Mesoscopic Physics and Electronics

Structures of integrated circuits are approaching fundamental limits and the next generation of devices might show unexpected properties due to quantum effects and fluctuations. A long history of research has shown that when features in electronic materials approach the size of electronic wavelengths, quantum effects manifest themselves in their properties. A new research field has been developed in which we pursue understanding of basic physics associated with such quantum structures, explore their controlability, and propose new devices. The purpose of this book is to give a comprehensive description of developments in this new field of mesoscopic physics and electronics.

One important length scale characterizing mesoscopic systems is the Fermi wavelength λ _F = 2π/k _F , where k _F is the Fermi wave number. At zero temperature, electrons occupy states specified by the wave vector k with |k| ≤ k _F . The Fermi wave vector is related to the electron density n through 1.2.1$$ n = \left\{ {\begin{array}{*{20}{c}} {\frac{2}{{{{(2\pi )}^3}}}\frac{{4\pi }}{3}k_F^3 (d = 3)} \\ {\frac{2}{{{{(2\pi )}^2}}}\pi k_F^2 (d = 2)} \\ {\frac{2}{{(2\pi )}}{k_F} (d = 1)} \end{array}} \right. $$ where d is the system dimension (0D for d = 0, 1D for d = 1, 2D for d = 2, and 3D for d = 3) and the factor 2 comes from the electron spin. In typical metals such as Cu and Ag, the Fermi wavelength is of the order of a few angstrom and in semiconductors such as 2D systems realized in GaAs/AlGaAs heterostructures we have λ _F ~400 Å for the electron concentration n~3×1011 cm−2.

Landauer derived the relation between the conductance of a one-dimensional (1D) wire and the transmission and reflection probabilities at the Fermi level [1]. Let us consider the system schematically illustrated in Fig. 1.3.1. A wire is connected at both ends to an ideal wire which is infinitely long and eventually connected to an electron reservoir. The ideal reservoir satisfies the following two conditions: (1)All incident electrons are absorbed by the reservoir irrespective of their energy and phase.(2)It constantly provides electrons with energy below chemical potential µ. The energy and phase of these electrons are independent of those of absorbed electrons.

The most remarkable phenomenon resulting from the wave nature of electrons is the Aharonov-Bohm (AB) effect [1,2], which not only reveals that electrons are waves but also demonstrates that the vector potential, but not the magnetic field, plays basic roles.

Some years ago, van Wees et al. [1] and Wharam et al. [2] found that the conductance of electrons through a narrow channel is quantized in the unit of 2e2/h, where e is the absolute value of the electronic charge and h is the Planck constant.

Mesoscopic physics has started with the study of various kinds of interference effects of the electronic wave, but the features of electrons as a particle also show up in various interesting phenomena. For example, consider a small tunnel junction connected to an electrical source of constant voltage V (Fig. 1.6.1a). Suppose an electron tunnels from one electrode to the other through the insulator between them. If the junction is very small, its capacitance is small too, unless the insulator between the electrodes is extremely thin. Then the charging energy e2/2C can be as large as temperature times the Boltzmann constant, and is not negligible. Therefore, the tunneling is not realized unless the voltage V is large enough for this energy to be compensated by the energy eV.

Landauer’s formula discussed in Sect. 1.3.3 gives the quantized conductance in an ideal quantum wire without impurities, 2.1.1$$ G = \frac{{2{e^2}}}{h}N $$ where N is the number of subbands which cross the Fermi energy EF. This formula was derived in noninteracting case. In one-dimensional systems, it has long been known that the mutual interaction has strong effects. In fact, each order of perturbation calculations has a divergence and thus the ground state is a non-perturbative one. Such a state is described as a Tomonaga—Luttinger liquid [1], which differs from the conventional Fermi liquid. It exhibits several anomalous behaviors such as a power-law decay of physical quantities and a spin-charge separation [2–5]. It is thus very interesting and important to study quantum wires and to find out such anomalous phenomena characteristic of the Tomonaga—Luttinger liquid.

The conductance of a perfect quantum wire in which N subbands (≡ modes) are occupied is 2e2N/h, as discussed in Sect. 1.5.2, but it will be reduced because of the scattering of electrons by impurities or boundary roughness (fluctuations of the width of the wire). Magnetoresistance provides useful information on the scattering mechanism in quantum wires. For instance, when a magnetic field is applied perpendicular to the wire, the electrons are pushed toward the boundaries and hence we can expect a considerable change in the scattering probability if the boundary roughness is a dominant scattering mechanism.

