Fano resonances in electronic transport through a single-electron transistor

Fano Resonances in Electronic Transport through a Single Electron Transistor J. Göres† , D. Goldhaber-Gordon‡, S. Heemeyer, and M.A. Kastner∗ arXiv:cond-mat/9912419v2 [cond-mat.mes-hall] 23 Dec 1999 Department of Physics, Massachusetts Institute of Technology Cambridge, MA 02139 Hadas Shtrikman, D. Mahalu, and U. Meirav Braun Center for Submicron Research Weizmann Institute of Science Rehovot, Israel 76100 We have observed asymmetric Fano resonances in the conductance of a single electron transistor resulting from interference between a resonant and a nonresonant path through the system. The resonant component shows all the features typical of quantum dots, but the origin of the non-resonant path is unclear. A unique feature of this experimental system, compared to others that show Fano line shapes, is that changing the voltages on various gates allows one to alter the interference between the two paths. scattering channel. Fano resonances have been observed in a wide variety of experiments including atomic photoionization [15], electron and neutron scattering [16,17], Raman scattering [18], and photoabsorption in quantum well structures [19,20]. I. INTRODUCTION When a droplet of electrons is confined to a small region of space and coupled only weakly to its environment, the number of electrons and their energy become quantized. The analogy between such a system and an atom has proved to be quite close. In particular, these artificial atoms exhibit properties typical of natural atoms, including a charging energy for adding an extra electron and an excitation energy for promoting confined electrons to higher-lying energy levels [1,2]. Remarkably, the analogy goes even further and includes cases where an artificial atom interacts with its environment. A system known as a single electron transistor (SET), in which an artificial atom is coupled to conducting leads, can be accurately described by the Anderson model [3–5]. The same model has been used extensively to study the interaction of localized electrons with delocalized ones within a metal containing magnetic impurities. One of its subtle predictions is the Kondo effect, which involves many-body correlations between an electronic spin on an impurity atom and those in the surrounding metal. This effect has recently been observed in an SET when the artificial atom develops a net spin because of an odd electron occupancy [6–10]. In this paper we report that by changing the parameters in a single electron transistor we observe another phenomenon typical of natural atoms: Fano resonances. While several aspects of the Fano resonances in our SETs are easily understood, others are very surprising. Asymmetric Fano lineshapes are ubiquitous in resonant scattering [11–13], and are observed whenever resonant and nonresonant scattering paths interfere. The more familiar symmetric Breit-Wigner or Lorentzian lineshape [14] is a limiting case that occurs when there is no interference, for example when there is no non-resonant FIG. 1. Electron micrograph of an SET showing the split gates (I) that define the tunnel barriers and the additional gate electrode (II) that adjusts the potential energy on the droplet. The widely successful application of the LandauerBüttiker formalism [21,22] shows that electron transport through a mesoscopic system is closely analogous to scattering in the systems described above. Therefore, one might expect to also observe Fano resonances in the conductance of nanostructures [23–26]. Indeed, Fano lineshapes have been reported in a recent experiment by Madhavan et al . measuring tunneling from an STM tip through an impurity atom into a metal surface [27]. However, this characteristic feature of interference between resonant and non-resonant scattering has not been reported for more conventional nanostructures. In this paper we report the observation of Fano resonances in conductance through a single electron tran1 tance between the the droplet and its environment, primarily the nearby conducting leads and gates. Thus, an energy U is required to overcome Coulomb repulsion and add an electron to the droplet. This energy cost causes the number of electrons on the droplet to be quantized and electron transport through the droplet to be suppressed. However, lowering the chemical potential of the droplet by adjusting the voltage on the plunger gate makes it possible to add electrons one at a time. At a charge degeneracy point, where the states with N and N + 1 electrons on the droplet have equal energy, transport of electrons from one lead through the droplet to the