Measuring the energy level repulsion in quantum dots

Superlattices and Microstructures 33 (2003) 291–300 www.elsevier.com/locate/jnlabr/yspmi

Measuring the energy level repulsion in quantum dots T. Heinzela,∗, S. L¨uscherb, M. Furlanc, K. Ensslinc a Institut f¨ur Physik der Universit¨at Freiburg, 79104 Freiburg, Germany b Department of Physics, Stanford University, Stanford, CA 94305-4045, USA c Solid State Physics Laboratory, ETH Z¨urich, 8093 Zurich, Switzerland

Accepted 5 February 2004

Abstract Energy level repulsion is one of the remnants of classical chaos in quantum mechanics. Measurements of the distribution of nearest neighbor spacings in quantum dots reveal, in contrast to other classically chaotic systems, deviations from the predictions made by random matrix theory. Here, we survey possible contributions to these deviations from experimental peculiarities present in measurements on quantum dots, and discuss the methods to eliminate or reduce such distortions. © 2004 Elsevier Ltd. All rights reserved. PACS: 73.23.Hk; 02.40.Xx Keywords: Quantum dots; Energy level repulsion; Quantized chaos

1. Introduction Over the past 10 years, transport experiments on quantum dots have become a highly active discipline in the field of Nanoscience [1, 2]. The fascination for quantum dots is easily understandable: they provide miniaturized laboratories to study the properties of a small number of interacting electrons in a solid state environment. Moreover, the electron density, the coupling of the island to the leads, as well as the shape of the confined region, can be tuned over wide ranges. As a consequence, some phenomena, like the breakdown of Hund’s first rule in strong magnetic fields, [3] have become experimentally accessible in quantum dots, while others, for example the Kondo effect, [4–6] can now be examined ∗ Corresponding address: Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland. Fax: +41-1-633-1146.

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Fig. 1. Measurements of Coulomb blockade oscillations in a quantum dot at an electron temperature of 120 mK. The nearest neighbor spacings of 30 consecutive conductance peaks are shown in the upper inset. Lower inset: the dot is defined in a Ga[Al]As heterostructure by local oxidation with an atomic force microscope. The potential of the dot can be tuned via in-plane gates I (used here) and II, while gates A, B allow the adjustment of the dot’s coupling to source and drain.

as a function of external parameters. A significant fraction of the transport experiments on closed quantum dots, i.e. quantum dots coupled to the environment only via tunnel barriers, can be interpreted terms of the simple constant interaction model [7]. Within this model, it is assumed that the total energy of the confined electronic system can be separated into a purely electrostatic and a purely kinetic component. The justification of this approach has its origin in the weakness of electron–electron interactions in most quantum dot systems. Strong interactions render the application of the constant interaction model impossible. This is expected to occur at interaction parameters rs ≡ ad∗ > 2 (d denotes the average B distance between electrons and a ∗B is the effective Bohr radius) [8, 9], while typical electron gases in quantum dots have rs ≈ 1 [1, 2]. Within the constant interaction model, it is—in principle—very simple to extract a single-particle energy level spectrum of the quantum dot. Just measure the Coulomb blockade oscillations (see Fig. 1) and extract the energy difference of adjacent singleparticle levels (the so-called nearest neighbor spacing—NNS) from
 e2 ∆VG, j = α +  j −  j −1 (1) CΣ where ∆VG, j is the separation of Coulomb blockade resonances j and j − 1, while CΣ denotes the total capacitance of the quantum dot to its environment,  j is the kinetic energy of the j th level, and α = CΣ /eC G is a lever arm that transforms gate voltage into energy (C G is the capacitance the tuning gate forms with the dot). In many experiments, the energy spectra obtained this way agree well with analytical single-particle spectra, for example with the Fock–Darwin spectrum at low electron numbers [3] and at moderate magnetic fields [10]. In quantum dots with a large electron number, however, an apparently random

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distribution of NN spacings is observed, as shown the inset in Fig. 1. This property is usually interpreted in terms of quantized chaos [11, 12]. Within this explanation, it is presumed that the classical dynamics of the quantum dot is chaotic. In addition, the NN spacing distribution is supposed to be appropriately described by random matrix theory (RMT). If time inversion invariance is preserved (i.e. magnetic fields are absent), the NN spacing distribution in quantum dots is expected to form a bimodal Wigner–Dyson (WD) distribution for Gaussian orthogonal ensembles,  π 2 π 1 δ(s) + se− 4 s (2) p(s) = 2 2 see Fig. 2(a). Here, s denotes the normalized single-particle NN spacing. The δ-function at s = 0 represents the spin degeneracy. The probability density for finding a small NN

