination are observed on each conductance peak in the B range of 3.4–4.15 T. White circles in Fig. 2 trace the V_g positions of the new conductance-resonance peaks under illumination (I_em = 3.5 A ). The traces primarily form the lines that are shifted from the non-excited traces (black circles) in the negative direction of V_g by the same amplitude (V_g -0.6 mV ) as that of the steps exhibited by the non-excited traces. This supports our interpretation that the new conductance resonance arises from the state containing one extra electron in the inner core. Also, several data points are located at the doubly shifted positions (V _g -1.2 mV), corresponding to the doubly excited state.

When the gate voltage is fixed at the original peak position of a given conductance resonance, the effect is of telegraph-like switches between two conductance states, as shown in Fig. 3b. Each switching-off event corresponds to an individual process of photon absorption, and each switching-on event to recombination of an excited electron–hole pair. As shown in Fig. 3c, the rate of switching-off events increases with increasing the FIR intensity at a low-excitation level, while the average time duration over which the switched-off state is maintained (the lifetime of the excited state) is kept unchanged. The rate of switching-off events levels off and starts to decrease at high levels of excitation where the average time interval between successive photon absorptions becomes shorter than the lifetime of the excited state (data not shown). If an additional electron–hole pair is excited already before a previously created electron–hole pair recombines, the QD undergoes transition to the doubly excited state, with no observable switch.

The conductance switches described here take place only when the FIR radiation is in resonance to the inter-Landau-level excitation energy of the QD, as demonstrated in Fig. 4, where the number of switching-off events of conductance counted over ten seconds is recorded as a function of the FIR photon energy (B_em). The absence of background signal shows that conductance switches without illumination are negligible. The magnetic fields, B_em, at which the respective emitters yield the largest number of counts correspond to the "magneto-plasma resonance" frequency of the QD (ref. 11) that is slightly higher than the cyclotron-resonance frequency of the bulk 2DEG (by about 3%). The spectral line width is determined by that of the radiation from the emitters. Hence, the true spectral response of the QD is probably sharper than the resonance lines presented here. These results indicate that the conductance switches that we observed here arise from individual events of single FIR photon absorption by the QD.

Figure 4: Excitation spectra of the quantum dot.

The number of switching-off events of conductance over ten seconds, occurring when V_g is fixed at the original peak position, is shown versus B_em. The measurements are repeated while B _em is increased in steps of 20 mT. The FIR radiation is generated by a GaAs 2DEG emitter (_H = 20 m² V ^-1 s^-1 and N_s = 1.8 10¹⁵ m^-2) and an n-InSb emitter placed outside the mixing chamber. The radiation is guided through a heat-anchored metal light pipe, 80 cm in length and with a 3-mm bore, using sufficient filtering, and introduced into the mixing chamber through a sealed silicon window. The corresponding photon energies are indicated by the broken lines. To keep the intensity of FIR illumination unchanged at a sufficiently low level over the entire spectral range, the electrical input to the emitters is kept to be constant at 0.5 mW: the FIR power effectively incident on the QD is estimated to be of the order of 0.3 fW. The magnetic field (B = 3.6 T) for the QD is the same as that for Fig. 3c but different from that for Fig. 3a and b. It follows that the lifetime of the excited state of the QD is about 5 ms, the same as that for Fig. 3c but substantially shorter than that for Fig. 3a and b. This condition of a relatively short lifetime is intentionally chosen for the practical purpose of measurements in order to avoid a too-long integration time for accumulating a sufficient photon count.

High resolution image and legend (11K)

We briefly discuss the lifetime of the excited state of the QD. As a general trend, the lifetime markedly increases from the order of 1 ms up to the order of 1,000 s as B increases from 3.4 T to 4.0 T. Above 4.0 T, it sharply decreases. This overall behaviour is superposed with sharp saw-tooth-like oscillations that synchronize with the step structure of the ground-state peak position shown in Fig. 2. We interpret these features by considering the size evolution of the inner core and the outer ring of the QD with increasing B (ref. 12).

Thus, we have achieved single-photon detection in the wavelength range 175–210 m (6.0–7.1 meV) restricted by the magnetic-field range (3.4–4.15 T) where the lifetime of the excited state exceeds the time constant of measurements (1 ms). This can be expanded by faster measurements—the intrinsic limit on the speed of the single-electron transistor is greater than 10 GHz (ref. 13). In practice, using a high-frequency single-electron transistor¹³ will improve the time resolution up to a 10-MHz range. We have also successfully observed conductance switches owing to the single FIR photon absorption up to T = 0.4 K; the upper bound is limited by the charging energy of the QD (ref. 14) (about 0.4 meV). Fabricating QDs on a narrow 2DEG mesa structure with minimum use of nearby metal gates may expand the upper bound to T 0.8 K; this could make possible the use of more convenient ³He refrigerators. Very few conductance switches are seen if the device is not illuminated: in an optimum B range, the dark signal, in terms of the switch rate, was as small as 0.001 s^-1 at T = 0.05 K . Finally, the quantum yield, or the ratio of the detected photon number to the incident photon flux on the effective antenna area is roughly estimated to be 1%. The quantum yield crucially depends on the architecture of the optics, which has not been optimized and may be significantly improved. Here, the effective noise equivalent power (NEP) reaches the order of 10^-22 W Hz ^-1/2 in the optimum B range, which exceeds other FIR detectors reported in the literature by a factor of more than 10⁴.

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Acknowledgements

We thank D. Austing and M. Stopa for helpful discussion. This work was supported by the Core Research for Evolutional Science and Technology (CREST) of the Japan Science and Technology Corporation (JST).