The ability to make electrically conducting structures of ever smaller size by nanofabrication techniques (the playground of mesoscopic physics) has brought with it entry into a wonderful new range of unexpected quantum phenomena. Interpretation of these phenomena requires full recognition of the wave nature of electrons and requires keeping track of the phase coherence of the electron wave functions and/or the discreteness of electron energy levels in samples of interest. Happily, many of the phenomena can be observed through the use of very straightforward experimental probes—commonly the dc electrical conductivity or conductance, and the Hall effect. The phenomena are observed in samples with one or more dimensions comparable to either the electron wavelength (up to 40 nm for carriers at the Fermi energy in some semiconductors) or the inelastic scattering length of the carriers (as large as many microns in some systems at low temperatures). Ohm’s law is no longer a firm guide to current–voltage relationships, and the Drude–Sommerfeld picture of electrical conduction is superseded. Many of the interesting phenomena are seen in samples of either two-dimensional (i.e., a third dimension is of the order of or less than the electron wavelength) or one-dimensional nature (either a tight, short constriction in the conductor or a longer “quantum wire”). In certain one-dimensional structures, one may have ballistic transport between input and output connections, and the quantum character of the electron motion is fully displayed. Planck’s constant h appears in the characteristic quantum of electrical conductance, In two dimensions, the addition of a large magnetic field produces the remarkably deep and still somewhat mysterious Quantum Hall Effect, characterized by the quantum of resistance, Other examples of the observation of electron interference and diffraction phenomena within solid materials are briefly highlighted. This short tutorial treatment emphasizes observed phenomena rather than details of the theoretical structures used to interpret them.
REFERENCES
1.
K.
von Klitzing
, G.
Dorda
, and M.
Pepper
, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance
,” Phys. Rev. Lett.
45
, 494
–497
(1980
).2.
B. J.
van Wees
, H.
van Houten
, C. W. J.
Beenakker
, J. G.
Williamson
, L. P.
Kouwenhoven
, D.
van der Marel
, and C. T.
Foxon
, “Quantized conductance of point contacts in a two-dimensional electron gas
,” Phys. Rev. Lett.
60
, 848
–851
(1988
).3.
D. A.
Wharam
, T. J.
Thornton
, R.
Newbury
, M.
Pepper
, H.
Ahmed
, J. E. F.
Frost
, D. G.
Hasko
, D. C.
Peacock
, D. A.
Ritchie
, and G. A. C.
Jones
, “One-dimensional transport and the quantisation of ballistic resistance
,” J. Phys. C
21
, L209
–L214
(1988
).4.
C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1996), 7th ed.
5.
Most topics contained in this paper are covered in detail in one of two monographs—The Quantum Hall Effect, edited by Richard E. Prange and Steven M. Girvin (Springer-Verlag, New York, 1990), 2nd ed.;
M. Janßen, O. Viehweger, U. Fastenrath, and J. Hajdu, Introduction to the Theory of the Integer Quantum Hall Effect (VCH, Weinheim, 1994);
or in the extensive article listed as Ref. 6.
6.
C. W. J. Beenakker and H. van Houten, “Quantum Transport in Semiconductor Nanostructures,” in Solid State Physics: Advances in Research and Applications, Vol. 44, edited by Henry Ehrenreich and David Turnbull (Academic, San Diego, 1991).
7.
C. T.
Van Degrift
and M. E.
Cage
, “Resource Letter QHE-1: The integral and fractional Quantum Hall effects
,” Am. J. Phys.
58
, 109
–123
(1990
).8.
Richard A.
Webb
and Sean
Washburn
, “Quantum interference fluctuations in disordered metals
,” Phys. Today
41
(12
), 46
–55
(1988
).9.
Paul L.
McEuen
, “Artificial Atoms: New Boxes for Electrons
,” Science
278
, 1729
–1730
(1997
). This short article serves as an introduction to several more extensive papers in the same issue of Science (pp. 1784–1794) which focus on energy levels in quantum dot systems.10.
L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R. M. Westervelt, and N. W. Wingreen, “Electron Transport in Quantum Dots,” in Mesoscopic Electron Transport, edited by L. L. Sohn, L. P. Kouwenhoven, and G. Schon (Kluwer, Dordrecht, 1997), p. 105.
11.
For those readers who might be motivated to explore in depth the detailed dynamics of electron motion in the presence of electric and magnetic fields as pictured in the Drude–Sommerfeld model, an illuminating resource is the pair of interactive computer simulation programs entitled “DRUDE” and “SOMMERFELD” which are contained in a guidebook and CD by Robert H. Silsbee and Jörg Dräger, Simulations for Solid State Physics (Cambridge U.P., Cambridge, 1997). The fortuitous two-dimensional limitation of the computer screen turns out to favor examples and exercises which are played out in 2D and, thus, directly applicable to the systems of interest in this paper.
