Quantum Point Contacts
The quantization of ballistic electron transport through a constriction
demonstrates that conduction is transmission.
Published in abbreviated form in Physics Today, July 1996, page 22.
Henk van Houten & Carlo Beenakker
Punctuated equilibrium, the notion that evolution in nature is stepwise rather than continuous, sometimes applies to evolution in science as well. It happens that the seed of a scientific breakthrough slumbers for a decade or even longer, without generating much interest. The seed can be a theoretical concept without clear predictions to test experimentally, or an intriguing but confusing experiment without a lucid interpretation. When the seed finally germinates, an entire field of science can reach maturity in a few years.
In hindsight, this is what happened ten years ago, when the authors (newly hired PhD's at Philips Research in Eindhoven) ventured into the field of quantum ballistic transport
. Together with Bart van Wees, then a graduate student at Delft University of Technology, we were confronted with some pretty vague challenges. On the experimental side, there was the search for a quantum-size effect on the conductance, which would reveal in a clear-cut way the one-dimensional density of states of electrons confined to a narrow wire. Experiments on narrow silicon transistors
(at Yale University and AT&T Bell Labs., Holmdel) had come close, but suffered from irregularities due to disorder. (These irregularities would become known as ``universal conductance fluctuations
'', see Physics Today
, December 1988, page 36.) We anticipated that the electron motion should be ballistic, i.e.
without scattering by impurities. Moty Heiblum (IBM, Yorktown Heights) had demonstrated ballistic transport
of hot electrons, high above the Fermi level. For a quantum-size effect one needs ballistic motion at the Fermi energy
. Our colleague Thomas Foxon from Philips Research in Redhill (UK) could provide us with heterojunctions of GaAs and AlGaAs
, containing at the interface a thin layer of highly mobile electrons. Such a ``two-dimensional electron gas'' seemed an ideal system for ballistic transport
On the theoretical side, there was the debate whether a wire without impurities could have any resistance at all.
Ultimately, the question was: ``What is measured when you measure a resistance?'' The conventional point of view (held in the classical Drude-Sommerfeld or the quantum mechanical Kubo theories) is that conduction is the flow of current in response to an electric field. An alternative point of view was put forward in 1957 by Rolf Landauer (IBM, Yorktown Heights), who proposed that ``conduction is transmission''.
Landauer's formula, a relationship between conductance and transmission probability, had evolved into two versions. One gave infinite conductance (= zero resistance) in the absence of impurity scattering, while the other gave a finite answer. Although the origin of the difference between the two versions was understood by at least one of the theorists involved in the debate,
the experimental implications remained unclear.
Looking back ten years later, we find that the seed planted by Landauer in the fifties has developed into a sophisticated theory, at the basis of the entire field of quantum ballistic transport
. The breakthrough can be traced back to experiments on an elementary conductor: a point contact. In this article we present a brief account of these developments. For a more comprehensive and detailed discussion, we direct the reader to the reviews in the bibliography. Quantized conductance
The history of ballistic transport
goes back to 1965, when Yuri Sharvin (Moscow) used a pair of point contacts to inject and detect a beam of electrons in a single-crystalline metal.
In such experiments the quantum mechanical wave character of the electrons does not play an essential role, because the Fermi wave length (
) is much smaller than the opening of the point contact. The two-dimensional (2D) electron gas in a GaAs–AlGaAs heterojunction has a Fermi wave length which is a hundred times larger than in a metal. This makes it possible to study a constriction with an opening comparable to the wave length (and much smaller than the mean free path for impurity scattering). Such a constriction is called a quantum
Figure 1: Schematic cross-sectional view of a quantum point contact, defined in a high-mobility 2D electron gas at the interface of a GaAs–AlGaAs heterojunction. The point contact is formed when a negative voltage is applied to the gate electrodes on top of the AlGaAs layer. Transport measurements are made by employing contacts to the 2D electron gas at either side of the constriction.
In a metal a point contact is fabricated simply by pressing two wedge- or needle-shaped pieces of material together. A quantum point contact
requires a more complicated strategy, since the 2D electron gas is confined
at the GaAs–AlGaAs interface in the interior of the heterojunction. A point contact of adjustable width can be created in this system using the split-gate technique developed in the groups of Michael Pepper (Cambridge) and Daniel Tsui (Princeton).
