Effective Hamiltonian for two-electron Quantum Dots from weak to strong parabolic confinement
EEffective Hamiltonian for two-electron Quantum Dotsfrom weak to strong parabolic confinement
Torsten Victor Zache and Aniruddha Chakraborty
School of Basic Sciences, Indian Institute of Technology, Mandi 175001 (Dated: April 27, 2017)We model quasi-two-dimensional two-electron Quantum Dots in a parabolic confinement potentialwith rovibrational and purely vibrational effective Hamiltonian operators. These are optimized bynon-linear least-square fits to the exact energy levels. We find, that the vibrational Hamiltoniandescribes the energy levels well and reveals how relative contributions change on varying the con-finement strength. The rovibrational model suggests the formation of a rigid two-electron moleculein weak confinement and we further present a modified model, that allows a very accurate transitionfrom weak to strong confinement regimes.
I. INTRODUCTION
Since the discovery of Quantum Dots (hereafter QDs)in 1981 , a lot of research has been carried out inthe direction of semiconductor nanostructures. Few-electron QDs, especially two-electron QDs are of hugetheoretical interest, because they provide a model sys-tem for studying electron-electron correlation and prop-erties of electronic spectra in general (e.g. Hund’s rule ).Experiments suggest, that the confinement can be mod-eled by a parabolic potential as a good approximation inmany cases. In this paper we only examine quasi-two-dimensional two-electron QDs in the absence of exter-nal fields. For external magnetic fields, the reader isreferred to other work. Nevertheless the energy lev-els in two-electron QD exhibit a complicated behaviour.In the strong confinement regime, electron correlation ef-fects can be neglected to a good degree of approximation and the spectrum exhibits some degeneracies as a resultof the generealized Kohn theorem . For weak confine-ment the two electrons behave similar to a rigid moleculewith characteristical rotational and vibrational energyspectra .In this paper, we present possible unified descriptionsof two-electron Quantum Dots from weak to strong con-finement regimes. Section 2 gives different approaches toconstruct an effective Hamiltonian operator for the en-ergy spectrum. One is based on the rovibrational ”molec-ular” model and the other one is purely vibrational usingthree coupled Harmonic Oscillators. Several non-linearfits are performed in order to improve the accuracy ofthe operators. We give our results for both weak andstrong confinement in Section 3 and interpret them interms of electronic correlation. Section 4 summarizes anddiscusses our findings. II. ROVIBRATIONAL AND VIBRATIONALEFFECTIVE HAMILTONIAN
We consider two electrons in a QD with parabolic con-finement potential. The exact Hamiltonian (using effec- tive atomic units) reads H = −∇ − ∇ + 14 ω r + 14 ω r + 2 | (cid:126)r − (cid:126)r | (1)Numerically the energy levels can be calculated exactly and we use these for fitting our effective energy opera-tors. The data we have used is given in TAB. I. Varyingthe confinement strength ω basically changes the rel-ative importance of the confining and the Coulomb re-pulsion potential. We will use this to test the validity ofour models and to interpret the effect of electron-electroncorrelation.In the case of weak confinement the energy spectrumexhibits a molecular-like picture consisting of rotationaland vibrational contributions as can be seen in FIG. 1.Therefore it seems reasonable to use a standard effectiveHamiltonian of the following form (we have ignored an TABLE I. Exact energy levels for (1) given in Ry ∗ . Thelabelings correspond to the rovibrational and purely vibra-tional effective Hamiltonian operators. The missing labelsare discussed at the end of section 3. Quantum numbers forcentre-of-mass, relative-motion and spin can be found in thereference paper. ( N, M ) ( n , n a , n b ) ω = 1 . ω = 0 . a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r FIG. 1. Selected states from TAB. I showing a molecular-like