quantum transport open nanostructures › journals › vlsi › 1998 › 024813.pdfsimplicity, hard...

VLSI DESIGN1998, Vol. 8, Nos. (1-4), pp. 179-184Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1998 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach SciencePublishers imprint.

Printed in India.

Quantum Transport in Open NanostructuresI. V. ZOZOULENKO* and K.-F. BERGGREN

Department of Physics and Measurement Technology, Linkfping University, S-58183 Linkfping, Sweden

Electron transport was studied in an open square quantum dot with a dimension typicalfor current experiments. A numerical analysis of the probability density distributioninside the dot was performed which enabled us to unambiguously map the resonant stateswhich dominate the conductance of the structure. It was shown that, despite of thepresence of dot openings, transport through the dot is effectively mediated by just a few(or even a single) eigenstates of the corresponding closed structure. In a single-moderegime in the leads, the broadening of the resonant levels is typically smaller than themean energy level spacing, A. On the contrary, in the many-mode regime this broadeningtypically exceeds A and has an irregular, essentially non-Lorentzian, character. It wasdemonstrated that in the latter case eigenlevel spacing statistics of the correspondingclosed system are not relevant to the averaged transport properties of the dot. Thisconclusion seems to have a number of experimental as well as numerical verifications.

Keywords: Quantum dots, resonant states, conductance fluctuations

In nanoscaled semiconductor quantum dots,electron motion is confined in all spatial dimen-sions and the lateral shape of the dot can becontrolled by an applied gate voltage [1-6]. Inhigh quality samples at low temperatures electrontransport is ballistic, i.e., large-angle elastic scat-tering events occur only at the boundaries of thestructure and the phase coherence length wellexceeds the dimension of the device. During recentyears a great deal of effort has been focused on thetransport properties of ballistic microstructures. Inparticular, both the statistical properties of theconductance fluctuations [1-4, 7-10] as well asgeometry-specific, non-averaged features of the

magnetoresistance of quantum dots [1, 4, 5, 9, 11]have been extensively studied.

Transport properties of open microstructuresare often analyzed on the basis of the knowncharacteristics of the spectrum of the correspond-ing isolated system. However, when the dotbecomes open, eigenlevels interact and acquire afinite broadening due to the possibility forelectrons to escape from the dot via lead openings.With several propagating modes in the leads, thisbroadening might well exceed the mean energylevel separation, A, resulting in an overlap of avast number of resonances. Also, the presence ofdot openings may cause a significant distortion of

* Corresponding author.

179

180 I.V. ZOZOULENKO AND K.-F. BERGGREN

corresponding eigenstates. Under these circum-stances it is not a priori evident whether adiscussion of transport through an open dot onthe basis of the properties of the Hamiltonian ofthe closed structure is still meaningful. To the bestof our knowledge, up to now no direct theoreticalcalculations on the actual broadening of theresonant levels for the open dots in the transmis-sive regime are available.

Besides, in the current literature there exists anumber of conflicting reports on the effects ofleads on the character of electron dynamics inopen systems (chaotic vs regular). In particular,[12] shows that the statistics of the spectra for opendots are exactly the same as those of thecorresponding closed system. At the same time,results [9, 10, 13] suggest that the leads attached tothe dot may change the level statistics, so thattransition to chaos can occur in a nominallyregular system. On the contrary, Wang et al. [8]conclude that the openness of the dot makeschaotic scattering non-essential.

