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1
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84927826393
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For reviews see Nanostructure Physics and Fabrication, edited by M. A. Reed and W. P. Kirk (Academic, New York, 1989) and C. W. J. Beenakker and H. van Houten in Solid State Physics: Advances in Research and Applications, edited by H. Ehrenreich and D. Turnbull (Academic, New York, 1991), Vol. 44, pp. 1 228.
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13
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0000618786
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C., and G.
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(1988)
Phys. Rev. B
, vol.38
, pp. 8518
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Ford, J.B.1
Thornton, T.J.2
Newbury, R.3
Pepper, M.4
Ahmed, H.5
Peacock, D.C.6
Ritchie, D.A.7
Frost, J.E.F.8
Jones, A.C.9
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23
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84927826392
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In the calculation with weakly coupled voltage probes, the hopping matrix element connecting the voltage probes to the current channel is reduced by a factor of 100. In this situation, then, the current and voltage probes are not identical, so Eq. ( refeq:rhall) is not valid and one must go back to Eq. ( refeq:rgen) to derive an expression for RH appropriate for this case.
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28
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0000730614
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(1988)
Solid State Commun.
, vol.68
, pp. 1051
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Takagaki, Y.1
Gamo, K.2
Namba, S.3
Ishida, S.4
Takaoka, S.5
Murase, K.6
Ishibashi, K.7
Aoyagi, Y.8
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38
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0001391109
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C. J.
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(1990)
Phys. Rev. B
, vol.41
, pp. 1274
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Molenkamp, L.W.1
Staring, A.A.M.2
Beenakker, C.W.J.3
Eppenga, R.4
Timmering, C.E.5
Williamson, J.G.6
Harmans, P.M.7
Foxon, C.T.8
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40
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0010281666
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edited by, M. A. Reed, W. P. Kirk, Academic, New York
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(1989)
Nanostructure Physics and Fabrication
, pp. 331
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Timp, G.1
Behringer, R.2
Sampere, S.3
Cunningham, J.E.4
Howard, R.E.5
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44
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84927826389
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While the main issues in this paper are the classical chaotic dynamics and the role of random scattering in the junction, the authors also address the nature of the bend resistance anomaly in the fully ballistic case. In this regard, Roukes and Alerhand argue that the peak in RB (B=0) near R/W approx 3 results from a combination of shadowing, the lack of direct paths between adjacent leads for R/W geq 2.4;
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45
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84927826388
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collimation, the fact that particles which have scattered a few times in the injecting lead contribute substantially to the forward transmission;
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46
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84927826387
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and scrambling, the increasing complexity of trajectories as R/W gets large. Shadowing suppresses the turning probability and hence enhances RB;
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47
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84927826386
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collimation enhances RB by the simple forward enhancement mechanism;
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48
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84927826385
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and scrambling acts to equalize the transmission into the different leads, decreasing RB. We point out that collimation may have other more subtle but equally important effects on the bend resistance in such structures, such as a suppression of the turning probability because the cone in momentum space of collimated injected particles does not overlap with the acceptance cone in momentum space of the collecting adjacent lead (Ref. onlineciteBvH89). Such an effect would become important when the widening of the leads becomes sufficiently adiabatic which also occurs at about R/W approx 3.
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57
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84927826382
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J. Sone and S. Ishizaka, in Nanostructures: Fabrication and Physics (Extended Abstracts), edited by S. D. Berger, H. G. Craighead, D. Kern, and T. P. Smith III (Materials Research Society, Pittsburgh, 1990), p. 91.
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59
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84927826381
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L. Kouwenhoven (private communication); S. Washburn (private communication).
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69
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84927826379
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and references therein;
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75
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84927826378
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A. Szafer, Ph.D. thesis, Yale University, 1990; A. M. Kriman, A. Szafer, A. D. Stone, and D. K. Ferry (unpublished).
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83
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84927826377
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M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer Verlag, Berlin, 1990), pp. 184 190 and 283 287; in Chaos and Quantum Physics, edited by M. J. Giannoni, A. Voros, and J. Zinn Justin (Elsevier, London, 1990), and references therein.
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88
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84927826375
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The hard wall classical simulations involve typically 106 particles for the resistance traces and 107 particles for the trajectory length analysis. The soft wall classical simulations involve integrating a differential equation for each trajectory; we used half as many particles as in the hard wall structures.
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90
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84927826373
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See, e.g., Chaos in Classical and Quantum Mechanics (Ref. onlineciteGutz), pp. 149, 236, and 251.
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91
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84927826372
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The scaling used in a soft wall potential U(y) is as follows. First, we retain the relation between our resistance scale and the experimental scale, namely, that one multiplies our scaled value by h/e2 times the number of modes. Using the definition of Eq. ( refeq:C16) for the effective width of the wire Weff, the resistance is scaled by R0 = (h/e2) [π/kF(0)Weff] (times mvF/ hbar kF in the lattice case). Second, we require that the slope in the two dimensional case be 2/π. This is accomplished by using Weff in the magnetic field scale, B0 = mc vF(0) /eWeff.
