Invitation to a Contemporary Physics (2004)

of nanostructures and their assembly into materials, into a single process.3 A few of the methods may also be rightly considered biomimetic in the sense that they seek to imitate some process in nature, for example in its use of seeds, templates or self-assembly for nucleation and growth. A completely di erent kind of tool is the computer, which now pervades all disciplines. But it is more than just a tool to analyze data or test models; it has become an integral approach to problem solving: it can predict outcomes for experiments too di cult to carry out, test structural designs, or do molecular simulations for materials.

The third factor contributing to the rise of nanoscience is a better understanding of the precise way the macroscopic properties and the functionality of materials and structures are related to the size and arrangement of their components. For example, it is crucial that we know exactly how the size and location of metallic or semiconducting nanocrystals in a material a ect its optical and electronic properties if we intend to implant atoms into appropriately sized dots at precisely controlled sites in order to obtain a product with the desired attributes. The new, finely engineered material may lead us to observe novel properties that we would not have otherwise suspected, which in turn may point us to other inventions. It is this kind of cross-fertilization of the fundamental and the applied that makes nanoscience and nanotechnology such an exciting and fast-moving field.

5.2.1Summary

The recent rise of nanoscience stems from the confluence of three important new scientific and technological developments: novel characterization tools (microscopes capable of atomic-scale resolution), advanced fabrication techniques (deriving from physics, chemistry and biology) and a better understanding of the basic science (aided by mathematics and computing science).

5.3 Confined Systems

In order to gain some understanding of nanostructures, we will find it fruitful to regard them as confined systems, that is, systems where the motion of the relevant microscopic degrees of freedom is restricted from exploring the full threedimensional space. The confining spatial dimensions need not and, in fact, cannot vanish, because of the quantum uncertainty principle (Problem 5.1). An electron or a photon in a semiconducting film with a thickness of the order of the particle’s characteristic wavelength is practically confined to a plane, having no freedom of motion in the transverse direction. Particles confined in an ultrathin wire are free

3Examples of chemical methods include chemical reduction, thermal decomposition, electrochemical processes, use of crystalline hosts, molecular-cluster seeding, and self-assembly. Among the physical or physico-chemical methods, we have lithography, ball-milling, scanning-probe methods and gas-phase condensation. See the Glossary for the definitions of some of these terms.

to evolve in only one dimension, whereas those confined in a nanosized box are constrained in all three directions.

5.3.1 Quantum Effects

Consider a piece of crystalline solid: it consists of a very large number (like 1021) of atoms sitting very close to one another in regular arrays. Each discrete energy level for the electrons in an isolated atom now broadens, by interactions with all other electrons and atoms in the solid, into a band of some 1021 closely packed energy levels. The bands become wider with increasing energy and are usually separated from one another by gaps (the remnants of the original atomic energy spacings), although in some materials a pair of adjacent bands may overlap. An electron may have an energy that lies in one of the bands but not in any of the gaps. The occupied band highest in energy is called the valence band (and the top filled level is the Fermi level), and just above it lies the conduction band, which contains allowed but very sparsely occupied levels. The relative locations and the degrees of occupation of these two particular bands determine many of the physical properties of the material. When an electron has an energy near the bottom of the conduction band or near the top of the valence band, it is a good approximation in the absence of applied fields to regard it as a free particle propagating with an e ective mass, m , smaller than its actual mass, i.e., with less inertia than it does in vacuum. As the parameter m takes di erent values with di erent bands, we have two distinct continuous energy spectra associated with the valence and the conduction electrons, just as experimentally observed.

Now, what happens when space shrinks, say, to an interval d of a few nanometers in the z direction? The electrons move freely as before in the xy plane, but are now confined in the third direction between two energy barriers separated by d, forming what is called a potential well. (The conduction and valence electrons evolve in separate potentials.) If we assume the barriers to be infinitely high, the wave functions that represent the particles in the z direction must fit snugly within those confines, vanishing at the interfaces, and hence can only have an integral number of half-wavelengths across the width of the quantum well. It follows that the only allowed de Broglie wavelengths are λ = 2d/n, where n = 1, 2, 3, . . . . So the motion of an electron across the layer is quantized, with allowed momenta pz = h/λ = nh/2d and corresponding energies En = p2z/2m = n2h2/8m d2, where h is Planck’s constant. As the energy for the in-plane motion remains essentially the same as in bulk volume, the total energy of an electron in the thin film is given by

p2 E(n, p ) = En + 2m ,

where p denotes the in-plane component of the momentum. This energy spectrum consists of a series of overlapping continuous bands (technically, subbands, i.e., small

Figure 5.3: Energy spectra and densities of states of systems with dimensionalities 3, 2, 1 and 0. The symbol EF stands for the Fermi energy, the energy of the top occupied level.

found at some energy per unit ‘volume’ per unit energy. It turns out that in three dimensions the density of states starts from zero at zero energy and increases slowly and smoothly with energy; by contrast, in the two-dimensional (2D) case it increases in equal steps, staircase fashion, rising at the successive quantized energies of the confined motion (Fig. 5.3). The DOS is a key factor in calculating transition rates. As it is non-vanishing in 2D even at the bottom level, many dynamical phenomena, such as scattering, optical absorption and gain, remain finite at low energies and low temperatures.