In this paper, on the basis of our direct mappingof resonant states performed for an open squaredot, we hope to contribute to the clarification ofsome fundamental issues in this context.The system under investigation is a relatively

large square dot with the side L lm which istypical for current experiments. It is connected toreservoirs by quantum point contact (QPC)-likeopenings (leads), see Figures 1- 3. For the sake ofsimplicity, hard wall confinement and a fiatpotential profile inside the dot are assumed whichseems to be a good approximation for large dots[11]. We disregard effects of the soft impuritypotential due to remote donors as well as inelasticscattering events.Conductance through the dot in perpendicular

magnetic field B at finite temperature is calculatedwithin the Landauer-Biittiker formalism [14],where a transmission probabilities and wavefunctions were computed by making use of therecursive Green function technique [15]. Analyzingthe probability density distribution inside the dotwe are in a position to identify resonant energy

02.41 2.43

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," (15,13

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(16,1,2),

2.45 2.4 2.5n, (1015 m-2)

FIGURE Lower left: The conductance of the square dot(schematically depicted in the upper panel) in the tunnelingregime as a function of the sheet electron density EFm /TrhTemperature T=0. Shadow regions in the leads representtunneling barriers with the height exceeding the Fermi energy.Lower right: Eigenenergy levels of the isolated dot. Height ofthe peaks represents the degree of degeneracy (1 or 2). Upperpanel shows the calculated probability density distributionItYmnl inside the dot for one of the tunneling peaks (left) andthe corresponding numerical results for the coefficients ICmnlcalculated on the basis of Eq. (1) (right). Dashed lines indicatethe circle with the radius kFL/Tr. A similar analysis has beendone for the rest of the peaks and the correspondence betweeneigenstates of the isolated square and resonant levels of the dotis indicated by the arrows. Quantum numbers of the resonantstates, (m, n), are shown in the parenthesis. In the case underconsideration the side of the dot was chosen to be L 0.5 tm.

states which effectively mediate transport throughthe dot at a given EF. To do this, we numericallyexpand the solution of the scattering problem inthe open dot, (x, y; E), in the set of eigenstates ofthe closed dot, .3mn - sin 7rmx sin ny (with eigen-energies Emn h2/2m*(k2m + kn); km -E-’Trm kn -)2 (E)sinTrmx 7my(x,y;E) -ZZ Cmn -E-sin-z-. (1)m nCoefficients Cmn represent the contributions of theeigenstates m, n in the total wave function.

QUANTUM TRANSPORT IN NANOSTRUCTURES 181

*.,

(a)5

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I 1006

FIGURE 2 The conductance of the square dot (schematicallydepicted in the inset) as a function of the sheet electron densityns EFm*/zrh2. The side of the dot L pm; temperatureT= 0. The lead openings support one propagating mode. Insetsshow calculated [m,[ inside the dot for two representativevalues of ns and the corresponding numerical results for thecoefficients ICml calculated on the basis of Eq. (1). Dashedlines indicate the circle with the radius r=kFL/zr. (b)Dependence of the coefficients [Cmn[ identifying the dominantresonant states with the quantum numbers (m, n) on the sheetelectron density ns. A horizontal bar indicates the mean energylevel spacing A. (c) Dwell time of the dot.

In what follows we focus on the three differenttransport regimes, namely the tunneling, thesingle-mode, and the many-mode regimes.

Tunneling regime Figure l(a) shows conductanceof the square dot in a regime when the dot isweakly coupled to the leads. Each conductancepeak corresponds to an excitation of one singleresonant energy level which effectively mediatestransport at the given Fermi energy. Near itsmaximum, each peak is characterized by the

// / \ (31,20)

/ I \/’,!,’, (24,28)

/ .,:=.L.................1

(28,24) (d).......... /t !! (14,34).....’,/lt (7,36)

2.004 2.006 2.008 2.010 2.012 2.014 2.016 2.018n, (1015 m-2)

FIGURE 3 The conductance of the square dot (see inset) in amany-mode regime in the lead openings, N 5. The side of thedot L Ixm; temperature T=0. (b) Coefficients ICmn[identifying the dominant resonant states with the quantumnumbers (m, n). A horizontal bar indicates the mean energylevel spacing A. (c), (d) The same as (b) but in a refined scale.(The contribution from the three dominant states indicated in(b) is not shown). (e) Dwell time of the dot.