other lead is allowed. This effect is known as Coulomb blockade [28] because transport is suppressed everywhere except close to the degeneracy points. Because of the small size of the electron droplet in our SETs, the energy spacing between the discrete levels ∆ǫ, that is, the energy to excite the droplet at fixed N , is only a few times smaller than the charging energy U . Depending on the transmission of the left and right tunnel barriers, characterized by tunneling rates ΓL /h, ΓR /h, respectively, we observe different transport regimes in our SETs at very low temperature. If the thermal energy kB T is much smaller than the coupling energy Γ ≡ ΓL + ΓR , quantum fluctuations dominate so that the resonances have width Γ. When Γ ≪ ∆ǫ the coupling is weak and one observes narrow quasi-periodic peaks (Fig. 2(c)). As the coupling is increased a new regime emerges, in which transport away from the resonances is enhanced by the Kondo effect [6] when there is an odd number of electrons on the droplet (Fig. 2(b)). Surprisingly, if we increase the coupling beyond the Kondo regime we observe asymmetric Fano resonances on top of a slowly varying background (Fig. 2(a)). Before discussing these data in detail, we present the analytic form predicted for such line shapes. Scattering theory, specifically the S-matrix approach, predicts Fano lineshapes as the general form for resonances in transport through quasi-1D systems [23]. The S-matrix, which relates the outgoing and incoming scattering state amplitudes, is a unitary matrix at real energies ǫ because of probability conservation. Therefore, for real energies the eigenvalues of S can be written in the form λ(ǫ) = e2iδ(ǫ) where δ(ǫ) is the scattering phase. Resonant scattering occurs when one of the eigenvalues of the S-matrix develops a pole at the complex energy ǫ0 − iΓ/2. Near this pole the resonant eigenvalue is given by λ(ǫ) = e2iδ0 e2iδres (ǫ) [23]. Here δres (ǫ) is the resonant contribution to the phase shift and is given by tan δres (ǫ) = −Γ/2(ǫ − ǫ0 ). It varies from zero to π as the energy is moved through the resonance from below. The background or nonresonant contribution to the scattering phase shift δ0 is, in this approximation, a constant independent of the energy. The total cross section is directly related to the scattering phase shifts through sistor (SET). This structure has the advantage over the tunneling experiment of Madhavan et al . [27] and over conventional scattering experiments that we can tune the key parameters and thus study the interference leading to Fano resonances in greater detail. This paper is organized as follows: In section II we describe the SET and the measurements we have made on it; we also summarize there the standard theory for Fano line shapes. Our results are presented in section III. In section IV we discuss the results, draw conclusions and point out issues which require further research. II. EXPERIMENTAL DETAILS AND THEORETICAL BACKGROUND An SET consists of a small droplet of electrons coupled to two conducting leads by tunnel barriers. Gate electrodes (shown in Fig. 1) are used to control the electrostatic potential of the droplet and, in our structures, the heights of the tunnel barriers. The SETs used in these experiments are the same ones used for studies by Goldhaber-Gordon et al . of the Kondo effect [6,7]. The electron droplet is created in a twodimensional electron gas (2DEG) that forms at the interface of a GaAs/AlGaAs heterostructure with a mobility of 100, 000 cm2 /Vs and a density of 8.1 × 1011 cm−2 . Applying a negative voltage to two pairs of split gate electrodes depletes the 2DEG underneath them and forms two tunnel barriers. The barriers separate the droplet of electrons from the 2DEG regions on either side, which act as the leads. The heights of the two barriers can be adjusted independently by changing the voltages on the respective constriction electrodes (I), and the potential energy of the electrons on the droplet can be shifted relative to the Fermi energies in the leads using an additional plunger gate electrode (II) near the droplet. Our SET’s are made with a 2DEG that is closer to the surface (≈ 20 nm) than in most other experiments, allowing the electron droplet to be confined to smaller dimensions. This also makes the tunnel barriers more abrupt than in previous structures. We estimate that our droplet is about 100 nm in diameter, smaller than the lithographic dimensions because of depletion, and contains about 50 electrons. For conductance measurements we apply a small alternating drain-source voltage (typically 5µV) between the leads and measure the pre-amplified current with a lock-in amplifier. The conductance is then recorded as a function of the plunger gate voltage Vg . For differential conductance measurements we add a finite offset drainsource voltage Vds and measure the response dI/dVds to the small alternating drain-source voltage as a function of both Vg and Vds . In an SET the Coulomb interaction among electrons opens up an energy gap U in the tunneling spectrum, given classically by e2 /2C, where C is the total capaci- σ(ǫ) ∝ sin2 (δres (ǫ) + δ0 ). 