Microwave spectroscopy on quantum dots

Chapter 6 MICROWAVE SPECTROSCOPY ON QUANTUM DOTS Robert H. B lick Max-Planck-lnstitut fiir FestkOrperforschung, 70569 Stuttgart, Germany

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. CoupledQuantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 4. 5. 6. 7. 8.

QuantumPoint Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photon-AssistedTunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ElectronSpin Resonance in Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectroscopyon a Double Quantum Dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Microwave Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

309 310 316 322 327 332 336 341 342 342

1. I N T R O D U C T I O N Transport through small electronic islands in semiconductor nanostructures at low temperatures reveals new effects such as Coulomb blockade (CB) and single electron tunneling (SET) [ 1]. Such islands can be obtained by using nanofabrication techniques. Typical dimensions of these devices are (500 nm) 2. They contain roughly 10-100 electrons and are called quantum dots or "artificial atoms" [2], because of the discrete level spectrum. CB governs transport through these devices, which results in an oscillating behavior of the conductance through the dot as a function of gate voltage. Quantum dots can function, for example, as extremely sensitive electrometers, and they can also be used as photon detectors, which will be discussed here. We will present measurements of the induced photocurrent through single and double quantum dots. We applied different antenna types to couple the millimeter wave radiation in the range of 30-200 GHz into the quantum dots. As we will show, photon-assisted tunneling through the quantum dots can be employed for spectroscopy on the few-electron system. Naturally coupling two quantum dots will result in the formation of an "artificial molecule." As a consequence of two strongly coupled dots, Rabi oscillations will occur. These can be probed directly in time-dependent measurements, where the electromagnetic field interacts with the oscillating valence electron. Complex photoconductive measurements of the current through a double quantum dot enable us to monitor effects of coherent

Handbook of Nanostructured Materials and Nanotechnology, edited by H.S. Nalwa Volume 2: Spectroscopy and Theory Copyright 9 2000 by Academic Press ISBN 0-12-513762-1/$30.00 All rights of reproduction in any form reserved.

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electron transport in the suppression of the Rabi oscillations of this artificial molecule. The current is induced by a new broadband millimeter-wave source functioning as a heterodyne interferometer, which consists of two nonlinear transmission lines generating harmonic outputs in the range of 2-400 GHz and, being coherent, allows tracking of the induced current through the sample in both magnitude and phase.

2. COUPLED QUANTUM DOTS Electronic transport through quantum dots is governed by the mechanism of Coulomb blockade and single-electron tunneling [1]. By coupling two quantum dots in series, a double quantum dot is formed. This system gains new quality compared to single quantum dots, as electrons can be shared between the two sites, thus forming an artificial molecule [3]. Recent experimental [2, 4a-d] and theoretical [5] approaches have mainly addressed the electrostatic interaction between the two dots. Detailed calculations have then been given, introducing the model of an artificial molecule and its implications for the experiment [6a-d]. We will present measurements on such an artificial molecule. The two dots are set up in a lateral geometry and are connected in series, coupled by a tunneling barrier, thus allowing electrostatic interaction and a finite coupling of the wave functions of the two systems. The two-dimensional electron gas (2DEG) used in this experiment forms in an AlxGal_xAs/GaAs heterostructure grown by molecular beam epitaxy; it is located 90 nm below the evaporated Schottky gates. The carrier density at a temperature of 4.2 K is 2.05 • 1011 m -2 with a mobility of 80 m2/V s. A Hall bar structure is etched into the substrate, and ohmic contacts are fabricated. The quantum dots in our experiments are realized by electron-beam written split gates on top of the heterostructure. The conductivity is measured within the common lock-in and preamplifier setup in a dilution refrigerator at a base temperature of 25 mK. The excitation voltage applied has an amplitude of 5/zV at a frequency of 14 Hz. In biasing the split gates on top of the heterostructure, two quantum dots of different sizes are formed. The different sizes are supported by different potentials at the gates and the specific geometry of the gates. The small dot will called A and the large one B. In a detailed numerical study of the electrostatic potential induced by the gate voltages [7], we found good agreement in comparing the calculated dot sizes and the experimentally determined capacities. (A self-consistent, spatially resolved simulation was carried out for the Poisson equation of the electrostatic potential landscape of the two dots with the gate configurations shown in Figure 1. The calculations were based on the algorithm developed by S. E. Laux et al. See also [7].) The absolute number of electrons in dots A and B obtained from these calculations are NA ,~ 12 4- 2 and N8 ~ 33 4- 2, which agrees with the capacitance ratio between the two dots of cA/Cz8 ~ 1/3. In Figure 1 a scanning electron microscope picture of the double quantum dot is shownmthe width of a single gate is 80 nm. Below, the calculated density of the electron gas and the quantum dots are presented. The small quantum dot is shaped asymmetrically, because of the confinement potential of the split gates. The charging energies for the two dots are determined to be E A -~ 3 meV and E~ = (0.9 4- 0.11) meV. These are obtained by measuring the drain source dependence of the CB regime of the two dots [8]. In addition, a back gate (VBc), situated 0.5 mm below the 2DEG, is operated to shift the electrostatic potentials of the two quantum dots. Operating only the large quantum dot B, the oscillating conductance shown in Figure 2 is found. When the small quantum dot A is tuned in, the conductance is reduced and we find a typical resonance pattern, depicted in Figure 3. Clearly, the two different periodicities 8 V~, v,A representing the charging energies of dot A and B, are found. G' The double quantum dot is best characterized by measuring its charging diagram, that is, by varying the electrostatic potentials of two independent gates while the conductance

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Fig. 1. The asymmetric double quantum dot. The upper part shows an SEM micrograph of the device; the gate width is 80 nm. The lower part is a representation of the electron density distribution in the two-dimensional electron gas, induced by biasing the split gates.

Fig. 2. Differential conductance of the large quantum dot B at T = 23 mK and B = 0 T. The small quantum dot A is not biased.

through the dot system is traced. In our case we varied a top gate and stepped the backgate voltage in increments of 0.5 V, as shown in Figure 4a. The coupling o f the dots to the leads as well as the interdot tunneling barrier were adjusted to be well in the regime of tunneling. In the gray-scale plot of Figure 4b the conductance resonances form a hexagonal

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lattice with different slopes, according to the different electrostatic coupling between the gates and the dots. The distance between adjacent (parallel) resonance lines reflects the different charging energies of the large (B) and the small (A) quantum dots (8 V~G, 8 VAG, where BG indicates the back gate.) The hexagons are strechted since the total capacities of the two dots differed strongly. Because the dots are connected, the charging of one dot influences the electrostatic potential of the other one, which leads to a repulsion of the resonance lines close to the points where the resonances of the two dots intersect and hence to the hexagon lattice [4a--d]. This displacement is proportional to the interdot capacity Cid, as indicated in the schematic charging diagram of Figure 4b. An increase in the value of Cid lifts the degeneracy of the crossing points. While this formation of a hexagonal lattice in Figure 5, reflecting the regions of stability for a given charge distribution (NA, NB), is already well known for coupled metallic islands, we find--in contrast to those systems--a finite conductance far from the triple points of the charging diagram (see Fig. 4b). Keeping in mind that only at the triple points do the conductance resonances of the two dots coincide, a finite conductance all along the boundaries of the charging diagram implies that we can--far from the points of degeneracy--detune the resonance condition considerably and still measure transconductance. This observation can be explained by a finite overlap between the wave functions of electrons located in the different dots, even if the corresponding energies are different. The total number of electrons in the double quantum dot does not change when the line connecting two displaced triple points is traversed, as for example, in the transition NA, NB + 1 ~ NA + 1, NB, where an electron simply moves from dot B to dot A (Fig. 4b). In this region the "topmost" electron is not localized in one of the two dots, but the particle's wave function is distributed across the whole double dot. This spread-out wave function characterizes a valence electron. Because of tunneling the energy of the valence electron is lowered compared to that of localized electrons [3]. We find a Lorentzian-shaped variation of the conductance amplitudes due to this molecular binding (data not shown here). First we want to focus on the effect of the tunnel splitting, which occurs when the two discrete states 8rA and eB overlap, forming a valence state. The magnified lower part in Figure 5, where different resonance lines are crossing, shows the influence of the tunnel

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MICROWAVE SPECTROSCOPY ON QUANTUM DOTS

Fig. 4. (a) Measured charging diagrams of the double quantum dot in surface plot representation: the top (TG) and back gate (BG) voltages define the diagram axis. (b) Charging diagram in gray-scale representation. The differently stretched hexagons represent separate charging states with a discrete number of electrons (NA, NB), as indicated. The gray-scale minimumis at trmin = 0/zS (white) and the maximumat trmax~<2.5/zS (black). (Source: Reprinted with permission from [41]. 9 1998 American Physical Society.)

splitting on the charging diagram: the electrostatic interaction Cid mainly determines the splitting at the two resonance points as shown. In addition, the interdot tunnel coupling 7~ leads to an enhanced splitting on the order of 27~. Hence there are four states available for transport. The rounded solid lines mark the ground or symmetric states, and the dashed lines indicate the antisymmetric states. (For simplicity we assume the ground state to be symmetric, although from a general point of view it might be antisymmetric.) These antisymmetric states can be monitored by taking different traces out of the charging diagram, as will be shown below. In Figure 6 a conductance resonance out of the charging diagram is shown in logarithmic representation, corresponding to the horizontal line through the charging diagram in Figure 4b (VBG = --38.5 V). The open boxes are the measured conductance values, and the solid line corresponds to a fit according to the derivative of the Fermi-Dirac function at an electron temperature of 100 mK. For comparison, the upper fight inset shows the same trace in a linear representation. Obviously we find in the crossing point of the resonance lines (marked by the arrow in the center) a strong asymmetry of the conductance peak structure, which cannot be fitted. This asymmetry is well pronounced in the log plot,

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Fig. 5. Theoretical charging diagram with a nonzero interdot capacitance Cid, resulting in the hexagon lattice. The black dots indicatemfor serial coupling the positions of conductance resonances. The magnified lower part, where different resonance lines are crossing, shows the influence of the tunnel splitting onto the charging diagram. The electrostatic interaction Cid mainly determines the splitting at the two resonance points as shown. In addition, the interdot tunnel coupling leads to an enhanced splitting on the order of 27~. (Source: Reprinted with permission from [41]. 9 1998 American Physical Society.)

