Surface Science 361/362 (1996) 677-681
Spectroscopy of quantum dots in the few electron limit Z. B o r s o s f o l d i a, I.A. L a r k i n b A.R. L o n g "'*, M . R a h m a n b M . C . H o l l a n d b, J . M . R . W e a v e r b, J . H . D a v i e s b, J.G. W i U i a m s o n b • Department of Physics and Aatronomy, Urdoersityof Glasgow, Glasgow G12 8Q~ UK b Department of Electronics and Electrical Engineering, Unit~,rsityof Glaagow, Glasgow G12 8QQ, UK Received 15 June 1995; accepted for publication 30 September 1995
Almract We have fabricated quantum dots with diameters of around 150 nm which contain less than 10 electrons. The conductance as a function of d.c. bias shows that the energy for adding successive electrons is around 3 meV, in good agreement with numerical modelling.
Keyworda: Electrical transport moaaurmnents; Electron denmty calculations; Semiconductor-scaniconductor heterosmLetur~
1. Introduction We have studied the characteristics of small surface-gated quantum dots containing ten electrons or less to examine the relative contributions of single particle energies and Coulomb interactions to the level spacings. An account of related work on dots containing around 10 electrons has recently been published [ 1].
2. Techniques employed The dots were formed on shallow GaAs/A1GaAs heterostructures with the electrons at an effective depth of 35 nm from the surface . They were defined by 4 separate gates (Fig. 1), were between 100 and 150nm in nominal diameter and were accessed through quantum point contacts around * Corresponding author. Fax: +44 141 334 9029, e-mail, arlong~elec.gla.ac.uk.
Fig. 1. Layout of a typical 150 nm 4-gate dot showing gains G1 to G4.
100 m wide. There are theoretical and experimental reasons for believing that this size maximi.~s the single particle energy level spacing when few electrons are present. Approximating the dot by a single circular electrode and using the method of Davies, Larkin and Sukhomkov , the curvature of the electrostatic potential in the centre of the dot at complete depletion is a maximum for a ratio
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Z. Borsosfoldt et aL/Surface Science 361/362 (1996) 677-681
Table 1 Parameters of the two quantum dots described in the text Sample
Carrier density (x 10tSm-2)'
~ Vs (mV)
of dot radius to electron depth of ~/3, assuming a frozen semiconductor surface. For our layers this suggests an optimal diameter of approximately 120 nm. Transport measurements through the dots were made at 1.3 K and above, as the spacing of conductance peaks is adequate for them to be resolved at this temperature (when the dots are almost fully depleted). The differential conductance between the regions either side of the dot (source and drain) was studied for different gate voltage combinations (defined with respect to the source), and for finite standing d.c. source drain voltages.
3. Experimental results The parameters of two samples whose gate voltage characteristics revealed only a few peaks are given in Table 1. In Fig. 2 we show the gate voltage characteristics of sample 2. The electrostatic potential is swept by means of the "plunger" gate G2, and the whole characteristic is shifted in both cases 0.03
VI,V3 •-.435V, V4=-.455V . . . . . . . V1,V3=-.45V, V4ffi-.47V
(32 voltage Fig. 2. Conductance of sample 2 a£,ain~t G2 voltage, measured at 1.3 K. The parametzra are the :voltages applied to the other thrce gates.
by varying the voltages on the other three gates. The characteristics show 2 broad conductance peaks with some indication of a third. Sample 1, fabricated on a different layer, was broadly similar in its behaviour. At more negative values of the G2 voltage, little sign of any further peaks above the noise floor (around 10 -4 e2/h) was found for either of the two samples. Although we cannot definitely exclude the presence of further electrons remaining in the dot at large negative gate voltages, they must be tightly bound and undetectable in the tunnel current. From the voltage shift data, the voltage period between the major conductance peaks for applying bias to different combinations of gates can be determined. The values A Vs quoted in Table 1 are for varying bias being applied to all gates in the dot simultaneously. We carried out further measurements to verify that these peaks arise from Coulomb blockade. The energy parameters for the dots were determined 14] by applying a variable d.c. bias between source and drain and measuring the differential conductance characteristics. The d.c bias causes peak shifts which may be Used to determine the energy differences AE between energy levels in the dot. In all this biased data, the peaks corresponding to the electron encrgy levels in thc dot passing through the Fermi level in the drain are dominant, and the entry in Table 1 marked slope (s) gives the rate of change of this peak voltage with V0.. For sample 2, two values, st and s2, are given corresponding to the interchange of source and drain connections. Using these slopes, and assuming the capacitance model for the response of the dot r5],
,~E =eAV,/Cst +s2-- 1).
