Fate of Spin Doublets in Quantum Dot with Many Interacting Electrons

Using the Single Electron Capacitance Spectroscopy, we study the energies required to add electrons to a quantum dot in a broad range of electron occupancy N. Following evolution of these energies as a function of magnetic field, we unveil the nature of the states occupied by electrons. For small N electrons fill states, which resemble oneelectron energy levels in a parabolic confinement. When N is large these states are drastically altered by increased Coulomb repulsion between electrons. We present a new and unified picture of the evolution from small to large N regimes.. The behavior of a single electron in a potential well is rather simple. It can be described in basic terms of single electron energy levels. However, for many electrons confined together the strong Coulomb repulsion between them makes this new many body problem enormously complicated. The rich physics in this regime is governed by the interplay of the two energies scales: the single particle level spacing and Coulomb interaction. The two scales depend on the size of the dot and the electron density, and typically do not vary significantly for small changes in the number of electrons N. For a small N in a sufficiently steep potential well (or small quantum dot) the confinement energy dominates the interaction energy. Single particle energies in this case resemble spatial quantization levels of one electron in the potential well. Noticeably, two electrons with opposite spin consecutively occupy the same spatial state. For that reason the energies required to add an odd μ(2k-1) and a consecutive even μ(2k) electron show similar behavior in magnetic field, specific to that spatial state. First, this behavior was predicted (1) and observed in several experiments in small vertical (2-4) and more recently small lateral dots (5-7) containing just a few electrons. While most features in the small dot spectra can be roughly described within the framework of the one electron levels picture (8), some experiments indicate on importance of electron interactions at least under certain conditions. Several interaction-driven singlet-triplet transition were observed in small dots (2,4,9). Coulomb interactions become comparable to single particle spacing for larger dots with substantial N. In these dots consecutive electrons do not necessarily fill the same spatial state with anti-parallel spins (10,11). Instead the small single particle level spacing allows for a ground state with a significant spin build-up (12-14). At high magnetic field the addition energies μ(B) demonstrate a saw-tooth behavior, as the electrons swing between the two lowest orbital Landau levels to minimize their repulsion energy (15,16). From this behavior, no similarity between energies μ(B) for adding odd and even electrons can be discerned. Recently, it has been shown that when electrons in a quantum dot completely fill the two lowest Landau levels, the interactions cause a significant spin build-up above some critical N (17). Our investigation bridges the gap between the studies of small and large dots. The experimental technique – Single Electron Capacitance Spectroscopy (18) permits examination of dots’ addition spectra while tuning the electron occupancy N from one to hundreds in one sample. To probe the nature of the states occupied by electrons, we follow the evolution of the addition spectrum with perpendicular magnetic field B. First few electrons fill well-known Darwin-Fock (DF) single-particle states as is typical for a small dot with parabolic confinement (3,8). These levels exhibit multiple crossing as B grows. We find that in the vicinity of every crossing, the electron interaction becomes the dominant energy scale, even in a small dot for a small number of electrons. Thus, four successively added electrons redistribute between two different crossing spatial states. We describe the observed resulting patterns by simple Hartree-Fock arguments. By gradually filling the dot with electrons we decrease the single particle level separation, hence increasing the relative importance of the Coulomb repulsion. As the result, the regions of the spectrum strongly affected by the interactions expand, until at high enough N no signature of DF states remains observable. Our study presents a unifying picture for the two limits: the small N regime with dominant confinement and the large N regime with prevailing interactions. The dots were fabricated within an AlGaAs/GaAs heterostructure as described in previous work (18). The essential layers (from bottom to top) are a conducting layer of GaAs serving as the only contact to the dots, a shallow AlGaAs tunnel barrier, a GaAs active layer that contains the dots, and an AlGaAs blocking layer. On the top surface, we produce a small AuCr top gate using electron beam lithography. This top gate was used as a mask for reactive ion etching that completely depletes the active GaAs layer in the regions away from the AuCr gate. A larger overlapping metal electrode then provides electrical connection to the gate (a schematic of our