Fig. 6. Measured differential conductance (open boxes) through the double quantum dot and fit according to the derivative of the Fermi-Dirac function 8fFD/8 E (solid line) in logarithmic representation. The curve is a trace out of the charging diagram as depicted in the upper left inset. The right inset shows the conductance for comparison in linear representation. (Source: Reprinted with permission from [41]. 9 1998 American Physical Society.)

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Fig. 7. (a) The two conductance resonances at the crossing point, magnified. The additional (tunnel splitting) resonance is found in the left shoulder of the right peak. In (b), with a differently tuned back-gate voltage, the additional resonance shifts toward the valley between the two main peaks. (Source: Reprinted with permission from [41]. 9 1998 American Physical Society.) whereas the peaks to the left and right (arrows) reveal symmetric flanks. As a reminder, the upper left inset shows the magnified part of the charging diagram out of the previous figure. The curve in the log plot corresponds to trace 1, as indicated in the inset--it crosses the left ground state and then the two states on the right side, which are the antisymmetric and symmetric states. Hence the shoulder in the right conductance peakmthe tunnel splitted state--is formed. To confirm this picture, we choose different traces taken out of the charging diagram crossing the region of interest at different points. The data obtained are given in Figures 7 and 8: we focus on the two main peaks in the center of the conductance traces--the data are plotted on a logarithmic scale again. In Figure 7a the resonance of Figure 6 is shown again; clearly the asymmetry and thus the inefficient fitting by the distribution function are found. In detuning the back gate voltage to - 3 8 V, we observe a diminishing of the conductance found in the shoulder of the right peak. In Figure 7 the conductance resonances are shown when the back gate is detuned to - 3 9 V and - 3 9 . 5 V--these traces refer to the sequentially numbered cuts through the charging diagram, as indicated in the left inset of Figure 6. In accordance with these traces, we find in Figure 8a that the additional resonance structure has now moved from the right peak's shoulder into the left one. Hence the tunnel splitted resonance of the left peak now contributes to the overall conductance. With further detuning we observe that this tunnel splitting is damped out and merely a background conduction remains. From these measurements and data from other crossing points, we determine the tunnel splitting to be on the order o f f e -~ 120/zeV. This value is

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an agreement with the fact that the excitation energies of quantum dot A are found to be much higher, on the order of 400/zeV. We have seen that a coherent tunneling mechanism or molecular mode evolves in a coupled dot system. This coherent state is found as a tunnel splitting in addition to the Coulomb interaction in the charging diagram of the double dot. This state can be detuned within the charging diagram. Although these data were obtained in pure transport measurements, we will see in the following sections how microwave spectroscopy can be applied to monitor the tunnel splitting.

3. QUANTUM P O I N T CONTACTS Before considering the spectroscopy of individual dots and double dots, we focus on the high-frequency characteristics of single and coupled point contacts. Quantum point contacts (QPCs) [9a, b] reveal the quantization of conductance, due to the one-dimensional subbands emerging in the constriction. The idea of photon-assisted transport at the limits of quantum detection with QPCs is very attractive for applications in radio astronomy [ 10]. Despite several predictions that photon-assisted transport (PAT) in QPCs should be possible [1 l a, b], mainly bolometric response was found in previous measurements [ 12a, b]. Here we will present measurements of the millimeter wave response of a complex QPC configuration. This setup allows us to study the response with different electrostatically tunable potentials in the energy range of some 100/zeV. As will be shown in the

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MICROWAVE SPECTROSCOPY ON QUANTUM DOTS

Fig. 9. The measurement setup. In the center the sample with a modified bow-tie antenna is shown, connected to voltage supply VACand VDCon the left and current amplifierand lock-in on the right. Data is then acquired by the computer (Dat).

measurements, we obtain a similar photon-electron pump, as introduced in the theoretical work by Hekking and Nazarov [ 13]. The QPCs applied in this experiment are realized by split gates on an inverted heterostructure. The samples were patterned with a small mesa (30/zm • 750/zm) and then immersed in the dilution refrigerator. A schematic representation of the setup is shown in Figure 9. In the sample center the Hall bar can be seen; radiation is fed in via the modified bow-fie antenna. The antenna leads also serve as DC voltage probes for the top-gate voltages Vri. In addition, the back-gate VB can be biased--AC and DC voltages are added and then applied to the sample (Vs), the resulting current is then preamplified and phase sensitively detected by a lock-in. Two different antenna structures commonly used in these measurements are depicted in Figure 10a: the upper image shows a so-called bow-fie antenna, and the lower images show a double-log periodic antenna [14]. Both antenna structures allow broadband coupling of radiation, which was verified by numerical simulations [ 15]. The double-log antenna is sensitive up into the THz regime, which can be seen in the magnified lower part of Figure 10a. In the center of the antennas the QPCs are defined. A scanning electron microscope (SEM) micrograph of the QPCs written by electron beam lithography is shown in the inset of Figure 10b. According to their position, the gates were deliberately connected as marked in the figure ("top," "center," and "bottom"). The center QPC's electrostatic potential is thus symmetric, whereas the top and bottom QPCs reveal asymmetric potentials, due to the plunger gates. The advantage of this layout is the flexibility of tuning the potentials of the QPCs and comparing the response of the individual potential shapes. Calibrating the QPCs in nonlinear transport measurements, we found the 1D subband level spacing to be in the range of three to four times the photon energy of the radiation applied (80-170 GHz). The measurements were performed in a top-loading 3He/4He dilution refrigerator with a sample holder, allowing quasi-optical transmission through a

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Fig. 10. (a) Two different antenna structures used. The upper image shows the bow-tie and the lower image the double-log antenna with a magnification of the center on the right side. (b) Conductance characteristics of the quantum point contacts (QPCs) as indicated. The best quantization is found for the center QPC, whereas the effect of the asymmetric potential for the other gates is seen as additional kinks in the characteristics (different curves are offset for clarity). The inset shows a SEM micrograph of the different QPCs coupled in series.

circular waveguide [ 16]. The radiation coupling was determined by measuring the induced photo current. A finite drain source bias voltage of VDS - - - 2 7 0 / z V was applied to define the forward direction of the current. The broadband microwave sources employed in these measurements are backward wave oscillators (BWOs) with center frequencies of 90, 110, and 170 GHz and a typical cw output power of 20 mW. For the determination of the QPC characteristics, the conductivity was measured by applying a finite AC excitation voltage of 1 0 / z V to the sample and varying the different gate voltages (see Fig. 10b). The temperature for all measurements shown was 78 mK, and the magnetic field was fixed at B = 0 T. As seen in Figure 10b, the best 1D quantization is found for the center QPC. The pinch-off value is close to - 0 . 8 5 V, and the plateaus are well pronounced and reveal deviations from the quantization only at higher numbers of occupied subbands. The bottom and top QPCs have slightly higher pinch-off values, because these contact tips are not as close as for the center contact. Moreover, deviations from the quantization are found close to ne2/h, because of the electrostatic potential of the plunger gates, which is manifested by the kinks in the characteristics. The induced photocurrent through the center QPC is shown in Figure 11 for different frequencies. As seen, the polarity strongly depends on the frequency applied. This in turn is caused by the sensitivity of the local electrostatic potential rectifying the radiation and not by the antenna, because it couples the radiation over a broad band. The QPC's

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MICROWAVE SPECTROSCOPY ON QUANTUM DOTS

Fig. 11. Photocurrentthroughthe centerpoint contact for different frequencies. The polarityof the signal changes accordingto slight deviations in the radiation coupling to the local electrostatic potential. The dotted line represents the cr-VG characteristics of the individualpoint contacts.

