Advances in Quantum Dot Structures

Advances in Quantum Dot Structures

SEMICONDUCTORS AND SEMIMETALS, VOL. 73 CHAPTER 5 Advances in Quantum Dot Structures S. Kim and M. Razeghi CENTER FOR QUANTUM DEVICES, ELECTRICAL AN...

1MB Sizes 1 Downloads 90 Views

SEMICONDUCTORS AND SEMIMETALS, VOL. 73

CHAPTER

5

Advances in Quantum Dot Structures S. Kim and M. Razeghi CENTER FOR QUANTUM DEVICES, ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, NORTHWESTERN UNIVERSITY, EVANSTON, ILLINOIS, USA

1. INTRODUCTION

199

2. PHYSICAL PROPERTIES

201

2.1. 2.2. 2.3. 2.4.

Density of States Energy States Optical Absorption and Transition in Quantum Dots Devices Based on Zero-Dimensional Quantum Structure

3. STATE OF THE ART

201 202 204 207 209

REFERENCES

212

1.

Introduction

Semiconductor quantum dots (QDs) represent one of the rigorous ongoing research areas for next generation optoelectronic devices. The strong interest in low-dimensional semiconductor structures originates from their exciting electronic properties that have an important impact on the performance of highspeed electronic and photonic devices and, moreover, on the development of novel device concepts such as the single electron transistor. The quantum dots known as quantum boxes are nanometer-scale islands in which electrons and holes are confined in three-dimensional potential boxes. They are expected to exhibit a zero-dimensional, S-function density of states and are able to quantize electrons free motion by trapping it in a quasi-zero-dimensional potential confinement. Due to these peculiar characteristics, quantum dots are expected to have superior characteristics for device performance in semiconductor lasers, detectors, and modulators. The condition for new electronic properties to occur in such device structures is that the lateral size of their active region must be smaller than the coherence length and the elastic scattering length of the carriers. Additional quantum-size effects require the structural features to be reduced to below 50 nm, that is, the range of the de Broglie wavelength. Therefore, the reproducible fabrication of these nanometer-scale quantum structures requires methods with atomic 199 Copyright © 2001 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752182-8 ISSN 0080-8784/01 $35.00

200

S. KIM AND M. RAZEGHI

scale precision, which is a major challenge for today's microstructure materials science. As a result of the strong confinement imposed in all three spatial dimensions, quantum dots are similar to atoms. They are frequently referred to as artificial atoms, superatoms, or quantum dot atoms [1]. What makes quantum dots such unusual objects is, first of all, the possibility to control their shape, their dimensions, the structure of energy levels, and the number of confined electrons. It is possible, for instance, to create and investigate, as a rectangular or parabolic potential well binding, one or several particles, as well as the Landau quantization of motion of a single electron, the radiative recombination from a few-particle system, and so on. Quantum dots were first realized by scientists from Texas Instruments Incorporated. Reed et al. reported the creation of a square quantum dot with a side length of 250 nm, etched by means of lithography. Since then, quantum dot and quantum wire structures have been fabricated by means of subsequent lateral patterning of two-dimensional heterostructures with lithographic techniques followed by chemical etching or selective crystal growth on prepattemed and masked substrates. However, although many fundamental properties of lowdimensional semiconductors can be demonstrated in these structures, it turns out that lithographic patterning processes and chemical etching always introduce defects that degrade the crystal quality and cause irregularities in size and shape of the quantum structures that are detrimental for practical applications in semiconductor devices. Especially to reduce the defect density, several methods for the direct fabrication of quantum dots and quantum wires based on the epitaxial growth process itself have been exploited. Quantum dots and wires have been grown by using the periodic step structure on vicinal surfaces, the generation of supersteps, and the breakup of high index surfaces into arrays of nanometer-scale facets. Recently, first breakthrough for growth of high quality highly strained epitaxial layers since the Stranski-Krastanow growth mode [2] was rediscovered during highly strained layer epitaxy; it is called self-assembled quantum dots. This discovery has great meaning for a new era of optoelectronic devices, providing new techniques for low-dimensional quantum structure [3-5]. When the lattice constants of the substrate and the crystallized material differ considerably, only the first deposited monolayer crystallizes in the form of epitaxial strained layers, where the lattice constant is equal to that of the substrate. When the critical thickness is exceeded, a significant strain that occurs in the layer leads to the breakdown of this ordered structure and to the spontaneous creation of randomly distributed islets of regular shape and similar sizes [6]. The small sizes of the self-assembled quantum dots (diameters in the range of 30 nm or even smaller), the homogeneity of their shapes and sizes in a macroscopic sample, the perfect crystal structure (without edge defects), and the fairly convenient growth process, without the necessity to precisely deposit electrodes or etching, are among their greatest advantages. Thus there is great hope with regard to their future applications in electronics and optoelectronics.