The magnetophonon resonance (MPR) effect has been used since the mid-1960s to investigate effective mass, optical-phonon energy and electron-phonon interaction in bulk III–V compounds [1–5]. Several authors have also studied MPR effect in two-dimensional (2D) systems, such as GaAs/AlGaAs, GaInAs/InP, and GaInAs/AlInAs heterostructures [6]. The MPR effect manifests itself as an oscillatory behavior of transverse-conductivity σ _xx as a function of applied magnetic field B. At high magnetic fields, electrons move perpendicularly to both electric and magnetic fields, and the transverse-current is carried by electron-hopping motion induced by some scattering mechanisms. The MPR effect arises from the resonant absorption or emission of longitudinal-optical (LO) phonons by electrons when the Landau level is sharp and well defined, and scattering by LO phonons makes a significant contribution to limiting electron mobility. As the magnetic field increases, a small part of σ _xx resonantly increases each time the integral multiple of the Landau level spacing becomes equal to the LO phonon energy, because the scattering of the electrons takes place with resonant absorption or emission of LO phonons. In bulk materials and 2D systems, the oscillatory part of σ _xx is therefore proportional to the Fröhlich coupling constant α, and the resonance condition where σ _xx becomes maximal is written as 2.3.1$$ \hbar {\omega _{LO}} = P\hbar {\omega _c}(P = 1,2,3, \cdots ),$$ where ħω_LO is the LO phonon energy, P the resonance index, and ħω_c = ħeB/m the cyclotron energy. From the measurement of the MPR effect, effective mass in or LO phonon energy ħω_LO can be deduced, provided that one of the two is known. In order to observe the MPR effect, the temperature must be high enough to have a sufficient phonon population, but not too high, since the thermal broadening of the Landau levels reduces the oscillation amplitude. Optimal temperatures are known to be usually in the range 100–250 K for III-V materials.

When a small dot is weakly coupled to reservoirs via small tunnel junctions, addition of an extra electron into the dot raises the electrochemical potential of the dot. The one-by-one change of the number of electrons N in the dot leads to a conductance oscillation as a function of a gate voltage (called the Coulomb oscillation or Coulomb blockade oscillation). The oscillation period is usually constant for a system containing many electrons. However, in a small dot containing just a few electrons, both electron-electron interactions and quantum confinement effects become sufficiently strong to cause a significant modification of the Coulomb oscillation [1–3]. Such a system can be regarded as an artificial atom [4]. There have been several experiments on quantum dots containing only a few electrons. These include transport through a two-terminal asymmetric double-barrier tunneling structure [5–8] and through a gated double-barrier tunneling structure [2,3,9–11], and capacitance measurements of a vertically gated modulation-doped heterostructure [1,12]. In this section we describe transport measurements on a sub-micron gated double-barrier structure.

The two-dimensional (2D) system modulated by a periodic strong repulsive potential is called an antidot lattice. Various interesting phenomena have been observed in antidot lattices in uniform perpendicular magnetic fields. They are the quenching of the Hall effect [1,2], Altshuler-Aronov-Spivak oscillation near vanishing field [3–6], so-called commensurability peaks in magnetoresistance [1,7–16], and fine oscillations around them [17,18].

One can impose artificial potential on a two-dimensional electron gas (2DEG) at the heterointerface of GaAs/A1GaAs by depositing a suitably patterned gate electrode on the surface. A periodic lateral potential can give rise to a rich variety of transport phenomena in 2DEG, in which the electron mean free path can be much larger than the lateral superlattice period. Magneto-transport in 2DEG under a periodic potential shows what can be generally called “commensurability effects,” which arise from the geometrical resonance depending on the commensurability condition between the lattice periodicity and the cyclotron orbit. An example of this class of transport phenomena, discussed in the preceding section, is magnetotransport in 2DEG subject to a 2D potential modulation. In this section, we discuss a related phenomenon, called Weiss oscillation [1,2], which involves 2DEG in a 1D superlattice potential.

