Superlattices: problems and new opportunities, nanosolids
 Raphael Tsu^{1}Email author
DOI: 10.1186/1556276X6127
© Tsu; licensee Springer. 2011
Received: 11 August 2010
Accepted: 10 February 2011
Published: 10 February 2011
Abstract
Superlattices were introduced 40 years ago as manmade solids to enrich the class of materials for electronic and optoelectronic applications. The field metamorphosed to quantum wells and quantum dots, with ever decreasing dimensions dictated by the technological advancements in nanometer regime. In recent years, the field has gone beyond semiconductors to metals and organic solids. Superlattice is simply a way of forming a uniform continuum for whatever purpose at hand. There are problems with doping, defectinduced random switching, and I/O involving quantum dots. However, new opportunities in componentbased nanostructures may lead the field of endeavor to new heights. The all important translational symmetry of solids is relaxed and local symmetry is needed in nanosolids.
Introduction
Of all the thousands of minerals as jewelry, only a few are suitable for electronic devices. Silicon, in more than 95% of all electronic devices, GaAsbased IIIV semiconductors, in the rest of the optical and optoelectronic devices, and less than 1% used in all the rest such as lasers, capacitors, transducers, magnetic disks, and switching devices in DVD and CD disks, comprise a very limited lists of elements. For this reason, Esaki and Tsu [1, 2] introduced the concept of manmade superlattices to enrich the list of semiconductors useful for electronic devices. In essence, superlattice is nothing more than a way to assemble two different materials stacked into a periodic array for the purpose of mimicking a continuum similar to the assemble of atoms and molecules into solids by nature. Although it was a very important idea, the technical world simply would not support such activity without showing some unique features [3]. We found it in the NDC, negative differential conductance, the foundation of a high speed amplifier. In retrospect, manmade superlattice offers far more as well as branching off into areas such as soft Xray mirror [4], IR lasers [5], as well as oscillators and detectors in THz frequencies [6]. The very reason why such venture took off is because the availability of new tools such as the molecular beam epitaxy, MBE, with in situ RHEED, better diagnostic tools such as luminescence and Raman scattering, the all important TEM and SEM, etc. After the introduction of scanning tunneling microscopy, STM; and atomic force microscopy, AFM, stage is set for further extension of quantum wells, QWs, into threedimensional structures, the quantum dots, QDs. The demand of nanometer regime is due to the requirement of phase coherency: the electrons must be able to preserve its phase coherency at least in a single period, on reaching the Brillouin zone in kspace. However, we shall see why new problems developed in reaching the nanometer regime. First of all, when the wave function is comparable to the size, approximately few nanometers in length, it is very similar to a variety of defects. Strong coupling to those defects results in random noise, the telegraph switching [7]. Thus we are facing great problems in pushing nanodevices. However, some of the new frontiers in these nanostructures are truly worthy of great efforts. For example, chemistry deals with molecules largely governed by the symmetry relationship within a molecule. In solids the symmetry is governed by the translational symmetry of unit cells. Now, with boundaries and shape to contend with, we are dealing with a new kind of chemistry, involving the symmetry of surfaces and boundaries as well as shapes. For example, we know that it is unlikely a tetrahedralshaped QD may be constructed with individual linear molecules. Catalysis is still a matter of mystery even today. Now we are talking about adding boundaries and shape for nanochemistry. The possibility of crossing over to include biological research of nanostructures is even more spectacular, which will ultimately lead mankind into the physics of living things.
Problems
Response of a superlattice: DC and AC
When are the full Bloch waves needed?
Therefore, for Type III, the traditional reflection coefficient R and transmission coefficient T involve the U and V Bloch functions. In general, Bloch waves should be used. The smaller the period, the larger is the interaction resulting in coupling and larger bandgap. Therefore, Type III gives rise to bandgap by design. However, if the period > coherent length, the system returns to semimetallic [12].
Resonant tunneling in a single quantum well with double barriers
Some important issues in resonant tunneling
where ϕ is the total phase shift and θ is the phase of the transmission amplitude through the DBRT structure. The delay time τ for a structure obtained from solving the timedependent Schrodinger equation using Laplace transform is in fact close to the approximate values in Equation 9.
There is an important point. The computed transmission time generally oscillates during the initial time, reaching several orders of magnitude down from the delay time. If the energy is at resonance, the delay time rises and overshoots approximately 8% [14] and settles down to the delay time using Equation 9, something quite familiar with most transient analyses. In timedependent microwave cavity with E & M waves, there is generally similar delay in time response at resonance. And at off resonance, a small fraction does get through quickly, but many orders of magnitude down. There is no need to argue about tunneling time as during 1960 s. If we really need to know, particularly with special circumstances, we should solve the timedependent wave equation using Laplace Transform.