In the preceding discussion, we have assumed ideal confinement, i.e., the walls of the potential to be infinitely high. When they are finite, the results change little, at least qualitatively, for the lowest levels; but whatever changes there are, they may prove significant in certain situations. Then, the particle wave functions do not vanish at the interfaces, but rather continue on and penetrate the barriers, diminishing rapidly in an exponential decay, exactly the e ects exploited in scanning tunneling microscopy. The confinement energies decrease with the heights of the potential walls, and the lower the barrier is or the higher the energy level, the deeper the penetration. And, if the barrier walls are thin enough, the wave functions may even go all the way through the walls, providing a mechanism for electrons and holes to escape from the well. Quantum wells and quantum barriers can be combined to build interesting quantum structures and devices.

Quantum wire is the generic name for quasi-one-dimensional (1D) systems with one direction for free motion and two directions of confinement. Its energy spectrum and DOS are qualitatively similar to those for quantum wells, with more subbands in the spectrum and a sawtooth profile for the DOS. The situation changes substantially for quantum dots, the quasi-zero-dimensional (0D) structures whose sizes are of the quantum scale all around. Here, the energy spectrum consists entirely of discrete levels, like rungs on a ladder, and the DOS of discrete sharp peaks, like posts in a fence (Fig. 5.3). In general, as the dimensionality is reduced in steps from three to zero, the energies become more and more sharply defined and the densities of states progressively more singular, more condensed at particular energies.

5.3.2How to Make them

The most common method of making ultrathin films of crystalline solids, usually the first step in the fabrication of many lower-dimensional systems, is a high-vacuum evaporation technique called molecular beam epitaxy (MBE). Think of it as spraypainting atoms onto a flat surface. In an ultrahigh-vacuum chamber, place a heated millimeter-thick crystalline (e.g., GaAs, InP) layer; spray it with jets of gases of the substances (Si, Ga, As, In, etc.) you want to deposit. The substrate provides a template for the arriving atoms and mechanical support for the final structure; the high vacuum eliminates unwanted impurities; and the use of regulated jets provides flexibility in the choice of the materials and control of the composition and thickness of the film. In this way, you can produce single or multiple (semiconducting, as well as insulating or conducting) layers of atoms of precise thicknesses and compositions.

To complete the construction of a semiconductor nanostructure in this method, it is necessary to follow growth with lithography. This advanced etching technique, similar to that applied in the fabrication of state-of-the-art integrated circuits, is used to transfer a desired pattern from a master copy (mask) into a protective polymer layer (resist) coating the semiconductor surface. After the resist has been irradiated by focused X-rays, electron or ion beams through the mask, its weakened exposed regions are washed o with solvent, and the semiconductor is etched away where it has now been exposed by the removed resist. The patterned semiconductor structure can be further engineered (doped, metallized or etched further) for targeted applications. Insulating tunnel barriers and metallic electrodes, or ‘gates,’ are often integral parts of the structure and fabricated with the same techniques.

Take, for example, the growth of a layer of aluminum-gallium arsenide (AlcGa1−cAs) on a gallium arsenide (GaAs) substrate. This combination of two lattice-matched semiconductors, having nearly identical atom-to-atom spacing, together with a careful crystal growth will guarantee a defect-free, stress-free, and hence high-quality interface. The alloy AlcGa1−cAs consists of periodic arrays of arsenic atoms together with a fraction c of aluminum and 1 − c of gallium. The aluminum fraction, controlled during growth, determines the energy band gap of the

alloy as well as the shape of the confining potential. Keeping it constant throughout the deposition leads to an abrupt interface and hence a square energy barrier, whereas varying it spatially during the fabrication process produces a compositionally graded material and hence a z-dependent potential well.

A quantum-well structure is built by sandwiching a thin (say 10 nm) layer of GaAs between two thicker layers of AlcGa1−cAs (Fig. 5.4). These two latticematched semiconductors di er slightly in their energy band gaps and so also in the energies of their free electrons. An aluminum concentration around 30% gives AlGaAs a band gap larger than that found in GaAs by 360 meV, split roughly 2:1 between the conduction and valence bands, so that electrons see a 240 meV energy barrier and holes see 120 meV. With potentials that deep, quantum size e ects are easily observable even at room temperatures.