characteristic is plotted as a dotted line. The main origin of the photocurrent is a nonresonant bolometric absorption process, leading to a net photo current. In principle, this effect allows the measurement of thermopower on QPCs [ 17]. However, here we want to discuss the resonant response of the QPCs only. The photo response of the top (upper graph) and bottom (lower graph) QPC induced by the radiation at different frequencies are shown in Figure 12. For comparison we plotted in the figure the conductance of the individual point contacts (dotted lines)--the curves for the different frequencies are offset for clarity. As seen, the way in which the polarity of the induced current changes depends on the contact chosen; this effect is independent of frequency. The insets indicate the contact operated (dark shaded). A variation of the bias across the QPCs (VDs) varied the overall slope of the cr-VG characteristics and could be applied to compensate for the induced current. Hence it is possible to balance and effectively null the resulting photo-pumped current. As shown in Figure 11, the polarity of the photocurrent through the center QPC was very sensitive to variations in the microwave frequencies and the bias on the other QPCs, whereas the polarity of current through top and bottom always remained the same. Apart from the bolometric effect, it is expected that PAT will contribute to the induced current. A closer inspection of the data shown in Figure 12 reveals clearly that at v = 149 GHz and at 165 GHz a strongly enhanced response in the lowest subband is found compared to 105 GHz. This amplification in the current at the transition into the regime of tunneling cr < e2/h is a clear sign for PAT. Obviously the response of the second and third subbands in relation to the lowest subbands is not as large. This deviation is another indication that toward the lowest subbands PAT enhances electronic transport through the QPCs. In Figure 13a the derivative of the current through the bottom QPC is shown with respect to the gate voltage (dI/dVc) for two different frequencies (dotted, v = 148.0 GHz;

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Fig. 12. Photocurrentthrough top (upper curve) and bottom (lower curve) point contacts at different frequencies. Depending on the contact chosen, the polarity of the signal changes. The dotted lines represent the tr-VG characteristics of the individual point contacts.

solid, v = 148.6 GHz). The dotted curve is even enhanced by a factor of 3. It is evident that the strongest response in this representation is found in the transition of the lowest subband at 148.6 GHz, i.e., PAT supports the transport through QPC for certain frequencies. For further verification, this amplification can be compared to common thermopower measurements or by simply taking the temperature-dependent cr-VG characteristics. In the inset of Figure 13a these temperature-dependent measurements were performed for the bottom QPC. The conductance here is given by , ~ ( d I / d V ) = &r = cr (1.6 K) - cr(78 mK) and shows oscillations similar to those in thermopower measurements. Obviously the response for the different subbands decays in a much weaker fashion than for resonant PAT. In Figure 13b the broadband frequency dependence of the top and bottom QPCs is plotted. The response Smwhich is a measure of the pumping efficiency--is determined from the experimental data by simply taking the difference of induced current in the third and first subbands (11Is=3 - I s = l II, all curves are finally normalized). For the bottom contact (filled triangles), the response increases nonmonotonically when the frequency is raised. The top contact (open dots) shows the onset of an increased response at the highest frequencies only. As we have seen, the antenna couples symmetrically; hence the response

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MICROWAVE SPECTROSCOPY ON QUANTUM DOTS

Fig. 13. (a) Derivative of the current with respect to the gate voltage dI/dVG is shown for two different frequencies applied at the bottom contact: v = 148.0 GHz (dotted) and v = 148.6 GHz (solid). In the inset the difference in conductance for this contact for two temperatures is given: 6(dI/dV) = a(1.6 K) - cr(78 mK). (b) Normalized frequency dependence of the response S of the bottom (filled triangles) and top (open dots) point contacts. The increase toward higher frequencies indicates an effective electron photon-pumping mechanism.

depends on the contact chosen and on the local shape of the electrostatic potential only. This increase or amplification can be explained as an effective electron pumping mechanism by the photons applied, as proposed by Hekking and Nazarov [ 13]. Their prediction gives a linear dependence of the current on the frequency applied. Although we do find a nonmonotonic increase in response, this can be understood as effective electron pumping by the photons. The additional contribution to the current is caused by PAT. Furthermore, it has to be considered that the effect is strongest in the limit of cr < e 2 / h . Here, a stronger response to the total current by PAT is expected, as found for quantum dots [ 16]. At higher subbands a strong bolometric response masks the resonant coupling of the photons. This is in agreement with calculations, comparing PAT for QPCs and quantum dots [ 18]. So far, we investigated measurements of the frequency-dependent photocurrent through a series of QPCs with different local potential shapes. The photocurrent response shows an oscillating amplitude dependent on the subband energies, whereas the sign of the photo

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current depends on the geometry of the gate structure. The main signal contribution is due to a nonresonant (bolometric) coupling of the millimeter waves. In the limit of the lowest subband population, resonant coupling is found, leading to a response amplification toward higher frequencies. This is attributed to photon-assisted transport, leading to a new form of electron pumping by photons.

4. PHOTON-ASSISTED TUNNELING We now investigate the influence of high-frequency microwave radiation on single electron tunneling through a single quantum dot. Such a single dot is shown in a schematic representation in Figure 14. The leads (drain/source) allow tunneling of electrons, and the high-frequency modulation is applied via gate VG. Effective coupling of the radiation to the quantum dot is achieved by an on-chip integrated broadband antenna. As we will show, radiation with a frequency of v = 155 GHz, which corresponds to half of the bare charging energy Ec/2, results in an additional conductance peak within the Coulomb blockade regime. This additional resonance is attributed to photon-assisted tunneling. The detection mechanism is based upon an absorption of photons by the electrons in the leads (see Fig. 15a), leading to the so-called photon-assisted tunneling (PAT). The energy acquired by the electrons allows them to tunnel through states of higher energy above the Fermi energy that are otherwise not accessible. This mechanism of tunneling is well known from the tunneling process between two superconductors separated by an insulator (SIS devices) [ 19, 20] and is now commonly used for high-frequency detection [21 ]. A similar effects is observed in the single-electron tunneling (SET) in quantum dot devices. This combination of SET and PAT is theoretically discussed for a single quantum dot [22]. It is found that ordinary CB oscillations are modified by the application of high-frequency electromagnetic radiation [23]. For the detection of high-frequency radiation (v > 100 GHz), a single quantum dot as a device offers a very high sensitivity and a very low total capacitance Cx ~ 10-16 F, which could surpass existing detection methods at these frequencies. Furthermore, such a quantum dot detector offers the possibility to tune the discrete states of the dot easily by varying electrostatic potentials, thus allowing a frequency-selective detection. We will demonstrate in transport experiments on a single quantum dot how microwave radiation at high frequencies (100 GHz < v < 200 GHz) influences electron tunneling through a single dot. The quantum dot is electrostatically defined by split gates, and the leads of the gates serve as an integrated antenna. The antenna allows us to couple mil-

Fig. 14. Diagramof the dot system:Electronstunnelfromsourceto drainvia the dot; microwavescouple via top-gate VG.

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Fig. 15, (a) Schematic representation of photon-assisted transport through a single quantum dot in finite bias regime (VDs # 0), An electron excited in the leads (hv) acquires energy and tunnels through an excited state of the N electron system or the ground state of the N + 1 electron system. The energetic difference between the N and N + 1 state corresponds to the charging energy in the constant interaction model. Photon absorption (hv f) in the dot can lead to similar excitations or to an electron being pushed out of the dot, At magnetic fields B > 2 T, Landau levels form, with a typical separation in energy of hvc > hr. (b) Representation of the possible excitation mechanism in the VG-VDs characteristics. Regions of Coulomb blockade (CB) are white, and the single electron tunneling (SET) regime is gray (see text for details).

limeter waves into the system under investigation. Our measurements demonstrate that a single quantum dot can be operated as a quantum mechanical detector [21], sensitive to the absorption of photons by the electron tunneling through the quantum dot. The measurements were performed in a top-loading 3He/4He dilution refrigerator with a sample holder, allowing quasi-optical transmission through a circular stainless steel waveguide (diameter 10 mm). The lowest temperature reached with this waveguide diameter was T = 55 mK, whereas the measurements presented here were conducted at 180 mK. The microwave source employed is a B W O with a center frequency of 170 GHz and a maximum output power of ~ 10 mW. Again the definition of the quantum dot is obtained by applying negative voltages to the gate fingers, thus depleting the electrons beneath the gates. The voltages we used at the top gates VTi w e r e V1 = V2 = - 6 2 5 mV and V5 = V6 - - 6 4 2 mV. The center gates V3 and V4, corresponding to VTG, are used to tune the electrostatic potential of the quantum dot (VTG = V3 = V4). As is sketched in Figure 15a, only one quantum dot is used in the experiments discussed in this section. A finite bias (VDs 5~ 0) is applied, leading to electron transport under radiation when an electron is excited in the source contact by radiation h v. This electron then tunnels through the N-electron system or the ground state of the N + 1 electron system. The energetic difference between the N and N + 1 state corresponds to the charging energy in the constant interaction model. A similar process in the photon absorption (h v') in the dot, which leads to similar excitations or to an electron being pushed out of the dot. Here, we consider magnetic fields B > 2 T, so that Landau levels evolve with a typical separation in energy of h vc > h r . This reduces the effects of spurious heating by the microwaves,