2 (1) resonant, for which the phase changes by π in an energy interval ≈ Γ around the resonance energy ǫ0 . The other is nonresonant and has a constant background phase shift δ0 . If there is no nonresonant channel or the background phase shift between the channels is zero, symmetric Breit-Wigner resonances are observed. In all other cases Fano resonances result. In his original work Fano treated the case of inelastic scattering in the context of autoionization and derived the so-called Fano formula for the scattering cross section [11] σ(ǫ) ∝ (ǫ̃ + q)2 , ǫ̃2 + 1 (2) where ǫ̃ ≡ (ǫ − ǫ0 )/(Γ/2) is the dimensionless detuning from resonance and q is called the asymmetry parameter. The asymmetry parameter is related to the background phase shift of the S-matrix treatment by q = cot δ0 . The magnitude of q is proportional to the ratio of transmission amplitudes for the resonant and non-resonant channels [11]. The limit q → ∞, in which resonant transmission dominantes, leads to symmetric Breit-Wigner resonances. In the opposite limit q → 0 non-resonant transmission dominates, resulting in symmetric dips. According to the Landauer-Büttiker formalism, conductance through any mesoscopic system can be expressed in terms of an S-matrix. Hence, Fano resonances should also be observed in conductance if a resonant and a nonresonant transmission path coexist [23]. Analogous to the scattering case Eq. (2) the conductance is then given by G = Ginc + G0 (ǫ̃ + q)2 , ǫ̃2 + 1 (3) where Ginc denotes an incoherent contribution to the conductance, which is often observed [29]. Note that the Breit-Wigner limit q → ∞ implies G0 → 0 leading to a finite conductance maximum of Ginc + G0 (1 + q 2 ) at ǫ̃ = 1/q. III. RESULTS FIG. 2. Comparison of conductance measurements in the (a) Fano regime, (b) Kondo regime [6] and (c) weak coupling regime [6]. Fits to the Fano formula (3) are shown for the center and right resonances in (a). The respective asymmetry parameters are q = −.03 and q = −.99. During the sweep through the left resonance a sudden shift in effective gate voltage occured, presumably caused by charge motion in the vicinity of the droplet, leading to the unusual shape of this resonance. As mentioned above, Fig. 2(a) shows three consecutive, well-separated and relatively narrow resonances on a background that varies smoothly in the range 0.11 – 0.22 e2 /h. The conductance does not vanish at resonance, as would occur if the destructive interference between the transmission paths were complete, presumably because of an incoherent component. The resonances in the center and right are of the Fano form Eq. (3) with asymmetry parameters given in the figure caption. We might imagine that we could calculate the combined transmission through the resonant and nonresonant channels simply by adding the complex amplitudes Varying the value of the background phase shift produces the entire family of Fano lineshapes. It is important to emphasize that two interfering channels are necessary for Fano resonances to arise. One is 3 FIG. 3. Conductance resonances as a function of the gate voltage for various strengths of the tunnel couplings (a) ΓL and (b) ΓR . The data sets labeled 0 mV are identical for both (a) and (b) and correspond to the case where −145 mV is applied to both the left and the right split gate electrodes (labeled I in Fig. 1). The couplings are increased in the direction of the arrows by successively increasing the voltages on the respective electrodes in 10 mV steps. All resonances are displaced horizontally to account for the capacitative shifts of the resonance positions caused by the different applied gate voltages. The resonances in (b) are displaced vertically for clarity in 0.1 e2 /h steps. This was not necessary in (a) because of the substantial variation of the background. The dashed curves are obtained by fitting the Fano formula (3) to the resonances. The best fitting parameters for the resonances in (a) and (b) are given in (c) and (d), respectively. The parameter Γ is given in mV of gate voltage. Multiplication by α ≈ 0.06 converts these into energy units (meV). for transmission through the two channels, each considered individually. In