PH3710 Semiconductors and Superconductors

Transcription

1 PH3710 Semiconductors and Superconductors James Nicholls Physics Department, Royal Holloway, University of London These notes cover most of the material on semiconductors. Contents Course Description 1 Lectures 1 Syllabus: Semiconductors 1 Teaching, Learning and Assessment Strategy 2 Books 2 I. Introduction 3 A. Four Classes of Conductor 3 1. Metals/semimetals 3 2. Semiconductors 3 3. Insulators 4 4. Superconductors 4 B. Which elements/compounds are semiconductors? 4 C. Crystal structure of semiconductors 5 D. Doping in semiconductors 6 II. Electrons in Semiconductors 7 A. Free electron model 7 B. Electrons in a periodic potential 7 C. The effective mass approximation 8 D. Metals vs Semiconductors 9 III. Carrier concentrations 10 A. Intrinsic semiconductors 10 B. Extrinsic semiconductors Intrinsic saturated regions Saturated freeze-out regions 11 IV. Conduction in Semiconductors 14 A. The Drude Model 14 B. The mobility µ Charged impurities Lattice vibrations 14 C. Hall effect and magnetoresistivity tensor (Drude model) 14 D. The Hall effect for two carriers 15 E. Electrical measurements Four-terminal measurements Four-probe measurements Corbino disk 17 V. Optical Properties of Semiconductors 18 A. Optical properties 18 B. Optical constants 18 C. A microscopic model 18 D. Phonon absorption 20 E. Shallow donors and acceptors 20 F. Band-gap absorption 21 G. Excitons 22 H. Free carrier absorption 22 I. Cyclotron resonance 23 VI. Solid State Electronics 25 A. Inhomogeneous semiconductors 25 B. Diffusion currents 25 C. Diffusion length 25 D. The p-n junction in equilibrium 26 E. A biased p-n junction 27 F. Applications of p-n junctions Varactor diode Photodiodes (or p-i-n diodes) Solar cells Light-emitting diodes (LEDs) 29 G. Appendix: Calculation of capacitance of pn-junction 30 VII. Growth and Doping of Semiconductors 31 A. Growth of bulk crystals 31 B. Epitaxial growth 31 C. Diffusion doping of semiconductors 32 D. Ion implantation 33 E. Chip production 33 VIII. Heterostructures and Nanoelectronics 34 A. Introduction to low-dimensional systems 34 B. The two-dimensional electron gas (2DEG) Density of states (DOS) in 2D 34 C. Quantum wells 34 D. High electron mobility transistors Nanostructures Quantum dots and the single-electron transistor Low-dimensional physics 36 E. Density of free electron states in 1D, 2D, and 3D D DOS: g 3D (E) D DOS: g 2D (E) D DOS: g 1D (E) Summary 38 Course Description Lectures The second part of this course consists of approximately six double lectures and three problem classes. The lectures are scheduled for 12:00-14:00 Friday T125, and the problem classes will take place 15:00-16:00 Monday T125, but not every week. Problem sheets will be issued every two weeks. The aim of this half of the course is to present the physics of semiconductors, and to give some examples of applications. Syllabus: Semiconductors Elemental and compound semiconductors; doping; extrinsic and intrinsic semiconductors; carrier concentration; transport properties; optical properties; the p-n junction; MBE semiconductors; applications.

2 2 Teaching, Learning and Assessment Strategy 1. There is no set book for this course which, as a third year optional course, contains some advanced material and some topics related to recent research. Therefore an extensive set of notes has been produced covering all of the syllabus. These notes will handed out, usually a few lectures in advance, thereby giving you the opportunity to study the material at your own pace, if you desire, before the lectures. The notes themselves are selfcontained, however in addition an extensive book list follows and you are encouraged to read around the subject (in order to experience a different approach to the material and to aid your learning process). This notes-based teaching method is appropriate for a subject where a single text book containing all of the required information is not available. 2. All the basic material on the semiconductor part of the course will be provided in the handouts. The handouts do not give all steps in a particular derivation, and you will be expected to work these out yourself, or by looking in reference books. A course summary will be handed out at the end of the course as an aid to revision, with the same structure as the notes. 3. Problem sheets will be distributed every two weeks. The three problem sets on semiconductors are designed to bring the course together as a whole. They (a) reinforce the course material, (b) contribute to assessment and feedback, (c) act as a focus for private study. A coursework mark of 10% is allocated on the basis of answers submitted. The numerical answers will be posted on moodle after marking. 4. There is a two-hour examination for PH3710. The exam paper is made up of two sections: Section A will consist of three questions on semiconductors, and, Section B will consist of three questions on superconductors. You will be asked to answer three questions, including at least one from each of sections A and B. No credit will be given for attempting any further questions. The exam result accounts for 90% of the marks for the course JEN (4 copies - all one week loan). This is an excellent book about the properties of real semiconductors and their growth and characterisation. It also includes a good treatment of the physics of semiconductors and the principles and techniques used. 3. Introduction to Solid State Physics by C Kittel, Wiley, 7th edition, Library class: KIT (3 copies - all one week loan). There is also an 8th edition of this classic text (3 copies - all one week loan) which is 60. It covers a much wider range of solid state physics than in PH371 and is very useful for more advanced courses and as a reference book should you work in this area after graduation. Books There are many good general books on solid state physics and specialised books on semiconductors and superconductors. The Library classifications are: Solid State Physics Semiconductors A good starting point is the book for PH2710: Solid State Physics J R Hook and H E Hall, 1st Ed., J Wiley, Library class: HOO (6 copies: 2 normal loan, 4 one week loan). 2. Semiconductor science: growth and characterization by Tudor E Jenkins, Prentice Hall, Library class:

3 3 I. INTRODUCTION A. Four Classes of Conductor To answer the question what is a semiconductor? we look at the resistivity ρ of materials, which is observed to vary by almost 30 orders of magnitude. Four classes of conductor can be identified: 1 1. Metals/semimetals Resistivity at room temperature 10 8 < ρ(t ) < 10 6 Ωm. Example: copper ρ = Ωm. Impurities increase ρ(t ). ρ(t ) decreases as T decreases. ρ(0) = 0 for an ideal (pure) metal. ρ(0) > 0 for a metal with impurities (residual resistivity). Carrier concentration about 1 electron/atom (> m 3 ). Metals are classified as conductors because their outer electrons are not tightly bound. FIG. 1 Evidence for a gap: the variation of the absorption coefficient α with photon energy. Note that GaAs has a direct gap, whereas Si and Ge are indirect gap semiconductors. A semimetal has a small overlap in the energy of the conduction band and valence bands. However, the bottom of the conduction band is typically situated in a different part of momentum space than the top of the valence band. Typical example, graphite: ρ = Ωm ω g E conduction band edge valence band edge Ω E ω valence band edge 0 k -k c 0 k c k 2. Semiconductors Adsorption Adsorption Resistivity at room temperature ρ(t ) = Ωm. Transparent region Impurities decrease ρ (in general). onset of direct photon transition onset of indirect photon transition onset of direct transition ρ(t ) increases (usually) as T decreases. ħω g photon energy ħω E g + ħω E vert photon energy ħω ρ(0) = for lightly doped semiconductor. Carrier concentration 1 electron/atom (< m 3 ). Carriers can be +ive (holes) or -ive. Basic characteristic is an energy gap. Evidence for a gap: Variation of optical absorption coefficient with photon energy - see Figs. 1 and 2. Some energy gaps are given in Table I; a typical value is 1-3 ev. 1 Hook and Hall Chapter 5, Kittel Chapter 8. FIG. 2 Direct and indirect energy gaps. TABLE I Band gaps (in ev) of common semiconductors. Group IV Group III-V Group II-VI Group IV-VI Si 1.11 GaAs 1.43 ZnTe 2.15 PbS 0.43 Ge 0.66 InP 1.35 CdTe 1.50 PbTe 0.29 SiC 2.31 GaP 2.26 ZnSe 2.67 AlAs 2.16 CdSe 1.70 GaSb 0.73 ZnO 3.2 InSb 0.81 CdS 2.42 GaN 3.39

4 4 3. Insulators + No appreciable conduction at room temperature (like semiconductors with a very large gap). Most materials are insulators, because even the outermost electrons are so tightly bound that there is essentially zero electron flow through them with ordinary voltages. Examples: quartz (fused), ρ = Ωm; glass, ρ = 1012 Ωm, teflon (PTFE), ρ = Ωm. 4. Superconductors + Zero resistivity below a transition temperature Tc. + Energy gap. + Meissner Effect (exclusion of magnetic flux). + Many special properties (see other part of PH3710). FIG. 3 Periodic table with outer electron configurations. B. Which elements/compounds are semiconductors? Group I, II, III: metals (and superconductors). Group VII and VIII: insulators. Semiconductors: Group IV elements: III-V compounds: II-VI compounds: Ternary compounds: C (diamond), Si, Ge, α-sn GaAs, GaP, InSb, InAs CdTe, CdS, ZnSe, etc. Alx Ga1 x As

5 5 TABLE II Crystal structure and lattice constant a (in nanometres) of some common semiconductors. Diamond Zinc blende Si Ge GaP AlAs InP GaAs GaSb InSb FIG. 4 (a) Diamond lattice; (b) zinc blende lattice. In (a) the atoms marked R make up a puckered hexagonal ring. C. Crystal structure of semiconductors Table II shows that the elemental Group IV semiconductors adopt the diamond structure. The diamond structure, see Fig. 4(a), is a cubic (Bravais lattice is facecentred cubic, fcc) with a basis of 2 atoms. There are 2 atoms per primitive unit cell, and 8 atoms per cubic unit cell. The centre-to-centre distance between atoms is d = a 3/4, where a is the side of the cubic unit cell. From the Periodic Table in Fig. 3, we see that the electronic structure of Si, a typical semiconductor with Z = 14, is 1s 2 2s 2 2p 6 3s 2 3p 2. The group IV elements are covalent bonded with 4 valence electrons/atom. Electrons are shared between neighbouring atoms to form covalent bonds, see Fig. 5, hence the electrons are tightly bound and there are no free carriers at low temperatures. Because of the covalent bonding, each atom has four nearest neighbours in a tetrahedral coordination. The diamond structure is very open, the densities of Si and Ge are 2.33 g/cc and 5.32 g/cc, respectively. The zinc blende structure 2 shown in Fig. 4(b) is adopted by many of the compound semiconductors. In GaAs the atoms occupy positions similar to those in Si, except they alternate between Ga and As. The III-V and II-VI compounds are partly covalent and partly ionic in character, this can be seen in the valence charge density contours of GaAs compared to those of Si, both shown in Fig. 6. Other semiconductors, for example II-VI compounds such as ZnS, CdS, and CdSe adopt the wurtzite structure, which is a hexagonal rather than cubic crystal system. 2 Hook and Hall p. 18, Kittel p. 20. FIG. 5 (a) Face-centred cubic unit cell of silicon (2 atoms per cell). Each atom is bound to four neighbours, e.g., (0) to (1), (2), (3) and (4). (b) Diagrammatic representation of the four covalent bonds each with a pair of electrons. FIG. 6 Top: valence charge density in Si - experiment and theory. Bottom: valence charge density in GaAs.