spectrum. (a) equally spaced vibrational excitations; (b)rotational excitations (cid:72) a (cid:76) vibrationalacgm (cid:72) b (cid:76) rotationalb cd e gi k mpoa0.250.300.350.400.45 E n e r gy (cid:72) R y (cid:42) (cid:76) irrelevant constant) H (1) eff = w (cid:18) N + 12 (cid:19) + BM − DM (2)Here the vibrational ( N ) and angular momentum ( M )quantum numbers have been assigned in a natural way(see TAB. I). (2) is known to describe molecular spec-tra like the one shown in FIG. 1 very well, even with D = 0, but the latter correction will be necessary forstronger confinement. We also want to mention that asimilar formula has been derived analytically from theexact Hamiltonian. Note that we are dealing with atwo-dimensional system, so that M should be used in-stead of the three-dimensional J ( J + 1) term. However,we also consider another form leading to surprising re-sults. H (1 (cid:48) ) eff = w (cid:18) N + 12 (cid:19) + BM ( M +1) − DM ( M + 1) (2 (cid:48) )Schwinger pointed out, that it is possible to describeangular momentum with two uncoupled harmonic oscil-lators. Since correction terms for angular momentumare necessary here, we use coupled oscillators instead.Empirically we arrive at the following purely vibrationaleffective Hamiltonian H (2) eff = w (cid:18) n + 12 (cid:19) + w ( n a + n b + 1) (3)+ α (cid:32)(cid:18) n a + 12 (cid:19) (cid:18) n b + 12 (cid:19) + (cid:18) n a + 12 (cid:19) (cid:18) n b + 12 (cid:19)(cid:33) In this case, the labels have been chosen, so that therelations n = N (4a) n a + n b = M (4b) | n a − n b | =min. (4c) TABLE II. Results of the fits for weak ( ω = 0 .
05 Ry ∗ ) andstrong ( ω = 1 . ∗ ) confinement given in units of 10 − Ry ∗ and Ry ∗ , respectively. ω = 50 . ω = 1 . H (1) H (1 (cid:48) ) H (2) H (1) H (1 (cid:48) ) H (2) ω, ω B D ω - - 3.0 - - 0.443 α - - 4.6 - - 0.065rms 1.7 0.9 3.0 0.177 0.035 0.039 are fulfilled. The condition (4c) allows to describe thestates with the given effective Hamiltonian (3) by cou-pling the different oscillator modes. The parameter α determines the strength of the coupling between the ”an-gular” vibrations n a and n b . III. RESULTS AND INTERPRETATION
We have fitted the energy levels in TAB. I to the oper-ators (2) and (3) by means of non-linear least-square fitsin order to optimze their parameters. Each fit consists of11 independent levels and is used to determine 3 parame-ters. The results are given in TAB. II. Both models fit theenergy levels reasonably well, as can be seen from the lowrms values. The rovibrational model (2) is more accuratethan the vibrational model (3) in the case of weak con-finement, which is not very surprising, because the spec-trum obviously shows a rovibrational structure (see FIG.1). But for strong confinement it does not describe theenergy levels as well as the vibrational model. We alsoperformed fits for the modified rovibrational model (2 (cid:48) )yielding surprisingly low rms values in both confinementregimes. The results are also pictured in FIG. 2. We wantto emphazise, that all effective Hamiltonians correctlydescribe the level-crossing of the different states. In therovibrational picture, this crossing can be explained bya broadening of the single rotational spectra (compareFIG. 1 and FIG. 2).The calculated values of ω, ω coincide with the con-finement strength ω as they are expected to. We find,that D = 0 for the simple rovibrational model in caseof weak confinement, where electron repulsion is strongenough to fix the distance between the two electrons andleads to the formation of a rigid two-electron molecule.For stronger confinement D (cid:54) = 0, so that the rigidity be-comes less accurate. D can be thought of as centrifugaldistortion - a common interpretation in molecular spec-troscopy. In this semi-classical picture, the stronger con-finement forces the electrons to stay closer together andincreases their effective moment of inertia.At first sight, the modified model looks very similar tothe rovibrational one. Although we also call it ”rovibra-tional”, one has to be very careful