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"Tsu follows the development of superlattices and quantum wells from their inception in 1969. He expects readers to have working knowledge in basic mathematics such as complex variables and partial differential equations; some skill in computer programming; and intermediate to advance courses in electromagnetics, quantum mechanics, and solidstate and semiconductor physics. Starting with superlattices, he progresses through resonant tunneling with artificial quantum well states; optical properties and Raman scattering in artificial quantum systems; dielectric function and doping of a superlattice; quantum step and activation energy; semiconductor atomic superlattices; silicon quantum dots; capacitance, dielectric constant, and doping quantum dots; porous silicon; some novel devices; the quantum impedance of a electrons; and why super and why nano."Reference and Research Book News
"This book is an update of a volume by the same name first published in 2005. It does form one of the most definitive descriptions of the physics underlying these new materials. It is also more than that, because it gives readers a lot of fresh insight to the behaviour of electrons in crystalline solids. Much of this book is ideal for assisting lecturers and tutors in putting across some of the more difficult concepts to advanced students… Overall some of the new additions make fascinating reading because Tsu relates to the reader in a very personal style…."Contemporary Physics
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Superlattice to Nanoelectronics
By Raphael Tsu
Elsevier
Copyright © 2011 Elsevier Ltd.All right reserved.
ISBN: 9780080968148
Chapter One
Superlattice1.1 The Birth of the ManMade Superlattice
To appreciate why manmade superlattices were conceived, we need to understand why of all the elements in the periodic table, only a handful of elements are suitable for electronic devices. Before the advent of semiconductor devices, vacuum tubes started the electronic revolution, whereby electrons are emitted into the vacuum from a heated filament, intercepted by a controlled grid, and collected by an anode. Even with all the wonderful functions that semiconductor devices can offer, particularly in integrated circuits (ICs), even today many highpower devices—like magnetrons used as electromagnetic generators for high frequency and traveling wave tubes used as highpower and highfrequency amplifiers—are still in use. However, hot filament is replaced by the search for a cold cathode. Other endeavors consist of getter for adsorption of the residual gas inside a vacuum tube and metals capable of stable operation at higher temperatures. After the introduction of solidstate devices such as the transistor, a search for appropriate materials launched a new discipline, material science, a term introduced by Gerald Pearson. He joined Stanford University from Bell Telephone Laboratories (BTL) after contributing to the first transistor. Even though incredibly rapid development has led to the present IC, a planar structure consisting of literally millions of circuits known as the chip, only a handful of materials are used, and of these mainly silicon.
Let us briefly summarize what types of material are involved in the modernday electronic revolution: metals for contacts, semiconductors for active components of a device, and high bandgap materials for insulation. When I first started my career at BTL, I was told that we should all look for new materials rather than inventing new schemes for devices. Group IV covalent materials like Si and Ge are good for basic transistors; Group IIIV semiconductors, GaAs for example, are used for detectors and photonic devices; Perovskite structures such as barium titanate provide high dielectric constants; lithium niobates are good for nonlinear optical devices; and rareearthdoped materials such as Nddoped YAG garnets are used for highpower lasers, and so on. Specifically, injection lasers were limited to GaAs, the best LED was GaP utilizing nitrogendoped bondexcitons, highfrequency transducers used quartz, and the best photoconductor used CdS, a highly defective Group IIVI compound semiconductor. Some crystals, such as hexagonal SiC, have a number of different structural forms known as polytypes (Choyke et al., 1964), forming a natural superlattice structure with a period ranging from 1.5 to 5.3 nm; however, these polytype structures cannot be controlled and the resulting energy gaps are too small to provide any useful electronic novelty. These considerations were the leading reasons that drove me and Esaki to contemplate manmade solids.