The gate is a negatively charged electrode on top of the heterojunction, which depletes the electron gas beneath it. (See figure 1
.) In 1988, the Delft-Philips and Cambridge groups reported the discovery of a sequence of steps in the conductance of a constriction in a 2D electron gas, as its width
was varied by means of the voltage on the gate.[6, 7]
(See Physics Today
, November 1988, page 21.) As shown in figure 2
, the steps are near integer multiples of
(after correction for a small gate-voltage independent series resistance).
Figure 2: Conductance quantization of a quantum point contact in units of
. As the gate voltage defining the constriction is made less negative, the width of the point contact increases continuously, but the number of propagating modes at the Fermi level increases stepwise. The resulting conductance steps are smeared out when the thermal energy becomes comparable to the energy separation of the modes. (Adapted from ref. 
An elementary explanation of the quantization
views the constriction as an electron wave guide, through which a small integer number
of transverse modes can propagate at the Fermi level. The wide regions at opposite sides of the constriction are reservoirs
of electrons in local equilibrium. A voltage difference
between the reservoirs induces a current
through the constriction, equally distributed among the
modes. This equipartition rule is not immediately obvious, because electrons at the Fermi level in each mode have different group velocities
. However, the difference in group velocity is canceled by the difference in density of states
. As a result, each mode carries the same current
. Summing over all modes in the wave guide, one obtains the conductance
. The experimental step size is twice
because spin-up and spin-down modes are degenerate.
The electron wave guide has a non-zero resistance even though there are no impurities, because of the reflections occurring when a small number of propagating modes in the wave guide is matched to a larger number of modes in the reservoirs. A thorough understanding of this mode-matching problem is now available, thanks to the efforts of many investigators.
The quantized conductance of a point contact provides firm experimental support for the Landauer formula,
for the conductance of a disordered metal between two electron reservoirs. The numbers
between 0 and 1 are the eigenvalues of the product
of the transmission matrix
and its Hermitian conjugate. For an ``ideal'' quantum point contact
eigenvalues are equal to 1 and all others are equal to 0. Deviations from exact quantization
in a realistic geometry
are about 1%. This can be contrasted with the quantization
of the Hall conductance in strong magnetic fields, where an accuracy better than 1 part in
is obtained routinely.
One reason why a similar accuracy can not be achieved in zero magnetic field is the series resistance from the wide regions, whose magnitude can not be determined precisely. Another source of excess resistance is backscattering at the entrance and exit of the constriction, due to the abrupt widening of the geometry
. A magnetic field suppresses this backscattering, improving the accuracy of the quantization
Suppression of backscattering by a magnetic field is the basis of the theory of the quantum Hall effect developed by Marcus Büttiker (IBM, Yorktown Heights).
Büttiker's theory uses a multi-reservoir generalization of the two-reservoir Landauer formula. The propagating modes in the quantum Hall effect are the magnetic Landau levels interacting with the edge of the sample. (Classically, these magnetic edge states correspond to the skipping orbits discussed later.) There is a smooth crossover from zero-field conductance quantization
to quantum Hall effect, corresponding to the smooth crossover from zero-field wave guide modes to magnetic edge states. When 1 mode = 1 atom
Since the conductance quantum
contains only constants of nature, the conductance quantization
might be expected to occur in metals as well as in semiconductors
. A quantum point contact
in a semiconductor
is a mesoscopic
object, on a scale intermediate between the macroscopic world of classical mechanics and the microscopic world of atoms and molecules. This separation of length scales exists because of the large Fermi wave length in a semiconductor
. In a metal, on the contrary, the Fermi wave length is of the same order of magnitude as the atomic separation. A quantum point contact
in a metal is therefore necessarily of atomic dimensions.
If the initial contact between two pieces of metal is formed by a single atom, the conductance will be of the order of
. This was first observed in 1987 by James K. Gimzewski and R. Möller (IBM, Zürich),
in experiments where the Ir tip of a scanning tunneling microscope
was pressed onto a Ag surface. Upon making contact, the conductance jumped from an exponentially small value to a value of
. Later work, using mechanically more stable devices, showed that further jumps of order
in the conductance occur as the contact area is increased.
Figure 3: Quantized steps in the conductance of a Na point contact. A clean point contact is made at 4.2 K by breaking the wire at a notch cut across a thin wire, whereafter the two parts are brought into contact mechanically. The width of the point contact is adjusted by increasing the force of contact through a piezo element. Electrical measurements are made using four miniature brass bolts connected to the wire. The experiment is not fully reproducible, because of different atomic rearrangements in the contact region. (Adapted from ref. 
shows experimental data for a Na point contact measured by Martijn Krans and collaborators from the Kamerlingh Onnes Laboratory in Leiden.