There are no free electrons that move about in pure semiconductors at low temperatures, all of them being consumed by the bonds that hold the solid together. However, by adding during the crystal growth a small number of silicon atoms in the AlGaAs compound at a distance of about 0.1 µm from the interface (a process known as modulation doping), the outer-shell electrons of the Si atoms escape into the GaAs layer, which has lower-energy electron states, and move freely about in this highly pure crystal, unimpeded by their then-ionized parent impurities, which remain behind in the AlGaAs layer on the other side of the barrier. The dipolar

Ga As

(c)

Figure 5.4: (a) Modulation-doped heterojunction. (b) Quantum well in a three-layer doubleheterostructure. (c) Formation of a 2DES in the potential well at the Si:AlGaAs/GaAs interface.

Much of the initial research and development on quantum-well devices concentrated on the GaAs/AlGaAs and GaAs/AlInGaP combinations which operate e ciently in the red–yellow optical region. However, the versatility of MBE and other growth techniques allows us to explore other materials and engage in bandstructure engineering, where we tailor artificial electronic states and energy bands to our design by programming the growth sequence of the MBE machine. An interesting example is the long search for suitable blue-light emitting materials, which has led eventually to the successful development of e cient devices based on gallium nitrides. The new light-emitting diode is a structure consisting of a 3 nm thick layer of indium-gallium nitride (In0.2Ga0.8N) sandwiched between a p-type (hole-rich) layer of aluminum-gallium nitride (AlGaN) and an n-type (electron-rich) layer of gallium nitride (GaN), all grown on a sapphire substrate. The InGaN film forms a quantum well into which are fed electrons and holes by the surrounding materials. The charge carriers go into their respective bottom (n = 1) subbands, and recombine to emit blue and green light. By combining red-, blueand greenlight emitting diodes of comparable power and brightness, it is possible to make full-color displays and e cient white-light sources that work longer, use less energy and, best of all, can be coupled to computers for novel applications.

While the InGaN LED consists of a single quantum well, the corresponding laser is a superlattice, or system of 2–10 coupled quantum wells formed by alternating thin layers of InGaN and AlGaN or GaN. As with most other absorption devices based on quantum wells, it is necessary to couple several wells together in order to boost the absorption rate to a sizeable level. This type of laser can operate continuously at a single wavelength in the 390–500 nm range at room temperatures. These short wavelengths should give decisive advantages to many applications. For instance, as CDs and DVDs now on the market rely on red semiconductor lasers to read the stored information, their data packing density is always limited by the size of the focused laser spot, but it can be increased when shorter wavelengths become more widely available. The data storage capacity of single-layer DVDs, now at 4.7 Gbytes, could be improved at least threefold by moving to blue light, putting high-definition DVDs within the general consumer’s reach.

The presence of subbands in quantum wells gives us the possibility of having intersubband optical transitions when a conduction electron, excited into an upper subband, relaxes to a lower subband and emits a photon with an energy roughly equal to the energy di erence between the two subbands. We expect the photon emitted to be in the mid-to-far-infrared spectrum, which includes wavelengths to which the atmosphere is transparent. A device based on intersubband transitions would di er in a fundamental way from conventional LEDs, because it would rely on only one type of carrier (electrons) and on transitions between energy levels arising from quantum confinement (and not transitions across a band gap).

Such a novel device has been developed by Frederico Capasso and his collaborators at AT&T Bell Laboratories in the US. This beautiful piece of technology,

Active region

Figure 5.5: An active region of a quantum-cascade laser.

which they called a quantum-cascade laser, is a closely coordinated complex of 500 perfectly calibrated layers of GaInAs (gallium-indium arsenide) and AlInAs (aluminum-indium arsenide) grown by MBE (Fig. 5.5). The central region consists of 25 identical stages, acting like steps in an energy staircase, and each stage is in turn made up of an optically active region and an injection–relaxation region. Each active region consists of three quantum wells, which are nanometer-thick layers of GaInAs flanked by insulating layers of AlInAs. The wells di er slightly in thickness and hence in energy for corresponding levels. When an appropriate electric field is applied, electrons are injected by tunneling into the n = 3 excited state of the first well, where they stay for a relatively long time (like 4 ps), before tunneling through an AlInAs layer into the second well, dropping into the n = 2 level and emitting an infrared photon (of about 300 meV). The electrons then drop very quickly (after, say, 0.5 ps) to the n = 1 subband of the third well (releasing a quantum of heat of about 30 meV), from which they escape equally quickly into the injection–relaxation region, before being re-injected into the n = 3 state of the next well to start another stage all over again, and so on down the cascade. Thus, for each electron injected, 25 photons will be produced over the whole cycle. Population inversion between the two upper states responsible for the lasing action is ensured by the long relaxation time of the uppermost state and the extremely short tunneling escape times out of the two lower states. Each injection–relaxation region is a compositionally graded AlInAs–GaInAs superlattice designed so that the energy levels between the steps are arranged in a descending order; its role is to convey the electrons from one active