fact, the phase difference between the two transmissions, and hence the degree of asymmetry of the resonant lineshape, depends on the relative strengths of transmission through the two channels. This means that just by changing the strength of resonant transmission we can change the shape of the Fano profile in a way that would naively seem to require changes in phase of one or both transmissions. This effect can be achieved experimentally by varying the voltages on our point contact electrodes, thereby changing the strength of transmission through each of the two channels, and generally changing the ratio of their strengths as well. The influence of the strength of the tunnel couplings ΓL and ΓR on the resonances is shown in Fig. 3. All the resonances in this figure can be fit very well by the Fano formula Eq. (3). The fitting parameters for the data in Fig. 3(a) and (b) are plotted in Fig. 3(c) and (d), respectively. Increasing ΓL leads to a more symmetric line shape. At the same time, the incoherent back- ground grows strongly, while the difference between maximum and minimum conductance remains almost constant. This is reflected in a strong increase of the asymmetry parameter q accompanied by a decrease in G0 leaving the product G0 (1 + q 2 ) nearly unchanged. Additionally a slight decrease in Γ is observed. By contrast, increasing ΓR leaves the magnitude of the incoherent transmission constant and decreases the asymmetry parameter. At the same time the resonance is broadened, as reflected by an increase in Γ. Two consecutive Fano resonances with small asymmetry parameters, resulting in nearly symmetric dips, are shown for a variety of temperatures in Fig. 4(a). The increase in width of the resonances with increasing temperature is in good agreement with the linear 3.5 kB T dependence expected from Fermi-Dirac broadening (Fig. 4(c)). From this we obtain the conversion factor α = .059±.006 that relates gate voltage Vg to energy. In contrast to this the temperature dependence of the dip amplitude is not that expected from Fermi-Dirac 4 However, the results in Fig. 5 are different in important ways. The resonances are dips rather than peaks, and they appear on top of a continuous background conductance that is almost independent of the applied voltages. From the slopes of the diamonds’ boundaries it is possible to determine the parameters α = Cgate /Ctot = .049 ± .005 and β = Clead /Ctot = .66 ± .09. Here Ctot is the total capacitance coupling the electron droplet to its environment whereas Cgate is the capacitance only to the plunger gate and Clead the capacitance only to a lead. The value for α is in good agreement with the one obtained from the temperature dependence above. The bottom resonance in Fig. 5 is identical to the left one in Fig. 4 allowing us to determine the spacing in gate voltage of three successive peaks. We assume that the smaller spacing is given by U/α and the larger by (U + ∆ǫ)/α. Using this and the width from Fig. 4, we find U = 1.13 ± .12 meV, Γ = 105 − 120 µeV and ∆ǫ = .66 ± .07 meV. It is surprising that the charging energy is only about 40% smaller than what we find in the Kondo regime. It is even more surprising that Γ has decreased by 50% compared to the Kondo regime, rather than increasing as expected, even though we have opened up the tunnel barriers resulting in a sizable non-resonant conductance and resonant dips instead of peaks. Also reminiscent of the Kondo regime are features that remain pinned near Vds = 0 as the gate voltage is varied, seen as a faint vertical stripe in the center of Figure 5. In the Kondo regime this results from a sharp peak in the density of states at the Fermi energy caused by coupling of the localized spin on the artificial atom to the spins in the metallic leads. However, in Fig. 5 the features do not FIG. 4. (a) Temperature dependence of the conductance for two resonances. At 100 mK the asymmetry parameters for the left and right resonance are q = −.35 and q = −.13, respectively. The background has been adjusted for a slight increase with temperature, corresponding to only 0.06 e2 /h at the highest temperatures. (b) Dip amplitude with respect to the background conductance as a function of temperature. (c) Width Γ, measured in mV of gate voltage, of the right resonance as a function of temperature as determined from a fit to the Fano formula Eq. (3). The slope of the linear fit (dashed line) at low temperatures gives a value of α = .059 ± .006 relating gate voltage to energy. broadening but rather is reminiscent of that seen for peaks in the Kondo regime [7]. Indeed, the amplitude, measured with respect to the background, shows a