An adjustable contact of atomic dimensions, with a high mechanical stability, is made by bolting a notched wire of sodium onto a flexible substrate. As the substrate is bent, the wire breaks at the notch. The contact area can be controlled down to the atomic scale, simply by bending the substrate more or less. A statistical analysis of a large number of samples shows that, as the contact area is increased, steps in the conductance appear near 1, 3, 5, and 6 times
. (Figure 3
shows the conductance steps for representative single measurements.) The absence of steps at 2 and 4 times
is significant, and has a neat explanation: In a cylindrically symmetric potential the second and third transverse mode are degenerate, as are the fourth and fifth mode, while the first and sixth mode are non-degenerate.
The energy separation of transverse modes in a point contact of atomic dimensions is so large that the conductance steps are visible at room temperature. Nicolás Garcı́a and his group at the Autonomous University of Madrid have made use of this property to develop a classroom experiment of quantized conductance. (See Physics Today
, February 1996, page 9.) Photons and Cooper pairs
The interpretation of conduction as transmission of electrons at the Fermi level suggests an analogy with the transmission of monochromatic light.
The analogue of the conductance is the transmission cross-section
, defined as the transmitted power divided by the incident flux. Figure 4
shows the transmission cross-section
of a slit of variable width, measured by Edwin Montie and collaborators from Philips.
Steps of equal height occur whenever the slit width
equals half the wave length
of the light. Because
for large slit widths, the step height is also equal to
. Two-dimensional isotropic illumination was achieved by passing the light through a random array of glass fibres parallel to the slit. The isotropy of the illumination mimics the reservoirs in the electronic case, and is crucial for the effect. The two-dimensionality is not essential, but was chosen because a diaphragm of variable area of the order of
is difficult to fabricate. (For a diaphragm, the steps in
Figure 4: Equidistant steps in the optical transmission cross-section of a slit of adjustable width. A 2D isotropic illumination is obtained by shining a beam from a diode laser (
, polarization parallel to the slit) onto a diffusor consisting of a random array of parallel glass fibers. (Adapted from ref. 
It is remarkable that this optical phenomenon, with its distinctly nineteenth century flavour, was not noticed prior to the discovery of its electronic counterpart. There is an interesting parallel in the history of the discovery of the two phenomena. In the electronic case, the Landauer formula was already known before the quantized conductance of a point contact was discovered. Yoseph Imry (Weizmann Institute, Israel) had made the connection with Sharvin's work on point contacts.
The reason that the conductance quantization
came as a surprise, was that the relation
for ballistic transport
was regarded as an order-of-magnitude estimate. To have quantization
, the relative error in this estimate must be smaller than
, which is not obvious. The equivalent of the Landauer formula for the transmission cross-section
has long been familiar in optics,
but also in this field it was not noticed that
holds with better than
One can speak of the optical analogue as a quantum point contact
for photons. Can the analogue be extended towards a quantum point contact
for Cooper pairs
? The answer is ``Yes'': The maximal supercurrent through a narrow and short, impurity-free constriction in a superconductor
is an integer multiple of
the energy gap of the bulk superconductor
A superconducting quantum point contact
has been realized by Hideaki Takayanagi and collaborators (NTT, Japan),
but the superconducting analogue of the quantized conductance remains to be observed experimentally. Thermal analogues
The conductance is the coefficient of proportionality between current and voltage. The additional presence of a small temperature difference
across the point contact gives rise to a matrix of coefficients:
The thermal conductance
relates heat current to temperature difference. The thermo-electric cross-phenomena are described by coefficients
. As first deduced by Lord Kelvin
, time-reversal symmetry requires that
(at a temperature
The two new transport coefficients
can be expressed in terms of the transmission probabilities, just like the electrical conductance
. (See sidebar.) Approximately,
is the total transmission probability at the Fermi energy
. (The proportionality of
, and hence to
, is the Wiedemann-Franz law of solid-state physics.) The stepwise energy dependence of the transmission probability through a quantum point contact
implies two types of quantum-size effects: steps in
and peaks in
. Both effects have been observed by Laurens Molenkamp and collaborators from Philips.