5

ADVANCES IN QUANTUM DOT STRUCTURES

2. 2.1.

201

Physical Properties

DENSITY OF STATES

Quantum confinement of charge carriers in semiconductors takes place when the carriers are trapped within potential wells of sufficiendy small dimensions. This quantum confinement can give rise to significant modification of the energy band structure and density of states (DOS) distribution in these materials. In the discussion that follows, we treat the quantum-confined structure in a simple, single-band model. Although this picture is adequate for the conduction band case, more elaborate multiband models, including the effect of band mixing, have been developed for the valence band. In the effective mass approximation, the energy spectrum E of the carriers is obtained by solving Schrodinger's equation 2

.^y^V{x.y^z)

^{x.y.z)

=

E^{x,y,z)

where "^ is the carrier envelope wave function, nf is the carrier effective mass, and y(jc, y, z) is the potential distribution. For potential wells of rectangular shape, one-, two-, and three-dimensional quantum confinement can be achieved in film-, wire-, and boxlike geometries by successively reducing the well dimensions r^, ty, and t^. For infinitely deep potential wells, the energy of the confined carriers (with respect to the band edge) is given by

'

lm*tl

2m*

one-dimensional (ID) confinement

Ei.n. = ^r-\-^ n* \tl \tl 2m*

+ t] -r\ + tl I) 2m*

two-dimensional (2D) confinement s2^2 / /2

^2

^2^

three-dimensional (3D) confinement where /, m, n = 1, 2 , . . . are the level quantum numbers and ky, k^ are the wave vector components along the unconfined dimensions. Quantum confinement in such quantum well (QWL), quantum wire (QWR), or quantum dot (QD) structures thus results in charge carriers of a quasi-2D, ID, or zero-dimensional (OD) nature.

202

S. KIM AND M. RAZEGHI

The density of states functions, including spin degeneracy, are given by

P3D =



^^-^ 27r2

P2D X

E0(^-^/) I

*V/2 (2m*)

PlD

X y

EiE-Ei,J

-1/2

I, m

J:HE-E,„J

POD X y z

l,m,n

for 3D, quasi-2D, ID, and OD carriers, respectively. S{x) is the Heaviside function: ^ = 0 for jc < 0; ^ = 1 for jc > 0. The DOS distributions (Fig. 1) acquire sharper features as the carrier dimensionality is reduced, particularly in the case of ID and OD structures. Note, however, that these sharp features can be significandy smoothed out by well size fluctuations, leading to inhomogeneous broadening of the energy spectrum.

2.2.

ENERGY STATES

To explain three-dimensional quantum confinement, the quantum mechanical problem of the motion of a particle in a box was recalled. This problem turned out to be very complicated, because all the possibilities of electronhole Coulomb interaction—valence band structure or nonparabolic bands—were considered.

>

3D

A

2D

0

Q

•L

El

E2

>

\ )D

ID

y

Ell

^ — >

^12 Ei3

FIG. 1. Density of States.

— > •

11

b ji ^21

^r,1 1

^21

5

ADVANCES IN QUANTUM DOT STRUCTURES

203

The total Hamiltonian H for an electron-hole pair in the QD is given by

^h,kinVh)

