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Physical Review B
Volumn 46, Issue 3, 1992, Pages 1439-1450
Theory of two-dimensional quantum-dot arrays in magnetic fields: Electronic structure and lateral quantum transport
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Indexed keywords

EID: 0000845726     PISSN: 01631829     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevB.46.1439     Document Type: Article
Times cited : (37)
References (77)
  • 7
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    • 33, 387 (1975) [6, 299 (1957)].
  • 42
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    • Some preliminary results were reported at EP2DS-9, Nara, Japan, 1991
    • (1992) Surf. Sci. , vol.263 , pp. 330
    • Kirczenow, G.1
  • 47
    • 84931517576 scopus 로고
    • in Proceedings of the European Physical Society Meeting, 1991, edited by J. L. Beeby, P. A. Maksym, and J. M. McCoy
    • (1991) Phys. Scr. , vol.39 T , pp. 288
  • 49
    • 0026820320 scopus 로고
    • R. P. Taylor, A. S. Sachrajda, J. A. Adams, P. Zawadski, P. T. Coleridge, and M. Davies, Proc. EP2DS-9 Nara, Japan, 1991
    • (1992) Surf. Sci. , vol.263 , pp. 247
  • 51
    • 84931504529 scopus 로고    scopus 로고
    • For a review of the Coulomb blockade, see D. V. Averin and K. K. Likharev, in Mescoscopic Phenomena in Solids, edited by B. L. Altshuler, P. A. Lee, and R. A. Webb (Elsevier, Amsterdam, 1991). It should be noted that in the recent transport experiments of Kouwenhoven et al. (Ref. 44) on one-dimensional quantum-dot chains where the conductance app e2/h implies relatively strong transmission (similar to that considered in the present paper), Coulomb blockade effects did not appear to be important.
  • 53
    • 84931504526 scopus 로고    scopus 로고
    • This follows from the fact that in the gauge A vec prime = (-By prime /2, Bx prime /2,0), where x prime =x-xm and y prime = y- ym, the Hamiltonian is H prime = ( p vec - qeA vec prime )2/2mstar+V ( | r vec prime | ), where r vec prime = r vec - r vecm. H prime has eigenfunctions of the form PSI primenlm= eil φ Rnl( | r vec prime |) (Ref. 46). A vec can be obtained from A vec prime by the gauge transformation A vec = A vec prime + del chi, where χ = B(yxm- xym)/2. Hence, PSInlm= eiqeχ/hbar PSI primenlm.
  • 56
    • 84931504527 scopus 로고    scopus 로고
    • Measurements of the transmission probabilities of individual constrictions coupling a single quantum dot to different leads have been reported in Refs. 39 – 41.
  • 58
    • 84931504524 scopus 로고    scopus 로고
    • For a treatment of multiple radial modes in a single dot, see Ref. 42.
  • 59
    • 84931504525 scopus 로고    scopus 로고
    • In general, for given n, is possible for eigenmodes PSInlm having differing values of l to have the same energy eigenvalue. However, the different l values will correspond to different radial functions Rnl in (2). Provided that the magnetic field is not too low, only the mode having the largest value of | l | at the given energy is of interest, since is the only one that has appreciable amplitude near the periphery of the dot and thus an appreciable probability of transmission into other dots. The presence of the other degenerate modes will be ignored in this paper.
  • 60
    • 84931504522 scopus 로고    scopus 로고
    • For isolated dots, the one-electron eigenstates are labeled by the quantum numbers n and l in Eq. (2), where n and l are integers. Thus, if only one radial mode n=0 is populated by electrons, and l ranges from a negative value l- to a positive l+ for the populated one-electron orbitals at T=0 K, then 2(l+- l-+1) is the number of electrons in the dot in the limit of zero transmission between dots (and in the absence of spin splittings). At the magnetic fields considered here, for quantum dots containing many electrons, |l-| >> | l+|, so that we can consider l- as a filling parameter for zero transmission. However, here we are interested primarily in the case of moderate and strong transmission. Even for strong transmission, in the case where only the n=0 mode is populated, is reasonable to assume that the geometrical size of each constriction between dots is much smaller than the size of a dot. (This is the case in reality, and is necessary for the description of the system as an array of quantum dots to make sense.) It then follows that the expectation value of the total electron charge in a quantum dot is roughly independent of the transmission through the constrictions, provided that the Fermi energy, magnetic field, and confinement potential away from the constrictions are all kept fixed. Within the present model, as explained in the Appendix, lF, the value of l at the Fermi energy (lF= l-), is given by the solution of EF= EnlF for lF, and the functional form of Enl does not involve the transmission at the constrictions. Thus, if the Fermi energy is held fixed, and the transmission through the constrictions is increased from zero to strong transmission, lF will not change and the number of electrons per dot will not change by much. Thus, at integer lF, -lF will still be an approximate filling parameter for the quantum dots for strong transmission. For strong transmission, is plausible that the expectation value of the charge per dot will vary smoothly, although not necessarily linearly, with the Fermi energy as lF sweeps between integer values, so that -lF will be a useful (although possibly nonlinear) filling parameter also for noninteger values of lF.
  • 61
    • 84931504523 scopus 로고    scopus 로고
    • Assume the quantum-dot diameter and the nearest-neighbor center-to-center spacing of the dots both to be d. For an N times N array, assuming perfect transmission through the constrictions, the total phase shift PHI accumulated in one edge-state orbit of the whole array as in Fig. 3(a) is then Φ = Bqed2(N-1)2/ hbar - 2 π l(2N-1), where the first and second terms are obtained from the first and second terms on the RHS of (4), respectively. The phase shifts of π /2 that occur each time the orbit passes through a constriction [due to the second term on the RHS of Eq. (9)] contribute a total phase shift that is an integer multiple of 2 pi and is ignored. Setting Φ = 2 n pi, and applying the definitions of beta and lambda given in the first paragraph of Sec. III yields Eq. (10). Equation (11) is obtained similarly for an orbit of a single antidot.
  • 62
    • 84931504532 scopus 로고    scopus 로고
    • These spectra are somewhat reminiscent of those in Ref. 20. However, the model of Ref. 20 was an array of pointlike tight-binding ``atoms'' with no internal structure, and was meant to represent a single quantum dot, not an array of finite quantum dots. Because of this there are significant differences between the topologies of the present spectra and those of Ref. 20, and some structures, such as Z in Fig. 4, have no analog in Ref. 20. The effective-mass theory used to interpret the results of Ref. 20 is not applicable here, since the present model is not based on a tight-binding Hamiltonian and the underlying physics is different.
  • 63
    • 84931504538 scopus 로고    scopus 로고
    • This is different from the results in Ref. 20, where the transmission peaks do not match the eigenstates of the isolated system in detail even in the edge-state region, having a somewhat different spacing in magnetic field.
  • 64
    • 84931504539 scopus 로고    scopus 로고
    • Some differences should be expected because coupling to the leads disturbs the array. Also, not all quantum transport anomalies are necessarily associated with resonant states; some can be due to other interference effects as shown in Ref. 42. See also Ref. 38.
  • 65
    • 84931504540 scopus 로고    scopus 로고
    • Several theoretical studies have previously shown that Hall-resistance minima occur when the Fermi level crosses resonant states in quantum-wire junctions.
  • 72
    • 84931504541 scopus 로고    scopus 로고
    • It is now evident that, in more complex systems, quantum resonances can result in Hall-resistance maxima as well.

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