Many of the characteristic energy scales in semiconductor quantum nanostructures such as quantized subband energies are typically between 1 meV and 100 meV. On the other hand, characteristic time scales in nanostructures such as scattering times and tunneling times are typically 0.1-10 ps. All these energy and time scales, if converted to frequencies of electromagnetic radiation, lie in the terahertz (THz) range. Therefore, the investigation of interaction between electrons and THz electromagnetic radiation gives us important information on the electronic states and dynamical transport properties of electrons in quantum nanostructures, which is not accessible by dc transport experiments. In this section, basic techniques of THz spectroscopy are briefly introduced and some recent topics on the THz spectroscopy of quantum nanostructures are discussed.

Since the first prediction by G.H. Wannier, the energy spectrum of a crystalline solid in an electric field, which is called a Wannier—Stark state or a Stark ladder, has attracted much interest. In 1960, Wannier studied electronic states in the presence of a uniform electric field theoretically and found that eigenstates are localized along the direction of the electric field and have quantized energy levels 2.8.1$$ {E_n} = neFd, $$ where n is an integer, F the electric field, and d the lattice period along the electric field [1]. The localized Stark ladder states are associated with Bloch oscillation. In an electric field F, temporal motion of an electronic states is described by $$ \dot k = eF/\hbar $$. At the edge of the Brillouin zone, electrons are Bragg reflected by the periodic crystal potential and their motion becomes periodic in space, as well as in the k space.

The quantum Hall effect is a manifestation of quantum effects in macroscopic transport phenomena. In particular, the integer quantum Hall effect has provided various fundamental problems in solid state physics such as localization in high magnetic fields, a topological winding number, edge channels, etc. It has been used also as a resistance standard and has opened up a way to examine the validity of quantum electrodynamics through comparison of the fine structure constant.

Various phenomena have been successfully analyzed by an edge-channel formulation of integer quantum Hall effect (IQHE) [1–3], which describes the conduction only in terms of the “edge current” that flows along the edge of a two-dimensional electron gas (2DEG) layer. On the other hand, a number of experiments strongly suggest that the conduction is dominated by “bulk Hall current” [4–8]. Consideration is given to an implication of the edge current in this section.

The spatial distribution of edge states at quantum Hall (QH) plateaus has been studied both experimentally and theoretically. In the classical simple picture, the width of the edge states is expected to be of the order of the cyclotron radius, since the electrons move along the sample edge by a skipping motion as shown in Fig. 3.3.1. In the one-electron quantum mechanical picture, the width of the edge states is also of the order of the magnetic length $$ l = \sqrt {\hbar /e{\rm B}} $$.

A reduction of the dimensionality of the electron motion in quantum wells, wires, and dots is expected to realize new device concepts or improvements in the performance of existing devices, such as transistors and lasers. In 1982, Arakawa and Sakaki proposed the use of quantum wires or dots as an active medium in semiconductor lasers, suggesting a significant improvement of lasing characteristics [1]. The quantum wire effects were realized by placing double heterostructure lasers in a high magnetic field [2]. This report is the first in which the concept of three-dimensionally confined electrons in semiconductor quantum dots is discussed. The target is still distant, because sufficiently small and uniform structures must be achieved.

First, the theory for the bulk crystals [1-3] is described. According to Bloch’s theorem, a single-body state of an electron in a solid crystal of volume V takes the following form: where b labels bands (conduction band, heavy-hole band, and so on), and Ubk is the cell-periodic function. The spin index is omitted for the sake of simplicity.

Spontaneous emission is a physical phenomenon observed not only in laboratories but also in daily life where various materials emit light spontaneously. The typical phenomenon is the fluorescence of materials excited in some way. Spontaneous emission is known as an elementary process which is required, together with the process of stimulated emission, to establish thermal equilibrium between matter and a light field. In free space, spontaneous emission is an irreversible process, i.e., once a photon is emitted by an atom, it never returns to the emitting atom and, ultimately, escapes to remote space or is dissipated in obstacles. Spontaneous emission occurs when an atom is excited to a high-lying level and then releases the energy in the form of light. The emission of light is accompanied by a decay to a lower-lying level.

