Probing Periodic Properties of "Artificial Elements" Assembled in a Quantum Wedge
Dongmin Chen and I. B. Altfeder The Rowland Institute for Science, Cambridge, MA 02142
chen@rowland.org
Abstract
	As an extension to the notion of an artificial atom, artificial elements are appropriate descriptions of artificial atoms with incremental number of occupied quantum states, for a succession of which can exhibit periodic properties in resemblance to Mendeleev's Periodic Table of natural elements. As an example we show that an assembly of successive artificial elements can be realized through the fabrication of a quantum wedge. The energy level of the highest occupied quantum state in these elements display a two fold periodic shift which gives rise to a binary electron interference fringes, as imaged with a low temperature scanning tunneling microscope.

An artificial atom[1] is a pool of electrons confined in a sufficiently small metal or semiconductor either by an energy potential barrier existing at the boundaries, or by a electrostatic potential introduced through a set of properly arranged electrodes. This is so named because both the charge and energy of such a small electronic system are quantized just like those of a natural atom. Artificial atoms, however, have their own unique properties, which often reveal atom-like features. A natural atom is a chemical element with a characteristic electron shell configuration. The periodic repetition of electron configuration of a succession of elements give rises to the periodic repetition of their chemical properties as revealed most elegantly in Mendeleev's periodic table. It is tempting, therefore, to ask whether an artificial atom could have its own identity that can be characterized by an artificial atomic number just like in the case of a natural atom? In this paper we shall argue that artificial elements are a useful extension to the notion of artificial atoms for describing the latter with a small but distinct number of filled quantum states (QS). As a concrete example, we show that a succession of artificial atoms can be assembled in a metallic quantum wedge[2], which, indeed, exhibits periodic properties. The number of filled QS and their energy spectrum of an artificial atom depend on its physical dimension (or size of the confining potential well). Consider for simplicity a one-dimensional metal artificial atom of size H = N·d0, where N is the number of atomic sites and do the interatomic spacing. The total number of QS in the partially filled conduction band is equal to N, and		increases by one with the addition of each new atomic site in the artificial atom. Within a good approximation, the energy separation between two successive QS near Fermi level, EF, is[3]
			,	(1)
		where vF is the Fermi velocity and Planck's constant. For the highest occupied quantum state (HOQS) and the lowest unoccupied quantum state (LUQS), their energy shift when the size of the artificial atom changed from N to N+1 is
			,	(2)
		where l F is the Fermi wavelength of the conduction electrons.  Eq. 2 asserts that the position of HOQS or LUQS relative to EF shall exhibit a pattern of quasiperiodic repetition with a successive increase of N, in resemblance to the periodic properties of natural elements with N playing the role of atomic number Z. The period, however, is material dependent as reflected by the ratio of l F/2d0. An example for a period of two is shown schematically in Fig.1.

FIG. 1. Schematic of the energy level configuration for four successive number of total QS in an ideal half-filled band of a metal. It shows a quasi two fold periodic repetition near the Fermi level, i.e., l F/2d0=2.   With the advanced nano fabrication technology, it is possible to tailor the size of an artificial atom so that the number of QS can be adjusted by unity, hence create a periodic table of artificial elements. In particular, an array of artificial atoms with an incremental number of QS can be realized through the fabrication of a quantum wedge, i.e. a nanoscale metallic wedge whose thickness changes monotonically by discrete atomic planes. Each increase in the thickness by an atomic layer adds a new QS into the energy band as a result of matching the boundary conditions. Thus a quantum wedge is an assembly of artificial atoms with incremental sizes, or a succession of artificial elements. To create the nano wedge described above, Pb islands of nanometer sale is grown epitaxially on a Si(111) surface with monatomic steps. A topographical image of a typical Pb wedge taken in situ by a scanning tunneling microscope at 5K is shown in Fig. 2a. The island expands several atomic terraces of the substrate but its top is atomically flat except for a small vertical lattice mismatch. Thus as the Si substrate descends down from left to right, the number of Pb layer on each terrace increases successively from 8 to 12. This unique geometry results from the balancing act between the minimization of stain energy vs. the surface energy in this heteroepitaxy system. Because of the large aspect ratio, the energy quantization in each element in the Pb wedge takes place only in the direction normal to the surface. Namely, the normal component of the wave vector k^ = np /H, where n, being an integer, is the index of the nth QS. The energy spectrum for a given element is a set of discrete subbands for the electrons are free to move in planes.