There were issues concerning onestep resonant tunneling through a DBQW and twostep process [12] pointed out that Luryi's twostep process is almost indistinguishable from resonant tunneling when the loss factor is fairly high, which is the case for most DBQW.
There are many issues concerning resonant tunneling. For example, some sort of intrinsic instability was suspected. However, at the end, it was resolved by recognizing that the very scheme for setting off the heavily doped contact away from the DBQW structure introduces an extra QW under bias [15].
Conductance from tunneling
where the sum is over the transverse degree of freedom (n, m), or integration in dk _{t} of the transverse channels, and $T\equiv T*T({{k}^{\prime}}_{l}/{k}_{\ell})$. The conductance per transverse channel is ${G}_{nm}=\frac{{e}^{2}}{h}{T}_{nm}$. If in each transverse channel T _{ nm } = 1, then ${G}_{nm}={G}_{0}=\frac{{e}^{2}}{h}$, the socalled quantum conductance. This last assumption is frequently made; however, it is noted that the condition T _{ nm } = 1 gives zero reflection, which happens near resonance and is contrary to the assignment of a contact conductance. In transmission line theory the only reflectionless contact is one with the input impedance exactly equal to the characteristic impedance, a wave impedance of the line. Let us discuss more in detail. First of all, an impedance function is merely a special case of a response function or transfer function for the input/output. Therefore, there is no such thing unless two contacts are involved serving as input and output. The impedance or conductance has been referred to as contact conductance by Datta [16]. In reality it is not a contact conductance. If T = 1 is taken, then Equation 10 applies to reflectionless. The real issue is why experimentally equal steps of G _{0} appear? I think the answer lies in the fact that all the transverse modes are not coupled with a planar boundary...More precisely, G _{0} is the conductance of the quantum wire with matched impedance at the input end and terminating in the characteristic impedance, the wave impedance for electron, therefore also matched at the output end. Letting a wave bounce between two reflectors adopted by Landauer [17] for the conductance is a special case of the general timedependent solution [12].
Noise and oscillation in coupled quantum dots
Capacitance, dielectric constant, and doping of QDS
Due to complexity, the quantum mechanical computation was carried out only up to two electrons. I have been trying to find a student with strong computational skill to expand the QM computation to at least 12. Nevertheless, I can say something about. Capacitance is monophasic, i.e., each additional electron defines a single phase. And since the dielectric constant is much reduced in quantum mechanically confined systems, primarily because dielectric screening requires electrons or dipoles to move to cancel the applied electric field. Highly confined system reduces the movement, thereby reducing the screening. Now doping is basically possible because high dielectrically screened systems have very low binding energy, allowing carriers to be thermally excited at room temperatures. With drastic reduction of screening, the binding energy is too high, leading to carrier freezeout at room temperatures. This is apart from the problem involving statistical factor due to the drastic reduction in size of the QDs. Doping is impossible.
Summary of problems
In fact these problems discussed are serious, however, the most serious problem is I/O [14, 20]. We reduce size to minimize realestate. However, contacts are equal potentials, which call for metals. Nanosize metallic systems may be insolating, apart from the problem in lithography. Most of the benchtop demonstrations of Nanoelectronics have inplane device configurations, not a real device. At this point I can conclude that with all the talk of nanoelectronics, the merit is perhaps due to special features, such as the THz devices, the QCL, and the new expectations in graphenebased electronics. It is true that MOSFET has been reduced to below 30 nm for the sourcedrain length, but there are still approximately 400 electrons in the channelgate system, according to Ye [21]. Quantum computing is a somewhat unrealistic dream, because binary system makes computing possible, with the unique feature that on or off represents timeindependent permanent states.
Opportunities
Quantum cascade laser with superlattice components
Quantum Cascade Laser was first succeeded at BTL under F. Capasso [5]. The idea was even patented before BTL succeeded. However, the patented version would not work because when many periods are in series, any fluctuation can start domain oscillation as pointed out by Gunn many years ago. Therefore, I shall single out QCL as an example how the problem is checked by introducing components each controlled separately as in QCL, with the three major components, the injector, the optical transition from the upper state to the lower state, and the collector. That is the direction of the superlattice, divided into components, together functioning as a device. With the exception of resistive switches, almost all devices such as MOSFET, flash memory, detectors, etc. involve components. In fact, the first optically pumped quantum well laser using very thin GaAsAlGaAs QWs constitutes a step in the direction of utilizing QWs as components in forming a quantum device [22].
THz sound in stark ladder superlattices