The thermal conductance
of a quantum point contact
exhibits steps when the gate voltage is varied, aligned with the steps in the electrical conductance
. Each step signals the appearance of a new mode at the Fermi level which can propagate through the constriction. A step in the transmission probability leads to a peak in the thermo-electric coefficient
. Pavel Str̆eda
(Prague) has calculated that, at zero temperature, the height of the peaks in
times the conductance quantum
. The unit
is the entropy production per Coulomb of charge transferred through the point contact, or
times the entropy
carried by a single conduction electron, which is on the order of Boltzmann's constant
Figure 5: Thermopower oscillations in a quantum point contact. The peaks in the thermovoltage (proportional to the thermopower) coincide with the steps in the conductance. (Adapted from ref. 
In figure 5
we show measurements of the thermopower
of a quantum point contact
(The thermopower is proportional to the voltage produced by a temperature difference for zero electrical current.) The coincidence of peaks in the thermopower with steps in the conductance (measured for the same point contact) is clearly visible. Joule
heating was used to create a temperature difference across the point contact in this work. Local heating by means of a focused beam of far-infrared radiation has been used in a more recent experiment.Shot noise
The electrical current through a point contact is not constant in time, but fluctuates. The conductance determines only the time-averaged current. The noise power
is the Fourier transform of the correlator of the time-dependent fluctuations
in the current at a given voltage
. One distinguishes equilibrium thermal noise (
) and non-equilibrium shot noise (
). Both types of noise have a white power spectrum (i.e.
the noise power does not depend on frequency over a very wide frequency range). Thermal noise is directly related to the conductance through the fluctuation-dissipation theorem (
). Therefore, the thermal noise of a quantum point contact
does not give any new information.
Shot noise is more interesting, because it contains information on the temporal correlation of the electrons which is not contained in the conductance. Maximal shot noise (
) is observed when the stream of electrons is fully uncorrelated. A typical example is a tunnel diode
. Correlations reduce
. One source of correlations, operative even for non-interacting electrons, is the Pauli principle, which forbids multiple occupation of the same single-particle state. A typical example is a ballistic point contact in a metal, where
because the stream of electrons is completely correlated by the Pauli principle in the absence of impurity scattering.
Figure 6: Periodic suppression of the shot-noise power of a quantum point contact, measured with an applied voltage of 0.5, 1, 1.5, 2, and 3 mV. (Adapted from ref. 
A quantum point contact
in a 2D electron gas has a different behavior. Using a Landauer-type formula (see sidebar), Gordey Lesovik (Moscow) has predicted peaks in the shot noise at the steps in the conductance.
The peak height
is half the maximal value for uncorrelated electrons. The shot noise vanishes in between the steps. Michael Reznikov and collaborators from the Weizmann Institute in Israel have recently presented a convincing demonstration of this quantum-size effect in the shot noise.
(See figure 6
.) By going to microwave frequencies (8–18 GHz) they could avoid the ubiquituous ``1/f noise'' at lower frequencies. Solid-state electron optics
The effects discussed so far refer to properties of the quantum point contact
itself. A wealth of new phenomena has been discovered using a quantum point contact
as a spatially coherent point source and detector, and specially formed electrodes as mirror, prism, or lens.
The basic experiment,coherent electron focusing
, is shown in figure 7
. A point contact injects electrons with the Fermi momentum
into the 2D electron gas, in the presence of a perpendicular magnetic field
. The electrons follow a ``skipping orbit'' along the boundary, consisting of circular arcs of cyclotron
. Some of the electrons are collected at a second point contact, at a separation
from the first. The voltage measured at the collector is proportional to the transmission probability between the two point contacts. V. S. Tsoı̆ (Moscow) first used this focusing technique in a metal.
The magnetic field acts as a lens, bringing the divergent trajectories at the injector together at the collector. The collector is at a focal point of the lens when
is a multiple of
, hence when
is a multiple of
(arrows in figure 7
). For reverse magnetic fields the injected electrons are deflected away from the collector, so that no signal is generated. The observation of peaks at the expected positions demonstrates that a quantum point contact
acts as a monochromatic point source of ballistic electrons, and that the reflections at the boundary of the 2D electron gas are specular. The fine-structure on the focusing peaks is due to quantum interference of trajectories between the two point contacts. Such fine-structure does not appear in metals. It demonstrates that the quantum point contact
is a spatially coherent source and that the phase coherence is maintained over a distance of several microns to the collector.
