Introduction

In the previous chapter, we laid the foundation for quantitative understanding of the performance of heterojunction bipolar transistors (HBTs) with a particular emphasis on fT and fmax. In particular, the connection of these quantities to microscopic physics such as carrier transport was described, as was their connection to higher-level parametric descriptions of the transistor, as embodied in an equivalent circuit.

In this chapter, we will discuss the practical constraints imposed by real material systems, epitaxial growth techniques and fabrication processes. The effect of these constraints on fT and fmax will be studied using the results of Chapter 18. The overall goal of the chapter is to develop physical insight into three areas: (i) the key elements which distinguish material technologies that are suitable for HBTs, and the relationship between the choice of material system and device performance; (ii) the interplay of process development issues and device performance; (iii) other factors than transport, doping behavior, and device geometry (such as reliability) that further constrain device performance. In particular, we will cover the history and evolution of material systems, from AlGaAs/GaAs to InP/GaAsSb, which have been used for HBTs, and we will describe a representative fabrication sequence. Armed with this background knowledge, we will apply some of the theoretical results of Chapter 18 to a state-of-the-art production HBT, and examine the prospects for improving its performance. We will also look at some examples of other problems that arise when a device is scaled and operated for maximum fT and fmax.

High-speed heterostructure devices is a textbook on modern high-speed semiconductor devices intended for both graduate students and practising engineers. This book is concerned with the physics and processes involved in the devices’ operation as well as some of the most recent techniques for modeling and simulating these devices. Emphasis is placed on the heterostructure devices of the immediate future: namely the MODFET, HBT and RTD. The principle of operation of other devices such as the Bloch oscillator, RITD, Gunn diode, quantum cascade laser and SOI and LD MOSFETs is also introduced.

This text was initially developed for a graduate course taught at The Ohio State University and comes with a complete set of homework problems. MATLAB programs are also available for supporting the lecture material. They can be used to regenerate a number of the pictures in the book and to assist the reader with some of the homework assignments.

This book should also prove useful to researchers and engineers, as it presents research material which is disseminated throughout the research literature and has never before been presented together in a book.

This text starts with two chapters reviewing the semiclassical theory of heterostructure devices. Five chapters are dedicated to presenting a realistic picture of heterostructures, introducing quantum devices and developing practical tools for analyzing quantum transport in these devices in the presence of scattering, and at high frequencies. One chapter is focused on the Boltzmann equation and its application to the derivation of moment equations for high-field transport.

Introduction

In Chapter 8 we studied the charge control of the 2DEG (two-dimensional electron gas). In Chapter 9 we studied high-field transport models applicable to horizontal transport in the 2DEG. The motivation for these studies was the application of the 2DEG as the channel of a field-effect transistor (FET). The resulting FET is referred under the various names of MODFET (modulation doped FET (University of Illinois, USA)), HEMT (high electron mobility FET (Japan)), TEGFET (two-dimensional electron gas FET (France)), and SDHT (segregation doping heterojunction transistor (Bell Lab., USA)) depending on the different laboratories which simultaneously developed it. The AlAs–InGaAs–InP lattice-matched MODFET [2] which provides power gain at millimeter frequencies (fmax = 405 GHz) is presently with the HBT one of the fastest semiconductor transistors. The microwave characteristic of the MODFET will be discussed in Chapters 11, 12, 13, 15 and 16.

We compare in Figure 10.1 the layout of an AlGaAs–GaAs MODFET (c) with that of that of a silicon MOSFET (a) and a GaAs MESFET (b). Although the layout of the MODFET is similar to that of the MESFET, its normal principle of operation (control of a 2DEG with a gate voltage) is similar to that of the MOSFET. Other semiconductors can be used to fabricate MODFETs (see [1] for a review). A semiconductor with a bandgap much wider than that of AlGaAs can also be used to simulate an insulator.

Introduction

Modern technology has made possible the growth of thin crystalline epitaxial layers of different semiconductors. These epitaxial layers can be as small as a few lattice parameters. For small heterostructures (100 Å or less) a quantum treatment of heterostructures becomes necessary. In this chapter we will attempt to build a quantum picture of heterostructures. Note that the conventional quantum picture of a semiconductor crystal cannot be applied to rapidly varying semiconductor heterostructures since crystals are defined as periodic structures extending up to infinity. New theoretical techniques are thus required to describe these ‘spatially-varying crystals’.

Our quantum picture will be based upon an envelope model. An envelope model focuses on calculating the relative distribution of the wave-function from atomic cell to atomic cell rather than on the detailed distribution of the electron wave-function in each atomic cell.

The particular envelope picture we shall use is the so-called generalized Wannier picture. The generalized Wannier picture is capable of handling both the concept of band structure and the concept of its variation in space in a rigorous fashion. This model will therefore permit us to understand the impact of the interface upon the band structure in a heterojunction.

Other envelope pictures have been developed such as the Ben Daniel Duke Hamiltonian (effective-mass model [10], see also [11]), the k · p envelope model ([11]), and the tight-binding model ([8]).

Introduction

Modern growth technologies have made possible the growth of new semiconductor devices with unprecedented control on the atomic level. In this chapter we shall briefly introduce the molecular beam epitaxy (MBE) growth technique and discuss its application to the growth of materials (alloys, pseudomorphic, modulation doped) for new device structures. The chapter will conclude with a review of the cubic crystal structure and its reciprocal lattice, as these concepts are used extensively in Chapters 2 and 3.