Lorentzian shape, in accordance to the Breit-Wigner formalism. The positions of the peaks areshifted with respect to the corresponding eigen-energy levels of the isolated square (cf. Figs. l(a)and (b)).Single-mode regime Figure 2(a) shows the con-ductance of the square dot where lead openings areadjusted to support one propagating mode. Herethe pattern of the probability density distribution

Imnl2 exhibits a complicated structure, whereeigenstates of the isolated square are not easilyrecognized. Thus, a numerical analysis on the basisof Eq. (1), in contrast to the tunneling case, isessential. Calculating the expansion coefficients


Cmn we find that, at the given kF, only thosecoefficients associated with the circle in the k-spacewith the radius R ke ,. v/2m*emn/l --Trv/m2 -+- n2/L are distinct from zero, see Figure 2(a). (Notethat non-vanishing contributions from other coef-ficients would indicate that eigenstates of theisolated dot are essentially distorted by the leadopenings such that a discussion of the transport inopen structure in terms of eigenstates of theisolated dot does not make sense). Typically, wefind that only a few (and sometimes even a single)coefficients give a dominant contribution. Abroadening of the resonant levels due to the effectof the dot openings in the k-space is less than thedistance between neighboring eigenstates whosequantum numbers differ by one, Ak Ikn- kn+ll

Ikm km+/-ll 7r/L. Therefore, we conclude thatdespite the presence of dot openings, transportthrough the structure is still effectively mediatedby afew eigenstates of the corresponding closed dotwith the eigenenergies lying in close proximity to theFermi energy, emn EF.

Calculating the coefficients Cmn as a function ofthe Fermi energy, we extract information aboutthe lead-induced broadening of the energy levels ofthe dot. A contribution of the dominant resonantenergy states is shown in Figure 2(b). The meanenergy level spacing, A 27rh2/m*L2 70 mK, isindicated by a horizontal bar. In contrast to thetunneling regime, the lineshape of ICm, n(E)l2 can benon-Lorentzian. Moreover, different states arecharacterized by different broadenings and theymay overlap with each other. However, a broad-ening (half-width) of the resonant energy levels istypically less than A. Therefore, transport mea-surements at very low temperatures in a single-moderegime in the leads may probe a single resonantenergy level of the dot.A comparison between Figures 2(a) and (b)

shows that features in the conductance of the dotare related to excitations of the particular eigen-states of the square. However, this correspondenceis rather complicated: different eigenstates can beresponsible for opposite features in the dotconductance (dips and peaks).

Many-mode regime As the lead openings becomewider, a number of the resonant states excited inthe dot increases. Nevertheless, like in the single-mode regime, at the given Fermi energy, a nonvanishing contribution comes only from the coeffi-cients which lie in the closest proximity to the circlewith the radius kF in the k-space. Therefore, even ina many-mode regime, transport through an openstructure is still effectively mediated by eigenstatesof the corresponding closed dot. A broadening ofthe resonant energy levels increases with anincrease of the lead openings. In Figure 3 wavefunction patterns are analyzed on the basis of Eq.(1) and contributions from dominant states areshown in a representative interval of the Fermienergy. Typically, several states dominate trans-port at a given EF. A broadening of the energylevels has, as a rule, a complicated essentially non-Lorentzian character with half-width being differ-ent for different states. In contrast to the singlemode regime, a half-width of the resonant energylevels is typically larger than the mean energy levelspacing, A. Comparing the conductance of thedot, Figure 3; and the dependence t?mn Cmn(EF),one can trace a certain correspondence betweenthe two. However, because many eigenstatestypically contribute to the conductance at a givenEF, a detailed explanation of the features of thedot conductance is not possible.