logarithmic dependence on T over almost an order of magnitude as shown in Fig. 4(b). In addition, the symmetric dip on the right shows a shift of the resonance energy with temperature suggesting that the energy is renormalized at low temperatures just as for conductance peaks in the Kondo regime. These data resemble curves obtained from a mean-field treatment of quantum interference in the Kondo regime [30]. The situation becomes even more intriguing when one examines the differential conductance as a function of both gate voltage and source-drain voltage in the vicinity of two dips. The results of this measurement are shown on a gray-scale plot in Fig. 5, where a clear diamond-shaped structure is traced out by the resonances as one varies the two voltages. This behavior is familiar from many experiments in the Coulomb blockade regime [31–33]. Indeed, close scrutiny reveals additional dips moving parallel to those forming the main diamonds, analogous to subsidiary peaks seen in the Coulomb blockade regime, which result from excited states of the electron droplet. FIG. 5. Differential conductance (dI/dVds ) at T = 100 mK as a function of both the bias voltage Vds across the SET and the gate voltage Vg . Notice that the features are produced by dips in the conductance rather than peaks and that there are weak features near Vds = 0 5 and the diamond structure and additional features associated with excited states in the differential conductance. It is surprising that we see such strong evidence for charge quantization (Fig. 5) coexisting with a continuous open channel through our SET. We can resolve this seeming contradiction if two paths for an electron through the SET have traversal times long and short respectively compared to h̄/U . The path which spends a long time in the region of the electron droplet (passing through a well-localized state) must respect the charging energy, so it exhibits Coulomb blockade resonances in conductance. The other path can temporarily violate energy conservation, adding a charge to the electron droplet during a rapid traversal of the SET [34]. We have calculated based on a suggestion by Brouwer et al. that the time required for ballistic traversal of our electron droplet should be comparable to h̄/U [35]. A puzzle remains: How can the resonances be quite narrow (Fig. 4(a)) even though the point contacts are more open than in the Kondo regime (Fig. 2(b)), as demonstrated by a nonresonant background conductance larger than e2 /h. Conductance through a point contact in the lowest subband cannot exceed 2e2 /h, corresponding to a transmission probability of unity. When electrons can be transmitted across a point contact into two distinct states, the sum of the two transmissions can be no greater than unity. So a large transmission across the first point contact into the delocalized state which ballistically traverses the electron droplet implies a correspondingly reduced transmission into the localized, resonant state. The same analysis holds for electrons traversing the second point contact to exit the droplet. The width Γ of a resonant state is given by the rate of escape of electrons from that state, which is in turn proportional to the sum of the transmissions across the two point contacts into or out of that state. This explains how the presence of a nonresonant transmission channel can actually make the transmission resonances sharper [36]. We have considered alternative explanations for the origin of the nonresonant background conductance, and find them unlikely. A path that circumvents the electron droplet might lead to a continous contribution to the conductance. However, a parallel conduction path in the dopant layer above the 2DEG has been ruled out by Hall and Shubnikov-de Haas measurements on samples from the same wafer with large-area gate electrodes. Since we observe Fano resonances in each of several devices we have studied, it also seems unlikely that the effect is a consequence of channels bypassing the quantum dot in the plane of the 2DEG, caused by lithographic defects or non-uniform charge distributions. Most importantly, should a path circumvent the electron droplet the resulting background would be incoherent. For interference it is important that both paths include the two point contacts since only they can act as coherent source and detector. Detailed measurements of the evolution from the Kondo regime to the Fano regime are underway, with the FIG. 6. Conductance as a function of gate voltage for various magnetic fields applied perpendicular to the 2DEG. depend on whether an even or odd number of electrons is on the droplet as evinced by their spanning