Application of an electric field to a weakly coupled semiconductor superlattice gives rise to an increase in the coherent folded phonon, generated by a femtosecond optical pulse [23]. The condition is whenever the stark energy eFd > energy of the phonon, in this case, the FP phonon. Why did it take 35 years after the first article by Tsu and Döhler [24], to realize a phonon laser using superlattices? I want to make a comment from my years of doing research. Nobody is so brave in doing research in a relatively new field, although the instruments to fabricate devices involving superlattices are widely available. However, the complexity involved is sufficient in deterring most researchers. This study represents a step jump in the sophistication and careful design of the superlattice structure. I cannot fail to make a comment in regard to what Mark Reed told me about his study with pulling a gold wire while obtaining quantized conductance of the wire before it snapped. Some success is due to hard work, and others might be due to clever ideas and good timing. I would like to add from what happened today when Hashmi and I were jumping up and down for making a discovery. I said, "If you do something everybody else does, it is highly unlikely you would get anything new." The name of the game is to do something quite different!
Cold cathode and graphene adventure
Cold cathode using resonant tunneling involving GaN [25] and using a layer of TiO_{2} [26] seem very different, but in fact are very similar, because both involve storing electrons in a region close to the surface by raising the Fermi level to effectively lowering the work function and resonantly tunneling out into the vacuum. Such schemes can be readily achieved when nanosized regions are created. And in a broad sense, creation of a region, or a component in general, as a section with electrons coherently related to the boundaries containing them.
Some new opportunities
Beyond chemistry
As we know, chemistry deals with point group symmetry in the formation of molecules. When dealing with QDs, the boundary and shape of the QD provide extra symmetry relationships. Therefore, we are dealing with something new, which reminded me of the complexity of catalysts. Most catalysts have delectrons, because the hybridization of dp orbitals provides wide range of new possibilities to deal with symmetry configuration offered by the catalytic processes. I cannot help to imagine how a wide range of possibility opens up with nanosize QDs offering new shapes and boundaries to the wave functions. I for one am extremely interested in experiments enriching the understanding of the symmetry role in these quantum dots. For example, we can use ebeam lithography to produce arrangements of dots representing various symmetry to study catalysis (nucleation in material growth). As we know that RPA, random phase approximation, introduced by Bohm and Pines as a catch phrase, no more than the recognition of not being able to take into account of phase relationship in totaling an interacting system. Most engineers would simply acknowledge the approximation by adding squaremoduli to avoid cancellations. We do that in most constitutive relationships such as dielectric function, elastic constants, etc. We should be seriously considering the alternatives to adding squaremoduli, or simply put, not using RPA. We know that the most powerful amplifier is the parametric amplifier where we cannot simply add oscillator strength. Ed Stern told me once why EXAFS is so powerful, because, with a giant computer, one can account for multiple scattering without resorting to the use of RPA, or nearest neighbor even next nearest neighbor interactions. As we pursue the nanoscience with ever increasing vigor using modern instruments such as AFM and STM having piezoelectric control of distance measured in nanometers, I think we should be seriously considering 'beyond RPA'.
Beyond solid state physics
When we are working with a macroscopic entity, nature shows us the waytranslational symmetry, normally referred to as solid state physics. As we know that nothing is perfect so that we resort to statistics to arrive at an average such as current, flow, etc. for the description of causeeffect as voltagecurrent, so useful for the description as well as the design of devices. As the size shrinks to dimensions in nanometers, the defects may be no more than zero or one in such way that statistical average does not apply. Many of the benchtop experiments I mentioned depends on what and where the device is, and whether we can control them or not. We can use statistics if there are many such devices in an ensemble average, but not summing and averaging the individual scatterings! In simple term, translational symmetry does not play a part, and therefore, it is not solid state physics, but perhaps we should use the term nanosolid. Moreover, if the size is still represented by several unit cell distances, superlattice definitely is the only definable entity. In fact, even in the very first article [2], we pointed out that all one need is three periods in forming a superlattice, a QD in threedimension.
Beyond composite
We shall go beyond electronics and optoelectronics to include the consideration of mechanical composites without glue. I envision a new kind of composite material consisting of components such as nanoscale entities dispersed in a matrix forming a composite instead of using nanorivets or glue, bonded together chemically as in superlattices, e.g., amorphous carbon as matrix, with embedded QDs of silicates. This recipe is not far from coal put together by nature.
Beyond biology
Superlattices have already broken into organic substances. It is only time to get involved with living organisms such as chlorophyll. Basically, now we have the tool to do it. I conclude here with one thought: Survival of the fittest for biological evolution should not be impeded by 'intelligent human technological advances' in nanoscience. However, we must be supervigilant to avoid possible disasters to mankind.
Abbreviations
 ATM:

atomic force microscopy
 MBE:

molecular beam epitaxy
 NDC:

negative differential conductance
 QDs:

quantum dots
 QWS:

quantum wells
 RPA:

random phase approximation
 STM:

scanning tunneling microscopy.
Declarations
Authors’ Affiliations
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