about this interpreta- FIG. 2. Results of the fits for weak and strong confinement on left and right side, respectively; (a) exact data ; (b) rovibrationalmodel (2); (c) vibrational model (3); (d) modified rovibrational model (2 (cid:48) ); the labels are explained in TAB. I (cid:72) a (cid:76) exact dataabcdeiogkpm (cid:72) b (cid:76) rovibrational (cid:72) c (cid:76) vibrational (cid:72) d (cid:76) modified E n e r gy (cid:72) R y (cid:42) (cid:76) (cid:72) a (cid:76) exact dataabcdeiogkpm (cid:72) d (cid:76) modified (cid:72) c (cid:76) vibrational (cid:72) b (cid:76) rovibrational E n e r gy (cid:72) R y (cid:42) (cid:76) tion. The modified model describes a three-dimensionalrotor-vibrator, which is not expected here. Another fitfor this Hamiltonian with fixed D = 0 for weak con-finement reveals a rms value of 5.6, which proves thatthe ”centrifugal distortion” term is needed to correct thepredictions of the model, even in the small regime. Wealso want to mention, that the sign of D changes fromweak to strong confinement, while the relative contribu-tion (as compared to B ) stays the same. The change insign makes a classical interpretation as centrifugal dis-tortion rather difficult. Still both models have a smallerangular contribution for weak confinement and a strongerangular contribution for strong confinement in common,as can be seen by comparing the relevant values of B (and D ) to ω .The purely vibrational Hamiltonian (3) is not as ac-curate as the modified rovibrational model, but its com-ponents make this operator even more intersting. Theangular contribution is implemented by two harmonic os-cillator modes n a and n b and a term describing a couplingbetween them. The fact, that the operator is invariantunder exchange of n a and n b suggests, that electron cor- relation might be partly described by these oscillators.Closer examination of the coupling term supports an in-terpretation as contribution to electon correlation energy.Indeed, this part looses relative importance (again com-paring the calculated parameters) in the large regime.This is in agreement with the expectation, that electroncorrelation has less influence in strong confinement. How-ever, it cannot be neglected completely.The strongest contribution stems from the ”main” vi-bration n , but it decreases for increasing confinementand the ”angular” vibrations n a/b gain importance, justlike the angular contribution in the rovibrational model.To show the effect of the confinement strength on thedifferent components of our model we have plotted theirrelative contributions in FIG. 3. Since the different termsenter with very different strengths depending on the con-sidered state, we have calculated average contributions.Still one has to be careful interpreting FIG. 3. Considerthe state o: (1,1,2) as an extreme example. It is easy tocalculate, that the relative contribution of the ”angular”vibrations (the w term) increases by a factor of 7 fromweak to strong confinement. The fact, that it becomes FIG. 3. Average relative contributions of the parameters ofthe effective vibrational Hamiltonian (3) normalized to unity.The weak and strong confinement regimes are shown in green(light grey) and black, respectively. w1 w2 a0,00,20,40,60,8 r e l a t i v e c on t r i bu t i on parameters even stronger than the ”main” vibrational contribution(the w term) explains the occurance of level crossings inthe vibrational picture.Examining FIG. 3, one might think, that the couplingterm does not play an important role at all. Another fitwith fixed α = 0 reveals the rms values 23.9 and 0.343 forweak and strong confinement, respectively. This provesthe importance of the corresponding term in (3). How-ever, it also shows, that the coupling term contains moreinformation than only electron correlation. Otherwise itcould approximately be neglected in strong confinement.We want to give some additional thoughts about thelabeling of states for our models. Note, that we havebeen very careful with the use of the expression ”quan-tum number”. Since the corresponding operators