In attempting to create a manmade solid, at the onset we recognized that mimicking the translational symmetry can best be physically realized using a planar structure that has a modulation of potential energy only in one direction, the direction normal to the planar layers. This modulation can be achieved either by a periodic pnjunctions or layers of A/B, two different materials arranged in a periodic way. To form an artificial energy band structure, the distances involved must be smaller or at least no greater than the junction width of a tunnel diode (Esaki, 1958); more precisely, the mean free path of the electron must be at least greater than the period of modulation in order to preserve phase coherence. Let us take a closer look at the comparison between the pnjunction and the heterostructures in forming a modulated potential variation. Almost all semiconductor devices, transistors, detectors, transducers, and switches were based on pnjunctions, a junction separating two different forms of doping, n for negative owing to the surplus electrons in the conduction band, and p for positive, owing to the deficiency of electrons in the valence band. Doping, such as phosphorus on silicon sites, results in ionization, at normal operating temperatures, of the extra electrons (not needed for fourfold coordination) into the conduction band, contributing to free carrier transport. A junction is formed when dopings on both sides develop a junction voltage caused by the alignment of the Fermi levels. Extra electrons ionized into the conduction band from the phosphorus dopant sites in ndoped Si fall into the boron dopant sites in pdoped Si, leaving positive charges in the vicinity of the nside and negative charges in the vicinity of the pside, producing an electric field across the junction. At equilibrium, the potential developed across the junction prevents further transfer of charges from the ndoped side to the pdoped side. We must recognize that the potential difference is spread over the depletion width of the pnjunction, because transferring charges involves a monopole with a range extending over a fairly long distance. Heavier doping reduces the distance needed to acquire sufficient charges to move the potential. However, the solid solubility limit prevents the excessive doping needed to reduce the depletion width below the mean free path, or in general, the coherence length of the electrons. Simply put, the solid solubility limit arises because whenever two P atoms are next to each other, they would rather be bonded metallically than covalently, resulting in no contribution to the extra electron in silicon in conventional ndoped silicon. The same reason applies to Bdoping in Si, which results in extra holes, a deficiency in electrons for pdoped silicon. Before we touch on the heterostructures used in superlattices and quantum wells, we need to develop an appreciation of why heterojunctions that have a potential difference on each side maybe well defined or "sharp" to within a few tenths of a nanometer. We know that the bandedge offset between two materials cannot be predicted using only an argument involving the alignment with respect to the vacuum level, which is based on the work functions of the two materials. Usually, one is left with measurements for the bandedge offset, or some complicated abinitio calculation such as the use of the density functional theory. What I want to convey is a simple rule as to why the heterojunction maybe very sharp. We know that in a multipole expansion of a potential functions, the monopole term falls off more slowly than the dipole term and the dipole term falls off more slowly than the quadrupole term, and so on. Whenever positive and negative pairs are neutralized in a given region, more precisely, in a unit cell of the solid, it is the multipole potential that an electron sees, thus, a much sharper potential profile is experienced by an electron. Heterostructures are neutral and therefore are quite similar to multipoles. In a manmade solid, the period must be less than the mean free path, which is no more than few tens of nanometers at room temperature. Except at temperatures close to 0 K, doping the superlattice can barely make the grade. The need to make contact to a given active region of a device calls for a planar structure with alternating layers of two materials that have a sufficient bandedge offset, mimicking a periodic variation along the direction perpendicular to the layers. In short, the criterion of a manmade superlattice introduced by Esaki and Tsu (1969, 1970), is to develop alternating layers of A/B, having sufficient bandedge offset at a distance well below the period of the layers. Being a planar structure, input/output, has as usual, contacts capable of handling large current density.
Figure 1.1 shows the original drawings depicting the two cases of modulation. Because of the reasons elaborated herein, only the heterojunction scheme was pursued at IBM Research.
1.2 A Model for the Creation of ManMade Energy Bands
We represented the periodic potential V(x) = V(x + nd) with a period d typically 10–20 times greater than the lattice constant a in the host solid. We assumed that the wave equation is that of a free particle Schrödinger equation with an effective mass m and a potential term V(x). With variable separable, we only deal with the xdirection,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
A general solution for a periodic V(x) results in the Bloch state,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)
For the sinusoidal case, V(x) = V_{1}(cos 2k_{d}x 2 1), Eq. (1.1) becomes the Mathieu's equation, well described by McLachlan (1947):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)
where the reduced energy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the dimensionless momentum [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with k_{d} = π/d. We used a sinusoidal variation of the modulation potential instead of a periodic squarewell potential at the very beginning because we thought that the potential variation even with heterojunctions maybe quite "soft" owing to interdiffusions. The comparison of the sinusoidal potential modulation and the periodic squarewell potential is shown in Figure 1.2, for the parameters shown in the figure. Note that the difference for the dispersion relation was not significant leading us, exclusively in later work, to use the periodic square potential, calculated with the KrönigPenney potential (Smith, 1961), for subsequent model calculations. Note that the Brillouin zone (BZ) boundary is reduced from π/a to π/d, resulting in the formation of the minizones. For d ~ 20a with a being the lattice constant of the host solid, the minizone is 20 times smaller allowing electrons reaching the minizone boundary to create some intriguing transport under tolerable values for the electric field.