This gives the first AE value for sample 2 quoted in Table 1. In these two samples, the separation of the
Z. Borsosfoldi et aL/Surface Science 361/362 (1996) 677-681
energy levels in the dot is large enough that the weaker peaks associated with the coincidence of the energy level in the dot with the Fermi level in the source, can also be seen even at 1.3 K. Hence the complete conductance "diamond"  can be deduced and is shown for sample 2 in Fig. 3. A further value for ,~E can be derived; those in Table 1 correspond to the gap between the first two conductance peaks. For sample 2, the value obtained by this method, the second entry in this column, is in good agreement with that derived by reversing source and drain. The calibrations of the energy scales derived from d.c. bias were checked by looking at the temperature dependences of the conductance minima and were found to be consistent in all cases. The observations of the peak splittings under d.c. bias, and the internal consistency of the values deduced for peak energy separations, show unambiguously that Coulomb blockade dominates the conductance of these dots with large energy separations even at 1.3 IC The final column in Table 1 is the ratio of energy separation to gate period (again for the first two major peaks). Two points may be noted about these values. Firstly, they are similar for each of the samples (and for all the other 150nm dots studied), as would be expected for dots of the same
size near depletion. Secondly, the values are considerably less than unity. In the capacitance model for Coulomb blockade, such a low value suggests that the capacitances of the leads to the dots dominate the gate capacitances. A comparison of the data of Fig. 2 with the lineshape expected for tunnelling through a single level  shows that the observed lines are much wider than expected. We ascribe this tentatively to a number of resonant levels lying under a particular peak. Such sub-structure is apparent in Fig. 2, especially in the first peak. Lower temperature studies will be necessary to resolve this structure fully, one interesting feature of the substructure is that it varies with the voltages applied to the main gates 1, 3 and 4. The most likely reason for this lies in the observation, deduced from our simulations, that the dots change shape when the relative voltages on the gates are varied, becoming more or less elliptical. If the substructures are associated with excited states of the dot, then it is not surprising that their relative weights vary with the dot's shape. We find that the exact energy spectra of these small dots depend critically on the local potential environment, and can change radically even during the course of a measurement in response to movement of trapped charge. The two dots reported here have particularly straightforward conductance peak structures with the largest energy separations observed in this type of dot, However we have studied other dots of the same nominal size and shape and have observed larger numbers of closer spaced conductance peaks. In addition to the potential imposed by the gates, we expect for dots containing few electrons that the local potential fluctuations generated by remote ionised donors  will have a significant effect on the energy spectrum. Because the dots contain few electrons, these fluctuations will be unscreened and may in certain circumstances be of a similar magnitude to the potentials generated by the gates.
G2 voltage Fig. 3. The positions of the conductance peaks in sample 2 as a function of d.c. source drain bias, measured at 1.3 IC • = electron energy level in the dot coincident with the Fermi level in the drain (dominant peaks). + =coincidence with Fermi level in source.
4. M o d e l l i n ~
We have modelled this device self-consistently by solving numerically the three-dlmensional
Z. Borsosfoldl et al./Surface Science 361/362 (1996) 677-681
Poisson equation and using the Thomas-Fermi approximation for the density of the 2DEG in the channel. The calculation shows that the cut-off voltage and number of electrons in the dot are strongly dependent on the boundary condition on the free surface . The results for a frozen surface are in excellent agreement with experimental measurements of cut-off voltage and variation of numbers of electrons in a previous dot [ 11. For sample 2 our calculated cut-off voltage is -0.55 V, some 10% higher than experimenL This suggests that, if there are additional electrons in the device beyond the first conductance peak, then there are only very few. The calculations also show that the shape of the dot is sensitive to the position of the "plunger" gate; the confining potential is almost circular for thin dot. Independent variation of the Fermi level inside the dot permits the total capadtance C and the Coulomb blockade energy e2/C to be evaluated. For 10 electrons in the dot, then capacitance to the gates is around 15 aF, corresponding to an all gate period of 10 mV, and e2/C~2.8meV. Both of these are in good agreement with the values in Table 1, if one ignores the single electron contributions to the energy spacings.
In small dots there are two contributions to the energy spacings between levels. The Coulomb energy due to interactions with the surrounding electrodes controls large dots; very small dots are dominated by the "single-particle" terms which depend on the size and shape of the dot and the interaction between electrons within the dot. Most dots described in the literature are larger than ours, containing many electrons, and one expects the Coulomb contribution to dominate. The conductance peaks, each resulting from the addition of a single electron, are then nearly equally spaced as is observed. However our dots are smaller and opfimised for maximum single-particle spacing. It is no longer clear that Coulomb energies dominate. Assume for simplicity that the dot is empty beyond the first peak. If Coulomb energies dominate, the first peak arises from the first electron,
with substructure due to single-electron excited states. The second peak corresponds to the second electron entering the dot. In contrast, if singleparticle energies are dominant, the first peak involves two electrons of opposite spin entering the ground state, and its width reflects the Coulomb interactions. The gap between first and second peaks then reflects the energy spectrum of the dot. At present the experiment cannot distinguish unambiguously between these two limits. However, our semiclassical simulations suggest that the Coulomb spacing between the levels for the first few electrons is greater than 2.8 meV for this size of dot, much larger than the width of the first peak. A rough estimate also shows that the single-particle energy levels should be about 1 meV apart. Thus the single-particle energies are not negligible, and may affect the position and shape of the observed peaks, but it is likely that Coulomb energies dominate even in these very small dots. Several quantum mechanical calculations (e.g. Ref. ) of the energy spectra in small dots have recently been published. These are however not easy to apply to our data because (a) they generally assume circular symmetry and ignore any ellipticity in the dot, (b) they use fixed parabolic confining potentials, and (c) they ignore any interaction with the surrounding gates and bulk 2DEG. Clarification of the exact model to apply to dots of this size in the few electron limit is only likely when further studies with samples of different geometries are performed.
We have presented measurements on two fully gated 150 nm quantum dots which contain a very small number of electrons. The energy for adding successive electrons to the dots was around 3 meV in good agreement with our modelling.
Acknowledgement This work was supported by EPSRC (UK) and by the University of Glasgow.
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