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because only a limited number of free carriers can be excited out of the source contact. A somewhat more explanatory picture is found in the VG-VDs characteristics shown in Figure 15b. Regions of Coulomb blockade form the common diamond-shaped structure, flanked by the regions of single electron tunneling. Excitations can occur via ground states, for example, of the N to the N + 1 electron system or via the excited states of the (N - 1)* and N* systems. We focus on the absorption of photons for simplification, but in principle the emission processes have to be considered as well (+h v). The scaling of the center gate voltage to the appropriate energy is given by the dependence of the resonance pattern on the drain source bias voltage. In this case, the scaling factor is c~ = (0.311 -t- 0.023) meV/mV, resulting in a total charging energy of Ec = e2/C~ ~ (1.24 4-0.088) meV (C~ is the total capacitance). To obtain good coupling of the applied microwave field at the quantum dot, we used an integrated broadband antenna, as described before. This method of coupling is useful at high frequencies, because the quasi-optical waveguide allows transmission of frequencies larger than 30 GHz, in contrast to coaxial transmission lines. Coupling and polarization depend on the antenna structure chosen. The antenna was designed with principles known from microwave technology. We chose a modified broadband "bow-tie" antenna structure with an impedance fairly well matched to that of free space, Z = 377 ~. The antenna itself is defined by optical lithography, then a contact sheet of Ni/Ge with a thickness of 10 nm is evaporated, followed by 150-nm-thick Au metallization. To verify the properties of the antenna we performed numerical simulations of the high-frequency current distribution at different frequencies (Fig. 16a-c). Using a commercially available program [ 15], we were able to simulate the coupling of the radiation, taking into account the 2DEG, the heterostructure itself, and the metallic back-short. A slight offset from beam center is taken into account, but does not qualitatively change the current distribution on the antenna tips near the microstructure. As seen from Figure 16, a clear maximum in the high-frequency current density is obtained in the tips of the bow-fie arms for all frequencies (calculations are shown for v = 100 GHz, 150 GHz, and 200 GHz), confirming the broadband characteristics of the antenna. The dark regions represent the maximum current density (16 A/m) and the minimum density is in white (0 A/m). In Figure 17a, the conductance through our quantum dot without microwave radiation is shown at VDS = 0; the CB oscillations can be seen clearly and are well separated. A magnetic field of B = 1.5 T was applied perpendicular to the plane of the 2DEG. Obviously a rectified current is obtained even without the application of microwaves due to EMF noise, as the dashed curve in Figure 17a indicates. This rectification at low frequencies in the adiabatic limit is discussed elsewhere [24] in detail. In Figure 17b the conductivity is shown under the influence of microwave radiation of v = 155 GHz at different intensifies. This frequency corresponds to an energy of Erad = 0.64 meV. As seen, additional conduction peaks evolve between the SET resonances as the intensity of the radiation is further increased. The peaks at A and B are somewhat less pronounced, but peaks D and E can be identified clearly. These additional peaks are induced by the photon absorption process discussed above. The energy position coincides with the additional peaks. The effect of the excitation of electrons in the leads is shown in Figure 15: the quantum dot is filled with N - 1 electrons in the ground state (solid line). The next higher Coulomb ground state with N electrons in the dot is empty (dashed line). If the dot is turned into the CB regime, no conductance is found. In this case, the absorption of a photon in the leads by an electron with the appropriate energy would allow the electron to tunnel through the N electron ground state, thus overcoming the CB gap with e2/(2C~:). It should be noted that this mechanism only leads to a net current when the barriers of the quantum dot are asymmetric or if a small bias is applied. In our case, we used barriers with slightly different transparencies and, in addition, the leads to the tunneling barriers are asymmetric because of other split gates nearby. It is important to note that the main SET resonances (e.g., the peaks marked by C) are not shifted by increasing intensity of the radiation. Furthermore,

324

MICROWAVE SPECTROSCOPY ON QUANTUM DOTS

Fig. 16. Simulationsof the high-frequencycurrent distributionon the antennaused. Blackregions correspond to high current density; low density regions are white. The frequencies are (a) v = 100 GHz, (b) 150 GHz, and (c) 200 GHz. (Source: Reprinted with permission from [16]. 9 1995 American Institute of Physics.)

the additional peaks in the resonance pattern are not shifted. Thus, there is no DC voltage drop at the different gate contacts or at the leads, which could influence the tunneling through the barriers. Obviously the peaks are surrounded by a region of finite conductivity, which is probably due to induced transport through excited states in the dot and to nonresonant heating of the reservoirs. With very low radiation power, the Coulomb blockade and the SET resonances are well defined (solid curve), and only the maximum conductance of the peaks is decreased by A a ~ 1 /zS (see peak C). With increasing intensity of the microwaves, the conductance resonances are broadened and the Coulomb blockade is weakened. In Figure 18 measurements of the DC current through the device under microwave irradiation are shown. As in Figure 17a peaks B, D, and E show the induced peak structure. At very high intensifies another fine structure is seen (marked by X), which cannot be explained by simple PAT. Most likely this peak results from excitation of electrons in the dot, which then tunnel out of the dot, leaving a vacancy to be filled by an electron out of the source contact. Varying the frequency of the microwaves slightly in the range of 4-2 GHz around 155 GHz, we found peaks similar to those in Figure 17. At larger variations, we found a strong increase in the background conductance (145 GHz) and asymmetric peak shapes (162 GHz), which incorporate the additional peak structure found at 155 GHz and are probably a result of exited states opening additional channels for transport through the quantum dot (see Fig. 18b). The strong response at 155 GHz can be explained as a

325

BLICK

,

(a)

(b)

12

0.2

C-.~

~

,_

9

hv b

0.0

I~ 6 A

-0.2

0

B !

L~.

6

0

0.36

0.37

i , i , i 0.355 0.360 0.365

-VG(V)

i 0.370

"VG(V)

Fig. 17. (a) Resonance pattern without radiation applied. The dotted line shows the rectified current due to low-frequency EMF noise. (b) Pattern identical to that in (a) under irradiation of microwaves with increasing power level at v = 155 GHz. The radiation induces peaks A,B,D, and E, at half of the Coulomb gap. (Source: Reprinted with permission from [ 16]. 9 1995 American Institute of Physics.)

0.3

9

145 GHz

0.2

A

'< c

=L6

0.1

b

~ m

'

:

v

i

,,

0.0

3

I -0.1 ( I

0.355

.

I

i

0.360

I

0.365

,

I

I

0.370

-v G (v)

l

0.360

,

I

,

0.365

I

0.370

-VG(V)

Fig. 18. (a) Rectified current due to 155-GHz millimeter wave radiation: Induced resonances are found at B, D, and Emin addition we observe a feature X, indicating a different absorption mechanism at a lower energy. (b) Differential conductance through the dot under different millimeter-wave freqUencies applied, as indicated. Dotted curve, without radiation applied. The strong increase in background conductance covers distinct excitation peaks. (Source: Reprinted with permission from [16]. 9 1995 American Institute of Physics.)

326

MICROWAVE SPECTROSCOPY ON QUANTUM DOTS

superposition of two replicas of conductance peaks corresponding to an absorption and an emission process [22]. In these first experiments on a single quantum dot, we have seen that it is possible to couple microwave radiation in the high-frequency regime via an integrated antenna structure into quantum dot microstructures. When the quantum dot is irradiated with microwaves at v = 155 GHz, we find additional resonances, which are attributed to photon-assisted tunneling through the quantum dot. Hence, photon-assisted transport can be employed for spectroscopy of quantum dots, Furthermore, it demonstrates the feasibility of using quantum dots as sensitive and frequency-selective detectors in the microwave frequency range.

5. ELECTRON SPIN RESONANCE IN QUANTUM DOTS Let us now focus on electron spin resonance (ESR) in a single quantum dot at filling factor v ~< 2 as an example for applications of microwave spectroscopy on single dots. ESR is a well-known experimental technique. One of the most intriguing applications was the observation of the Lamb shift in hydrogen [25]. In 2DEGs, ESR was first found in transport experiments by Stein et al. [26]. The optical detection of single spin resonance in a molecule has recently been reported [27]. We will first give a brief summary and will then present a theoretical explanation for the strong enhancement of the effective ge*ff factor, which causes spin splitting in a few electron system. We will focus on magnetic fields corresponding to a filling factor v ~< 2, where v is defined as v = hns/eB (where ns denotes the sheet electron density). Thereafter I will discuss the experimental approach and give the first results. There are only a few publications in which the effect of the electron spin on electron transport through a quantum dot is considered. Some authors have explained the variation of the Coulomb blockade (CB) oscillations at moderate magnetic fields (perpendicular to the plane of the dot), where in many cases the quantum dots are analyzed at v > 4 and with a comparatively large number of electrons within the edge-channel model [28a-c, 29a, b]. Although up to now several studies on transport through quantum dots at finite magnetic fields exist [30-32] it is not understood in detail how the few-electron system behaves at magnetic fields corresponding to filling factors close to and below unity, where spin effects become important. So far, the internal spin density structure of single quantum dots was suggested [33], and recently the detection of compressible and incompressible states in quantum dot lattices in far-infrared spectroscopy was reported [34]. Here we will combine these two ideas of transport and optical spectroscopy by presenting the ESR in a single quantum dot. The typical implementation of a single quantum dot for our transport measurements is shown in the SEM micrograph in Figure 19. Applying a negative bias voltage to these gates, a well-defined quantum dot is formed, connected to the 2DEG reservoirs by tunable tunneling barriers. Transport through such a single quantum dot is only possible when, for example, the two gate voltages G1, G2 vary the electrostatic potentials in such a way that electrons from the lead can overcome the Coulomb blockade of transport (see Fig. 19). At B = 0 the density of electrons is smooth, whereas at finite magnetic fields discrete Landau levels (LLs) in the dots are formed. For example, for v ~< 2, two compressible LLs are separated by an incompressible region, defining a dot structure, which consists of an outer ring and a core. The CB oscillations of this ring/core dot are best analyzed in a charging diagram, as was shown in measurements on a double quantum dot [35]. This model is visualized in Figure 19, where the smooth spin density in the quantum dot is deformed to such a ring/core dot structure; the compressible tings are shown. We have to emphasize the" in our case at v ~< 2, the ring and the core dot represent different orientations of the electron spins. The question arising is how such a strong spin-split state can be explained and if the transition from the spin-up ring to the spin-down core is induced by microwave radiation.

