Sektion Physik, LMU München, Geschwister-Scholl Platz 1, 80539 München, Germany
G. Medeiros-Ribeiro and P.M. Petroff
Materials Department / QUEST, University of California, Santa Barbara, CA 93106, USA
Electronic coupling effects in self-assembled InAs quantum dots are investigated using capacitance and far-infrared (FIR) spectroscopy. Coupling between different dot-layers is studied using capacitance spectroscopy on samples with double-layers of vertically aligned quantum dots. A strong electrostatic dot-dot interaction reveals itself in distinct shifts of the many particle ground state energies. The fact that the line widths of the FIR resonances are typically a factor of three narrower than expected from the inhomogenous broadening due to size fluctuations of the dots is attributed to lateral coupling between the dots within the plane of the dot ensemble.
Keywords: GaAs/InAs; self assembled quantum dot, coupling
Correspondence address :
Sektion Physik, LMU München, Geschwister-Scholl Platz 1, 80539 München, Germany Fax +49 89 2180 31 82; email: firstname.lastname@example.org
Recently, self-assembled quantum dots have gained particular interest as they provide for well defined, nm-size quantum dots with sizes in the 10 nm range [1,2,3]. For possible device applications it is of fundamental interest to know the effects of dot-dot interactions as they could on the one hand limit the integration density, but on the other hand might offer the possibility to implement new computational concepts .
Here we study the effects of the static and dynamic Coulomb interaction in large-scale dot ensembles. The samples are grown by molecular beam epitaxy using the Stranski-Krastanov growth procedure to create self-assembled InAs dots . The dots are embedded into a suitably designed MISFET (metal-insulator-semiconductor-field-effect-transitor) type GaAs/ AlGaAs heterostructure as described in Refs. [2,3]. Fig. 1 shows the essential layer sequence and a sketch of the conduction band edge for the investigated double-layer samples. The dots in the second layer are aligned along the growth direction to the dots of the first layer as the strain field of the first dot acts as a nucleus for the growth of the second one . Except for the second dot layer, the single layer samples are identical. From atomic force micrographs of similar grown samples we estimate the InAs dots to be approximately 20 nm in diameter and 6 nm in height .
The samples are provided with Ohmic contacts and a semi-transparent gate. They are then mounted in a liquid He cryostat equipped with a superconducting solenoid, capable of producing fields of up to B=15 T perpendicular to the plane of the dots. The FIR transmission of the sample is recorded using a rapid-scan-Fourier-transform spectrometer, the capacitance-voltage (CV) spectra are measured by standard lock-in technique.
Vertical coupling is investigated on double-layer samples with vertically aligned quantum dots with separation of 10 and 20 nm. In order to investigate possible quantum mechanical coupling effects we first calculate the single-particle energy splitting between two levels of different dots using a simple model of a double quantum well. With the given values for our system we obtain even for the 10 nm sample an energy splitting of only 100 ueV. This is far beyond the limit of resolution due to inhomogenous size broadening of our dot ensembles. Therefore, the dominant type of interaction observed in the CV spectra of vertically aligned double quantum dots is the Coulomb interaction between the dots of different layers.
Fig. 2 (a) shows a CV trace of the sample A ( 20 nm dot-dot distance). For comparison, CV traces from a single-layer sample and a sample with only two wetting layers are also shown (b),(c). For negative gate voltages the spectra of the double -dot sample is the same than for the single layer one. As described in detail for single-layer samples in Ref.  the double peak at Vg -1V corresponds to the filling of the lowest (s-) state of the quantum dots. The s-state is doubly spin degenerate and the difference in the gate voltage between loading of the first and the second electron is a direct consequence of the Coulomb blockade caused by electron-electron interaction within one dot. In our oblate, nearly circular dots the second shell is fourfold degenerate and, because of Coulomb repulsion, we expect four peaks at higher gate voltage, when the p-state is gradually filling. In contrast to earlier investigations we could resolve for the first time loading of distinct p-states within such a large ensemble of dots (approx. 108), which is depicted in the inset of Fig. 2. We attribute this to improved growth conditions which result in a further narrowed dot size distribution.