T V.—i—v. + v — i — V

— ^

-e^

1

where ^^jcin and ///j^in are the kinetic energy of electrons and holes, and V^ and Vyj are the potential energy of electrons and holes, respectively. The H^ is the Coulomb interaction term, which is the only term dependent on coordinates of both electrons and holes, and couples of their motion. In spherical quantum dots, taking into account the Coulomb potential, there is a break in symmetry, because the Coulomb interaction depends on the spatial distance between the electrons and the holes. The simplest approach takes into account that the Coulomb energy scales like the inverse of the electron-hole distance (~ 1//^), whereas the kinetic energy scales like the square of the inverse radius (~ l/R^)One possible description for small dot radii in the so-called strong confinement range (R <^ag, where a^ is the excitonic Bohr radius) [7] is the neglect of all Coulomb interaction effects. Even the Coulombic effect cannot be completely neglected, and it is solved by numerically using the perturbation theory. To solve the hole energy states, more complicate models for the kinetic energy term of the hole H^^ have been introduced in the Hamiltonian. The hole Hamiltonian H^ can be expressed for cubic materials with strong spin-orbit coupling by

- —[lp.p,WM niQ

+ ip,p.WAl + tp.pMV-.)]

where JTIQ is the electron mass, 7i,72'73 are the parameters introduced to describe the valence band dispersion, P is the hole momentum operator, and J is the 3/2 angular momentum operator. The preceding expression can be simplified by considering the spherical symmetry. Small contributions of terms of cubic or hexagonal symmetry are neglected and parameter m is introduced to give the strength of spherical spin-orbit interaction. This can be described as p2_^(p(2)j(2))

2mo

204

S. KIM AND M. RAZEGHI

where the coupling parameter m is defined by M =

673+472 57,

The involvement of the full valence band structure results in a qualitative new description of the electron and hole energy states (Fig. 2).

2.3.

OPTICAL ABSORPTION AND TRANSITION IN QUANTUM DOTS

2.3.L

Absorption Coefficients

We consider the absorption coefficient of an ensemble of quantum dots 1

C , ^ 4477 77,

«av(w) = — / dR—R'P{R)a^^{(o,

R)

which is one of the important optical properties. The averaged absorption spectrum aav(^) c^^ ^^ expressed by 1

C

477

«ave = TT / dR—R'PiR)aQ^ico,

R)

where VQD is the average quantum dot volume, R is the radius, P{R) is a characteristic distribution function for the dot sizes, and aQ^{(x), R) is the absorption coefficient of a single quantum dot. The absorption spectrum is given by a series of Lorentzian lines for the ground and excited states at energies £QD = hoj^, with homogeneous line widths Yj and oscillator strengths ff.

To solve these two equations, we need information about the radius R and the size distribution P(K), and we need to determine suitable relationships for the size dependence of energy E{K), homogeneous line broadening r(/?), and oscillator strength /(/?). 23.2.

Absorption Process

To discuss the optical absorption spectrum a(a)) of a quantum dot, the probability for the dipole allowed optical transitions between single electron and hole states has to be evaluated. According to a simplified model, the most important optical transitions between electron and hole states are Is^ ^- 1^;^, Ipe -> IPh^

5

ADVANCES IN QUANTUM DOT STRUCTURES

70%

12nm (a) without Piezoelectric Potential " . . . |ioo> —ig^N,

1

d

LF!:

^^^ *^i^

1110)

with Piezoetectric Potential

(b) FIG. 2. Energy state of the InAs QDs in GaAs. Reprinted with permission from [8].

205

206

S. KIM AND M. RAZEGHI

IE

two-pair s

electron states I P

AE

-lSel(P,F)3/2

-TT-

;; IS.

i

hole states

l(S,D)3^1(P,F)3„ • 2(S,D)3^ • 3(S,D)3^

^II



r

1

'

;

A

1 2(P-F)3,

ls,l(S,D) ls,l(P,F) ls,l(S,D)

1 : I i III ;:i i

1

,

*^

one-pair s IPe 1(P,F

1

ls^2(S,D 1i

)r

ls,l(S,D

3f groi

'. .T. y.

1S„ FIG. 3. Simplified scheme of the optical transitions plotted for a quantum dot in the strong confinement regime (a) in the picture of independent electron and hole states and (b) in the one-pair and two-pair picture and considering Coulomb effect. Reprinted with permission from [9].