327

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Fig. 19. Image of the quantum dot structure in a perpendicular applied magnetic field. Plotted are the compressible stripes in the quantum dot at v = 1. The dot is split into a core and a ring state in the lowest Landau level; the arrows depict the spin-up and -down states. The inset shows a SEM micrograph of the split-gate structure. In the center of the left dot the spin distribution in the lowest Landau level is depicted. The width of one gate finger is 80 nm.

* factor, well known in 2DEGs [36], is found in sysTheoretically the enhanced gefl~ tems of reduced dimensionality such as quantum dots [37]. The effects of the exchange interaction lead to an enormous spin splitting in these dots, which strongly influences electronic transport. In our model, we consider Ns strictly two-dimensional electrons to model qualitatively a real heterostructure in which the 2DEG is confined to the lowest electrical subband. It is confined to a disk of radius R in the 2D plane by a potential step Vconf(r) -- U0 exp

4Ar

+ 1

(1)

where zXr = 22 ~ . To ensure charge neutrality of the system, a positive background charge nb resides on the disk: nb(r)

= hs[exp( r - R

+ 1] -1

(2)

with the average electron density of the system given by hs = Nb/(ztR2). In the H a r t r e e - F o c k approximation, the state of each electron is described by a single-electron Schr/Sdinger equation:

{/10 +

VH(r)+

Vconf(r)}~r

fd2r'A(~,;')~rr162

(3)

nb(r r I F - F']

(4)

for an electron moving in a Hartree potential, VH(r) - - tc

d2r 'ns(r')

and a nonlocal Fock potential with e2

~r~ (~')r

;3 = 7

- ")

328

j ; - ;'l

(~)

(s)

MICROWAVE SPECTROSCOPY ON QUANTUM DOTS

(a)

30 ~ / ~ 20

~-~

~ ~,, /

~ ;

~

~.

25

. . . . . . .

~'

!,

~

(b)

~ 20

~" Io

( ,

0

15

o

_1ol +-+~../~ v = l

~=

~" ~o o

0

'

2 -lo

o

~o

20

M

3'o

Filling factor

Fig. 20. (a) Hartree-Fock Landau bands in a quantum dot at different numbers of electrons. Upper graph shows filling factor v = 1 (Ns = 22) and lower graph v = 2 (Ns = 42). Crosses mark the spin-up s = +1/2 state, and the diamonds mark spin-down s = -1/2 states. (b) Calculated r . m . s , g e*f f factor for all states in the lowest three Landau bands in the quantum dot at different filling factors (parameters: m* = O.067me, r -- 12.4, geff = -0.44).

where f(e~ - lz) is the Fermi distribution at the finite temperature T. The electron density is ns(r) = ~~___, " [#C (F)12f(e( ' -/ZDot) C with the chemical potential//,Dot

(6)

label ( represents the radial quantum number nr, the angular quantum number M, and the spin quantum number s = 4-89 H ~ is the single particle Hamiltonian for one electron with spin in a constant perpendicular external magnetic field [38, 39a, b]. A Landau band index n can be constructed from the quantum numbers nr and M as n = (IM! - M ) / 2 + nr. The Landau levels of H 0 with energy En, M,s = ha)c(n Jr 1) -Jr"sg* (/ZB/h) B are degenerate with respect to M with the degeneracy --

t ~ S / D 9 The

no = (2rr12) -1 per spin orientation./za is the Bohr magneton and (Oc denotes the cyclotron frequency. The Hartree-Fock energy spectrum e~ and the corresponding wave functions are now found by solving (3)-(6) iteratively on the basis of H ~ [39a, b]. The chemical potential/z is recalculated in each iteration to preserve the total number of electrons Ns. The number of basis functions used in the diagonalization is chosen such that a further increase of the subset results in an unchanged density ns(r). The specific confinement potential (Eq. (1)) and the positive background charge (Eq. (2)) are used here to induce a region inside a small quantum dot with fiat Landau bands, thus enhancing the exchange effects. Similar effects can be found in parabolically confined quantum dots with many electrons. In assuming a quantum dot with a total number of electrons Ns = 22 (v ,~ 1), we obtain at a magnetic field B = 3 T a large spin splitting, which is shown in the upper graph of Figure 20a. The crosses mark the spin-up s -- + 1/2 state, and the diamonds mark the spin-down s = - 1/2 state. In the lower graph we consider 42 electrons (v ~ 2) and obtain a large separation of LLs, but no spin splitting, that is, the qualitative spin structure in quantum dots strongly depends on the even/odd number of electrons in the dot and the filling o f the Landau bands. In Figure 20b the large root mean square (r.m.s.) geff factor of the lowest three Landau bands is plotted vs. the filling factor v ,-~ Ns/23. It is clearly seen that the enhancement is at its maximum whenever the filling factor is odd. Close to odd integer u the effective geff factor within a quantum dot reaches a value at which the clear separation into a spin-up and a spin-down state (ring/core) in the quantum dot is possible. Considering the existing experimental work [33], it is reasonable also to

329

BLICK

(a)

hv

NR

K

(b) NK+I

~

E

(c) VGI~,

NR ) B

s s

S

"'

,.,.

, s~-sJ ,I s s~-si.,~ s "' " Y

,'f,>,.,y;,

Fig. 21. (a) Energy level diagrams of the ring/core dot. With varying gate voltages G1 and G2, the numbers of electrons in ring and core dot are changed (NR,NK).(b) Microwaveradiation of energy/1o9induces spin transitions between ring and core dot at different magnetic fields. (c) A honeycomb pattern of lines of conductance is obtained in the case of the ring/core dot with variation of two electrostatic potentials VG1, VG2. The solid lines give the ground-state resonances, and the dashed lines indicate conductance resonances induced by microwaveradiation.

adopt the edge-channel picture in the integer quantum Hall regime for filling factors v < 2 for quantum dots. Hence, we can assume a self-consistent arrangement of the charge due to the exchange interaction, which leads to the formation of a compressible ring and core dot with different spin orientation and an incompressible quantum Hall liquid, separating the two dots. To transfer an electron from the ring with spin-up orientation to the core with spindown orientation, a spin-flip is necessary. The energy required can be supplied by a variation in the electrostatic potentials by a charge in the number of flux quanta connected to the ring/core dot or by millimeter wave radiation. It has been shown that photon-assisted tunneling can be used for millimeter-wave spectroscopy of single quantum dots [ 16]. In Figure 2 l a a spin-flip resonance is depicted in the simplest case: the photonic mode transfers energy to an electron in the ring R, which then tunnels into the core dot (K' spindown). This is more clearly seen in Figure 21b, which shows for clarity the common E vs. B representation of spin-up and -down states and the condition for ESR. Because of the large value of ge*ff around v - 1, we find a spin-split LL in the quantum dot, which evolves into the ring/core structure, sketched in Figure 19 (number of NR electrons in the ring and NK in the core). Keeping the magnetic field fixed, this structure can be analyzed within the typical charging diagram shown in Figure 2 lc. Here the solid lines mark finite conductance through the system, because the two dots are effectively connected in parallel. Obviously the resonances in the VG1-Vc2 plane form a honeycomb pattern, typical for a double quantum dot [35]. Considering for simplicity only absorption processes in the ring