After filling of the p-shell, the double-dot system behaves differently than the single-dot system: In addition to the charging of the wetting layer and the 2D electron gas at the GaAs/ GaAlAs interface we observe three additional peaks in the 20 nm double dot sample (labelled sdot2, pdot2, x in Figs. 2,3), which show the charging of the second dot system. In order to determine coupling effects between the two layers we convert the gate voltage into an electron-energy (the electron energy in the dot measured from the GaAs band edge) using the simplifying lever arm model given. e.g. in Refs. [2,3]
where 650 mV is the value for the Schottky barrier between the gate and the GaAs cap. It is important to note that the lever arm for each of the two dot layers is different (see Fig. 1): In our 20 nm-samples we have a lever arm of 1/7 for the first dot layer and 1/3.9 for the second one. Therefore the CV spectra of a single layer sample with the same leverage than the second dot layer would be compressed compared to the spectra of the first dot-layer alone. In addition, as the gate voltage required to reach the charging threshold for the dot scales linearily with the distance from the back contact to the dot (t2 in Fig 1) it is also shifted to higher gate voltages. Fig. 3b shows such a (calculated) spectrum. However, in the double layer sample we expect the charging of the second dot layer to be influenced by electron-electron interaction. In analogy to the Coulomb blockade within a dot (which is responsible for the s and p-level splitting) we have in this case a Coulomb blockade between dots of different layers. Using an electrostatical approach this electron-electron interaction is modeled in the following way
Ee-dot-dot is the additional Coulomb energy of an electron in dot 2 when dot 1 is filled with 6 electrons taking into account the image charges in the back contact (s. inset of fig. 3). With d= 25 nm and a = 20 nm we obtain a value of 24 meV. In addition, we take the next neighbours of dot A into account. From an analysis of AFM data we estimate 4 next neighbours to be at distances of 40 nm and thus the total Coulomb energy in this model adds up to 48 meV. The interaction with dots that are further apart laterally is effectively screened out by the metallic back contact and can be neglected. With the lever of 3.9 we would expect an energy shift of 0.19 V. Fig. 3c shows the same trace as in Fig. 3b, but shifted by this additional inter-dot Coulomb energy. As it matches very well with the 2 additional peaks in Fig. 3a, we conclude that the additional peaks are indeed the s- and p-states of the second dot layer and that the shift is a direct consequence of the dot-dot interaction.
Magneto-capacitance on this sample confirms the energy diagrams, but as there is no intermixing between the levels of the double-dot system (contrary to the 10 nm sample discussed later), the spectra correspond to those of two single layer samples. It is to note that the energy level at Ug=0,85 V (labelled x in Fig. 2a) is not yet understood. A possible scenario would be an additional energy level in the wetting layer of the second dot system, created by the strain field of the first dot-layer.
Fig. 4 shows the magneto-capacitance of the 10 nm sample. In contrast to the 20 nm sample deviations from the spectra of a single-layer sample already occur within the charging of the p- shell. In this system the lever for the second dot layer is 5 and the energy shift of 230 meV can again be explained quite accurately using the model for electrostatic dot-dot interaction described above. However, in this case the filling of the dots in the second layer will add an additional Coulomb blockade term to the unfilled p-states of the first dot layer. A sketch with energy levels that correspond to this situation is shown in Fig. 5a. For B=0 charging of the second dot (s21) occurs after the p-level of the first dot is filled with 3 electrons. The second electron of dot 2 cannot be resolved, probabely due to charging of the wettinglayer of dot 1. In single-layer samples the s-state is only little affected by magnetic field, whereas the p-state shows characteristic Zeeman splitting with two states increasing with increasing field and two decreasing . Applying this to the energy levels of the double-dot system we expect p1 and p2 to decrease and p3 and p4 to increase with increasing magnetic field. When p3 reaches the level of s21, the influence of the inter-dot Coulomb blockade changes the charging sequence: For B=0 the p3 level of dot 1 will be charged before the s21 level of the second dot, whereas at high field, the s21 state will be charged first. Thus the combination of orbital Zeemann splitting and Coulomb interaction between the dots results in a magnetic-field-induced filling of the second s-state. This picture is confirmed by Fig. 5b, which shows the difference between the magneto-capacitance of the 10 nm double-dot sample and a single-layer sample at various magnetic fields. The development of the peaks shows that the cross-over takes place at 6 T.
