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Studying the single--electron transistor by photoionization

We present theoretical results demonstrating that photoionization can be a useful tool for investigating single--electron transistors. Suggestions are given on how to conduct experiments using photoionization alone or in combination with transport measurements.


I. INTRODUCTION
In a single-electron transistor (SET), a quantum dot (QD) is coupled via tunneling junctions to metallic electrodes [1,2]. An isolated quantum dot (QD) behaves like an "artificial" atom, where the electron motion is confined within a small region with size of the order of nanometers [3,4]. Similar to atoms, the single electron levels of a QD are well separated energetically. Experiments on SET transport are often interpreted within the Anderson impurity model at equilibrium [5,6,7,8]. The QD is modeled by a single localized state. Its energy ε d can easily be tuned by varying the potential V g of a gate.
This approximation is particularly justified in small QDs, where single electron levels are well separated energetically. This state can accommodate n d = 0, 1, or 2 electrons. A key quantity for electronic transport through the SET is just this number n d of ("valence") electrons.
Unlike in extended systems, a nonvanishing charging energy U = e 2 /2C has to be paid to add one electron to a QD (C is the total capacitance). At sufficiently low temperatures, for a small QD weakly coupled to electrodes, the charging energy U represents the dominant energy scale; it exceeds the thermal energy and the finite width Γ of the dot level resulting from its hybridization (t d ) with electron states of the electrodes.
Therefore, charge fluctuations are largely suppressed and the number of electrons of the dot is an integer in broad ε d -ranges (plateaus): n d ≃ 0 for ε d > ε F + Γ, n d ≃ 1 for ε F − U + Γ < ε d < ε F − Γ, and n d ≃ 2 for ε d < ε F − U − Γ. The transitions between the states with well defined dot charge occur within narrow ε d -ranges of width ∼ 2Γ. In these ranges, the QD state is a combination of two nearly degenerate states of well defined charge (mixed valence regimes). There, the dot charge can fluctuate and, thus, electron transport becomes possible. This gives rise to the well known Coulomb blockade peaks, separated by ∆ε d ≃ U in the curve of the conductance G(ε d ) at low temperatures (but larger than the Kondo temperature T K ) [4]. Although charge fluctuations are suppressed within the plateaus, spin fluctuations through virtual intermediate state are allowed within the (Kondo) plateau where the QD is occupied by one electron (n d ≃ 1). At T < T K , they yield a narrow peak (Kondo resonance) in the QD density of states, which enhances the conductance up to the value of the ideal point contact G 0 = 2e 2 /h (unitary limit) [9].
The experimental observation of the Coulomb blockade phenomenon and the Kondo effect in electric transport of SETs is remarkable [9,10]. It indirectly confirms the occurrence of the charge plateaus in broad V g -ranges. This evidence is indirect, because what one directly measures in experiment is the conductance G, and not the dot charge n d . Besides, the charging energy U is also indirectly estimated as ∆ε d ≃ U, while the experimentally measured quantity is V g and not ε d .
In the present paper, we shall propose an alternative method to investigate phenomena related to the charge plateaus in a SET, namely the photoionization. Photoionization is known as a very useful tool to study strong electron correlations [11,12]. As we shall show, from the investigation of the ionization one can extract more direct information on the QD charge than from transport experiments.
The remaining part of the paper is organized as follows. In Section II, we describe the model and present the theoretical results on photoionization. Based on these theoretical results, in Section III we suggest possible experiments to employ photoionization as a tool for investigating SETs. Conclusions are presented in the final Section IV.

II. MODEL AND THEORETICAL RESULTS
We shall also describe the SET by the Anderson model where a l,σ (a † l,σ ) denote annihilation (creation) operators for electrons of spin σ in the left and right leads (L, R), d σ ≡ a 0,σ (d † σ ≡ a † 0,σ ) destroys (creates) electrons in the QD, and n d,σ ≡ d † σ d σ are electron occupancies per spin direction. The QD-electrode coupling is characterized by the hopping integral t d , which can be experimentally controlled. In the experimental setup of Ref. 10, t d can be changed by varying the potential of the gates that form the constrictions, and the dot energy ε d can be tuned by varying the potential of a "plunger" gate electrode V g ε d = αV g + const. (2) The number of electrons will be assumed equal to the number of sites (N = M L + M R + 1).