330

MICROWAVE SPECTROSCOPY ON QUANTUM DOTS

dot with an energy of the radiation corresponding to ~ 1/5 of the charging energy of the ring, additional lines of conductance resonance in the charging diagram will be found. These are marked as dashed lines in Figure 2 l c, crossing each hexagon. The thin dashed lines cannot be found in such a parallel geometry and are only shown as a guide to the eye. The energetic conditions for resonance are extracted by referring to the level diagram in Figure 2 lb. The states of the ring with NR electrons and the core with NK electrons can be tuned by the gate voltages VG1, VG2 or the magnetic field B and are given by the different charging energies. The spin orientation representing the influence of the ge*ff factor in the dot is then easily obtained. In the current experiment we applied circularly polarized microwave radiation from a background wave oscillator of o9 = 2zr x 73 GHz corresponding to an energy Ev -ho9 = 300/zeV to the ring/core dot structure. This additionally supplied energy results in the spin-flip transition of an electron from the ring to the core dot. Our split-gate device, on top of an AlxGal_xAs/GaAs heterostructure (x = 0.33, # -- 80 m2/V s and ns = 2 • 1015 m -2) is biased, and the quantum dot formed contains roughly Ns ~ 50 electrons (see inset, Fig. 19). A magnetic field of B = 8.47 T is applied corresponding to a filling factor v < 2. The charging energy of the quantum dot is obtained by the relation Ec = e 2 / C z (Cz: total capacitance of the system). The quantum dot in our experiments has a diameter of d = 350 nm and a total charging energy of Ec ~ 1.4 meV at B = 0. At a finite magnetic field, where the ring/core dot is formed, we determined the individual charging energies to be E~ ~ 1.10 meV of the ring and Ec~ ~ 0.29 meV of the core. The calibration is obtained using the drain source dependence [8]. In the following we will refer to the charging energies and the resulting gate voltage periodicities as or,/3, and y. The energy E~ for the ring is surprisingly larger than that for the core Ec~. This can readily be explained by the fact that the effective area and hence the "charging capacity" of the ring are smaller compared to those of the larger core dot. The schematic ring/core structure in Figure 19 is not to scale; more likely the ring itself is very thin, resulting in a larger charging energy. As seen in Figure 22 we determined charging diagrams of the ring/core dot structure depending on one gate voltage (Vg = VG1) and the magnetic field. The step size of the magnetic field variation of the different traces is 6B = 15 mT. The two diagrams show the same part of the charging diagram and differ only in that (a) is measured without radiation applied and (b) is taken under irradiation at co = 2zr x 73 GHz. In (a) the two characteristic resonance periodicities ot and/3 refer to the charging energies of the dots from Figure 21 c. Adding those two charging energies, one obtains y, the charging energy for the whole quantum dot at B -- 0. The variation in conductance amplitude is due to the different strengths of coupling to the leads. The application of radiation in (b) leads to a new structure of the conductance resonances in the charging diagram: the left peak shows an additional resonance, and the fight peak structure of (a) is replaced under irradiation by a single peak. This redistribution of resonances is a clear sign of the photon-assisted spin-flip transition, because the radiation energy and the charging energy match Ev -- Ec~. Hence, electrons are pumped from the ring into the core dot. In the case of the left peak this means a new transport channel is opened by pumping electrons through the ring and core structure. In the case of the right peak the electrons are pumped out of the core dot, that is, there is one transport channel missing. Computing from the microwave energy and the magnetic field the effective geff* factor (ho9 -- ge*ff#BB), we obtain a value of geff -- 0.61 which is considerably larger than the geff factor for electrons in a 2DEG of A1GaAs/GaAs heterostructures (geff-- -0.44). We have given an explanation of the enhancement of the g*eft factor and have shown that it has considerable influence on the transport properties of a quantum dot, because it reflects a certain spin orientation in the dot. The enhanced ge*fffactor leads to the formation of Landau levels at v ~< 2 in the dot, which evolve as a ring/core dot similar to a double-dot geometry measured in a charging diagram. This ring/core dot consists of regions with different spin orientations, which show a spin-flip under irradiation with circularly polarized

331

BLICK

8

6

::

.,._.

A

:::L

4

>

"O

~

"O

.....

"0

2

...

_ j \

0.02

....... 0.()3

0

0.()4

0.02

"

0.03

0.04

-v (v)

-v (v) g

g

(a)

(b)

Fig. 22. Measuredcharging diagrams of a single quantum dot at a magnetic field of B = 8.47 T. The conductance is shown, depending on one gate voltage Vg. The magnetic field is varied in steps of gB = 15 mT. (a) Resonances of the ring/core dot with the characteristic periodicities ot and fl without microwave radiation. (b) The same part of the charging diagram, with microwave radiation of 73 GHz. The periodicity y represents the total dot charging energy, y = ot +/3.

microwaves of 73 GHz. From these measurements we determined the enhanced effective * factor is smaller than the g*eft factor to be 0.61 That the measured value of the effective geff value predicted by the Hartree-Fock approximation has two explanations: the exchange is too strong in the approximation, and a model including correlations would produce a more appropriate value. Second, it is still an open question that might be answered by further experiments what g factor, the bare or the enhanced, is actually measured in experiments on microwave-induced tunneling in few-electron systems. The determination of the complete frequency dependence will give further insight into the complex spin structure within quantum dots.

6. S P E C T R O S C O P Y O N A D O U B L E Q U A N T U M D O T We have demonstrated the combination of SET and PAT through single quantum dots as suggested [22] and then reported by Blick et al. and Kouwenhoven et al. [16, 23]. We will now apply this as a tool for millimeter wave spectroscopy to two coupled quantum dots [40]. The aim is to study the different possible excitation mechanisms and especially the influence of coherence on electronic transport. The energy scales dominating the transport properties of quantum dots are the thermal energy ks T and the Coulomb energy Ec, competing with the radiation energy hr. The frequencies corresponding to the charging energies of the individual quantum dots A and B are on the order of E A/2 ,~ 390 GHz and E~/2 ~ 150 GHz. In the measurements presented here we will concentrate on the excitation range of quantum dot B. The ratio Y = hv/kBT gives an estimate of how far from thermal equilibrium electrons are excited. Applying the experimental parameters, v = 150 GHz and T -- 200 mK, we obtain a value of y = 36, that is, excitation creates a prominent nonequilibrium electron distribution.

332

MICROWAVE SPECTROSCOPY ON QUANTUM DOTS

Fig. 23. Excitation spectrum of the conductance through a double quantum dot in a gray-scale plot (trmax) 3/zS, black; Crmin ~<-0.25/zS, white). The superimposed modulations of the two different Coulomb charging energies are nicely seen. For clarity, a white and a black mesh were added, representing the typical diamond shapes of dots A and B.

To define the quantum dots, QPCs were used. We investigated their properties as photon detectors, before the quantum dot measurements in the second section. We will now focus on the double quantum dots. By biasing all of the gates, we obtained an asymmetric dot system, that is, dots of different sizes. This is seen by the two different periodicities of the CB oscillations corresponding to the small quantum dot A (large period) and the large dot B (smaller period) in the current vs. gate voltage characteristics of Figure 23. This plot gives the energetic calibration of the dot system by determining the different diamond-shaped regions of Coulomb blockade for dot A (white grid) and dot B (black grid). Moreover, microwave spectroscopy can be compared to the results from this nonlinear transport measurement. Obviously, the great advantage of applying microwaves is the possibility of exciting the electronic states of the "artificial molecule" far from equilibrium. To understand the measured signal in the photocurrent it is necessary to classify the different mechanisms under which electrons can be excited by irradiation. As seen in Figure 24, the radiation hv can either be absorbed in the leads/zs or/ZD or within the double dot. An absorption in the leads allows electrons to tunnel through excited states of the quantum dots and can be thought of as an outer photo effect. Absorption in the dots (the inner photo effect) also leads to the transfer of electrons in excited states, from which they decay into the ground states (GS) or into the leads. A special type of such excitations between states that are extended over the whole double quantum dot are the Rabi oscillations of this "artificial molecule." This is depicted by the "circulating state" from e A A,B to es. As seen in Figure 24a, the Coulomb charging energies E C are much larger than the tunnel splitting. In our case we found in measurements of linear transport in a detailed resonance shape analysis the tunnel splitting of the double quantum dot to be on the order of ~e = e2 - e l -'~ 1 1 0 - 140/zeV [41 ]. This energy corresponds to a Rabi frequency of

333

BLICK

Fig. 24. Level diagram of the double quantum dot. (a) The radiation can either be absorbed in the leads /xS,/z D or in the quantum dots as shown. In the dots transitions between ground states (GS: eB, eA) and excited states (ind.: e~ and e~) can be found. Transitions between the tunnel split discrete states in the dots eA, eB are possible. (b) The way spectroscopy is performed. Photons are absorbed by an electron in one dot, which tunnels into discrete states of the other dot.