and so forth. For an optical transition within the ladder of either an electron or hole state (m^raband transition), we obtain the selection rules rif / n^, If — l^ = 0, ± 1 , and mf — nii = 0, ± 1 . Next, we briefly describe the possible optical transition between the calculated QD energy states. Valence band coupling gives rise to formerly forbidden optical transitions between electron and hole states with An ^0 and significant oscillator strength is acquired. In Figure 3a, we illustrate the energy label scheme for the energetically lowest electron and hole states and the allowed optical transition between them. The relevance for the nonlinear spectra is illustrated as a one-pair-two-pair picture in Figure 3b. Transitions 1 and 2 are the same as the dominant one-pair transition between s- and p-type electron and holes; transition 3 is the somewhat weaker one-pair transition with the hole in the first excited state. For the two-pair states in three dimensionally confined quantum dots, the theory predicts (i) a large binding energy, (ii) observation of both ground and excited two-pair states in the optical spectra because of the absence of the typical 3D continuum states, and (iii) transitions to excited twopair states that originally were forbidden and now occur due to the change of selection rules caused by the Coulomb potential. The existence of the two-pair states is important for the interpretation of differential absorption spectra. After population of the one-pair states by pump photons, the pair states can be populated by an photon absorption process. Then induced absorption is energetically low compared to the one-pair energy that appears in the spectrum due to the transition to the two-pair ground state. These transitions are only possible because the Coulomb interaction has changed the selection rules.

5 2.4.

2.4.1.

ADVANCES IN QUANTUM DOT STRUCTURES

207

DEVICES BASED ON ZERO-DIMENSIONAL QUANTUM STRUCTURE

ID and OD Electronic Devices

Rapid progress in fabrication technology and well controlled growth results are being realized for nanostructure quantum devices for future applications. The ultrasmall structure concept for speed and integration is clearcut and follows the mainstream approach to very large and ultra-large scale integration. Within this natural trend, quantum devices are just the result of the evolution of lithography. Whereas deterministic, zero defect interconnections seem out of reach to conserve reasonable fabrication yields, the error margins will decrease due to increased fluctuations, such as diminished carrier numbers for logic operation, or fabrication tolerances. Granular electronics are highly desirable devices, because single or a few electrons are used to perform digital operations instead of our present-day, wellknown transistors that operate at best with = 10"^ electrons. Such devices are based on concepts similar to the Coulomb blockade concept, where the single electron charge changes a local potential energy in a structure by ^V = e/C, where C is a local capacitance, determined by the device geometry and materials. The Coulomb blockade effect [10, 11] has been the subject of many investigations in the context of small, superconducting tunnel junctions at low temperatures and can be shown to allow digital electronic functions when several devices are combined. A recent implementation demonstrates a turnstile device activated by alternating gate voltages. Future issues are the demonstration of high-temperature, error-free devices (i.e., device energy e^/C much higher than operating thermal energy k^T), and circuit and systems architecture, including interconnections, and self-repair. Recent progress in ballistic deterministic motion [12] indicates that many large, interesting effects can be obtained in a nonquantized, classical situation where electrons travel without collisions. Some effects occur in unquantized modes, such as those due to coherent focussing effects or lateral hot-electron devices. Analog steering of an electron beam into spatially arranged collector electrodes through electrostatic lenses or split gates might allow analogto-digital conversion in the multi-gigahertz range. The ballistic electron device would be used as the critical element of the converter, and connections to the outside world would be made through standard devices, achieving an integrated electrooptics component. A tunneling hot-electron transfer amplifier could also be realized in the 2D electron gas plane thanks to narrow metal gates (^ 50 nm). Cellular automata machines [13, 14] (see Fig. 4), which consist of arrays of elementary digital processors that are located at the nodes of a regular lattice and are connected to a few neighbors, can be revolutionary, fault-tolerant architectures at low dimensions to alleviate difficulties associated with nondeterministic modes of operation of tens or hundreds of millions of integrated transistors. The processors can be as simple as logic gates, but can also be full

208

S. KIM AND M. RAZEGHI

c<-

« c~->

ail

ai2

All

322

Cil

C|2

C21

C22

+ab

FIG. 4. Schematic representation of a cellular automata machine. Left: Elementary node processor performing the elementary operation for the inner product. Right: Schematic of the motion of data through a cellular machine performing the matrix inner product c = a • b. Reproduced with permission from C. Weisbuch and B. Vinter, "Quantum Semiconductor Structures." Academic Press, San Diego, Copyright 19xx.

scale microprocessors. The main advantage of such machine architecture is the easy synchronization of operations and modularity. These systems can also be made robust against hard or soft failure by redundancy and verification. Such machines have been shown to be very efficient for implementing specialized functions to solve problems that rely on physical laws that are similar to their architecture (i.e., incorporating high degrees of locality and parallelism). The ongoing debate concerns whether the universal computer machine can be efficiently designed through cellular automata architecture. In any event, quantum devices could easily be associated in such architecture to yield standard cellular automata functions such as matrix inner products, Fourier transform, and convolution products. 2.4.2.