6 6 TABLE III Static relative dielectric constant of semiconductors. Crystal ε Crystal ε Diamond 5.5 GaSb 15.7 Si 11.7 GaAs Ge 15.8 AlAs 10.1 InSb 11.7 AlSb 10.3 InAs SiC 10.2 D. Doping in semiconductors A pure, or undoped semiconductor is known as an intrinsic semiconductor. If the semiconductor is doped with substitutional impurities, it is then known as an extrinsic semiconductor. Acceptors from Group III: Donors from Group V: B, Al, Ga, In P, As, Sb Electrons/holes are bound to these charged impurities and have hydrogenic energy levels and ionisation energies Ry m /m 1/ε 2, where the Rydberg energy (Ry) is given by Ry = me4 8ε 2 = 13.6 ev; 0h2 m is the effective mass, and ε is the dielectric constant. ε = 11.7 for Si and ε = 15.8 for Ge, and Table III gives the relative dielectric constant of other semiconductors. For P doped into Si (written as Si:P) the ionisation energy is 45 mev, which is much less than the band gap E g. This ionisation energy is equivalent to a temperature T = 540 K, and so at room temperature most of the impurity states will be ionised. N D is defined as the donor concentration, and N A as the acceptor concentration. Semiconductors with both donors and acceptors are compensated. If all the levels are ionised then the electron concentration will be n = N D N A (if N D > N A ) or the hole concentration will be p = N A N D (if N A > N D ). The compensation ratio K is defined as K = N D /N A. Majority carrier denotes the carrier which is primarily responsible for current transport in a piece of semiconductor. If an intrinsic semiconductor is doped with n-type impurities then the majority carriers are electrons and the minority carriers are holes. If the semiconductor is doped with p-type impurities then the majority carriers are holes and the minority carriers are electrons. Table IV gives the ionisation energies of some impurities in Si, Ge and GaAs - these impurities have small ionisation energies (< 0.1 ev) and are known as shallow FIG. 7 Left figure: (a) Qualitative temperature dependence of the concentration n of electrons in the conduction band of an n-type semiconductor for two different donor concentrations. (b) Qualitative temperature dependence of the Fermi energy E F (T ) in the same semiconductor. Right figure: The concentration n of the free electrons in n-type Ge measured using the Hall effect (Sec. IV.C). For samples (1)-(6), the donor concentration varies between and cm 3. The temperature dependence of the electron concentration in the intrinsic region is shown by the dashed line. TABLE IV Ionisation energies in ev of some impurities in Si, Ge and GaAs. The acceptor (a) ionisation energy is measured from the valence band edge E V, and the donor (d) from the conduction band edge E C. Si Ge GaAs Li (d) Li (d) Si (d) P (d) Sb (d) Si (a) As (d) P (d) Ge (d) Sb (d) As (d) Ge (a) 0.04 impurity levels. There are also deep levels which lie much closer to the centre of the gap - these can be group II or VI impurities or transition metal atoms such as chromium, Cr. These can play an important role in optical properties as recombination centres, and also can be used to make highly resistive substrates.

7 7 II. ELECTRONS IN SEMICONDUCTORS A. Free electron model Currents are carried by electrons, and the free electron model 3 (FEM) is a good starting point for discussing transport properties. In the FEM the electrons are noninteracting point charges in a box where the potential energy is zero (or constant). Each electron state is specified by its energy E (which is just kinetic energy for free electrons) and its momentum p = hk, where k = 2π/λ is the wavevector. For non-relativistic free electrons p = mv, where v is the velocity. Hence the energy E is FIG. 8 (a) Plot of E(k) for free electrons. (b) E(k) for an electron in a monatomic linear lattice of lattice constant a. The energy gap E g is associated with the first Bragg reflection at k = ±π/a. E(k) = mv2 2 = p2 2m = h2 k 2 2m, (2.1) and the dispersion curve E(k) is parabolic, as shown in Fig. 8(a). For free electrons in a sample of volume V, the normalised wavefunction is ψ k = 1 V e ikr. (2.2) The number of electron states (including spin) in an energy range from E E + de is g(e) de, where g(e) is the density of states per unit energy range per unit volume, and is given by g(e) = 1 π 2 h 3 (2m3 E) 1/2 = CE 1/2. (2.3) Electrons are fermions and the occupation probability for a state of energy E is given by the Fermi-Dirac distribution function f(e) = 1 exp[(e E F )/k B T ] + 1. (2.4) For N free electrons at T = 0 K, the electron states are occupied up to the Fermi energy E F = h2 k 2 F 2m = k BT F, (2.5) where k F = (3π 2 n) 1/3, the electron concentration is n = N/V, and T F is the Fermi temperature. The Fermi energy at any temperature T is defined by the condition (see Fig. 9) n = N(E) de = f(e) g(e)de. (2.6) FIG. 9 Calculation of the density of occupied electron states N(E). (a) The Fermi-Dirac distribution f(e), (b) the density of states g(e), and (c) N(E) = f(e) g(e). B. Electrons in a periodic potential Electrons in crystals move in the potential of the atoms, which is periodic. 4 The electrons interact with the lattice and can be Bragg reflected. The electron states are described by the dispersion relation (or band structure) E(k), which can depart from free electron parabolic behaviour (see Fig. 8(b)). One approach 5 to this is to consider the atomic energy levels for each atom. As a collection of atoms are brought together each energy level develops into a band of levels within an energy range or bandwidth. These bands may or may not overlap, and may be full or partly full. In semiconductors, there is an energy gap between the valence band and the conduction band. However, the actual band structure is quite complicated. The situation is similar to the phonon spectrum m(q), or dis- 3 Hook and Hall, Chapter 3, Kittel, p. l Hook and Hall, Chapter 4, Kittel, Chapter 7. 5 The tight binding approximation.

8 8 FIG. 10 The first Brillouin zone of the fcc lattice. The points of high symmetry, Γ, X, and L, are indicated. persion relation, for lattice vibrations. Just as for the phonons the band structure for a 1D crystal is plotted in the Brillouin zone, π/a < k < π/a, where a is the lattice spacing. On the Brillouin zone boundary, k = ±π/a, the wavelength of the phonon or electron is λ = 2a and strong Bragg reflection occurs. For the face-centred cubic (fcc) crystal structure, the three-dimensional Brillouin zone is the truncated octahedron shown in Fig At T = 0 K, the valence band states are all full and the conduction band states are empty. Therefore no conduction can occur and the conductivity is zero, σ = 0. The wavefunction of an electron in state E(k) can be written as ψ k (r) = 1 V u k (r) e ikr, (2.7) where u k (r) is a function which is periodic in the crystal structure and repeats in each unit cell. The result that the wavefunction can be written in the form of Eq. 2.7 for a periodic system is called Bloch s theorem. FIG. 11 Typical E(k) of a III-V semiconductor showing light hole, heavy hole, and split-off hole bands. The valence band structure for the elemental semiconductors is similar. FIG. 12 Dispersion curves E(k) for (a) GaAs and (b) Si. In GaAs, the minimum of the conduction band and the maximum of the valence band are at the centre of the Brillouin zone; it is a direct gap semiconductor. Si has an indirect band gap. C. The effective mass approximation Fortunately a very simple approximation can be used, based on parabolic energy bands. 7 For GaAs the energy E e of an electron state just above the conduction band edge at E C can be written as E e (k) = E C + h2 (kx 2 + ky 2 + k 2 m z) 2 = E C + h2 k 2, (2.8) e 2 m e where k = (k x, k y, k z ), and m e is the electron effective mass whose value depends on the band structure. The 6 See Kittel, p Hook and Hall p. 131, Kittel p contours of constant energy are spherical. The energy of an electron state just below E V, the top of the valence band, can be written as E e (k) = E V h2 k 2 2 m h (2.9) which is equivalent to a state with negative effective mass! Since a full band does not contribute to conduction it is better to work with the empty states, called holes, which can be regarded as positive carriers with positive effective mass. In the semiconductors considered here there are actually three hole bands, known as light holes, heavy holes and split-off holes - each with its own effective mass (m lh, m hh, and m so ). In silicon, the situation for electrons is more complicated, as the minimum in the conduction band occurs

