The measurement of phase in coherent electron systems—that is, ‘mesoscopic’ systems such as quantum dots—can yield information about fundamental transport properties that is not readily apparent from conductance measurements. Phase measurements on relatively large quantum dots1 recently revealed that the phase evolution for electrons traversing the dots exhibits a ‘universal’ behaviour, independent of dot size, shape, and electron occupancy2,3. Specifically, for quantum dots in the Coulomb blockade regime, the transmission phase increases monotonically by π throughout each conductance peak; in the conductance valleys, the phase returns sharply to its starting value. The expected mesoscopic features in the phase evolution—related to the dot's shape, spin degeneracy or to exchange effects—have not been observed, and there is at present no satisfactory explanation for the observed universality in phase behaviour4. Here we report the results of phase measurements on a series of small quantum dots, having occupancies of between only 1–20 electrons, where the phase behaviour for electron transmission should in principle be easier to interpret. In contrast to the universal behaviour observed thus far only in the larger dots, we see clear mesoscopic features in the phase measurements when the dot occupancy is less than ∼10 electrons. As the occupancy increases, the manner of phase evolution changes and universal behaviour is recovered for some 14 electrons or more. The identification of a transition from the expected mesoscopic behaviour to universal phase evolution should help to direct and constrain theoretical models for the latter.
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Van Houten, H., Beenakker, C. W. J. & Staring, A. A. W. in Single Charge Tunnelling - Coulomb Blockade Phenomena in Nanostructures (eds Grabert, H. & Devoret, M. H.) 167–216 (Plenum, New York, 1992)
Schuster, R. et al. Phase measurement in a quantum dot via a double slit interference experiment. Nature 385, 417–420 (1997)
Ji, Y., Heiblum, M., Sprinzak, D., Mahalu, D. & Shtrikman, H. Phase evolution in a Kondo-correlated system. Science 290, 779–783 (2000)
Hackenbroich, G. Phase coherent transmission through interacting mesoscopic systems. Phys. Rep. 343, 463–538 (2001)
Yacoby, A., Schuster, R. & Heiblum, M. Phase rigidity and h/2e oscillations in a single-ring Aharonov-Bohm experiment. Phys. Rev. B 53, 9583–9586 (1996)
Aronov, A. G. & Sharvin, Yu. V. Magnetic flux effects in disordered conductors. Rev. Mod. Phys. 59, 755–779 (1987)
Aharonov, Y. & Bohm, D. Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485–491 (1959)
Field, M. et al. Measurements of Coulomb blockade with a noninvasive voltage probe. Phys. Rev. Lett. 70, 1311–1314 (1993)
Sprinzak, D., Ji, Y., Heiblum, M., Mahalu, D. & Shtrikman, H. Charge distribution in a Kondo correlated quantum dot. Phys. Rev. Lett. 88, 176805 (2002)
Breit, G. & Wigner, E. Capture of slow neutrons. Phys. Rev. 49, 519–531 (1936)
Hackenbroich, G. & Weidenmüller, H. A. Transmission through a quantum dot in an Aharonov-Bohm ring. Phys. Rev. Lett. 76, 110–113 (1996)
Weidenmüller, H. A. Transmission phase of an isolated CB resonance. Phys. Rev. B 65, 245322 (2002)
Aharony, A., Entin-Wohlman, O., Halperin, B. I. & Imry, Y. Phase measurement in the mesoscopic AB interferometer. Phys. Rev. B 66, 115311 (2002)
Levy Yeyati, A. & Büttiker, M. Scattering phases in quantum dots: An analysis based on lattice models. Phys. Rev. B 62, 7307–7315 (2000)
Silvestrov, P. G. & Imry, Y. Towards an explanation of the mesoscopic double-slit experiment: A new model for charging of a quantum dot. Phys. Rev. Lett. 85, 2565–2568 (2000)
Silvestrov, P. G. & Imry, Y. Spin effects and transport in QD with overlapping resonances. Phys. Rev. B 65, 035309 (2002)
Silva, A., Oreg, Y. & Gefen, Y. The signs of QD lead matrix elements: The effect on transport vs. spectral properties. Phys. Rev. B 66, 195316 (2002)
Oreg, Y. & Gefen, Y. Electron scattering through a quantum dot: A phase lapse mechanism. Phys. Rev. B 55, 13726–13729 (1997)
Lee, H. W. Generic transmission zeros and in-phase resonances in time-reversal symmetric single channel transport. Phys. Rev. Lett. 82, 2358–2361 (1999)
Entin-Wohlman, O., Aharony, A., Imry, Y. & Levinson, Y. The Fano effect in AB interferometers. J. Low Temp. Phys. 126, 1251–1273 (2002)
Büttiker, M. 4-Terminal phase-coherent conductance. Phys. Rev. Lett. 57, 1761–1764 (1986)
Kouwenhoven, L. P., Oosterkamp, T. H., Tarucha, S., Austing, D. G. & Honda, H. Coulomb oscillations in few-electron quantum dot. Physica B 249–251, 191–196 (1998)
Zumbuhl, D. M., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Voltage-tunable singlet-triplet transition in lateral quantum dots. Phys. Rev. B 66, 035320 (2002)
Kyriakidis, J., Pioro-Ladriere, M., Ciorga, M., Sachrajda, A. S. & Hawrylak, P. Cotunneling spectroscopy in few electron QDs. Phys. Rev. Lett. 93, 256801 (2004)
Ashcroft, N. W. & Mermin, N. D. Solid State Physics Ch. 32 (Holt, Rinehart and Winston, Orlando, 1976)
Oreg, Y., Brouwer, P. W., Waintal, X. & Halperin, B. I. in Nano-Physics and Bio-Electronics (eds Chakraborty, T., Peeters, F. & Sivan, U.) Ch. 5 (Elsevier, Amsterdam, 2002)
The work was partly supported by the Minerva foundation, the German Israeli Project Cooperation (DIP), the German Israeli Foundation (GIF), and the QUACS network. We are grateful to Y. Levinson, Y. Oreg and A. Yacoby for discussions. We thank M. Popadic for collaboration in the experiments.
Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare no competing financial interests.
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Avinun-Kalish, M., Heiblum, M., Zarchin, O. et al. Crossover from ‘mesoscopic’ to ‘universal’ phase for electron transmission in quantum dots. Nature 436, 529–533 (2005) doi:10.1038/nature03899
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