12.
For those readers who wish to study in detail the nature of electron transport in the quantum realm, a thorough treatment is given on pp. 1–80 in the treatise Quantum Transport in Semiconductor Nanostructures, by C. W. J. Beenakker and H. van Houten, listed in Ref. 6.
13.
See the article “Quantum Point Contacts,” by
Henk
van Houten
and Carlo
Beenakker
, Phys. Today
49
(7
), 22
–27
(1996
) for a treatment which parallels this section.14.
R. M.
Landauer
, “Spatial variation of currents and fields due to localized scatterers in metallic conduction
,” IBM J. Res. Dev.
1
, 223
–231
(1957
);15.
A good discussion of the use of this wave transmission/reflection model to treat electrical conductance is given by Yoseph Imry in an article entitled “Physics of Mesoscopic Systems,” in the review volume Directions in Condensed Matter Physics, edited by G. Grinstein and G. Mazenko (World Scientific, Singapore, 1986).
16.
Reference 6, pp. 111–112.
17.
A.
Yacoby
, H. L.
Stormer
, Ned S.
Wingreen
, L. N.
Pfeiffer
, K. W.
Baldwin
, and K. W.
West
, “Nonuniversal quantum fluctuations in quantum wires
,” Phys. Rev. Lett.
77
, 4612
–4615
(1996
).18.
W. A.
de Heer
, S.
Frank
, and D.
Ugarte
, “Fractional quantum conductance in gold nanowires
,” Z. Phys. B
104
, 468
–473
(1997
).19.
J. L.
Costa-Krämer
, N.
Garcı́a
, P.
Garcı́a-Mochales
, and P. A.
Serena
, “Nanowire formation in macroscopic metallic contacts: Quantum Mechanical conductance [by] tapping a table top
,” Surf. Sci.
342
, L1144
–L1149
(1995
).20.
Stefan
Frank
, Phillippe
Poncharal
, Z. L.
Wang
, and Walt A.
de Heer
, “Carbon Nanotube Quantum Resistors
,” Science
280
, 1744
–1746
(1998
).21.
E. A.
Montie
, E. C.
Cosman
, G. W.
’t Hooft
, M. B.
van der Mark
, and C. W. J.
Beenakker
, “Observation of the optical analogue of quantized conductance of a point contact
,” Nature (London)
350
, 594
–595
(1991
).22.
E.
Abrahams
, P. W.
Anderson
, D. C.
Licciardello
, and T. V.
Ramakrishnan
, “Scaling theory of localization: Absence of quantum diffusion in 2D
,” Phys. Rev. Lett.
42
, 673
–676
(1979
).23.
S. V.
Kravchenko
, Whitney E.
Mason
, G. E.
Bowker
, J. E.
Furneaux
, V. M.
Pudalov
, and M.
D’Iorio
, “Scaling of an anomalous metal–insulator transition in a 2D system in Si at
” Phys. Rev. B
51
, 7038
–7045
(1996
);S. V.
Kravchenko
, D.
Simonian
, M. P.
Sarachik
, Whitney E.
Mason
, and J. E.
Furneaux
, “Electric field scaling at a M–I transition in 2D
,” Phys. Rev. Lett.
77
, 4938
–4941
(1996
).A follow-up paper organizes some earlier data in complementary fashion:
Dragana
Popovic
, A. B.
Fowler
, and S.
Washburn
, “Metal–insulator transition in 2D: Effects of disorder and magnetic field
,” Phys. Rev. Lett.
79
, 1543
–1546
(1997
).24.
See, for example, J. R. Hook and H. E. Hall, Solid State Physics (Wiley, Chichester, 1991), 2nd ed., pp. 400–412.
Good treatments are given by R. E. Prange in The Quantum Hall Effect (Springer-Verlag, New York, 1990), 2nd ed., pp. 22–30 and in the early part of an article by
L. J.
Challis
, “Physics in less than three dimensions
,” Contemp. Phys.
33
, 111
–127
(1992
). Several topics treated in the present paper overlap the subject matter of the Challis article.25.
I call attention to a more extensive treatment of various elements of theoretical background than is given here, in an article by
J. P.
Eisenstein
, “The quantum Hall effect
,” Am. J. Phys.
61
, 179
–183
(1993
).An even more complete treatment of theoretical background for a number of aspects of the IQHE is given in the very nice review paper by
H.
Aoki
, “Quantised Hall Effect
,” Rep. Prog. Phys.