ID and OD Photonic Devices

The impact of lower dimensionality in photonic devices is more prominent than in electronic devices. It relies on several effects, the most important of which is the progressive restriction of allowed states dispersed over E{k) bands toward more concentrated single-energy atomlike levels of quantum dots. The atomlike levels in quantum dots enormously sharpen resonant behavior and, therefore, energy selectivity. In addition to all the resonance effects, lower dispersion of optical properties over k states is expected due to the k selection rule that only vertical transitions are allowed in the E{k) representation of quantum

5

ADVANCES IN QUANTUM DOT STRUCTURES

209

states (/:-conserving transitions). Therefore, the occupancy of varied k states as required by statistics increasingly scatters the properties of injected carriers in ID, 2D, and 3D states. For injection lasers, the occupancy of the same number of electrons of 2D, ID, and OD states above inversion leads to higher gains due to the concentration of electrons and holes over fewer k states. Considering exciton effects, three scales are to be considered in quantum dots: When L <^ a^, the confining kinetic energy is much larger than the Coulomb interaction between electrons and holes. In this case, the latter is a perturbation and the wave functions are the exact quantum box wave functions. The oscillator strength per transition in the quantum box is the usual interband oscillator strength, because the transition matrix element can be factorized in the three directions, such as f{D^Dy)~^ per unit surface, where D^ and Dy are the center-to-center distances in the x and y directions. An oscillator strength per unit surface of quantum well of (8/7ra|)/at can be strengthened if the center-tocenter distance is less than (25(77/8)^/^. When L > a^, the oscillator strength is enhanced mainly due to the coherent excitation of the QB volume, which yields a transition matrix element / ' ^ /at ' V ^'^box / ''^exc /

where V^^ and V^^^ are the QB and exciton volume, respectively. In addition, nonlinear effects exhibit lower thresholds due to the smaller number of states that need to be filled to reach saturation. Multiple-particle interactions are strongly increased in ID and OD: Increasing number of electron-hole pairs in a confined volume leads to large Coulomb interactions, which can be described as biexciton or multiple-exciton effects.

3.

State of the Art

The earliest method of obtaining quantum dots was implemented by Reed et al, who etched them in a structure containing two-dimensional electron gas. In this case, the surface of a sample containing one or more quantum wells was covered with a polymer mask and then partly exposed. Several techniques can be used to etch uniform pillars with a few tens of nanometers. Low-energy electronbeam lithography (Fig. 5) can be used to realize uniform 30-nm patterns [15] over a large area. In addition, laser-induced surface electromagnetic waves has been used to enhance gas etching and fabricate quantum dots successfully [16]. Another method of creating quantum dots consists of the creation of miniature electrodes over the surface of a quantum well by means of lithographic techniques as shown in Figure 6 [17]. The application of an appropriate voltage to the electrodes produces a spatially modulated electric field, which localizes the electrons within a small area (Fig. 7).

210

S. KIM AND M. RAZEGHI

oft;

,::;J^^Si

••*••':•

ill*?"

8588

t'S.akV

X 9 « . en' ^ ' 3 a'jni

FIG. 5. Successful application of low-energy electron-beam lithography forms uniformly distributed quantum dots with a narrow distribution of dot size. Reprinted with permission from [15].

The lateral confinement created in this way shows no edge defects, which are characteristics of such etched structures. An electric gate can also be created around the etched dot, thus eliminating edge defects and additional squeezing of electrons. Quantum dots can also be created through the selective growth of a semiconductor compound with a narrower bandgap on the surface of another compound with a wider bandgap. The restriction of growth to chosen areas is obtained by first creating a pattern deposited with a mask (Si02 or SiN^) and then etching miniature triangles on it. The growth is carried out on the surface that is not covered by the mask (Fig. 8). This type of growth was also demonstrated with III-N based materials.

FIG. 6. Laser-induced surface electromagnetic wave etching used to etch uniform quantum dots of GaAs and InP. Reprinted with permission from [16].