Using site specific tunneling spectroscopy, a portion of the quantized spectrum for each of all the elements can be measured. Fig. 3 is a set of such spectra where the integer next to each of the curves indicates the number of Pb layers of that particular element. Note that due to the focussing effect of the tunnel junction, only states with a small in-plane wave vector k|| in a subband contribute to the tunneling spectrum. Together with the weak dispersion near the [111] symmetry point of Pb[4], it gives rise to the sharp steps in the tunnel spectra. In the region of negative tip biases, each step corresponds to an unoccupied QS, whereas in the positive region the dominant source of the tunnel current is from the HOQS. This asymmetric behavior is also an intrinsic property of the tunnel junction[2].  Fig,3 clearly shows the progression of the nth QS with successive N as marked by the traces. Fitting the data of Fig. 3 to Eq. 1, we obtain a Fermi velocity vF » 1.9 x 108 cm/sec, in agreement with the bulk value of Pb[5]. It is also evident that the level of QS near EF exhibits a two fold repetition. All the QS near EF for all the even elements are more or less aligned, and so are those of odd elements (except for number 9 and 10). But between these two groups there is a substantial energy shift, and for the HOQS or LUQS of two successive elements, the shift is approximately .     FIG. 2. (a) 5000Å x 5000Å STM topography of a Pb wedge grown on a stepped Si(111) surface. The thickness of the wedge increases successively from 8 to 12 layers. (b) STM image of the same wedge taken with an opposite polarity of the tip bias (+5V), revealing the interference fringes of the highest occupied quantum state near the Fermi level.

FIG. 3. A series of I-V spectra acquired at 5K over an individual quantum element assembled in a Pb wedge. The integer next to each spectrum indicates the number of Pb layers in that element. Short vertical bars trace the progressing of quantum states with the same index.   This is precisely what Eq. 2 predicts since for Pb, l F = 3.7d0 (d0 = 2.86Å), so that . It is even more remarkable that the pattern of the quasi two fold repetition is interrupted at N = 10, just as expected from the irrational relationship between lF and 2d0. While d0 is purely a material parameter, l F is associated with the percentage of the total QS that are filled, and therefore can depend on the size of the artificial atom as well as the valence electron density of the material. Since the energy dispersion is, in .

general, nonlinear away from the center of a band, a wide range of period for the repetition of the QS level can be expected for other materials and structures. It should be noted that Eqs. 1 and 2 are no longer applicable in the nonlinear regime. Like the natural elements, the quasiperiodic repetition of QS configuration in a succession of artificial atoms can lead to significant consequences when they interact with an external probe, or with each other. This will undoubtedly affect the transport characteristics of single electron transistors and coupled quantum dots. In the foregoing Pb wedge experiment, it is the large shift of the energy spectrum between two successive elements that make it possible to probe the QS spectrum of an individual element with great clarity even though they are physically connected. The periodic nature of the energy shift gives rise to a periodic oscillation of the tunneling current at positive tip biases as evident in Fig. 3. When electrons tunnel from the wedge to the tip, those at the HOQS make dominant contributions and the closer the HOQS to EF, the smaller the tunnel barrier. Thus all the odd elements yield much higher current intensity than the even elements for N>10, and the opposite is true for N£ 10. Imaging the wedge under such condition, therefore, yield a marvelous binary fringe pattern as shown in Fig. 2b. In summary, using a Pb quantum wedge as an assembly of artificial atoms with a successive number of QS, we have demonstrated the existence of a quasiperiodic repetition of the QS configuration, and have thus proven the worthiness of the notion of "artificial elements" as an extension to that of an artificial atom. We thank K. A. Matveev for many stimulating discussions. This work was supported by the Rowland Institute for Science References: [1]. M. A. Kastner, Physics Today 46, 24 (1993). [2]. I. B. Altfeder, K. A. Matveev, and D. M. Chen, Phys. Rev. Lett. 78, 2815 (1997). [3]. N. W. Ashcroft and Mermin, Solid State Physics (Saunders College, Philadephia, 1976). [4]. J. R. Anderson and A. V. Gold, Phys. Rev. 139, 1459 (1965). [5]. K. Horn et. al.,  Phys. Rev. B 30, 1711 (1984).