more than two resonances. Furthermore, the zero-bias peak in the measurements of Goldhaber-Gordon et al . [7] has an amplitude that depends strongly on the separation in Vg from the resonance, whereas the features in Fig. 5 are nearly independent of Vg . The effect of a magnetic field perpendicular to the 2DEG containing the electron droplet is dramatic (Fig. 6). A field of only 15 mT produces a clear effect on the line shape, and a field of 50 mT completely transforms it from an asymmetric resonance with a dip into a somewhat asymmetric peak. IV. DISCUSSION A. Nature of interfering channels The good fit of the Fano form to our measurements makes it clear that we are observing interference between resonant and non-resonant paths through our SET. Were electrons non-interacting, it would not be surprising for resonant and non-resonant transmission to coexist. We can see this with the help of a semiclassical noninteracting analog for the SET: a cavity with two small openings to reservoirs on the left and right sides. Electrons incident on the cavity from the left side at a random angle would bounce around the cavity, only achieving significant transmission to the right should their energy match an eigenenergy of the cavity. This process would give resonant transmission. In contrast, electrons incident on the left side at exactly the correct angle could traverse the cavity and leave through the righthand opening without suffering any bounces inside the cavity. This is a nonresonant process, independent of electron energy. However, in our case the resonant channel shows all the features typical of charging of artificial atoms: the nearperiodicity in gate voltage of the conductance resonances 6 hope of further elucidating the nature of the nonresonant background conductance. resonant process due to applied field are more important than destruction of coherence by that field. B. Magnetic field dependence V. ACKNOWLEDGEMENTS Changes in transport even at very small field scales are not unexpected given the droplet’s geometry and the 2DEG properties. For our nonresonant transmission, electrons traverse the droplet directly, so backscattered paths enclose approximately the area of the droplet. Assuming an electron droplet of 100 nm diameter, one flux quantum Φ0 ≡ h/e penetrates the droplet at approximately 530 mT applied field. Thus, at this field scale, changes in nonresonant conductance would result from the breakdown of coherent backscattering [37]. However, the resonant path through our droplet is less strongly connected to the leads, so electrons traverse the droplet by more roundabout paths, enclosing more net area than simply that of the droplet. This means that breakdown of coherent backscattering should occur p at much lower flux through the droplet, Φcorr = Φ0 / g(∆ǫ/Γ), where g ≡ G/(2e2 /h) is the dimensionless conductance of the droplet itself [37]. We saw earlier that ∆ǫ/Γ ≈ 5; and g should be comparable to the conductance of the 2DEG g2D ≈ 300, though somewhat suppressed because the electron density of the droplet is less than that of the 2DEG [38]. Taking g = g2D , we arrive at 12 mT as an estimate and lower bound for Φcorr , consistent with our empirical observations [39]. Empirically (Fig. 6), small magnetic fields produce dramatic alterations in the resonances, while leaving the nonresonant background essentially unchanged. The argument made above explains the changes in transport at small magnetic fields as resulting from the breakdown of coherent backscattering in the resonant channel, and the concomitant increase in forward transmission through that channel. Since non-resonant transmission is not affected at these low fields, the net result is an enhancement of q. The alternative explanation — that the magnetic field destroys the interference between nonresonant and resonant processes, transforming resonant dips into peaks — does not account for the extremely low field scale at which the change occurs, nor does it fully match the data. The breakdown of coherent backscattering indeed is caused by the loss of the special phase relation between pairs of time-reversed paths, but here we are concerned with interference between two distinct forward scattering paths (resonant and non-resonant) which do not form a time-reversed pair. In addition, interpreting the zero-field data as the simple interference between two paths would suggest that the resonant path has half the amplitude of the non-resonant path. Yet applying a field causes the resonant contribution of the right-hand peak to exceed the non-resonant background by a factor of three. 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