Before we proceed to show the transport property, current versus applied voltage in the next section, I want to discuss several points that led to the resolution of some initial concerns. We pointed out that we were not pursuing the doping superlattice, instead concentrating on the A/B alloys. The Group IIIV compounds of GaAs for the well and GaAlAs for the barriers were selected because these materials have been used for the double heterojunction (DH) lasers developed at BTL where GaAIAs was used to confine the charge carriers in the socalled DH lasers. We must distinguish between quantum confinement, which Esaki and I were seeking, and charge confinement, used in the DH lasers. The first problem developed because we were told that GaAs and GaAlAs form an ohmic junction. If this were true, it would not have been possible to serve as a superlattice potential modulation. I was very concerned, but Esaki set my concerns aside. He said to me, "Experts are not always right." The next big step was asking, short of measurements, what values can one use for the modulation V_{1} in Eq. (1.1)? Esaki called on Frank Herman, who was in charge of the Department of Large Scale Computation at IBM Research at San Jose. He used an LCAO calculation and presented us with his result that 80% of the band gap aligned with the conduction band and 20% with the valence band. In other words, we could take 80% of the difference of the band gap of GaAs and GaAlAs as the modulation parameter for the squarewell potential in the conduction band. However, he warned us that his calculation was based on alignment of the 1S state of the As atom in GaAs and the 1S state of the As atom in GaAlAs, so that his error bar could be as high as 1 eV. Again I was shocked, but Esaki quickly reassured me that his only concern was to ask Herman to wait before submitting his calculation for publication until we had had a chance to demonstrate our superlattice. As I pointed out in the Introduction, one of my goals in writing this book is to provide a fairly detailed discussion of the difficulties and stages we passed before this venture succeeded. The preceding concerns can provide some insight as to the extent that one needs to stand firm. Frankly, without Esaki's experience and forcefulness, I would have given up.
1.3 Transport Properties of a Superlattice
A simplification by Pippard (1965) of the path integration method of Chambers (1952) was used to obtain current versus applied field F. The equations of motion are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)
and the drift velocity is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)
Taking a sinusoidal Ek relationship, the socalled tightbinding dispersion relation,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)
in which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and k_{d} = π/d. We see in Figure 1.3 that the drift velocity versus the applied field F reaches a maximum value and beyond this point the slope is negative and the socalled negative differential conductance (NDC) appears. What happens is that electrons driven to the BZ boundary turn around because of the Bragg reflection. Without scattering, τ > ∞, oscillation results in zero constant current. With scattering, a constant current appears, but decreases with increase of ξ. Therefore, the source of the NDC is precisely what leads to oscillation, the socalled Bloch oscillation. It is probably unknown to most researchers today that Krömer (1958) proposed using the heavyhole band in semiconductors to create a negative effective mass amplifier. However, the scheme was not developed owing to difficulty in controlling the presence of transverse negative masses. In any case, this book deals with the manmade superlattice that has a negative effective mass only along the direction of the superlattice.
1.4 More Rigorous Derivation of the NDC
A more rigorous derivation involving the Boltzmann equation with a tightbinding Ek relationship was obtained 1 year after the simple derivation using the impulse method of Pippard (1965), presented in the last section as suitable for low carrier concentration. This was the version that formed the basis for launching the manmade superlattice program at IBM Research. Esaki and I made a survey of what properties maybe exclusively attributed to the formation of a manmade superlattice. We decided that the appearance of an NDC serves as the key criterion. The basic assumption involves essentially the use of twodimensional electron gas (2DEG), where the Ek in the transverse direction is parabolic, free electron energy momentum, and scattering time is assumed to be energy independent. The latter assumption is in fact quite acceptable owing to the cancellation, to some degree, of the primary scattering from acoustic phonons and impurities. In most elementary treatments of the Boltzmann transport equation, a simple shifted distribution results in Ohm's law. Lebwohl and Tsu (1970) found that for a onedimensional nonparabolic Ek relationship in the Boltzmann equation, an exact solution is possible for constant relaxation time. They showed that their result for FermiDirac distribution is identical to the proof by Budd (1963) of Chambers's (1952) path integral method for Maxwellian distribution.
Taking the electric field F in the xdirection, the Boltzmann transport equation without the time and spatial variation becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)
in which k_{0} = eFτ/[??] and f_{0}(k) is the equilibrium distribution function. The periodicity of the crystal along F is described by the reciprocal lattice vector K. The energy bands are assumed to be:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where k]??] is the component of the wave vector perpendicular to the direction of the superlattice and E_{x}(k_{x}) is periodic in k_{x} with period K. Eq. (1.8) has the general solution
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)
(Continues...)
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