9 9 FIG. 13 Constant energy surfaces in the conduction bands of Ge and Si. For Ge the dashed parts of the ellipses of revolution are outside the first Brillouin zone. For Si the six equivalent entire conduction band minima are inside the first Brillouin zone. TABLE V Electron and hole effective masses near band extrema for some common semiconductors. Si Ge GaAs InP GaP m L m T m e m hh m lh m so FD distribution and the DOS (see Fig. 9) describes how the states near E F become available. In an intrinsic semiconductor such a description is complicated by the presence of the gap E g between the valence and conduction bands. At T=0 K, all the states in the valence band are completely full, while those in the conduction band are completely empty; the Fermi energy lies mid gap, that is at E F = E V +E g /2 = E C E g /2. At finite temperature the picture (see Fig. 9) which shows how states are counted in the FEM fails because electron (hole) states become available close to E C (E V ), but far away from E F. Section III shows that in an intrinsic semiconductor electrons and holes are thermally activated in equal quantities. Note: in a semiconductor the band gap E g 1 ev is much bigger than the thermal energy (k B T 25 mev at 300 K). Intrinsic: for an undoped semiconductor the Fermi energy stays roughly mid gap for all temperatures. Extrinsic: for an n-type (p-type) semiconductor the Fermi energy lies close to the bottom (top) of the conduction (valence) band. away from the band centre, along the 100 directions. 8 We can write the electron energy near the conduction band minimum along 100 as E e (k) = E C + h2 2 m L (k x k min ) 2 + h2 2 m T (k 2 y +k 2 z), (2.10) where k min is the position of the energy minimum along the k x axis. The effective mass depends on the direction of motion of the electron and is anisotropic: m L is the longitudinal mass and m T is the transverse mass. Table V gives values for other semiconductors. In germanium, the minima in the conduction band energies lie along the 111 directions in k-space. 9 D. Metals vs Semiconductors The key difference between a metal and a semiconductor is the presence of a gap in density of states (DOS) in the latter. In metals the DOS is continuous and at T = 0 K electronic states are filled up to the Fermi energy E F, and for finite temperatures the product of the 8 Kittel p Kittel p. 215.

10 10 III. CARRIER CONCENTRATIONS A. Intrinsic semiconductors The effective mass approximation can be used to model a semiconductor; it is based on spherical conduction and valence bands, with effective masses m e and m h, and an energy gap E g = E C E V. 10 In this picture the free electron density of states (Eq. 2.3) is modified so that m m e, and the energy is measured from the bottom of the conduction band at E C : 3/2 2 m e g e (E) = π 2 h 3 (E E C ) 1/2 = C e (E E C ) 1/2. (3.1) The Fermi-Dirac function, which applies to electrons, is f e (E) = [exp[(e E F )/k B T ] + 1] 1 exp[ (E E F )/k B T ], (3.2) where the latter approximation is made in the nondegenerate regime, E E F k B T. The total electron concentration n is given by 11 n = f e (E) g e (E) de = C e (E E C ) 1/2 exp[ (E E F )/k B T )] de E C N C exp[ (E C E F )/k B T ], (3.3) where N C = 2 ( m e k B T 2 π h 2 ) 3/2 is the effective density of electron states. For the hole states, the probability of finding an empty state at an energy E in the valence band is given by f h, where f h = 1 f e = 1 exp[(e E F )/k B T ]. 1 exp[(e E F )/k B T ] + 1 Hence the hole concentration p is given by (3.4) p = N V exp[ (E F E V )/k B T ], (3.5) where N V is the effective density of hole states. Figure 14 shows a model where the conduction band is replaced by N C states at energy E C, and the valence band is replaced by N V states at energy E V, where ( ) 3/2 ( ) 3/2 me k B T mh k B T N C = 2 2 π h 2 N V = 2 2 π h 2 10 Hook and Hall p. 139, Kittel p Use x 2 = (E E C )/k B T and 0 x 2 e x2 dx = π/4. FIG. 14 Intrinsic semiconductor at T = 0 K, where E F lies exactly midway between the valence and the conduction band. The band gap is E g = E C E V. and ( ) 3 kb T N C N V = 4 2 π h 2 (m e m h ) 3/2. (3.6) Note that N C and N V are temperature dependent. The expressions for n and p in Eqs. 3.3 and 3.5 can be multiplied to obtain np = N C N V exp[ E g /k B T ]. (3.7) This is a very important equation, relating the electron and hole concentrations in a semiconductor. It is true for both intrinsic and extrinsic semiconductors (except at very high carrier concentrations in degenerate semiconductors) and is known as the law of mass action. At room temperature, T = 300 K, with m e = m h = m we find N C = N V = m 3. In any semiconductor there must be charge neutrality, for an intrinsic semiconductor this gives n i = p i = (N C N V ) 1/2 exp[ E g /2k B T ], (3.8) where n i and p i are the intrinsic electron and hole concentrations. Note that n i and p i depend on T, m e, m h and the energy gap E g. The intrinsic carrier concentrations for Si, Ge and GaAs at room temperature are: E g (ev) n i (m 3 ) Ge Si GaAs Making Eqs. 3.3 and 3.8 equal to each other, the position of the Fermi level in an intrinsic semiconductor is given by E F = 1 2 (E C + E V ) + 3 ( ) 4 k mh BT ln. (3.9) m e