In the photoionization of the QD, a photon of energy ω is absorbed by an electron, which is ejected from the QD into vacuum, and brings the ionized system in one of its eigenstates Ψ k .
The ionization process can be characterized by an energy threshold ω k = Ψ k |H|Ψ k -Φ|H|Φ and a spectroscopic factor f k,σ Here Φ stands for the neutral ground state (case T = 0). The spectroscopic factor is directly related to the weight of a line in the ionization spectrum, more precisely to the partial-channel ionization cross section [11]. From Eq. (3) one can deduce the following sum Eq. (4) represents an important result. It permits to directly relate the integrated weight of the ionization spectrum to the number of electrons on the QD in the neutral ground state.
From a technical side, Eq. (4) turns out to be very useful to test the numerical results. It allows to check whether the Lanczos algorithm targets all the relevant ionized states Ψ k .
To compute the ionization spectrum, we shall employ the method of exact numerical (Lanczos) diagonalization. Although this method can only be applied to small electrodes, the results are relevant provided that the number of sites of the electrodes are properly chosen to mimic a metallic behavior [13]. The single-particle energies of an isolated electrode lie Our numerical results for n d of Fig. 1 demonstrate the formation of broad, well defined charge plateaus (n d ≈ 0, 1, 2), separated by narrow mixed valence regions, as discussed above. The fact that the n d -curve (as well as those for the bright diabatic states, see below) for various electrode sizes and boundary conditions are almost identical within the drawing accuracy in Fig. 1 demonstrates that finite-size effects are not important and the study of small systems is physically relevant. Our extensive numerical results demonstrate that what is essential for the weak finite-size effects is also essential for the occurrence of the charge plateaus, namely a width parameter Γ ∝ t 2 d /t smaller than the charging energy U. On this basis, we argue that the charge and the ionization spectra of a QD connected to semi-infinite leads can be accurately deduced by computations for such small "metallic" electrodes.
Although it does not represent a central issue from the present perspective, the weakness of finite-size effects is also relevant for the Kondo effect, which we only mention here in passing. As is well known, essential for the occurrence of this effect is the Kondo screening cloud. The latter extends over a number of sites ξ K ∼ t/T K , which rapidly grows (exponentially for large U) far beyond electrode sizes treatable by exact numerical diagonalization.
Therefore, it is impossible to obtain the conductance within standard transport approaches by employing short electrodes. For semi-infinite electrodes, the zero-bias conductance G can be computed via the Friedel-Langreth sum rule [14,15,16] once the dot occupancy n ∞ d is known. Eq. (5) was deduced for a QD attached to semiinfinite leads with a continuum density of states, which is the real nanosystem of interest.
The pleasant thing is that the zero-bias conductance G for this case can be solely determined from n ∞ d , and a reliable estimated of the latter suffices for this purpose. Fortunately, to this we need not to carry out calculations for semi-infinite or very large electrodes: in view of the weak finite-size effect mentioned above, n d computed for short electrodes does represent an accurate estimate, n ∞ d ≃ n d . Indeed, the curve for G(ε d ) obtained by inserting the calculated n d calculated for short electrodes instead od n ∞ d , also presented in Fig. 1, has a similar appearance to that obtained by means of the numerical renormalization group [17,18] for semi-infinite electrodes and nicely reveals the occurrence of the Kondo plateau.
Let us now switch to the results for SET photoionization. To avoid misunderstandings, we emphasize that the photoionization considered in this paper is the qualitatively different from the widely-studied photo-assisted tunneling (see, for example, Ref. 19 and references therein). The process where a photon helps an electron to tunnel from one electrode to another through a Coulomb island might also be viewed as an "ionization" phenomenon.
Nevertheless, the scale of the corresponding "ionization" energies is determined by the charging energy U. For semiconducting QDs, it is of the order of ∼ 1 meV, that is, much smaller than ionization energies of the order of the work functions (∼ 1 eV) considered by us.
Typical exact numerical results for the ionization energies and spectroscopic factors of all the significant ionization processes are depicted in Fig. 2. To understand these results it is helpful to analyze first the limit t d → 0.