In-plane dipole coupling within a single plane of dots was investigated on single-layer samples. The FIR transmission of the sample shows absorption lines corresponding to the allowed dipole dot-excitations  that are too narrow to be explained by independent excitations in each dot of the ensemble. The observed line width of 2-4 meV in these spectra is well below the broadening that is related to size fluctuations, both in thickness and diameter of the quantum dots. The energy difference E1-E0 (which is measured in the FIR absorption) is predominantly affected by fluctuations of the dot-dot diameter, whereas fluctuations of the dot thickness contribute mainly to the ground state and thus broaden the charging threshold. AFM investigations revealed fluctuations of the dot diameter of approx. 10 % . Assuming a parabolic potential and 50 meV, which is confirmed by earlier investigations , we therefore estimate within a single -electron model a corresponding energy broadening of 10 meV. We attribute the narrowing of the lines observed in the experiments to inter-dot coupling effects. Then the FIR resonance can be seen as a collective response of the dot ensemble as a whole. Such a collective mode can be expected whenever the lateral coupling energy exceeds the energy broadening caused by size fluctuations. Applying a model proposed by Merkt  we can qualitativly explain this effect, but quantitative modelling of this system requires elaborate numerical simulation and is beyond the purpose of this paper.
We gratefully acknowledge financial support through BMBF grant 01BM623, through the Max Planck Gesellschaft and through QUEST, a NSF Science and Technology Center.
 L. Goldstein, F. Glas, J. Y. Marzin, M.N. Charasse, G. Le Roux,. Appl. Phys. Lett 47, 1099 (1985); Q. Xie, A.. Madhukar, P. Chen, N.P. Kobayashi. Phys. Rev. Lett. 75, 2542 (1995); M. Grundmann et al. Phys. Rev. Lett. 74, 4043 (1995); J.-Y. Marzin, J.M. Gerard, A. Izrael, D. Barrier, G. Bastard, Phys. Rev. Lett. 73, 716 (1994)
 H. Drexler, D. Leonard, W. Hansen, J.P. Kotthaus, P.M. Petroff, Phys. Rev. Lett. 73 2252 (1994); G. Medeiros-Ribeiro, D. Leonard, P.M. Petroff, Appl. Phys. Lett. 66, 1767 (1995)
 M. Fricke, A. Lorke, J.P. Kotthaus, G. Medeiros-Ribeiro, P.M. Petroff, Europhys Lett. 36 197 (1996)
 S. Benjamin, N.F. Johnson, Appl. Phys. Lett. 70 2321 (1997)
 D. Leonard, M. Krishnamurthy, C.M. Reaves, S.P. Denbaars, P.M. Petroff, Appl. Phys. Lett. 63 3203 (1993)
 J. Tersoff, C. Teichert, M.G. Lagally, Phys. Rev. Lett. 76, 1675 (1996)
 W. Hansen, T.P. Smith, K.Y. Lee, J.A. Brum, C.M. Knoedler, J.M Hong, D.P. Kern, Phys. Rev. Lett. 62 2168 (1989)
 U. Merkt, Phys. Rev. Lett. 76 1134 (1996)
Sketch of the conduction band edge of the investigated double-layer samples with respect to the Fermi level Ef along the growth direction. The indicated distances define the lever arm according to ti/ttot which converts voltage differences into energy differences. t1= 25 nm, t2 =45 nm (sample A) / 35 nm (sample B), ttot =175 nm.
Capacitance voltage (CV) traces of (a) a double-layer sample with vertically stacked quantum dots at 20 nm distance (sample A); (b) a single-layer sample; (c) a sample with two wetting layers The curves are offset for clarity. The inset shows the charging of the first dot-layer of (a) at higher magnification.
(a) CV trace of 20 nm double dot sample; (b) calculated CV trace of a single-dot sample with a lever arm corresponding to the second dot layer; (c) CV spectra of (b) shifted in gate voltage by the calculated inter-dot layer Coulomb energy. The inset shows schematically the applied electrostatic model used for the calculation.
Magnetocapacitance of a 10 nm double-dot sample (sample B) for magnetic fields B=0,2,4,6,8,10,12 T. The curves are offset for clarity.
(a)Schematic energy diagramm for the energy levels of dot 1 and dot 2 at B=0 T and B=12 T; (b) Difference in capacitance between 10 nm double-dot sample and a single-layer sample between B=0 T and B=12 T; the levels s21and s22 correspond to the levels of the second dot in (a).