Photonic crystals are artificial nanostructures constructed from optical atoms arranged in a background medium with a period on the order of half the optical wavelength [1]. They are of great interest since those made of semiconductors have the possibility of spontaneous emission control, which allows thresholdless operation of laser diodes. A large refractive-index contrast between semiconductor and air provides a wide photonic band gap, which means a frequency range that inhibits the existence of modes. Figure 4.4.1 schematically illustrates a photonic crystal of various dimensions, and the corresponding wavevector space of cavity modes and that of the emission spectrum, both inhibited by each photonic band gap.

In this section, we review progress on microcavity vertical cavity surface emitting lasers by featuring their materials and performance including ultimate thresholds, power output, efficiency, and so on. Advanced technologies for control of polarization, lasing wavelength, spontaneous emission, etc., will be introduced. We discuss potential applications such as parallel optical interconnects and links, opto-electronic equipments including laser printers, etc., and generation of the field of ultra-parallel opto-electronics.

One very feasible nanostructure optical device is a quantum dot (QD) laser. When quantum wire (QWR) and QD lasers were proposed in 1982 [1], a suppression of the temperature dependence of the threshold current in QWR and QD lasers was discussed theoretically. A significant improvement of lasing characteristics such as a low threshold current, fine modulation dynamics, and good spectral properties were predicted theoretically [2–4]. In this section, we focus on QD lasers with emphasis on lasing characteristics and microcavity effects [5].

The long history of the field of electronics shows that it takes more than a decade to mature a new technology after planting its seed. LSI technology is growing steadily at present but the growth cannot last forever and it is now time for the seed of new electronics and post-LSI technology to be planted. The wave nature of the quantum-mechanical electron motion gives rise to various phenomena such as interference, multiple reflection, Bragg reflection, diffraction, etc. The discrete electronic charge gives rise to fascinating phenomena like the Coulomb blockade, single-electron tunneling, etc. These quantum-mechanical aspects of the electron may be used for the control of electron transport and have high potential as new quantum-effect devices.

Metal/insulator ultrathin heterostructures are good candidates for high-speed electron devices, because the high carrier density of the metal and the low dielectric constant of the insulator are suitable for size reduction and high-speed operation [1,2]. In addition, due to a very large conduction-band discontinuity at the heterointerface, the interference of the electron wave is expected to become significant in, multilayer structures, which result in high transconductance and multifunctionality of the quantum-effect devices [3,4].

One of the most fundamental problems of quantum mechanics is the time evolution of a nonstationary wave packet [1–6]. The motion of such wave packet is purely coherent if phase-breaking interactions with a thermal bath are absent. Experimentally, we can observe such coherent quantum dynamics if the dimension is comparable to or less than the mean free path of the electrons.

By analogy with optics, for example, Fourier transforms can be performed by the electron wave. Important for the realization of such devices is the spread of the electron wavefront, which is the subject of this section [1]. Figure 5.4.1 shows a conceptual electron wave device where the electron wave from a source passes through an input layer where the amplitude and/or the phase of the electron wave is modulated according to the input signal pattern I(x). Then the electron wave forms a diffraction or interference pattern of the probability current density on the output layer. This conceptual device performs pattern transformation using the nature of the wave propagation. The wavefront spread is determined by the emitter and the propagation. Here we focus on the former using an effective-mass Schrödinger equation.

The size of VLSI devices is becoming smaller and smaller for higher integration and higher performance Figure 5.5.1 shows the trend for VLSI devices, taking DRAMs for example. The density of DRAMs has increased 4 times during every three years, while the gate length of MOSFETs has decreased 0.7 times during every three years. This device trend has been valid for last 25 years. If the trend continued in the future, the device size would be 0.1 μm in 10 years and 0.03 μm in 20 years. These sizes are almost comparable with the electron wavelength in semiconductors, suggesting that quantum effects or single-electron phenomena would play an important role in electron transport even in VLSI devices. Various problems and issues that limit device scaling are being discussed, including size fluctuations, statistical fluctuations of dopants in the channel, limited lithography resolution, and difficulty of manufacturing.

One of the goals in quantum electronics is to develop computing systems that can perform data processing by using quantum devices. There are two basic ways that we can use quantum-effect devices for data processing. One is to imitate existing silicon LSI circuit systems, using quantum devices as analogs of MOSFETs. The problem with this approach is that integrated circuits made of new transistor-type devices would have a hard time competing with silicon LSI, which is a well-established, mature technology. The more promising strategy is to reconsider the procedure for implementing data processing and to pursue a path different from that of ordinary LSI systems. This section reviews several approaches that are based on such a new-paradigm strategy.