In this limit, two ionization processes of the QD are possible. For ε d < ε F − U, the dot is doubly occupied (n d = 2), and the relevant ionization process consists of removing an electron from the upper Hubbard "band". The corresponding ionization energy is ω 0 where the creation of one hole in the upper or lower Hubbard band is accompanied by the excitation of a particle-hole pair in electrodes are represented by lines parallel to the ω 0 uor ω 0 l -lines, respectively. To mention only one more example, 2h-1p processes involving the creation of two holes in the upper and lower Hubbard bands and of one electron in electrodes at or above the Fermi level are represented by the straight line ω 0 l′ = −2ε d − U + ε F or parallels to it. At the intersection points of such lines avoided crossings arise rather than true intersections, the latter occurring only for t d → 0. As discussed recently [20], avoided crossings are very frequent in tunable QD systems.
Numerous avoided crossings are visible in Fig. 2a, but for practical purposes they are less important, and therefore only a few are shown in Fig. 2b. Parts thereof are not important because the spectroscopic factors are too small. But even if the signals are sufficiently intense, avoided crossings are less important: in the narrow regions around the critical points where the intensities are significant, the energy differences are usually smaller than the experimental resolution, and only the summed intensities of the participating states can be measured. What one measures is a smoothly varying spectroscopic factor, attributed to a bright "diabatic" state [20]. The constant spacings between different families of parallel lines in Fig. 2a can be understood by noting that the single-particle energies of the isolated electrodes considered there are ε F , ε F ± t, and ε F ± t √ 3. For long electrodes, bundles of dense parallel lines within a width 4t will appear, so of practical interest are just such bright diabatic states passing through a multitude of avoided crossings.
Therefore, besides the results for individual eigenstates, we also show in Figs. 2a and 2b the curves Ω u,l and f u,l for the two bright diabatic states that are significant. Deep inside the plateaus, the ionization signals corresponding to these two diabatic states behave similarly to the case t d → 0, but substantial deviations are visible in the mixed valence regimes. Noteworthy is not only that the spectroscopic factors penetrate the neighboring charge plateaus, where they gradually decrease, similar to n d (Fig. 1). Even more interesting is the fact that all these penetrations are accompanied by substantial changes in the slope of the Ω u,l -curves. are not quantitatively significant for the ionization spectrum, but are showed to better visualize the occurrence of the avoided crossings. Notice that the ionization energies are measured relative to the electrode work function (−ε F ), a quantity much larger than the charging energy U . Parameter values as in Fig. 1. See the main text for details.

III. PROPOSED EXPERIMENTS
In view of the present theoretical results, we propose to conduct the following experiments using an intense incoming flux of monochromatic photons, which is well focused on the QD of a SET. To avoid confusions we emphasize again that the photons should have energies of the order of the work functions, significantly higher than of the microwave and rf-fields considered in the previous studies on SETs and the Anderson model.
(i) The first type of experiments is a standard ionization study. To measure the absorption coefficient would be desirable but presumably a too hard task. Probably it would be easier to measure the zero kinetic energy (ZEKE) photoelectrons emitted from the dot at the ionization threshold, as in threshold ionization experiments in molecular physics. Just because the QD is very small, not only photoelectrons from the QD, but also from electrodes will be inherently ejected into vacuum. Therefore, it is important but fortunately easy to experimentally distinguish between the ionization of the QD and that of the electrodes: by varying the gate potential V g , the ionization energies ω i change in the former case and ω i ≈ −ε d (V g ) − U] but remain constant in the latter (ω i = −ε F ). Let us analyze the change in ionization by gradually increasing the dot energy ε d starting from a sufficiently low value, by making V g more and more negative. In the first stage, an ionization signal will be observed, which is characterized by a nearly constant intensity I u (I u ∝w u ≃ 1). Its intensity will drop to zero within a narrow range δV g1 ∼ 2∆V (∆V = −Γ/α > 0) centered . A little before this signal disappears, another ionization signal will appear at V g ∼ V g1 + ∆V , whose intensity rapidly rises to a