samples is shown in the Fig.1A). The measurements are carried out using on-chip bridge circuit described in (18,19). To register electron additions, we monitor the a.c. capacitive response to a small (<80μV) a.c. excitation applied between the top gate and the contact layer. At gate voltage values corresponding to the electron additions, an electron oscillates between the dot and the contact increasing the measured a.c. capacitance and resulting in a peak (18) in our measurements. The greyscale panel in Fig.1B expands a part of the addition spectrum between 9th and 15th electrons. Each successive trace corresponds to the energy for adding an extra electron to the dot. The data shown were taken on an elliptical dot with bare parabolic confinement with ωx=2.78mV; ωy=8.33mV. The lateral dimensions of the electron puddle (Lx≈200nm; Ly≈70nm for N≈20) are much smaller than the lithographic size of the dot. Such a large lateral depletion of the electron gas together with small lithographic dimensions ensure the parabolicity of the confining potential in both x and y. Thus, the overall spectral features can be described qualitatively within the constant interaction model for DF states, as is typical for small elliptical dots. Quantum numbers [nx,ny] can be assigned to the orbital DF states (1,20) and some are shown on Fig.1. Addition energies “wiggle” with magnetic field as different orbital states become the ground state of the dot. Pairs formed by each odd and consecutive even Coulomb blockade traces (marked by arrows), show mostly similar oscillatory behavior. This similarity arises because of the paired electrons entering the same spatial state but with opposite spins. However, more careful inspection of the data shows features that the simple model does not explain (marked by ovals). Each oval encases a region on Vg-B plane containing four electron additions to two crossing orbitals. Consider electron addition traces No11-14 in the magnetic field range of 1.8-2.5T (second lowest oval). The 11 trace forms a peak at B0=2.2T, as the electron jumps from the orbital [0,1] for BB0 (moving downwards with B). The 12 trace mostly follows this trend, except for a small vicinity of B0, where it forms a small asymmetrical V-notch. Intriguingly, the 13 trace develops an inverse feature at slightly different value of B: Λnotch at B0=2T. Finally, a dip at the 14 trace occurs again at the same B0=2T, slightly shifted from the peak at B0=2.2T in the 11 trace. Neither this misalignment nor the existence of the V and Λ notches is present in the single-electron non-interacting picture (21), and therefore both are the result of the Coulomb interactions. Note, that some of these features, namely V-notches at even traces occurring only for some crossing orbitals were reported in (9). To analyze the observed behavior in greater details we plot the addition energies of the four electrons No11-14 as a function of magnetic field in the range 1.8-2.5T in Fig.1C. Here, constant energies of 98.8; 103.6; 107.8; 111.9 meV were subtracted from the traces, so that to facilitate their comparison. Observe that the additional features in the 12 and 13 traces are complementary: the Λ-notch in the 12 trace perfectly matches the V-notch in the 13 trace. Moreover, the overall perfect point symmetry of the plot calls for an explanation involving all four electrons. So we adopt a simple Hartree-Fock arguments that were put forward in (22) to explain a mechanism for internal spin-flip transitions in armchair carbon nanotubes. We denote the bare single particle states [0,1] and [6,0] in our elliptical DF level sequence as a and b. At B0 =2.2T, the two levels Ea and Eb cross, and correspondingly the 11 electron fills the orbital a for BB0. The 12 electron follows the 11 away from B0. On the contrary, in the very vicinity of B0 the 12 electron avoids the Coulomb repulsion with the 11 by filling a different higher energy orbital. This becomes possible as the two orbital a and b are nearly degenerate, and thus the reduction of the Coulomb repulsion is greater than the loss in single particle energy |EbEa|. In particular, for B1B2) the orbital a (b) is doubly occupied, and this electron is inevitably added to the lowest empty orbital b (a). In the range of magnetic field B1B0). Hence, the 14 trace has a single dip centered at B0. All the matrix elements Vij and Jij can be directly measured from the data as shown on Fig.1C. Note that |B1B0|=|B2-B0| (B1=1.9T B2=2.3T). This equality explains the point symmetry of Fig.1C. These simple arguments describing rearrangement of four successively added electrons between two different crossing spatial states capture every feature of the experimentally observed behavior. We stress, that the single particle picture can account for none of these features (21). The traces described above are not the only set exhibiting notches. Several other pairs of complementary “V” and “Λ” notches are marked on Fig.1B. Notice the 10 and 11 traces at B=1.7T (note the deviation from 9 trace) and on the 14 and 15 traces at B=1.4T and 2.5T. In fact, similar pairs of notches appear at every orbital