Semiconductor quantum structures, such as quantum wires (QWRs) and dots (QDs), are attracting much interest in the field of fundamental physics and novel device application, as described in the previous chapters. The realization of such properties requires extreme control of fabrication tolerance, and nonuniformity in structural dimensions often fatally degrades the advantageous properties. For instance, inhomogeneous broadening of the gain spectrum of QD lasers will result from fabricational variations in the dot size and shape. Quantum dot lasers do not offer significant advantages over conventional bulk lasers unless fabricational tolerances are tightly controlled [1].

The use of monoatomic and multi-atomic steps formed on vicinal (001) GaAs substrates has been proposed and demonstrated to realize two-dimensional quantum confinement structures such as quantum wires (QWRs) grown by MOCVD [1–4] which can be used as a new type of electron wave interference device [5] Similar QWR structures have been observed in tilted GaAs/AlAs superlattices on vicinal (001) GaAs substrates grown by MBE [6], in a thin In-GaAs layer at the edge of InP multiatomic steps on vicinal (001) InP surfaces grown by metalorganic MBE [7], and in a thin AlGaAs composition modulation layer at the edge of AlAs multiatomic steps on vicinal (110) GaAs substrates grown by gas-source MBE [8]. In this section we review some of these structures.

To fabricate quantum wires, various techniques such as wet chemical etching [1], reactive ion etching [2], ion beam implantation [3], and ion beam milling [4,5] have been investigated. These methods suffer from free surface effects, creation of a damage field during implantation, or interface problems due to various disordering mechanisms. To avoid these problems, growth techniques on masked substrates [6,7] and nonplanar substrates [8–10] have been investigated. Pioneering work by Kapon et al. successfully fabricated quantum wires on V-grooved (100) oriented GaAs substrates [8] or submicron gratings [9] by MOCVD. Vertically-stacked quantum wires on a single V-groove were also achieved [10]. In these works wet chemical etching was used.

MBE growth of GaAs on GaAs (110) surfaces has been reported not to provide smooth surface morphology due to columnar growth of epitaxial layers [1]. Highly faceted surfaces are observed for MBE growth on (110) surfaces. Only vicinal (110) surfaces misoriented 6° toward (111)A have provided device quality GaAs layers [2]. However, when GaAs/AlGaAs superlattices are grown on vicinal (110) surfaces misoriented toward (111)A, quantum wire-like (QWR-like) structures have been observed [3,4]. These QWR-like structures are induced by the formation of coherently aligned giant growth steps and remarkable composition and thickness modulation of AlGaAs and GaAs layers at the giant step edges. Using these phenomena, A1GaAs and GaAs quantum wires (QWRs) have been formed on vicinal (110) surfaces misoriented toward (111)A.

We describe two kinds of high quality GaAs/A1GaAs quantum wires (QWRs) grown by MBE: (1) GaAs/Al_0.3 Ga_0.7As tilted T-shaped QWRs (T-QWRs) with extremely precisely controlled line width and cross section fabricated on reverse-mesa etched stripes on a (100) GaAs substrate by a two-step growth of glancing-angle MBE (GA-MBE) and normal MBE and (2) extremely high density (8x105 cm−1) of GaAs/AlAs QWRs naturally formed in a GaAs/AlAs quantum well (QW) with a regularly corrugated AlAs/GaAs interface and a flat GaAs/AlAs interface grown on a (775)B-oriented GaAs substrate by MBE. The former is favorably compared with conventional T-QWRs fabricated on a (110) cleaved surface of a GaAs/AlGaAs multi-quantum well (MQW) layer by cleaved edge overgrowth (CEO), because many tilted TQWRs can be made on a large area of usual (100) GaAs substrate surface. The latter shows high optical quality, high density, and high uniformity in addition to high reproducibility of fabrication, indicating its high application potential for optical devices.

Si-based heterostructures are being intensively studied since various new functionalities which cannot be obtained with Si alone are expected owing to the predictable and controllable modification of the band structure [1]. In particular, SiGe/Si heterostructures are of great importance and have had significant impact on Si-based device applications as well as on the basic science of semiconductors. The built-in strain is not an obstacle but is now regarded as an advantage since it can give additional flexibility in the band engineering.