5

ADVANCES IN Q U A N T U M D O T STRUCTURES

211

FIG. 7. Quantum dot at the intersection of electrodes. Four internal electrodes localize the electrons and four external electrodes serve as contacts for the electrons tunneling to and from the dot.

However, the damage introduced by the etching procedure used in the pattern transfer onto the semiconductor is such that it dominates all electronic properties at small dimensions. One very well documented effect of etching damage is carrier depletion, which extends, in the best cases, for standard carrier concentrations, over ^50 nm. The observations rely on the dependence of

X

r

FIG. 8. SEM images of (a) quantum dots created on the surface of GaAs in selective MOCVD growth and (b) GaN quantum pyramids created by selective growth. Reprinted with permission from [18, 19].

212

S. KIM AND M. RAZEGHI

FIG. 9. Evolution of self-assembled quantum dots grown by MBE. Reprinted with permission from [6].

the electronic population of quantum wires or dots on geometric lateral size: below '^100-nm, quantum wires or dots are empty. Another probe of the damage is the decrease in luminescence efficiency due to numerous nonradiative defects. To obtain less damaged structures, softer fabrication techniques have been sought. Direct semiconductor growth on vicinal surfaces, step-induced directional growth using the patterned substrate, or growth of cleaved and etched multiquantum well structures on sidewalls have been tried through a delicate two-step growth technique. In addition to the etching damage, overgrowth on the etched pillar and the acquisition of smaller size present other hard tasks for optical devices. Actually, the performance of quantum dot optoelectronic devices has been inhibited mainly due to these limitations in technologies. Therefore, the emerging growth technology of self-organization of the epitaxial semiconductor layer on different lattice constant substrates holds great promise for removing complicated technologies. This method, originally called the Stranski-Krastanow (S-K) growth mode, was discovered in 1939 [2] and rediscovered [6] during the epitaxy of InAs on GaAs, where islands form due to the strain of lattice mismatch between the epilayer and the substrate. These islands are small enough to expect quantum size effects, which we call zerodimensional quantum structure. Figure 9 shows the evolution of InAs quantum dots on GaAs grown by molecular beam epitaxy (MBE).

References 1. R. C. Ashoori, Nature 379, 413 (1996). 2. I. N. Stranski and L. von Krastanow, Akad. Wiss. Let. Mainz Math. Natur Kl lib 146, 797 (1939). 3. L. Goldstein, F. Glas, J. Marzin, M. Charasse, and G. LeRoux, Appl. Phys. Lett. 41, 1099 (1985). 4. R. Notzel, N. Ledentsov, L. Daweritz, M. Hohenstein, and K. Ploog, Phys. Rev. Lett. 67, 3812 (1991). 5. D. Leonard, M. Krishnamurthy, C. Reaves, S. Denbaars, and R Petroff, Appl. Phys. Lett. 63, 3203 (1993). 6. R Petroff, A. Gossard, R. Logan, and W. Wiegmann, Appl. Phys. Lett. 41, 635 (1982). 7. A. Efros, Sov. Phys. Semicond. 16, 702 (1982).

5

8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

ADVANCES IN QUANTUM DOT STRUCTURES

213

M. Grundmann, O. Stier, and D. Bimberg, Phys. Rev. B 52, 11,969 (1995). U. Woggon, Festkitper Problem 35, 175 (1995). J. Scott, S. Field, M. Kastner, H. Smith, and D. Antoniadis, Phys. Rev. Lett. 62, 583 (1989). M. Kastner, S. Field, U. Meirav, J. S. Thomas, D. Antoniadis, and H. Smith, Phys. Rev. Lett. 63, 1894 (1989). A. Palevski, C. P. Umbach, and M. Heiblum, Phys. Rev Lett. 62, 1776 (1989). G. Frazier, "Concurrent Computations." Plenum, New York, (1988). H. Wu and D. Sprung, J. Appl. Phys. 84, 4000 (1998). R. Steffen, Th. Koch, J. Oshinowo, F Faller, and A. Forchel, Appl. Phys. Lett. 68, 223 (1996). M. Ezaki, H. Kumagai, K. Toyoda, and M. Obara, Proc. SPIE 2125, 344 (1994). M. A. Reed, Sci. Am. March (1993).