VR = 3 e / h ~ 27-34 GHz. In another experiment, as we will find later, we were able to probe these Rabi oscillations in the double quantum dot by the radiation applied [42], as suggested by Tsukuda et al. [43] and Stafford and Wingreen [44]. In measurements performed on a single quantum dot, we observed conductance resonances due to PAT [ 17]. The resolution is quite enhanced in the setup presented here, where the small quantum dot A is coupled in series. This is seen in Figure 25a, where the induced DC current at a frequency of v = 168 GHz is plotted. The lowest line is taken without radiation applied, and the upper lines show the current at small microwave power. Obviously, additional resonances are induced, which are marked by the arrows (ind.). These resonances do not shift when the microwave power is increased. In Figure 24b it is shown how spectroscopy of one quantum dot by another under irradiation is performed: the electron in the discrete state eB (i.e., with little thermal broadening) absorbs a photon and can then tunnel through an excited state of the connected "artificial atom." As seen, the charging energies of the two dots are quite different; hence, the excitations of the small and the large dot can easily be distinguished. Here we only consider excitations of the large dot. It should be noted that we do not observe an influence of the displacement current, in accordance with the theoretical calculations of Bruder and Schoeller [22]. Applying radiation at different frequencies, we find a shift of the position of the induced resonances, in accordance with PAT. This can be seen in Figure 25b, where the current resonances appear at a different position 3 Vc corresponding to the frequency applied, v -87 GHz. The frequency dependence of ~ Vc is shown in Figure 26. The offset at v ~ 0 possibly is an artifact due to the limited resolution in calibrating the value or, which depends A,B on the total capacitance of the dots C z .

334

MICROWAVE SPECTROSCOPY ON QUANTUM DOTS

t

v = 168 GHz

v = 87 GHz

20

GS ind

0

)

E A c

=

.=

$

lo

0

0

I

I

0.30

0.25

,

o.3s -vg (v)

o. o

o.15

,

o. o

' 0.35

0.40

-vg (v)

(a)

(b)

Fig. 25. Induced photocurrent through the double quantum dot at two different frequencies. Clearly seen are the two periodicities reflecting the different charging energies of the dots (E A and E~). In (a) the lowest curve is taken without radiation, whereas under radiation applied additional peaks (ind.) are found (v = 168 GHz). In (b) these induced peaks shift according to photon-assisted tunneling. The position shifts with frequency here v = 87 GHz was chosen (with VDS = 20/xV). Note that under irradiation a strong backward current can be inducedm this is a signature of a coherent electron pumping process. (Source: Reprinted from [40], with permission of Elsevier Science.)

l Ec= A 3meV / Ec-B_ 1 14meV 1.0

A

>

. v

.."''""

E

u.I c~ II

>

0.5

,

0"00

I

50

i

I

,

100

I

150

,

I

200

v (GHz) Fig. 26. Frequency dependence of the induced peak positions 6 VG with respect to the ground state. A linear dependence is observed, as expected. The solid line shows an extrapolation of the data points, and the dotted line represents the slope h = 6.62 x 10-34 J s. (Source: Reprinted from [40], with permission of Elsevier Science.)

As w e have seen in F i g u r e 26, the i n d u c e d r e s o n a n c e s o b e y the linear relation E =

hv = otrVG. Interestingly, we o b s e r v e d that at certain f r e q u e n c y bands we hardly f o u n d r e s o n a n c e s , for e x a m p l e , in the large gap f r o m 100 to 130 GHz. This b e h a v i o r is c o m prehensive, c o n s i d e r i n g the discrete excitation spectra f o u n d for such artificial atoms or m o l e c u l e s [8, 41 ]. T h e r e are only a finite n u m b e r of excited states that can be p o p u l a t e d by m i l l i m e t e r w a v e radiation. This is an indication that the a b s o r p t i o n of the radiation occurs within the q u a n t u m dots.

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100

g -

75

50

-0.1

0.0

0.1

v G IV) Fig. 27. Charging diagram of the current through the double quantum dot under irradiation at v = 151 GHz. The back-gate voltage is varied in steps of 8VBG ---0.5 V from VBG = --30 V. The solid line indicates the ground state (GS) resonance of the large quantum dot B, and the dashed line shows the ground state of the small dot A. The induced (ind.) resonance (dashed-dotted) obviously follows the slope of the resonances of dot B. (Source: Reprinted from [40], with permission of Elsevier Science.)

We have to point out that in addition to PAT, a strong electron pump effect is observed, which was explained as electron pumping by spatial Rabi oscillations [44]. This is best seen in Figure 25, where a large backward current, ~ - 10 pA, even under forward bias, is induced. Such a coherent electron pumping mechanism might overcome the limitations of metrological devices that are based on sequential tunneling. We also performed millimeter wave spectroscopy on the double dot by determining the charging diagram under radiation of v = 151 GHz, as seen in Figure 27. This approach differs from the method applied before--where the frequency is varied--because it allows us to distinguish between absorption in the leads and that in the dots. With varying backgate voltage, the induced resonance line of the large dot can be tuned into resonance with the other dot, thus allowing spectroscopy of one dot by the other, as shown in Figure 25b. Here we measured the current under a small forward bias of VDS = 20/zV, depending on the top-gate voltage VTG, as in Figure 27. In addition, we changed the back gate in steps of VBG = 0.5 V from VBG = 30 V, thus shifting both quantum dot potentials. The different shifting of the CB resonances is caused by the different capacitive coupling CBG-A and CBG-8. This is seen in the slopes of the solid and dashed lines in Figure 27. These lines represent the ground states of the two dots. In the measurements shown, the resonance line (dashed-dotted) shifts parallel to the ground-state resonance of the large dot. This parallel shifting shows that the photon is absorbed in the large quantum dot. By varying the two potentials, it is now possible to match any specific resonance condition of the double dot. Regarding the possible application as a single electron/photon detector, it is obvious that such a quantum dot offers a very sensitive device for detection of millimeter- and submillimeter-wave radiation, which might even reach the quantum limit of detection [21].

7. T H E MICROWAVE I N T E R F E R o M E T E R PAT [ 16, 23, 45] through single quantum dots has been demonstrated and is now applied as a tool for millimeter-wave spectroscopy on these devices. Measurements with microwave radiation performed so far give evidence that the well-known effect of Coulomb blockade

336

MICROWAVE SPECTROSCOPY ON QUANTUM DOTS

Fig. 28. Schematicenergy diagram of the double quantum dot, showing the tunnel splitting of the discrete levels el, 82 and the resulting Rabi oscillations (hv = 3e). The millimeter-wave radiation at frequency v interacts with the oscillating electron. (Source: Reprinted with permission from [42]. 9 1998 American Physical Society.)

(CB) [ 1] can be overcome with this continuous-wave (cw) radiation. Time-dependent measurements reveal more detailed information about coherent electronic modes in quantum dots. Initial experiments were undertaken by Karadi et al. [46] to investigate the dynamic response of a quantum point contact and by Dahl et al. [47] in measurements on large metallic discs. Other measurements on superconductor tunnel junctions have shown the importance of electronic phase effects (also termed quantum susceptance) on electronic transport [48]. Recently measurements of the electron's phase were reported with quasiDC measurements of a single quantum dot [49]. In quantum dot systems to date, only cw frequency sources have been applied for spectroscopy in the range from some MHz up to 200 GHz [ 16, 50]. Moreover, the common method of detection gives only scalar information; that is, it shows signals in the magnitude of the induced current without information on the phase. Recent work demonstrated the ability of integrated broad-band antennas to couple cw millimeter- and submillimeterwave radiation to nanostructures, such as quantum dots and double quantum dots, showing the expected PAT resonances in transconductance [ 16]. The energy acquired by the electrons through the absorption of the photons allows them to tunnel through states of higher energy above the Fermi energy, which are otherwise not accessible. We now present transport experiments, performed on a coupled quantum dot structure under irradiation with a new type of broad-band millimeter-wave interferometer. This is of major interest, because the double quantum dot we investigate contains only a few electrons and should thus evolve molecular modes; that is, an electron's wave function can be spread out across the whole double dot. This is schematically illustrated by the level diagram in Figure 28. The discrete states El, ee in the two dots split due to the tunnel coupling, and coherent Rabi oscillations result. An externally applied frequency h v --- 6e on the order of the Rabi frequency then suppresses the coherent modes in the quantum dots. The new source we apply generates picosecond pulses of radiation, corresponding to harmonics in a frequency range of 2--400 GHz. Because these pulses are coherent, we can obtain the magnitude and phase of the induced signal. In Figure 29a such a source, consisting of nonlinear transmission lines in series, is shown: the transmission line is intersected by Schottky diodes, which cause a steepening of the sinusoidal input signal, leading to the generation of harmonics. The lower part of Figure 29a depicts the magnified center of the bow-tie antenna at the termination of the transmission line. The double quantum dot system used in these experiments is seen in the SEM micrograph in the fight part of Figure 29b. In the measurements we found CB energies of E A = e e / c ~ z ,~ 3.0 meV for the small dot A and for the large dot B E~ ~ (1.17 4-0.1) meV (C~" total capacitance of the dot). The complete characterization of such a double quantum dot is performed within the so-called charging diagram, discussed in detail elsewhere [35]. The

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Fig. 29. (a) Nonlinear transmission line used to generate the millimeter wave radiation. The upper part shows the whole circuit (4 x 5 mm); in the lower part the antenna apex is magnified. (b) The broad-band antenna of the double dot. The antenna arms serve as gate contacts for the quantum point contacts defining two quantum dots of different sizes (see right side). (c) Setup of the millimeter-wave interferometer: the radiation of two nonlinear transmission lines (A and B) is superimposed, radiating onto the sample and inducing a drain source photoconductance. The transmission lines are driven by two phase-locked frequency generators in the lower K band (v0: offset frequency). Current through the sample is amplified, which allows analysis of the amplitude and phase of the high-frequency signal. (Source: Reprinted with permission from [42]. @ 1998 American Physical Society.)

m e a s u r e m e n t s w e r e p e r f o r m e d in a t o p - l o a d i n g 3He/4He dilution refrigerator with a sample holder, a l l o w i n g quasi-optical t r a n s m i s s i o n [16]. T h e t e m p e r a t u r e in the m e a s u r e m e n t s p r e s e n t e d is a r o u n d 250 mK. T h e p e r f o r m a n c e of the a n t e n n a was verified with a detailed n u m e r i c a l analysis, using a c o m m e r c i a l l y available p r o g r a m [ 15].