value I l ≈ I u /2 (I l ∝w l ≃ 0.5) beyond Further on, this intensity remains nearly constant up to a point where it rapidly falls down to zero within a range δV g2 ∼ 2∆V centered on V g = V g2 . Except for the (mixed-valence) ranges δV g1 and δV g2 , the corresponding ionization energies, Ω u (V g ) and Ω l (V g ), are straight lines of the same slope. This slope is half the slope of the Ω l (V g )-curve in the range V g1 < V g < V g1 + ∆V . At the other end, V g2 − ∆V < V g < V g2 , this curve tends to saturate, Ω l (V g ) → −ε F . The same tendency of the other curve (Ω u (V g ) → −ε F ) can be seen in the opposite mixed-valence range V g1 − ∆V < V g < V g1 . By extrapolating the linear parallel portions of the Ω u,l -curves (Ω u,l →Ω u,l ), one can directly determine the charging energy from the difference of their ordinates taken at the same value of V g , [21]. Noteworthy, for this, the values of V g or ε d themselves are not needed. This represents an important difference from the determination of U in transport experiments. In the latter, what one can directly measure are the values V g1 and V g2 , from the positions where the Coulomb blockade peaks occur at temperatures T > ∼ T K or from the extension of the Kondo plateau observable for T < T K . The determination of U from the difference V g1 − V g2 alone is impossible, and to this aim the relationship between V g and ε d , Eq. (2), is necessary. There is still another important difference between ionization and transport measurements. As visible in Fig. 2a, Ω u (V g1 ) ≃ Ω l (V g2 ) ≃ −ε F . The Fermi energy can thus be estimated. This is also important because this quantity can alternatively be deduced from the ionization energy −ε F of the electrodes (electrode work function). To conclude, the V g -dependent ionization spectrum allows one to give direct evidence on the formation of the charge plateaus and to determine the quantities U, Γ, and ε F . The dependence ε d (V g ) can also be obtained, which is also important, as it allows to quantitatively investigate the capacitances of nanosystems.
(ii) A (ZEKE) ionization study might be difficult, but a mixed ionization-transport experiment as proposed below is presumably easier while still relevant. In a usual transport experiment, one can measure the conductance as a function of the gate potential, G(V g ), and determine V g1 and V g2 , say, from the Kondo plateau (T < T K ). At a fixed gate potential one can now shine the QD with radiation whose frequency ω is varied. At a certain value of the latter, ω = ω c (V g ), the incoming photon will ionize the QD. Because the single (unpaired) electron of the QD will be ejected, the Kondo effect will be suppressed, and this will be evidenced by the drop in the conductance (G ≈ 0). From the above considerations one expects a linear V g -dependence of ω c (V g ) ≡ Ω l (V g ). Again, the charging energy can be directly determined as U =Ω l (V g1 ) −Ω l (V g2 ). The dependence of ε d on V g is not needed, rather it can be deduced by this method, as well as the Fermi energy ε F = −Ω l (V g2 ).
(iii) A similar mixed experiment can be imagined at V g < V g2 , where the nonionized QD is doubly occupied. At the resonant photon frequency ω = Ω u (V g ) one electron will be removed. For a sufficiently intense irradiation, one can reach a stationary state where the ionized QD will be occupied by an unpaired electron. Because the latter is a prerequisite for the occurrence of the Kondo effect, it is tempting to speculate on a possible photoionizationinduced nonequilibrium Kondo effect; thence, an effect of photoionization opposite to that of suppressing the equilibrium Kondo effect discussed above.

IV. CONCLUSION
In this paper, we have showed that photoionization is a valuable method, which can be used along with transport measurements, to study SETs. To our knowledge, conductance measurements represent the only way to determine the charging energy U, which thus resembles very much a fit parameter. Therefore, the comparison with the result of a completely different method is quite meaningful, even more if U is obtained in a direct manner, and not via a supplementary relation between ε d and V g . For specific purposes, photoionization turns out to provide even richer information than transport studies.
The above considerations refer to a QD with a single level. However, from the physical analysis it is clear that the results are also relevant for more general situations.
Another important result is that QD-ionization can be accurately described by considering short electrodes (weak finite-size effects), and only very few ionized states are relevant.
Both aspects are very important, since they urge to carry out realistic ab initio calculations on photoionization, particularly for single-molecule based SETs.