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92
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For a nonzero magnetic field, the symmetry of the junction is essential in showing this result.
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93
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In order to reconcile our view that the ballistic anomalies are caused by short paths with the data of Roukes and co workers in Ref. 32, we suggest the following interpretation of their data. (1) Guiding of the particles around the corner at high magnetic field is less sensitive to disorder since does not depend on collimation. Only backscattering degrades guiding; hence is not surprising that guiding is less sensitive to disorder than quenching which depends on collimation and is sensitive to forward scattering. The greater sensitivity of quenching to disorder in no way implies that longer trajectories cause this effect than cause guiding. (2) The flared junction collimates only marginally. Thus RH is very sensitive to disorder and only shows quenching at the highest density when the mean free path is largest. RB decays as disorder increases, consistent with its ballistic origin. The enhancement of RB substantially above the square corner value indicates that some collimation is present, and the fact that RB is substantial even when quenching has vanished is consistent with our results in Fig. refQ2. (3) Collimation is better established in the grossly flared junction because of the greater widening. Thus, a small region of inversion insensitive to density is present in RH. RB is sensitive to the density in this region which suggests that longer trajectories contribute to RB than to RH, as stated by Roukes and co workers and in agreement with our trajectory length analysis in Fig. refC6 (note that these trajectories are still short in terms of scrambling). (4) The structure in RH for the straight junction is presumably caused by a completely different mechanism. A possibly related effect occurs in the quantum calculation of the concave cavity (Fig. refS4) in which interference effects in the cavity are the cause. A small potential well in the junction region, enhanced perhaps by impurity effects, may be sufficient to cause this type of effect.
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95
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84927826369
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and references therein.
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96
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84927826368
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The parameter values for the quantum calculations are as follows. (1) For the linearly widened and four disk structures, W=18 sites for which E1 /V =0.03. The results for up to 10 subbands mentioned in the text use W=24 sites. (2) For the concave cavity, W=24 for which E1 /V= 0.017. (3) For the soft wall structures, W=39 in the junction at the maximum energy which corresponds to the threshold for the sixth or seventh mode. Writing U=an xn yn /W2n, we use a2 =29, a4 =430, and a6 =7700 and match onto a yn cross section at x=W. For these values, E1 (n=2) = 0.14, E1 (n=4) = 0.062, and E1 (n=6) = 0.046. The widening of the wires is about a factor of 2 for the maximum energy used. Because the calculations are performed on a lattice, the dispersion relation is neither parabolic nor isotropic. The nonparabolicity is easily accounted for by distinguishing between hbar kF and mvF in scaling the data. In contrast, distortion of the Fermi circle leads to preferential propagation in certain directions and must be avoided; for instance, a ballistic bend resistance of the wrong sign can occur. Our calculations are done for kx a < 1.4 and EF/V < -2.4 for which the maximum deviation of kF (vF) from the mean is 3
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97
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84927826367
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There is some energy dependence to the value of f: f decreases as the energy increases. This presumably occurs because faster particles see a less adiabatic potential, as argued in the Introduction. The range of f for our structures is approximately pm 0.05.
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100
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84927826366
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While the calculation in Appendix B is carried out for zero magnetic field, can easily be generalized to include a magnetic field in the scattering region combined with the B=0 injection conditions. Because of the B=0 injection conditions, the continuum mode limit is the same as in Appendix B. If one fixes B/B0 as the average over k, Eq. ( refeq:B9), is performed, the paths which contribute to the semiclassical sum are independent of k, and one finds that the reduced action plus a term related to the enclosed area (see Ref. 38) must be equal for two paths to interfere. Because the sum over paths is discrete, as before, this condition is satisfied only if the two paths are the same, neglecting terms of order 1/N from exact symmetries. Thus, for fixed B/B0, the simple classical expression is recovered for a continuum of modes and after an average over an infinite energy window.
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101
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The mean free path is calculated numerically for a strip with a narrow disordered region through the Born approximation as in Ref. 21. For the values given here, the strip width is 36 sites, which corresponds to 3W for the R/W=4 structure, 2W for the gradually widened structure, and twice the classical width at the highest energy for the soft wall structure. (Note that LD, the direct length, is approximately the same in these three structures.) The mean free path varies substantially with energy, by as much as a factor of 2; the values given result from averaging over energies for which 6 13 modes are contributing ([-3.72,-2.72] in units of the hopping matrix element). In narrower strips 18 sites wide, the averaged mean free path is nearly the same (within 5
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108
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84927826364
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The continuum mode approximation is good up to order 1/N. This can be easily seen from the semiclassical approximation to the real space form of the conductance (Refs. 9 and 53). This approach provides an alternative way to discuss the limiting procedures leading to the classical results.
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