In what follows we critically examine severalapproaches [9, 12] to the analysis of the statisticsof the spectra of open dots. A statistical analysis ofthe distribution of the energies at resonances ofconduction fluctuations in chaotic stadium andregular circular billiards has been performed byIshio [9]. In both billiards the statistics followsWigner-type distribution which was taken as anindication of the transition to chaos. (The Wigner-type statistics of the nearest energy level spacingdistribution is characteristic of the classicallychaotic closed billiards). With regard to thisanalysis, the question immediately arises "doesthe resonance energy statistics of the conductancefluctuations reproduce the corresponding statisticsof the isolated system?" We have shown above

QUANTUM TRANSPORT IN NANOSTRUCTURES 183

that even in a single mode regime, resonantenergies in the conductance fluctuations onlyoccasionally correspond to resonant eigenstatesof the isolated dot. Instead, in most cases resonantenergies are related to those energies when morethan one state is simultaneously excited in the dotsuch that their mutual interference leads to theresonance behavior of the transmission coefficient.Therefore, in our opinion, Wigner-type statistics ofthe spacing of the conductance fluctuation reson-ances cannot be taken as an indication of thetransition to chaos in a nominally regular but opensystem.Wang et al. [12] analyzed statistics of the open

system on the basis of the calculated electron dwelltime, - fdslmnl2, which identifies the time anelectron spends inside the dot; in the abovedefinition an integration is performed within thedot area. Statistics of the spectra were found to beexactly the same as that of the correspondingclosed system. Again, a similar question arises"does the dwell time maxima (which is integratedcharacteristic) unambiguously identify individualresonant states of the open dot?".

Figures 2(c) and 3(c) show the calculated dwelltime in the square dot in the single- and many-mode regimes respectively. In a single-moderegime all maxima in the dwell time do correspondto the resonant eigenstates, (cf. Figs. 2(b) and (c)).However, dwell time does not identify resonantstates unambiguously because some eigenstates areoverlooked by this analysis. Nevertheless, since thenumber of missing states is usually small ouranalysis, as far as a single mode regime isconcerned, tends to support the conclusion [12]that the statistics of the dwell time spectra for opendots are the same as that of the correspondingclosed systems.

In a many-mode regime, our analysis stronglysuggest that the resonant level spacing statisticsbecome ill-defined. This is not only because thebroadening of resonant states typically exceedsmean level spacing (for Lorentzian broadening onecan still define statistics of the spacings between

peak maxima). This is due to the fact that thebroadening itself in many cases is essentially non-Lorentzian, see Figure 3(c). For most of theresonant states the concept of statistics of thespectra does not make any sense, since it is notpossible to introduce any reasonable definition ofthe spacing between resonances. Therefore, weconclude thatfor the quantum dots strongly coupledto the leads with several modes available in the leadopenings, eigenlevel spacing statistics of the corre-sponding closed system are not relevant to theaveraged transport properties of the structure.

This conclusion seems to have a number ofexperimental as well as numerical verifications.The difference between statistical properties of theconductance oscillations in a chaotic (stadium)and a regular (circular) dots has been studied byMarcus et al. [1]. Corresponding averaged auto-correlation functions are almost identical over thetwo decades of decay, although they exhibit quan-titative distinctions in the tail. The data, from thesimilar studies of Berry et al. [2] for chaotic (circu-lar with a bar) and regular (circular) dots, does notshow any significant discrepancy over the fourorders of magnitude in power. Numerical studiesof autocorrelations functions for a chaotic (sta-dium) and a regular (circular) dots [8] in a many-mode regime show the similar behavior, which is inaccordance with our arguments.To conclude, despite of the presence of dot

openings, transport through the open dot iseffectively mediated by just a few eigenstates ofthe corresponding closed structure. In a single-mode regime in the leads the broadening of theresonant levels is typically smaller than the meanenergy level spacing, A. On the contrary, in themany-mode regime the broadening exceeds A andhas essentially a non-Lorentzian character.

Acknowledgement

I.V.Z. acknowledges a grant from the RoyalSwedish Academy of Sciences.


References[1] Marcus, C. M., Rimberg, A. J., Westervelt, R. M.,

Hopkins, P. F. and Gossard, A. C., "ConductanceFluctuations and Chaotic Scattering in Ballistic Micro-structures", Physical Review Letters, July 1992.