Figure 7: Magnetic focusing in a 2D electron gas at
. The top panel shows the experimental arrangement. Electrons injected through one point contact (i) follow skipping orbits over a distance of
to a second point contact (c). The arrows indicate the positions of the focusing peaks expected when the point contact separation is a multiple of the cyclotron diameter. The fine-structure on the peaks is due to quantum interference. (Adapted from ref. 
Magnetic focusing has been used by several groups to obtain information on the dynamics and scattering of quasiparticles in the 2D electron gas. An intriguing application in the regime of the fractional quantum Hall effect is the focusing of composite fermions
which can be thought of as electrons bound to an even number of flux quanta. In the regime of the integer quantum Hall effect, the geometry
of figure 7
has been used to selectively populate and detect the magnetic edge states mentioned earlier.
The observation of plateaus in the Hall conductance at anomalously quantized values provides support for the edge-state theory of the quantum Hall effect.
Electrostatic focusing, by means of the electric field produced by a lens-shaped electrode, provides an alternative technique to focus the beam of electrons injected by a point contact.
Instead of focusing the beam, one can also deflect it — either by means of a magnetic field,
or by means of a prism-shaped electrode.
The building blocks of electron optics in the solid state have by now all been realized. Ultimate confinement
A quantum point contact
which is nearly pinched off (so that its conductance is less than
) is a tunnel barrier of adjustable height for electrons near the Fermi level. This property has been used to inject and detect electrons in a small confined
region of a 2D electron gas, called a quantum dot
. A quantum dot coupled to the outside by a pair of quantum point contacts
has provided an ideal model system for the investigation of the effects of Coulomb repulsion on resonant tunneling. (See Physics Today
, January 1993, page 24.)
The zero-dimensional quantum dot forms the logical end to the reduction of dimensionality of the two-dimensional electron gas. In this article we have reviewed the role played by the one-dimensional quantum point contact
in the conceptual development started by Landauer four decades ago. The concept of electrical conductance
was conceived in the nineteenth century, at a time when the electron was not even discovered. It is amusing that it required the sophisticated micro-electronics technology of the late twentieth century to demonstrate experimentally that ``conduction is transmission''.
Landauer's original 1957 formula,
expresses the conductance of a one-dimensional system as the ratio of transmission and reflection probabilities. As explained by Imry,
this formula gives infinity for unit transmission because it excludes the finite contact conductance contained in
Extension to higher dimensions
is achieved by replacing the transmission probability
by the eigenvalue
of the transmission matrix product
, and summing over
Generalizations of the Landauer formula have been found for a variety of other transport properties, besides the conductance.
At zero temperature, these expressions are of the form:
- conductance : , .
- shot-noise power : , .
- conductance of a normal-metal – superconductor junction: , .
- supercurrent through a Josephson junction with phase difference : , .
The expressions for the thermo-electric coefficients involve an integration over energies around the Fermi energy
, weighted by the derivative
of the Fermi-Dirac distribution function at temperature
If the energy-dependence of the transmission eigenvalues is small on the scale of the thermal energy
, one has approximately
(the Wiedemann-Franz law) and
the Lorentz number.
Each Landauer formula predicts a specific quantum-size effect in a ballistic constriction, for which
equals 0 or 1. The effect is a step-function dependence on the width of the constriction in the case of
, and an oscillatory dependence in the case of
Henk van Houtenheads a department at the Philips Research Laboratories in Eindhoven, and is a Professor of Physics at the University of Leiden (The Netherlands).Carlo Beenakkeris a Professor of Physics at the University of Leiden.
- ^ For an historical account of the controversy, from two different perspectives, see R. Landauer, IBM J. Res. Dev. 32, 306 (1988); A. D. Stone, A. Szafer, ibidem, page 384. For a more recent review and a comprehensive list of references, see Y. Imry, Introduction to Mesoscopic Physics, Oxford University Press (1996).
- ^ab R. Landauer, IBM J. Res. Dev. 1, 223 (1957).
- ^abc Y. Imry, in: Directions in Condensed Matter Physics, eds. G. Grinstein and G. Mazenko, World Scientific, Singapore (1986).
- ^ Yu. V. Sharvin, Sov. Phys. JETP 21, 655 (1965).
- ^ T. J. Thornton, M. Pepper, H. Ahmed, D. Andrews, G. J. Davies, Phys. Rev. Lett. 56, 1198 (1986); H. Z. Zheng, H. P. Wei, D. C. Tsui, G. Weimann, Phys. Rev. B 34, 5635 (1986).
- ^abc B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel, C. T. Foxon, Phys. Rev. Lett. 60, 848 (1988); Phys. Rev. B 43, 12431 (1991).