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spacing between adjacent orbital levels is suppressed, compared to the Poisson distribution p(s) = e−s found in many regular systems, like for example a rectangle. This quantum property of classically chaotic systems is referred to as energy level repulsion. The measurements of NN spacing distributions on quantum dots, however, deviate from the predictions of the RMT [13–17]. Although the results of these experiments, as well as the sample parameters, vary greatly, they share the absence of a bimodal distribution. Rather, a single peak is observed. In some experiments, a single Gaussian peak with an increased full-width at half maximum (FWHM), as compared to the FWHM of the WD distribution [13, 14, 16], gives the best fit to the data. In others, modified versions of the WD distribution have been used to interpret the results [15, 17]. In [17], for example, it was assumed that the spin degeneracy is lifted by exchange and correlation effects, resulting in a spin peak of Gaussian shape, centered at s > 0. As a consequence, the orbital NN spacings no longer form a monomodal WD distribution, convoluted with the spin peak, see Fig. 2(b). This phenomenological approach, of course, does not explain the position or the width and shape of the spin peak. The experimental observations have triggered a large body of theoretical work recently [11], which mostly orbit around the effect of interactions beyond the constant interaction model. Alternatively, parametric shape deformations have been discussed as a possible explanation [18, 19]. Below, we complement these models by discussing experimental issues which, besides effects intrinsic to the quantum dot, potentially influence the measured NN spacing distribution, and thus, within the phenomenological picture sketched in Fig. 2(b), may unintentionally contribute to the fit parameters. 2. Experimental sources of modified NN spacing distributions We begin with geometric considerations and proceed with statistical issues. Finally, influences of an imperfect environment are discussed. As a gate voltage is tuned, the potential landscape at and around the dot is changed. This has several implications for the measured NN spacing. Suppose the dot has a rigid confinement potential with walls of finite slope at the Fermi level, like the widely used Fock–Darwin potential provides. As the gate voltage is, say, reduced, the dot’s electronic area shrinks, and with it both C G and CΣ . Thus, CΣ as well as α in Eq. (1) are unknown functions of VG . In [14], for example, ∆VG, j increased by approximately a factor of 3 as j was changed from j0 to j0 + 200. In that experiment, the authors have assumed a linear dependence of C G on VG and corrected the determination of the NN spacings accordingly. However, even slight changes in the functional form of C G (VG ) can have a profound influence on the NN spacings, which are responsible only for a small fraction (typically 10%) of the measured peak spacings ∆VG, j . Parametric shape deformations are a related issue. Changing the gate voltage may deform the dot and thus alter the single-particle energy spectrum. [19] have studied the evolution of the energy levels as a function of the dot deformation. It was found that the bimodal WD distribution can be significantly modified by the shape deformations. Specifically, the shape of the NN spacing distribution approaches a Gaussian and its FWHM becomes larger as the “softness” of the dot, i.e. the deformation per unit gate

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voltage interval, increases. In soft dots, the extracted NN spacing distribution thus represents an average over level spacings of a variety of different dots of different shapes and sizes. Up to now, shape deformations have not been taken into account in interpreting experimental data. These effects of a geometrical character can be, in principle, reduced by using large dots (such that the capacitance changes and the shape deformations are negligible), or by designing hard-wall dots. The former concept is limited by the conflicting requirement to have NN spacings well above the thermal energy. The latter approach has been followed in [17], where the quantum dots have been fabricated by scanning probe lithography, a technique known to give steep confinement potentials [20]. This can be guessed from the inset in Fig. 1 as well: no monotonous change of ∆VI , and hence of C G , as a function of VI is observable. A further problem frequently encountered in such experiments is the small number of conductance resonances available. Numerical simulations [21] have shown that about 1000 data points are sufficient to obtain good agreement of NN spacing histograms with a WD distribution. While such a large number of energy levels can be measured in most classically chaotic systems like, for example, microwave cavities of chaotic shape [22], or hydrogen atoms in strong magnetic fields [23] without great difficulties, it poses a major challenge in quantum dots. This is mostly due to the limitations of the gate voltage tuning range. At typical electron densities of two-dimensional electron gases (say, 3×1015 m−2 ), a quantum dot designed to contain 1000 electrons has a diameter of the order of 650 nm. This corresponds to an average NN spacing of 10 µeV, which is comparable to thermal smearing at typical electron temperatures in dilution refrigerators. In addition, a gate voltage cannot tune the electron number all the way down to zero. In small quantum dots with higher electron densities, on the other hand, the gate voltage tunes the coupling to the leads as well. Hence, one is limited to a maximum of about 200 Coulomb blockade resonances in a single gate voltage sweep. A partial relief comes from the possibility of tuning the quantum dot by an additional parameter p, such as a second gate voltage, or a magnetic field, see Fig. 3. Clearly, the peak positions and thus their separations show a parametric dependence. Provided the system is ergodic, a single trace of a tuned NN spacing can provide several statistically independent data points. In order to ensure statistical independence experimentally, the autocorrelation function C(δp) of the peak position as a function of the parameter p is calculated. Once p has been changed by more than the autocorrelation parameter δp0, defined by C(δp0) = 12 C(0), the data are regarded as statistically independent. For example, in order to determine the NN distribution for the dot of Fig. 1, the voltage applied to gate II, VI I has been used, with δVI I,0 ≈ 30 mV. This way, the number of statistically independent data could be increased by a factor of 5 as compared to a single gate voltage sweep in that particular experiment. In case p is a magnetic field B, the autocorrelation magnetic field B0 is found to be of the order of the field required to change the number of magnetic flux quanta that penetrate the dot area by one. The magnetic field breaks the time reversal symmetry, which forces the random matrices to be unitary, and generates the slightly different WD distribution for Gaussian unitary ensembles. The available tuning range B for the purpose of getting statistically independent data while leaving the character of the energy spectrum unchanged