[2] Berry, M. J., Katine, J. A., Westervelt, R. M. andGossard, A. C., "Influence of Shape on Electron Trans-port in Ballistic Quantum Dots", Physical Review B,December 1994.

[3] Chang, A. M., Baranger, H. U., Pfeiffer, L. N. and West,K. W., Weak Localization in Chaotic versus NonchaoticCavities: A Striking Difference in the Line Shape",Physical Review Letters, October 1994.

[4] Persson, M., Pettersson, J., von Sydow, B., Lindelof, P. E.,Kristensen, A. and Berggren, K.-F., "ConductanceOscillations Related to the Eigenenergy Spectrum of aQuantum Dot in Weak Magnetic Field", Physical ReviewB, September 1995.

[5] Bird, J. P., Ferry, D. K., Akis, R., Ochiai, Y., Ishibashi,K., Aoyagi, Y. and Sugano, T., "Periodic ConductanceOscillations and Stable Orbits in Mesoscopic Semicon-ductor Billiards", Europhysics Letters, September 1996.

[6] Zozoulenko, I. V., Schuster, R., Berggren, K.-F. andEnsslin, K., "Ballistic Electrons in an Open SquareGeometry: Selective Probing of Resonant-Energy Levels",Physical Review B, April 1997.

[7] Jalabert, R. A., Baranger, H. U. and Stone, A. D.,"Conductance Fluctuations in the Ballistic Regime: AProbe of Quantum Chaos?", Physical Review Letters,November 1990.

[8] Wang, Y., Wang, J., Gou, H. and Roland, C., "TunnelingThrough Quantum-Dot Systems: A Study of the Magne-to-Conductance Fluctuations", J. Physics." CondensedMatter, August 1994.

[9] Ishio, H., "Quantum Transport and integrability in OpenBilliards", J. Statistical Phyics, January 1996.

[10] Albeverio, S., Haake, F., Kurasov, P., Kug, M. and ,eba,P., "S-matrix, resonances and wave functions for transportthrough billiard with leads", J. Math. Phys., October 1996.

[11] Reiman, S. M., Persson, M., Lindelof, P. E. and Brack,M., "Shell Structure of a Circular Quantum Dot in aWeak Magnetic Field", Z. Phys. B, October 1996.

[12] Wang, Y., Zhu, N., Wang, J. and Guo, H., "ResonanceStates of Open Quantum Dots", Physical Review B, June1996.

[13] Berggren, K.-F. and Ji, Z.-L., "Quantum Chaos in Nano-sized Billiards in Layered Two-Dimensional Semiconduc-tor Structures", CHAOS, November 1996.

[14] Datta, S., Electronic Transport in Mesoscopic Systems(Cambridge University Press, Cambridge, 1995).

[15] Zozoulenko, I. V., Maao, F. A. and Hauge, E. H.,"Coherent Magnetotransport in Confined Arrays ofAntidots", Physical Review B, March 1996.

Authors’ Biographies

Igor Zozoulenko is a senior scientist at theDepartment of Physics at Link6ping University.His current research deals with electronic aspectsof low-dimensional semiconductor structures likequantum dots, wells and related systems. Duringhis career he has awarded fellowships from RoyalNorwegian Council for Scientific and IndustrialResearch and Royal Swedish Academy of Scien-ces, and has held positions at Institute for Theore-tical Physics, Kiev.

Karl-Fredrik Berggren is Professor of Theore-tical Physics at the Department of Physics atLink6ping University and head of the Laboratoryof Theoretical Physics. In general terms hisresearch is focused on condensed matter physics.He is a member of the Swedish Council for HighPerformance Computing, a member of the Com-putational Physics Board of the European Phy-sical Society, and a member of the CondensedMatter Theory Committee at the Nordic Insti-tute of Theoretical Physics (NORDITA), Copen-hagen.

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