- ^ 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, G. A. C. Jones, J. Phys. C 21, L209 (1988).
- ^ An early theoretical paper is by L. I. Glazman, G. B. Lesovik, D. E. Khmel'nitskiı̆, R. I. Shekhter, JETP Lett. 48, 238 (1988). More references are given in the review by C. W. J. Beenakker, H. van Houten, Sol. State Phys. 44, 1 (1991); cond-mat/0412664
- ^ M. E. Cage, in: The Quantum Hall Effect, eds. R. E. Prange and S. M. Girvin, Springer, New York (1990).
- ^ M. Büttiker, Phys. Rev. B 38, 9375 (1988).
- ^ J. K. Gimzewski, R. Möller, Phys. Rev. B 36, 1284 (1987).
- ^ab J. M. Krans, J. M. van Ruitenbeek, V. V. Fisun, I. K. Yanson, L. J. de Jongh, Nature 375, 767 (1995).
- ^ H. van Houten, C.W.J. Beenakker, in: Analogies in Optics and Micro Electronics, eds. W. van Haeringen and D. Lenstra, Kluwer, Dordrecht (1990); cond-mat/0512611
- ^ab E. A. Montie, E. C. Cosman, G. W. 't Hooft, M. B. van der Mark, C. W. J. Beenakker, Nature 350, 594 (1991).
- ^ A. W. Snyder, C. Pask, J. Opt. Soc. Am. 63, 806 (1973).
- ^ C. W. J. Beenakker, H. van Houten, Phys. Rev. Lett. 66, 3056 (1991); A. Furusaki, H. Takayanagi, M. Tsukada, Phys. Rev. B 45, 10563 (1992).
- ^ H. Takayanagi, T. Akazaki, J. Nitta, Phys. Rev. Lett. 75, 3533 (1995).
- ^ P. Str̆eda, J. Phys. Condens. Matter 1, 1025 (1989).
- ^ A review of thermo-electric effects in quantum point contacts is: H. van Houten, L. W. Molenkamp, C. W. J. Beenakker, C. T. Foxon, Semicond. Sci. Technol. 7, B215 (1992); cond-mat/0512612
- ^ab L. W. Molenkamp, H. van Houten, C. W. J. Beenakker, R. Eppenga, C. T. Foxon, Phys. Rev. Lett. 65, 1052 (1990).
- ^ R. A. Wyss, C. C. Eugster, J. A. del Alamo, Q. Hu, M. J. Rooks, M. R. Melloch, Appl. Phys. Lett. 66, 1144 (1995).
- ^ G. B. Lesovik, JETP Lett. 49, 592 (1989).
- ^ab M. Reznikov, M. Heiblum, H. Shtrikman, D. Mahalu, Phys. Rev. Lett. 75, 3340 (1995).
- ^ab H. van Houten, B. J. van Wees, J. E. Mooij, C. W. J. Beenakker, J. G. Williamson, C. T. Foxon, Europhys. Lett. 5, 721 (1988); Phys. Rev. B 39, 8556 (1989).
- ^ V. S. Tsoı̆, JETP Lett. 19, 70 (1974).
- ^ V. J. Goldman, B. Su, J. K. Jain, Phys. Rev. Lett. 72, 2065 (1994).
- ^ U. Sivan, M. Heiblum, C. P. Umbach, H. Shtrikman, Phys. Rev. B 41, 7937 (1990).
- ^ L. W. Molenkamp, A. A. M. Staring, C. W. J. Beenakker, R. Eppenga, C. E. Timmering, J. G. Williamson, C. J. P. M. Harmans, C. T. Foxon, Phys. Rev. B 41, 1274 (1990).
- ^ J. Spector, H. L. Stormer, K. W. Baldwin, L. N. Pfeiffer, K. W. West, Appl. Phys. Lett. 56, 2433 (1990).
- ^ E. N. Economou, C. M. Soukoulis, Phys. Rev. Lett. 46, 618 (1981).
- ^ D. S. Fisher, P. A. Lee, Phys. Rev. B 23, 6851 (1981).
- ^ The formulas for and are due to Uri Sivan and Imry [Phys. Rev. B 33, 551 (1986)], for to Büttiker [Phys. Rev. Lett. 65, 2901 (1990)], and for and to Beenakker [Phys. Rev. Lett. 67, 3836 (1991); Phys. Rev. B 46, 12841 (1992)].