11 11 FIG. 15 Band diagram, density of states, Fermi function and carrier concentration as functions of energy for an intrinsic semiconductor. At T = 0 K, the Fermi energy lies exactly midway between the valence and conduction bands, see Fig. 15. If m e = m h, then E F = (E C + E V )/2 at all temperatures. For a semiconductor with heavy and light holes, ( ) 3/2 N V = 2(m hh + m lh ) 3/2 kb T 2π h 2. (3.10) For Si, with six conduction band minima (M C = 6) ( ) 3/2 N C = 2M C [m 1/2 L m kb T T ] 2π h 2 (3.11) B. Extrinsic semiconductors In a pure (undoped) semiconductor the intrinsic carrier concentration n i is very small, for example, for Ge at 300 K it is n i = m 3. The atomic density of Ge is m 3, therefore 1 ppm of impurities can give a carrier density of m 3, which is much greater than n i. Even for light doping, the impurities will dominate the electrical transport properties. In an n-type semiconductor, the pure semiconductor is doped with substitutional impurities that have one extra electron compared to the host; the donor concentration is N D. 12 The donor impurities create a new energy level at E D, which lies an energy E d = E C E D below the bottom of the conduction band. There are two equations that determine carrier densities of electrons (n) and holes (p) in an n-type semiconductor. From charge neutrality n = p + N D (3.12) And from Eq. 3.7 np = n 2 i (3.13) 12 In general, not all donors give up their extra electron, that is, N D = N + D + N D 0, where N + D is the ionised donor concentration and ND 0 is the neutral donor concentrations. For simplicity we will assume that all donors are ionised, therefore N D = N + D. FIG. 16 Extrinsic semiconductor; n-type doping introduces an energy level just below E C. Note that the donor activation energy E d = E C E D has the property E d E g. At T = 0 K, E F lies midway between the donor level and the conduction band. Summary: in an intrinsic semiconductor there is only one energy scale: the band gap E g. In an extrinsic semiconductor there are two energy scales, E g and E d, where E g E d. Therefore with temperature there are three different regimes to consider in a doped semiconductor: Intrinsic high temperature k B T > E g n n i Saturated room temperature k B T > E d n N D Freeze-out low temperature k B T < E d n N D 1. Intrinsic saturated regions From Eqs and 3.13 we obtain n = n2 i n + N D. (3.14) At high temperatures, many electrons (and holes) can be created by thermal excitation across the band gap E g. n i N D, and from Eq we obtain n = n i, this is the intrinsic regime. At intermediate temperatures, all the electrons are thermally excited out of the donor level into the conduction band. n i N D, and from Eq we obtain n = N D ; this is the saturated regime. 2. Saturated freeze-out regions At intermediate temperatures, k B T E g, the system can be analysed using the same model as before but with the energy levels at the donor energy E D and at the conduction band energy E C, as shown in Fig. 16. If the donor activation energy E d k B T then all the donors will be ionised and we again obtain the expressions for the saturation region: n = N D p = n 2 i /n = n2 i /N D Majority carriers Minority carriers At lower temperatures where k B T E d, then the semiconductor is in the freeze-out region. We can directly use Eq. 3.7 by replacing the hole concentration

12 12 Intrinsic Saturated Freeze-out T > 500 K 45 K < T < 500 K T < 45 K In a compensated semiconductor with both donors and acceptors, charge conservation gives n + N A = p + N D. n can be obtained by solving the following quadratic equation in n n = n2 i n + N D N A, (3.17) and p can be obtained from np = n 2 i. FIG. 17 Upper: The excitation of carriers in a semiconductor as a function of temperature (typical values for a specimen of doped germanium). Lower: The variation with temperature of majority and minority carrier concentration, and mobility µ and conductivity σ for a semiconductor. p with the concentration of ionised donors N D (though these are not mobile, of course) and by replacing E g by E d we obtain nn D = N C N D exp( E d /k B T ) Since n = N D we have at low temperatures n = (N C N D ) 1/2 exp( E d /2k B T ), (3.15) which is similar to Eq. 3.8 for excitations across the bandgap in an intrinsic semiconductor. The Fermi level at low temperature is given by E F = 1 2 (E C + E D ) k BT log(n D /N C ). (3.16) At T = 0 K, E F lies midway between the donor level and the conduction band. For Si:P ( m 3 donors) the three different regimes are:

13 13 FIG. 18 Band diagram, density of states, Fermi function and carrier concentration as functions of energy for an n-type semiconductor. FIG. 19 Band diagram, density of states, Fermi function and carrier concentration as functions of energy for a p-type semiconductor.

14 14 IV. CONDUCTION IN SEMICONDUCTORS A. The Drude Model The force on a charge q in an electric field E and a magnetic field B is F = qe + qv B m v τ = m dv dt, (4.1) where τ is a relaxation or scattering time; the term m v/τ represents a frictional or resistive force. The velocity v can be either the velocity of an individual electron or the average drift velocity of the carriers. At B = 0 and in the steady state, dv/dt = 0, the mean drift velocity v, the mobility µ, the conductivity σ and the current density J = n q v are v = q τ m E = µe J = σ E v where µ = E = q τ m = e τ m σ = n q µ = n q2 τ m (4.2) For electrons (charge -e) and holes (charge +e) the conductivities add J = (σ e + σ h ) E = σ E, where σ = n e µ e + p e µ h. (4.3) FIG. 20 Temperature dependence of the mobility µ for a semiconductor in which there is scattering from phonons and charged impurities. TABLE VI Carrier mobilities at room temperature, in cm 2 /Vs. Crystal Electrons Holes Crystal Electrons Holes Diamond GaAs Si GaSb Ge PbS InSb PbSe InAs PbTe InP AlAs AlSb SiC B. The mobility µ In a pure crystalline semiconductor (or metal) at T = 0 K there is no scattering. However in a real semiconductor at finite temperatures there is scattering due to lattice vibrations (phonons) and defects (ionised impurities). For N s scattering centres per m 3 with a crosssection A s, the mean free path l of electrons with a velocity v is l = vτ = 1/N s A s. The mean thermal velocity < v T >= 3k B T/m T 1/2 scattering rate: 1/τ 1/µ T 1/2 N s A s 1. Charged impurities At low temperatures scattering from charged impurities gives Rutherford scattering: A imp 1/v 4 T 1/T 2 1/τ T 3/2 N imp 2. Lattice vibrations At high temperatures there is scattering from the acoustic phonons (lattice vibrations). For non-polar semiconductors A ph k B T 1/τ ph T 3/2 = µ ph T 3/2, where µ ph is known as deformation potential mobility. In polar semiconductors (for example, GaAs) there is also a strong coupling to the optic phonons which can dominate the mobility at room temperature. Adding scattering rates using Matthiesen s law gives 1 τ = (4.4) τ imp τ ph Electrons and holes will have different mobilities, and usually µ e > µ h, see Table VI. µ imp T 3/2 /N imp. At very low temperatures, freeze-out will occur. C. Hall effect and magnetoresistivity tensor (Drude model) In an electric field E and a magnetic field B the equation of motion, Eq. 4.1, for the steady state drift velocity