do notcommute with the given effective Hamiltonians, we thinkit is more appropriate to speak of approximate quan-tum numbers or just labels. However, our assignmentfor the rovibrational model is in good agreement withthe center-of-mass and relative-motion description. Atleast for the lower rotational levels, M coincides with theangular quantum number of relative-motion. States withodd and even M are singlet and triplet, respectively. Thecareful reader might have noticed, that the energy of thestate l (in weak confinement) suggests, that it belongs tothe first rotational spectrum. However, we did not chooseto label it (0,4), because it is a singlet, not a triplet andthe state (0,3) is missing in the data. We note, that theaverage energy of f and h (in weak confinement) coincideswith the expected (0,3) energy. This could hint at a split-ting of these two states, which remains to be explained. We also mention, that the states n and l exhibit neardegeneracies with k and p, which are broken in strongerconfinement. Some of these observations could proba-bly be described by additional vibrational modes, if theelectrons are considered to form a ”triatomic” molecule(with the center-of-mass as the third component). How-ever, the data considered here is not sufficient to test thishypothesis. IV. SUMMARY AND CONCLUSION
In this paper, we have constructed different effec-tive Hamiltonians for quasi-two-dimensional two-electronQDs in parabolic confinement. The rovibrational modelstrongly supports the formation of a rigid two-electronmolecule in the weak confinement regime. We presentedanother modified rovibrational model, that describes theenergy spectrum surprisingly well for both weak andstrong confinement. The third model also allows a veryaccurate description in the large regime and is a little bitless useful for weak confinement. However, it is of greatinterest, because it is purely vibrational. Two coupledharmonic oscillators are used to replace the angular con-tribution of the rovibrational model allowing a transitionfrom weak to strong confinement.In conclusion, the presented effective Hamiltoniansallow an accurate description of the energy spectrumof two-electron QDs in weak and strong confinementregimes. It will be interesting to test the validity of themodels in even stronger confinement regimes and for alarger amount of excited states. The explanation of thesurprisingly high accuracy of the modified rovibrationalmodel remains the subject of further research. Further-more, we want to point out, that the vibrational modelcould be used in more complicated situations, like three-electron QDs, where a ”molecular” description as therigid rotor-vibrator has not been found yet. We stronglybelieve, that the examination of few-electron QDs witha purely vibrational description will lead to a deeper un-derstanding of electron correlation in general.
ACKNOWLEDGMENTS
Some of the figures for this article have been createdusing the LevelScheme scientific figure preparation sys-tem [M. A. Caprio, Comput. Phys. Commun. 171, 107(2005), http://scidraw.nd.edu/levelscheme].We would like to thank the German Academic Ex-change Service (DAAD) for providing scholarship forTorsten Zache during his internship at IIT Mandi. A. Onushchenko and A. Ekimov, JETP Lett. , 345(1981). N. F. Johnson, J. Phys.: Condens. Matter , 965 (1995). T. Ihn, C. Ellenberger, K. Ensslin, C. Yannouleas,U. Landman, D. C. Driscoll, and A. C. Gossard, Int. J.Mod Phys B , 1316 (2007). T. Sako, J. Paldus, A. Ichimura, and G. H. Diercksen, J.Phys. B , 235001 (2012). D. Heitmann and J. P. Kotthaus, Phys. Today , 56(1993). P. Maksym and T. Chakraborty, Phys. Rev. Lett. , 108(1990). T. Chakraborty, Comments Cond. Mat. Phys , 35(1992). G. W. Bryant, Phys. Rev. Lett. , 1140 (1987). W. Kohn, Phys. Rev. (1961). J. F. Dobson, Phys. Rev. Lett. , 2244 (1994). C. Yannouleas and U. Landman, Phys. Rev. Lett. , 1726(2000). J.-L. Zhu, J.-Z. Yu, Z.-Q. Li, and Y. Kawazoe, J. Phys.:Condens. Matter , 7857 (1996). A. Puente, L. Serra, and R. G. Nazmitdinov, Phys. Rev.B , 125315 (2004).14