MBE technology

One of the most versatile growth techniques available for research is the MBE. In this growth technique a semiconductor substrate is placed in a high-vacuum chamber (see Figure 1.1). Different components such as Ga, Al, As, In, P, and Si are heated in separate closed cylindrical cells. These components escape through an opening in the cylindrical cell and form a molecular beam. These beams are directed toward the substrate. A shutter positioned in front of each cell is used to select the desired molecular beams. By selecting a low temperature for the substrate growth and a slow growth rate (a few micrometers per hour), it is possible to grow high-quality crystals, while making abrupt changes in doping and crystal composition.

This growth technique can be used to grow semiconductor alloys such as AlxGa1–xAs, InxGa1–xAs, InxAl1–xAs, and Si1–xGex, where x, the mole fraction, specifies the composition of the alloy.

Quantized resistance and dissipationless transport

The Hall effect has long been a standard tool used to characterize conductors and semiconductors. When a current is flowing in a system along one direction, which we here take to be the y-axis, and a magnetic field H is applied in a direction perpendicular to the current, e.g., along the z-axis, there will be an induced electrostatic field along the x-axis. The magnitude of the field E is such that it precisely cancels the Lorentz force on the charges that make up the current. For free electrons, an elementary calculation of the type indicated in Section 1.8 yields the Hall resistivity ρH = −H/ρ0ec, and apparently provides a measure of the charge density of the electrons. For Bloch electrons, as we saw in Section 8.3, the picture is more complicated, but ρH is still predicted to be a smoothly varying function of H and of the carrier density. In some circumstances, however, the semiclassical treatment of transport turns out to be inadequate, as some remarkable new effects appear.

In a two-dimensional system subjected to strong magnetic fields at low temperatures, the response is dramatically different in two respects. First, the Hall resistivity stops varying continuously, and becomes intermittently stuck at quantized values ρH = −h/je2 for a finite range of control parameter, e.g., external magnetic field or electron density. In the integer quantum Hall effect, j is an integer, j = 1, 2, …, and in the fractional quantum Hall effect, j is a rational number j = q/p, with p and q relative primes and p odd.

Conductance quantization in quantum point contacts

In Chapter 8 we discussed the Boltzmann equation and the approach to describing transport properties, such as electrical conductivity, that it provides. In general, this approach works very well for most common metals and semiconductors, but there are cases where it fundamentally fails. This happens, for example, when the wave nature of the electron manifests itself and has to be included in the description of the scattering. In this case, interference may occur, which can affect the electrical conduction. We recall that the Boltzmann equation describes the electron states only through a dispersion relation of the Bloch states of an underlying perfect crystal lattice, a probability function, and a scattering function that gives the probability per unit time of scattering from one state to another. All these quantities are real, and do not contain any phase information about the electron states. Consequently, no wave–like phenomena can be described. The question then arises as to when the phase information is important. This really boils down to a question of length scales. We have earlier talked about the mean free path of an electron, which is roughly the distance it travels between scattering events. A simple example is given by scattering off static impurities that have no internal degrees of freedom. In this case the electron scattering is elastic, since an electron must have the same energy before and after a scattering event. Furthermore, in the presence of impurity scattering the phase of an electron wavefunction after a scattering event is uniquely determined by the phase before the scattering event.

Elementary excitations

The most fundamental question that one might be expected to answer is “why are there solids?” That is, if we were given a large number of atoms of copper, why should they form themselves into the regular array that we know as a crystal of metallic copper? Why should they not form an irregular structure like glass, or a superfluid liquid like helium?

We are ill-equipped to answer these questions in any other than a qualitative way, for they demand the solution of the many-body problem in one of its most difficult forms. We should have to consider the interactions between large numbers of identical copper nuclei – identical, that is, if we were fortunate enough to have an isotopically pure specimen – and even larger numbers of electrons. We should be able to omit neither the spins of the electrons nor the electric quadrupole moments of the nuclei. Provided we treated the problem with the methods of relativistic quantum mechanics, we could hope that the solution we obtained would be a good picture of the physical reality, and that we should then be able to predict all the properties of copper.

But, of course, such a task is impossible. Methods have not yet been developed that can find even the lowest-lying exact energy level of such a complex system. The best that we can do at present is to guess at the form the states will take, and then to try and calculate their energy.

The aim of this book is to make the quantum theory of condensed matter accessible. To this end we have tried to produce a text that does not demand extensive prior knowledge of either condensed matter physics or quantum mechanics. Our hope is that both students and professional scientists will find it a user-friendly guide to some of the beautiful but subtle concepts that form the underpinning of the theory of the condensed state of matter.

The barriers to understanding these concepts are high, and so we do not try to vault them in a single leap. Instead we take a gentler path on which to reach our goal. We first introduce some of the topics from a semiclassical viewpoint before turning to the quantum-mechanical methods. When we encounter a new and unfamiliar problem to solve, we look for analogies with systems already studied. Often we are able to draw from our storehouse of techniques a familiar tool with which to cultivate the new terrain. We deal with BCS superconductivity in Chapter 7, for example, by adapting the canonical transformation that we used in studying liquid helium in Chapter 3. To find the energy of neutral collective excitations in the fractional quantum Hall effect in Chapter 10, we call on the approach used for the electron gas in the random phase approximation in Chapter 2. In studying heavy fermions in Chapter 11, we use the same technique that we found successful in treating the electron–phonon interaction in Chapter 6.