50
, 665
–730
(1987
).26.
See J. R. Hook and H. E. Hall in Ref. 24. In particular, Ref. 25 of their book, pp. 402–412.
27.
See J. R. Hook and H. E. Hall in Ref. 24. In particular, Ref. 25 of their book, p. 408.
28.
D. B.
McWhan
, A.
Menth
, J. P.
Remeika
, W. F.
Brinkman
, and T. M.
Rice
, “Metal–insulator transitions in pure and doped
” Phys. Rev. B
7
, 1920
–1931
(1973
).29.
N. F. Mott, Metal–Insulator Transitions (Taylor & Francis, London, 1990), 2nd ed., pp. 36–39.
30.
See, e.g.,
Judy R.
Franz
, “Metal–insulator transition in expanded alkali metal fluids and alkali metal–rare gas films
,” Phys. Rev. B
29
, 1565
–1574
(1984
).31.
See, e.g.,
Shuichi
Ishida
and Eizo
Otsuka
, “Galvanomagnetic properties of n-type InIb at low temperatures. II. Magnetic field-induced Metal–Nonmetal Transition
,” J. Phys. Soc. Jpn.
43
, 124
–131
(1977
).32.
The fractional effect has been brought into high public prominence by the award of the 1998 Nobel Prize in Physics to experimentalists Horst Störmer and Daniel Tsui for its discovery and theorist Robert Laughlin for his elucidation of a theoretical model for the effect. A full treatment of the fractional QHE is given in the monograph The Quantum Hall Effect, edited by Richard E. Prange and Steven M. Girvin (Springer-Verlag, New York, 1990), 2nd ed. Comment: The present author finds it curious that the new plateaus associated with fractional quantum numbers appear innocently in the experimental data as though only a refinement of the integral plateaus. But it then turns out that they require a significantly more elaborate theoretical model for their explanation.
33.
See the chapter entitled “Experimental Aspects and Metrological Application,” by Marvin E. Cage, in the Prange and Girvin volume, Ref. 5, for a discussion of this fascinating dividend stemming from discovery of the QHE.
34.
The pathway to setting a local standard value of the ohm in terms of the experimental value of the Klitzing constant, is described in a paper by
B. N.
Taylor
and T. J.
Witt
, “New international electrical reference standards based on the Josephson and Quantum Hall effects
,” Metrologia
26
, 47
–62
(1989
).35.
J. E.
Furneaux
and T. L.
Reinecke
, “Novel features of quantum Hall plateaus for varying interface charge
,” Phys. Rev. B
29
, 4792
–4795
(1984
).36.
Most Hall effect measurements have been made with samples and electrical contacts arranged in some variation of the scheme shown in Fig. 1(b)—often loosely referred to as a “Hall bar.” While much less frequently used, the circular configuration known as the “Corbino disc” provides an alternate arrangement, with important differences in electrical characteristics. Recent measurements using the Corbino disc geometry, to be referenced subsequently, have provided valuable experimental information to supplement the Hall bar data. The Corbino disc configuration and its use is briefly described in Appendix B.
37.
The reader may find it of great value to refer to a recent article in this journal. (“Electric potential in the classical Hall effect: An unusual boundary value problem,” by
Matthew J.
Moeller
, James
Evans
, Greg
Elliott
and Martin
Jackson
, Am. J. Phys.
8
, 668
–677
(1998
).)38.
A. H.
MacDonald
, T. M.
Rice
, and W. F.
Brinkman
, “Hall voltage and current distribution in an ideal 2D system
,” Phys. Rev. B
28
, 3648
–3651
(1983
).39.
P. F.
Fontein
, J. A.
Kleinen
, P.
Henriks
, F. A. P.
Blom
, J. H.
Wolter
, H. G. M.
Lochs
, F. A. J. M.
Driessen
, L. J.
Giling
, and C. W. J.
Beenaker
, “Spatial potential distribution in heterostructures under Quantum Hall conditions, studies with the linear electro-optic effect
,” Phys. Rev. B
43
, 12090
–12093
(1991
).40.
Note that the direction of flow of these edge currents, formed from the skipping orbits, is such as to oppose the diamagnetic effect of cyclotron orbits within the sample. In a simple classical picture, these countervailing effects cancel one another out. But Landau, in 1930, showed that a correct quantum mechanical calculation for a free electron system recovers a net diamagnetism—which happens to be in magnitude equal to one-third of the Pauli spin susceptibility of the free electron system. A full quantum-mechanical treatment of this Landau diamagnetism is given in The Theory of Metals, by A. H. Wilson (Cambridge University Press, Cambridge, 1958), pp. 160–167 (and, no doubt, in other standard second-level solid state physics treatments.)