crossing. To examine the addition spectrum in its entirety we collapse addition traces for electrons from 7 to 23 by shifting them in the gate voltage (Fig.1D). The subtracted voltage values are chosen to allow neighboring traces touch. This means that the gaps marked as (Vab±Jab) on Fig.1C are eliminated. In shifting each trace we aimed to match the rightmost feature on that trace (near ν=2). It appears that all features can be fit with one number. The plot shows a very ordered pattern of single-particle DF orbitals. Each single-particle state can be followed through various pieces of different addition traces. The V-notch and Λ-notch on the traces No{12;13} discussed above form a diamond at B≈2.1T. More diamonds formed by similar pairs of notches are seen at every orbital crossings. Note three pairs of half-notches at zero field comprised by the pairs of traces No{8,9};{12,13};{18,19}. Two deviations from this rule occurs for the pairs of traces No{16,17} and No{20,21} at around B≈0.7T and B≈0.85T correspondingly. Instead of complementary notches these pairs of traces display smooth anticrossings. Careful examination of the picture actually reveals a duplicated set of DF orbitals. This is because the V and Λ notches prevent us from positioning even addition traces exactly on top of the odd ones when collapsing traces in the gate voltage. Two important conditions become obvious from the plot: the rearrangement of electrons appear near any degeneracy points of any two orthogonal orbitals. First, when two states are nearly degenerate the loss in single particle energy |Ei-Ej| is diminishing and become smaller than the gain in the Coulomb repulsion |Vii-(VijJij)|, which is large for orthogonal states. On the other hand, when the degeneracy is lifted for two mixed orbitals, the two states separate far in energy, while the would-be Coulomb gain approaches zero for mixed states (No{16,17} and No{20,21}). The single-particle states in our dot are orthogonal (and, therefore, can be potentially degenerate) either for the strong confinement potential (relatively small N) or sufficiently high magnetic field. Second, in the range shown on the plot only two states participate in a formation of notches. The slope in magnetic field allows us to identify precisely a given orbital state. Since, in the vicinity of every diamond structure, we see only two slopes, we conclude that only two orbitals are involved, i.e. four electrons rearrange for the optimal filling of only two orbitals. In our highly symmetrical dots the wavefunctions of different single particle states have very small overlap. Thus the direct Coulomb repulsion between two states i and j is much stronger than the exchange interaction (Vij>Jij). Still the exchange contribution to the interaction is measurable in our experiment. In Fig.2A, we plot the gate voltage values that we subtracted from addition traces (Vij±Jij) to collapse them on Fig.1D. All values are relative to that of the 9 trace. As electrons entering the dot gradually screen the confinement potential, each next electron added sees more relaxed confinement due to this screening. As a result electrons spread over the larger area reducing the Coulomb repulsion Vij. This explains the general decrease seen in the plotted spacings, which can be accounted for by subtraction of the smooth background. Fig.2B plots the spacings after a smooth quadratic background is removed. The spacing for even electrons are Vij-Jij while those for odd numbers are Vij+Jij. The apparent alternation of spacings on the plot indicate on observable nonzero contribution of exchange interaction, allowing us to deduce the spin configuration of the dot for various gate voltages and magnetic fields. The resulting color spin maps in Vg-B space are shown on Fig.2C and 2D. The data shown repeats that presented on Fig.1C and 1D. The spin maps show multiple but isolated regions of s=1 spin build-up. Spin never exceeds one, as it requires rearrangement of electrons on more than two orbitals, which does not happen in the Vg-B range shown. Finally, Fig.3 illustrates evolution of the addition traces with increasing N. The Vnotches on the even and the Λ-notches on the odd traces become more pronounced for higher N, eroding the similarity between even and odd traces. Essentially the notches expand to leave double the number of “wiggles” on addition traces occurring for magnetic fields corresponding to 2<ν<4. The even and odd traces become completely indistinguishable when N reaches about 60 in our dots. The shown traces No{61-65} appear remarkably similar to those of a highly populated dot in high magnetic field (15,16). And quantum dots in this regime are known to be spin polarized (15,16). We speculate that for N somewhere in between 25 B0, while electrons No13-14 complementary occupy (6,0) state for B B0. Therefore, the traces No12(14) would identical to No11(13). 22. Y. Oreg, K.Byczuk, B.I. Halperin, Phys.Rev.Lett. 85, 365 (2000) 23. We are grateful to H.U.Baranger, B.L.Halperin, K.A.Matveev and Y.Oreg for illuminating discussions. Expert etching of samples was performed by S.J.Pearton.