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is, however, quite limited as well. In order to avoid modifications of the energy spectrum by Landau quantization, the cyclotron frequency ωC should be small compared to the characteristic confinement strength ω0 . In addition, the Zeeman splitting should be small in comparison to the spin splitting at zero magnetic field. Since in nanostructures, the effect g-factor may be strongly enhanced, this condition may determine the available magnetic field range. Fig. 4 shows the NN separation of a strongly correlated pair of conductance resonances (supposedly a spin peak) as a function of B. While strongly fluctuating, the splitting shows an average tendency to increase as B is increased. Interpreting this slope as the signature of Zeeman splitting, a linear fit gives an effective g-factor of |g| = 0.45, a value almost identical to the GaAs bulk value. Thus, the parametric tuning of the quantum dot is able to increase the number of statistically independent NN spacings by less than an order of magnitude. Alternatively, combining the measurements on several quantum dots [15] gives, of course, access to an arbitrary number of NN spacings, and is only limited by measurement time and patience. We proceed by discussing the influence of possible background charge rearrangements on the measured NN spacing distribution. The quantum dot is, after all, embedded in a very clean, but nevertheless imperfect semiconductor environment. As a consequence, sweeping a gate voltage may induce charge rearrangements between traps located in the region where the electric field is significant. In two-level tunneling systems (TLTS), i.e. two weakly coupled states at separate locations [24], electrons can be transferred between the sites by the gate voltage. Such switching events in TLTS, which are typically hysteretic in gate voltage, can manifest themselves as gate-voltage dependent noise inside a conductance resonance. However, for most of the gate voltage interval, Coulomb blockade is established, the conductance is zero, and TLTS switching will remain

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B(T) Fig. 4. The NN separation (dots) of two strongly correlated conductance resonances as a function of magnetic field, measured on the sample of Fig. 1. A linear fit (line) can be interpreted in terms of Zeeman splitting, see the text.

unnoticed in a measurement such as shown in Fig. 1. The relevance of gate-voltage induced TLTS switching can be convincingly demonstrated by studying the Coulomb blockade oscillations of a metallic single-electron tunneling (SET) transistor. Here, the island’s NN spacing is well below 1 µeV and thus not resolvable. Fig. 5(a) shows some Coulomb blockade resonances as measured in a Al SET transistor [25]. One immediately notices, for example, a reduced spacing between some resonances. A histogram for 600 consecutive resonances (note that, in contrast to quantum dots, this poses no problem in metallic SET transistors, since the gate voltage does not modify the island’s coupling to the leads) reveals that a significant fraction (about 20%) of the peak spacings is reduced (Fig. 5(b)). This can be understood within a simple picture based upon gate voltage induced TLTS switching [25]. Suppose a TLTS is located in between the gate and the dot, as indicated in Fig. 5(c). Suppose further the electron sits at the site next to the dot, while the second site is empty. As the gate voltage is increased, the energy levels of the two TLTS sites are shifted with respect to each other, and at some gate voltage, the electron will hop towards the gate. This rearrangement, however, reduces the potential of the island and with it the effective gate voltage felt by the island, such that the peak spacing will be reduced. A similar argument can be made for down-sweeps of the gate voltage. A switching event increases the peak separation, provided the corresponding TLTS is located at the far side of the island, opposite to the gate. Here, however, the electric field is much weaker, and thus this kind of charge rearrangement is unlikely. Thus, from measuring just the Coulomb blockade resonances at low source–drain bias voltage, it is not a priori clear whether the deviation from the electrostatic part peak spacing reflects the intrinsic properties of the dot, or the imperfect environment. Distinguishing such events requires measurements of Coulomb blockade resonances at larger bias voltages, such that the conductance becomes everywhere non-zero, and the