41.
R. B.
Laughlin
, “Quantized Hall conductivity in two dimensions
,” Phys. Rev. B
23
, 5632
–5633
(1981
).42.
B. I.
Halperin
, “Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential
,” Phys. Rev. B
25
, 2185
–2190
(1982
). Halperin’s representation of the Laughlin argument uses a physical model in which the sample is in the form of a flat, circular “washer.” The edge currents flow smoothly in circular paths at the inner and outer edges of the washer. This configuration is, in fact, a variant of the Corbino disc. See Appendix B for a discussion of the Corbino disc sample configuration.43.
Reference 4, pp. 566–570.
44.
M.
Büttiker
, “Absence of backscattering in the Quantum Hall effect in multiprobe conductors
,” Phys. Rev. B
38
, 9375
–9389
(1988
).45.
R. J. F.
van Haren
, F. A. P.
Blom
, and J. H.
Wolter
, “Direct Observation of Edge Channels in the Integer Quantum Hall Regime
,” Phys. Rev. Lett.
74
, 1198
–1201
(1995
).46.
B.
Jeanneret
, B. D.
Hall
, H.-J.
Bühlmann
, R.
Houdré
, M.
Ilegems
, B.
Jeckelmann
, and U.
Feller
, “Observation of the integer Quantum Hall Effect by magnetic coupling to a Corbino Ring
,” Phys. Rev. B
51
, 9752
–9756
(1995
).47.
See Beenakker and Van Houten, Ref. 6, pp. 172–177, for extended discussion of the model which combines edge channels with the effects of nonconducting islands.
48.
S. H.
Tessmer
, P. I.
Glicofridis
, R. C.
Ashoori
, L. S.
Levitov
, and M. R.
Melloch
, “Subsurface charge accumulation imaging of a quantum Hall liquid
,” Nature (London)
392
, 51
–54
(1998
).49.
Dmitri B.
Chklovskii
and Patrick A.
Lee
, “Transport properties between Quantum Hall plateaus
,” Phys. Rev. B
48
, 18060
–18078
(1993
).50.
M. E.
Cage
, “Current distributions in Quantum Hall effect devices
,” J. Res. Natl. Inst. Stand. Technol.
102
, 677
–691
(1997
);M. E.
Cage
and C. F.
Lavine
, “Potential and current distributions calculated across a Quantum Hall effect sample at low and high currents
,” J. Res. Natl. Inst. Stand. Technol.
100
, 529
–541
(1995
).51.
G.
Ebert
, K.
von Klitzing
, K.
Ploog
, and G.
Weimann
, “2D magneto-quantum transport in heterostructures under non-Ohmic conditions
,” J. Phys. C
16
, 5441
–5448
(1983
).52.
An extended discussion of the conditions necessary for observation of various electron quantum interference effects is given by
Richard A.
Webb
and Sean
Washburn
, “Quantum interference fluctuation in disordered metals
,” Phys. Today
41
(12
), 41
–55
(1988
).53.
G.
Bergmann
, “Weak localization in thin films
,” Phys. Rep.
107
, 1
–58
(1984
).54.
R.
Corey
, M.
Kissner
, and P.
Saulnier
, “Coherent backscattering of light
,” Am. J. Phys.
63
(6
), 560
–564
(1995
).55.
Analogies in Optics and Microelectronics, edited by W. Van Haeringer and D. Lenstra (Kluwer, Dordrecht, 1990).
56.
R. A.
Webb
, S.
Washburn
, and C. P.
Umbach
, “Experimental study of nonlinear conductance in small metallic samples
,” Phys. Rev. B
37
, 8455
–8458
(1988
).57.
P. A.
Lee
, A. D.
Stone
, and H.
Fukuyama
, “Universal conductance fluctuations in metals: Effects of finite temperature, interactions and magnetic field
,” Phys. Rev. B
35
, 1039
–1070
(1987
).58.
Reference 6, pp. 91–98.
59.
O. M.
Corbino
, “Bahn der Ionen in Metallen
,” Phys. Z.
12
, 561
–568
(1911
). This circular configuration, infrequently used in the decades since the original paper, eliminates contact effects if the current is driven by an ac source, with Hall voltage picked up by a search coil placed concentrically around the disc. (Note: The journal which published Corbino’s paper is Physikalische Zeitschrift, not the more familiar Zeitschrift für Physik. This reference is included for reasons of historical interest. Knowledge of its content is not particularly relevant to uses of this configuration in studying the QHE.)
This content is only available via PDF.
© 1999 American Association of Physics Teachers.
1999
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