15 15 FIG. 21 Experimental temperature dependence of the mobility µ of free electrons in n-type Ge. For samples (1) to (6), the donor concentration N D varies between and cm 3. becomes (put v = J/nq) E = ρ 0 (J + sµb J) = ρj, (4.5) where s = +1 for positive charges and s = 1 for negative charges, ρ 0 = 1/σ 0 = 1/n q µ, and µ and ρ are always taken as positive quantities. For a magnetic field along the z-axis, the resistivity tensor ρ can be written as 1 sµb 0 ρ = ρ 0 sµb 1 0. (4.6) This relates the components of the current density to those of the electric field. In this simple model the magnetoresistivity is independent of magnetic field, since for a current along the x-axis only (as in a Hall bar or wire) we have E x = ρ xx J x = (1/σ 0 )J x, (4.7) where σ 0 = neµ is the zero-field conductivity (now using the electronic charge e = e ). The electric field perpendicular to the current, the Hall field E H, is given by E H = E y = ρ yx J x = sb ne J x. (4.8) FIG. 22 Experimental conductivity σ of n-type Ge as a function of temperature. For the samples (1) to (6) the donor concentration N D varies between and cm 3. The Hall coefficient is defined as R H = E H /J x B, hence R H = s ne. (4.9) Note the Hall coefficient R H is positive for holes and negative for electrons. The geometry of the fields and currents is shown in Fig. 23, where the electric field is at the Hall angle φ H to the current, which is given by tan φ H = ρ yx ρ xx = µb = ω c τ, (4.10) where ω c = eb/m is the cyclotron frequency. D. The Hall effect for two carriers In semiconductors with electrons (concentration n, mobility µ e ) and holes (concentration p, mobility µ h ), the conductivity is given by Eq The Hall coefficient is then given by R H = 1 e (p µ 2 h n µ2 e) (p µ h + n µ e ) 2. (4.11) If the mobilities are equal, then the Hall coefficient equals zero for n = p. Note that a negative Hall coefficient can

16 16 FIG. 23 Geometry of the Hall effect. FIG. 24 Schematic setup for a Hall effect measurement. B is the magnetic field, i is the current through the sample, and U H is the measured Hall voltage. FIG. 25 Temperature dependence of the Hall constant R H for (a) p-type and (b) n-type silicon. For p-type with a boron concentration of cm 3, intrinsic conductivity sets in at 1300 K. be obtained even in p-type semiconductors if the hole mobility is much lower than the electron mobility, which is often the case. This is also the reason why the Hall coefficient is usually negative in intrinsic semiconductors where the carrier concentrations are equal. The Hall coefficient is also modified if the electron relaxation time is energy dependent and for an anisotropic effective mass, as in Si and Ge. E. Electrical measurements Several techniques are used to measure the resistivity and Hall effect of semiconductors. Three common ones are four-terminal measurements, the four-point probe and the Corbino disk. 1. Four-terminal measurements A four-terminal arrangement is shown in Fig. 26. A DC current I is passed through the sample and the voltages between contacts 1 and 2 (V R ) and contacts 1 and 3 (Hall voltage V H ) are measured. The current is determined from the voltage V SR across a standard resistor R S. The FIG. 26 Four-terminal measurement setup. resistance R between contacts 1 and 2 is R = V R I = a ρ b c = a σ b c, (4.12) where ρ and σ = ρ 1 are resistivity and conductivity of the material. a is the distance between the contacts, b is the sample width, and c is the sample thickness. The direction of the magnetic field B is also shown in Fig. 26, allowing measurement of the Hall coefficient, which is R H = E H JB = V H b c c I B = V H b I B, (4.13) where E H is the Hall field (perpendicular to both the

17 17 Laplace equation. The non-uniformity is modeled 13 as a series of distinct layers, each having a uniform resistivity. The result is complicated; for the case of a finite layer thickness x on an insulating substrate V I = ρ(x) πx ln [ sinh(x/s) sinh(x/2s) ]. (4.18) In a thin layer of uniform resistivity ρ, where the separation of the probes is very much greater than the thickness of the layer, x s, this expression reduces to and the sheet resistance is given by V I = ρ ln 2 πx, (4.19) ρ s = ρ x = π V ln 2 I. (4.20) 3. Corbino disk A two terminal measurement of the conductivity σ xx is possible using the Corbino geometry shown in Fig. 28. FIG. 27 The linear four-point probe technique. current I and the field B) and J is the current density. The Hall voltage contacts may not be exactly opposite each other and the measured V H will have contributions from the longitudinal resistance as well as the transverse Hall voltage. There will also be a thermoelectric voltage V 0 even with no current flowing. Hence, the voltage between the two contacts can be written V (I, B) = V 0 + I R + I B R H /b. (4.14) The resistance in zero field is calculated from the change in V R when the current is reversed: R = [V R (I, 0) V R ( I, 0)]/2I. (4.15) The Hall coefficient is calculated from the change in V H when the field B is reversed: R H = [V H (I, B) V H (I, B)]b/2IB. (4.16) The resistance in a magnetic field R(B) is calculated from the change of V R, when both the current I and the field B are reversed: R(B) = [V R (I, B) V R ( I, B)]/2 I. (4.17) 2. Four-probe measurements The four-point probe resistance on the top surface of a semiconductor structure, see Fig. 27, having a vertically non-uniform structure can be obtained by solving the FIG. 28 Corbino disk. One electrode is at the centre of the disc, and the other is around the circumference. Current flows radially, so that a Hall field cannot be established. 13 P. A. Schumman, Jr. and E. E. Gardner, Journal of the Electrochemical Society 116, 87 (1969).