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Fig. 5. (a) A series of conductance resonances, measured in a metallic single-electron transistor. Arrows indicate reduced peak spacings. (b) Diagram of 600 consecutive peak spacings, measured in a single-gate voltage sweep. (c) Interpretation of the reduced peak spacing in terms of background charge rearrangements (see text). The full circle denotes a site occupied with an electron, the empty circle an empty site. (d) Measurements of Coulomb blockade resonances (see Fig. 1 for the sample geometry) at various source–drain bias voltages V between −1 mV and +1 mV. The ellipse indicates a noisy region at larger V which appears perfectly quiet for small bias voltages.

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TLTS switching can be identified by noise. In Fig. 5(d), this method is demonstrated at the dot of Fig. 1. While at small source drain bias voltages below 40 µV, thermally smeared Coulomb blockade resonances at a constant, low noise level are found, excess noise appears inside and in between two conductance resonances. Hence, the corresponding peak spacing should be excluded from the NN spacing distribution. It should be remarked that due to the more complicated electric field configuration in quantum dots, we expect that TLTS switching may also increase the measured peak spacing. 3. Summary and conclusion The NN spacing distribution in quantum dots continues to be a fascinating topic in mesoscopic physics. Explaining the deviations from the Wigner–Dyson distribution remains a challenge for theoretical considerations, in particular with respect to the validity of various ways of treating the residual electron–electron interactions theoretically. The experimental challenge consists in obtaining sufficiently many NN spacings with proved statistical independence, which are unhampered by unwanted influences due to the environment. We have surveyed the related issues and discussed the experimental and analytic techniques how to assure that the data are meaningful. Acknowledgement T.H. acknowledges many stimulating discussions with J.P. Kotthaus. References [1] L.P. Kouwenhoven, C.M. Marcus, P.L. McEuen, S. Tarucha, R.M. Westervelt, N.S. Wingreen, in: L.P. Kouwenhoven, G. Sch¨on, L.L. Sohn (Eds.), Mesoscopic Electron Transport, Proceedings of the NATO Advanced Study Institute, ser. E, vol. 345, Kluwer, Dordrecht, 1997, pp. 105–214. [2] L. Jacak, P. Hawrylak, A. Wojs, Quantum Dots, Springer, Berlin, 1998. [3] S. Tarucha, D.G. Austing, T. Honda, R.J. van der Hage, L.P. Kouwenhoven, Phys. Rev. Lett. 77 (1996) 3613. [4] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, M.A. Kastner, Nature 391 (1998) 156. [5] S.M. Cronenwett, T.H. Oosterkamp, L.P. Kouwenhoven, Science 281 (1998) 540. [6] J. Schmid, J. Weis, K. Eberl, K.V. Klitzing, Physica B 256 (1998) 182. [7] C.W.J. Beenakker, Phys. Rev. B 44 (1991) 1646. [8] K.-H. Ahn, K. Richter, I.-H. Lee, Phys. Rev. Lett. 83 (1999) 4144. [9] R. Egger, W. Husler, C.H. Mak, H. Grabert, Phys. Rev. Lett. 82 (1999) 3320. [10] A.A.M. Staring, B.W. Alphenaar, H. van Houten, L.W. Molenkamp, O.J.A. Buyk, M.A.A. Mabesoone, C.T. Foxon, Phys. Rev. B 46 (1992) 128693. [11] For a review of the statistical properties of quantum dots, see Y. Alhassid, Rev. Mod. Phys. 72 (2000) 895–968. [12] For a general survey of the application of random matrix theory in quantum physics, see T. Guhr, A. M¨ullerGroeling, H.A. Weidenm¨uller, Phys. Rep. 299 (1998) 189–425. [13] U. Sivan, R. Berkovits, Y. Aloni, O. Prus, A. Auerbach, G. Ben-Yoseph, Phys. Rev. Lett. 77 (1996) 1123. [14] F. Simmel, T. Heinzel, D.A. Wharam, Europhys. Lett. 38 (1997) 123. [15] S.R. Patel, S.M. Cronenwett, D.R. Stewart, A.G. Huibers, C.M. Marcus, C.I. Duru¨oz, J.S. Harris, K. Campman, A.C. Gossard, Phys. Rev. Lett. 80 (1998) 4522. [16] F. Simmel, D. Abusch-Magder, D.A. Wharam, M.A. Kastner, J.P. Kotthaus, Phys. Rev. B 59 (1999) 10441.

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