338

MICROWAVE SPECTROSCOPY ON QUANTUM DOTS

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v (GHz) Fig. 30. Responseof the quantum dot system to the interferometer radiation up to 400 GHz. The data points with the largest response values of the coupled dot system are marked by c~, r, and y, corresponding to frequencies of 28, 42, and 56 GHz. The inset showsthe Coulombblockade oscillations of the doublequantumdot, dependending on one top-gate voltage VTG.The arrowindicates the gate voltage at which spectra were recorded. (Source: Reprinted with permission from [42]. 9 1998American Physical Society.) The setup of the photoconductance experiment with the broad-band interferometer is shown in Figure 29c: the interferometer consists of two nonlinear transmission lines (NLTLs) with integrated antennas, generating high-frequency signals up into the THz regime [5 l a, b]. The NLTLs are driven by two frequency generators operating in the frequency range 1-20 GHz (HP 83711A). The interferometer superposes the two broad-band spectra from the NLTLs with an offset frequency 8v on the order of some 10 Hz (in this experiment v0 = 23 Hz). These spectra are optically coupled into the waveguide and transduced by the antenna of the "artificial molecule." The high-frequency radiation induces a photocurrent through the nanostructure, which is measured in both magnitude and phase because of the coherence of the interferometer. In locking onto one of the interferometer's harmonics, which are separated by the offset frequency times the number of the harmonic, the corresponding high-frequency conductance signal is obtained. In the measurements shown, the data were taken in amplitude/phase mode of the lock-in. With phase-sensitive detection, the frequency of interest can be selected in the signal at the microwave difference frequency nSv (where n is an integer). Hence this technique allows direct probing without application of the Kramers-Kronig relation. A more detailed description of the interferometer has been given elsewhere [52a, b]. In Figure 30 we present the response of the double-dot system to broad-band radiation with the gate voltages fixed (at the position of the arrow in the inset). In the inset of Figure 30 the conductance resonances of the double quantum dot system are shown. The two different periods corresponding to the small quantum dot A (large period) and the large dot B (smaller period) are easily recognized. Although the temperature in the measurements presented is around 250 mK, the CB resonz_.~es reveal a broadening due to the constant nonresonant heating of the electrons in the leads by the radiation applied. We chose a double-dot system for these measurements, because electron transport investigations of such devices have shown that the wave functions in the two dots effectively couple in a molecular-like fashion [35]. The amplitude of the signal induced by the radiation is plotted vs. frequency. Each square point of the discrete spectrum corresponds to a harmonic generated by the interferometer; the step width is 7 GHz. The best response is obtained in the lower frequency regime. At frequencies below the point marked or, the cutoff frequency of the waveguide damps the millimeter-wave signal. The overall sensitivity of the

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"artificial molecule" is comparatively high, because well-defined signals were measured up to 400 GHz. The spectrum shown can be divided into two parts: the low (a,/~, y) and the high ((1 and (2) energy excitations. The different energies of the excitations of the double dot result from the strong variation of the different confinement potentials of the coupled dots. In transport spectroscopy in the nonlinear response regime of the small dot, we found the excitation energies to be on the order of E ~ / 3 ~ 1 meV [8]. Considering the energies of the excitations (1 and (2, which are on the order of e~l ~ 1 meV and e~2 ~ 1.4 meV, we find very good agreement. We conclude that the excitations found are generally excitations of the coupled system, although (1,2 are dominated by the confinement potential of the small dot A. The resolution of the interferometer in its current configuration did not enable us to identify additional structures induced by PAT due to resonance broadening: the contributions of all frequencies applied by the interferometer are superimposed. Our main interest here, however, lies in the determination of the phase dependence of the induced current in the high-frequency regime. The term "phase" here is to be understood as the relative phase of the electron's wave function in the double quantum dot. We anticipate achieving better resolution and higher-frequency measurements with an in situ arrangement of the dual-source interferometer. To focus on different excitations, we select the appropriate harmonic and change the charging state of the artificial molecule by varying the gate voltage. In Figure 31 the highfrequency conductance of the dots is detected in amplitude/phase mode of the lock-in; the bias was fixed in the low-bias regime at Vds = --60/zV to enhance the resolution. The left axis gives the magnitude and the fight axis the relative phase of the conductance signal. From top to bottom the frequency increases, as marked. The individual curves correspond to the points a,/~, and y in Figure 30 as indicated, where the response of the dot system was strongest. In general the conductance signal can be understood as a phasor. The coherent signal from the millimeter-wave interferometer enables the detection of conductance (cr) and relative phase (q~) of the wave function in the high-frequency domain. The conductance amplitude resembles the periodicity of the conductance resonances shown in the inset of Figure 30. Its amplitude decreases toward the higher frequency shown (56 GHz), in accordance with the amplitude we found in the broad-band response. The phase data shown in Figure 31 are strongly influenced when the microwave harmonic measured varies from the lowest value of c~ (28 GHz) to the higher values (15: 42 GHz, y: 56 GHz). It must be noted that the variation of the phase in case c~ reflects charging processes of the large dot (B) and is seen as a modulation that corresponds to the periodicity of the charging process (see inset Figure 30). This can be seen as probing the admittance Y of this particular dot and the change from capacitive to inductive behavior and vice versa [53a, b]. With increasing frequency, the charging process of the small dot (A) becomes visible, because the change in phase is large and is dominated by the CB oscillation period of this dot. At the highest frequency, the phase switches between capacitive and conductive transport through the small dot, whereas the modulation from charging the large dot is no longer visible. Finally, the question is raised of how the strong variation in the phase signal towards higher frequencies relates to the electronic transport through the "artificial molecule": one approach is to state that the high-frequency radiation influences a "sloshing mode" of the electronic wave function; that is, the spread-out wave function is localized by an external disturbance of the system [21 ]. More precisely, this phenomenon can be described as Rabi oscillations in the double quantum dot, which are probed by the radiation applied, as has been shown theoretically [43]. In a detailed resonance shape analysis from measurements of linear transport, we found the tunnel splitting of the double quantum dot to be on the order of ~e = e2 - el = 110 - 140/zeV [42]. This energy corresponds to a Rabi frequency of VR = Be~ h ~ 27-34 GHz, which is on the order of the transition frequency

340

MICROWAVE SPECTROSCOPY ON QUANTUM DOTS

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observed. Hence, our externally applied disturbance with frequencies above 25 GHz tends to destroy these Rabi oscillations; that is, the collective electronic wave function is not able to follow the external driving force of the electromagnetic field. This can be regarded as a direct probing of this wave function and hence of the molecular coupling between the two quantum dots.

8. SUMMARY Millimeter wave spectroscopy on quantum point contacts and single and coupled quantum dots was investigated. In a detailed study of the response of quantum point contacts, the influence of potential asymmetries in such devices was shown. Photon-assisted transport through single and double quantum dots in the high-frequency regime could clearly be demonstrated. This method can be applied for spectroscopy of quantum dots far from

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equilibrium, which was demonstrated by electron spin resonance in a single dot at high magnetic fields. Finally, a millimeter-wave interferometer was applied to perform coherent spectroscopy on coupled quantum dots. The newly developed interferometer covers the whole millimeter-wave regime and allows both magnitude- and phase-sensitive detection, In the measurements shown, the high-frequency conductance through the double quantum dot was detected. A broad-band response (i.e., excitations of the coupled quantum dot) was found. Furthermore, strong variations in the relative phase signals were observed, when the frequency of the radiation was larger than the Rabi frequency of the "artificial molecule," This leads to the destruction of the coherent mode in the double quantum dot.

Acknowledgments I thank Frank Stern for performing the electrostatic model calculations of the nanostructures. I also acknowledge the support of Klaus von Klitzing, Daniela Pfannkuche, Rolf J. Haug, Jiirgen Weis, Vidar Gudmundsson, Jurgen Smet, Daniel van der Weide, Karl Eberl, and Gerhard Miiller during this work. And finally I thank Helga Ludwig, M o n i k a Riek, Frank Schartner, Susanne Demel, and Jtirgen Behring for their technical support. This work was funded in part by the Bundesministerium fiir Bildung, Wissenschaft, Forschung und Technologie (BMBF). All of the figures were taken from my thesis (available under ISBN 3-8171-1507-5; R. H. Blick, R e i h e P h y s i k 57 (1996)).

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