18 18 V. OPTICAL PROPERTIES OF SEMICONDUCTORS A. Optical properties The interaction of light with semiconductors gives an important and powerful technique for investigating their properties. Interactions occur over a wide range of frequencies. The main features shown in Fig. 29 are: ➀ The fundamental absorption region arises from transitions between the conduction and valence bands, with high absorption. A well defined absorption edge at typically 1 ev determines the band energy gap. ➁ Just below the band edge, structure is found in the spectrum due to bound electron-hole states known as excitons. ➂ At longer wavelengths the absorption increases again due to free carrier absorption. ➃ At energies between 0.02 and 0.05 ev absorption peaks occur due to interactions with the phonons. This is very strong in polar semiconductors (reststrahlen region). ➄ Impurity levels also give absorption lines, due to the excited states and ionisation of donor and acceptor atoms. ➅ Extra absorption lines are found, at low energies, in a magnetic field, due to cyclotron resonance of the carriers. In magnetic semiconductors, spin waves called magnons are also detected. FIG. 29 Absorption spectra of a hypothetical semiconductor. which defines a complex dielectric constant or relative permittivity ε = ε 1 + j ε 2, and a complex refractive index n = ε = n 1 + i n 2 = ck/ω. The reflectivity is measured when a plane wave of amplitude E i is normally (90 ) incident from vacuum (ε = 1, n = 1) onto a medium with dielectric constant ε. The reflected wave has amplitude E r and the fraction of power reflected is called the reflectivity R = E r 2 E i 2 = 1 2 ε n 2 ε = 1 + n 2 = (1 n 1) 2 + n 2 2 (1 + n 1 ) 2 + n 2. 2 (5.3) If the refractive index is purely imaginary, n = i n 2, then R = 1, and there is 100% reflection of the incident wave. B. Optical constants The dielectric function ε(ω, k) describes the response of a crystal to an electromagnetic field; it depends on the electronic band structure, and can be studied by optical spectroscopy. In the infrared, visible and optical regime k is very small compared to the shortest reciprocal lattice vector, k G, and is usually taken to be zero. An electromagnetic wave propagating along the z-axis in a semiconductor has a perpendicular electric field which can be written as E(z) = E 0 e i (ωt kz) = E 0 e iωt e ik 1z e k 2z, (5.1) where k = k 1 i k 2 is the complex wavevector of the light. The intensity of the light varies I(z) = E 0 E 0 = I 0 exp( az), where a = 2 k 2 is the optical attenuation coefficient. The speed of light in a non-magnetic semiconductor is given by v = c n = ω k = 1 εε0 µ 0 n 2 = c2 ω 2 k2 = ε, and hence (5.2) C. A microscopic model A simple model 14 can be used to gain insight into optical interactions. Consider a bound charge, mass m, with a natural resonance frequency of ω 0. In an applied electric field the equation of motion for the displacement r from equilibrium is Force = qe mγ dr dt mω2 0r = m d2 r dt 2, (5.4) where γ is a damping term representing an energy-loss mechanism, and mω 2 0r is the restoring force. For a sinusoidal electric field at an angular frequency ω we have 14 Jenkins, Chap. 5. r = qe/m (ω 2 0 ω2 ) iγω.

19 19 If there are N charges per unit volume 15 then the polarisation P (charge displacement) produced is P = (Nq2 /m) E (ω 2 0 ω2 ) iγω = χε 0E = (ε 1) ε 0 E. The dielectric constant ε(ω) is frequency dependent and given by ε(ω) = 1 + P ε 0 E = 1 + (Nq2 /ε 0 m) (ω0 2 ω2 ) iγω. (5.5) This simple model gives the following two limits for ω = 0 and ω = ε(0) = 1 + Nq2 ε 0 mω0 2 and ε( ) = 1, (5.6) where ε(0) is the static dielectric constant. At low frequencies the charges follow the field, and there is a resonant response at ω = ω 0. At high frequencies the charges can no longer respond. This is a classical model, and is known as the Lorentz oscillator. The equivalent quantum result is ε(ω) = 1 + (Nq2 /ε 0 m)f (ω 2 0 ω2 ) iγω, (5.7) where ω 0 = (E m E n )/ h is the frequency for the transition between two quantum states E m and E n, and f is the oscillator strength for this transition. In real materials there will be many contributions to the dielectric constant: FIG. 30 Components of the complex permittivity ε = ε 1 +iε 2 (top: left & right), and the complex refractive index n = n 1 +in 2 (middle: left & right), as a function of frequency ω for the case of a single oscillator: (a) ω 0 /γ = 25; (b) ω 0 /γ = 10; (c) ω 0 /γ = 5. The bottom figure shows reflection (top), attenuation (middle) and absorption (bottom) curves calculated for ω 0/γ = 25. The absorption curve is displaced for clarity. ε(ω) = 1 + i (Nq 2 /ε 0 m i )f i (ω 2 0 ω2 ) iγ i ω. (5.8) For any given mechanism this can be written as ε(ω) = ε( ) + (ε(0) ε( )) ω2 0 (ω 2 0 ω2 ) iγω, (5.9) where ε( ) expresses the contribution from all the other effects. Given the dielectric constant ε(ω) the refractive index n, the wavevector k, the absorption coefficient and the reflection coefficient R can be calculated from Eqs. 5.2 and 5.3. These quantities are shown in Fig. 30 for a single oscillator with different values of ω 0 /γ. Figure 30 shows reflectivity curves for a system of two oscillators with resonant frequencies ω 1 and ω 2. FIG. 31 Resultant reflectivity of system of two oscillators of resonant frequency ω 1 and ω 2. (a) ω 2 /ω 1 = 1.1; (b) ω 2 /ω 1 = 1.2; (c) ω 2 /ω 1 = In Chap. 3, N was the total number of electrons and n = N/V was the density. In this chapter n is the refractive index and N is the density.

20 20 FIG. 32 Infrared reflectivity due to lattice vibrations in GaP. The continuous line is a calculated fit to a classical damped simple harmonic oscillator. FIG. 34 Observed absorption spectrum for As impurities in Ge. FIG. 33 The different phonon modes as a function of the wavevector k = 2π/λ. There are transverse acoustic (TA), longitudinal acoustic (LA), transverse optic (TO), and longitudinal optic (LO) phonon branches. The frequency of the optic modes is typical 10 THz (50 mev). D. Phonon absorption The simplest application of this model 16 is in polar semiconductors, such as GaAs, where the bonding is partly ionic and the atoms are effectively charged. An electric field applied to an ionic crystal causes a displacement of the positive and negative ions - thus affecting the dielectric constant ε(ω). As a result there is a very strong coupling between light and the transverse optic phonon at an angular frequency ω T O. The dielectric constant, from Eq. 5.9, can be written as ε(ω) = ε( ) + (ε(0) ε( )) ω2 T O (ω 2 T O ω2 ) iγω, (5.10) where ε( ) allows for the contributions from all the other oscillators and interactions in the semiconductor. The 16 Jenkins p. 266, Hook and Hall p. 265 damping is usually very low for this mode and hence ε(ω) can be negative for ω > ω T O. The refractive index n = ε is then purely imaginary, which gives 100% reflectivity. As ω increases further, ε(ω) 0, and all electric interactions are switched off. This corresponds to the longitudinal optic phonon frequency at ω LO. The region between these two frequencies is one of very high reflectivity and is known as the reststrahlen region. It does not exist in covalent semiconductors such as Si or Ge. Figure 32 shows the reflectivity for GaP - the restrahlen region ω T O < ω < ω LO, (see Fig. 33 for the phonon dispersion curves) corresponds to a reflectivity of unity between µm. E. Shallow donors and acceptors Another application is in the absorption of infrared radiation by shallow impurity sites. In a simple hydrogenic model the energy levels are given by E n = R H p 2 R H = R Hm ε 2 R H = 13.6 ev, (5.11) where p is the Bohr quantum number. For Ge, this would give RH = 10 mev for all impurities, compared with 14 mev for Ge:As, 9.8 mev for Ge:Sb and 12.8 mev for Ge:P. Note that spin-orbit splitting (the interaction between the magnetic field produced by the orbital motion and the electron spin) lifts the degeneracy of the 2p state. The band structure of the Ge also modifies the

21 21 FIG. 35 Calculated (left, using hydrogenic model) and observed (right) energy levels for As impurities in Ge. E E FIG. 37 Optical absorption in pure indium antimonide, InSb. The transition is direct because both conduction and valence band edges are at the centre of the Brillouin zone (k = 0). ω g conduction band edge valence band edge Ω ω valence band edge 0 k -k c 0 k c k Adsorption Adsorption Transparent region onset of direct photon transition onset of indirect photon transition onset of direct transition ħω g photon energy ħω E g + ħω E vert photon energy ħω FIG. 36 Direct gap and indirect gap semiconductors, and their associated photon absorption characteristics. simple model. F. Band-gap absorption The absorption across the energy gap from the conduction band to the valence band is the strongest intrinsic absorption in most semiconductors and lies in the nearinfrared through the visible to the ultraviolet. There are two types of absorption: direct gap and indirect gap. In each case momentum and energy must be conserved. However the momentum of the photon, hω/c is very small and can be neglected. In direct gap materials (GaAs, InSb, etc.) a vertical transition occurs for hω E g. An FIG. 38 Variation of optical absorption coefficient with photon energy. GaAs is a direct gap material. electron-hole pair is created at a wavevector given by hω = E g + h2 k 2 2m e + h2 k 2 2m h = E g + h2 k 2 2m r, (5.12)

22 22 where m r is the reduced effective mass 1 = m r m e m h The absorption coefficient α d is proportional to the density of states and hence α d = C d ( hω E g ) 1/2. (5.13) In indirect gap materials a phonon is created or absorbed to conserve momentum: In this case hω = E g + hω q = k min. (5.14) α i = C i ( hω E g ) 1/2. (5.15) FIG. 39 An exciton is a bound electron-hole pair, usually free to move together through the crystal. In some respects it is similar to an atom of positronium, formed from a positron and an electron. G. Excitons Just below the band-edge, structure is seen due to the formation of excitons, which are hydrogenic-like bound states formed from an electron in the conduction band and a hole in the valence band. Hence we have where R H hω = E g + h2 k 2 2m r R H n 2, (5.16) is now the effective Rydberg energy R H = m re 4 8ε 2 ε 2 0 h2. H. Free carrier absorption FIG. 40 Exciton levels in relation to the conduction band edge, for a simple band structure at k = 0. For mobile charge carriers we can find 17 the optical response from the dielectric constant in Eq In this case we put ω 0 = 0 (the charges are not bound to any position) and γ = 1/τ, where τ is the relaxation or scattering time used in the conductivity to obtain ε(ω) = 1 + (Nq2 /ε 0 m) ( ω 2 iω/τ) = 1 ωp 2 (ω 2 + iω/τ). (5.17) Here n is the carrier concentration (for n-type semiconductors) and ω p = (Ne 2 /ε 0 m) 1/2 is the plasma frequency, which is the natural response frequency of the charge carriers and is typically Hz. At room temperature ω p τ < 1 and free carrier effects are relatively small. At low temperature (and in metals), ω p τ 1, and the dielectric constant is 17 Jenkins p. 276 ε(ω) = 1 ω2 p ω 2 ω p τ > 1. (5.18) FIG. 41 Energy levels of an exciton created in a direct process. If ω < ω p, then ε(ω) < 0 - which corresponds to an imaginary refractive index and 100% reflectivity. As ω > ω p, the reflectivity falls rapidly. In semiconductors we must add the free electron response to the other mechanisms and the plasma frequency becomes ω p = (Ne 2 /εε 0 m) 1/2, where ε can be taken as the static dielectric constant.

23 23 FIG. 42 Effect of an exciton level on optical absorption in GaAs at 21 K. The band gap E g is ev and the exciton binding energy is ev. FIG. 43 Reflectivity of a semiconductor in the plasma frequency regime. I. Cyclotron resonance In a magnetic field B, free electrons move in circular orbits in a plane at right angles to the field; the angular frequency is ω c = eb/m, where ω c is the cyclotron frequency. In semiconductors, the conduction electrons behave similarly with the cyclotron frequency defined as ω c = eb/m, where m is the cyclotron effective mass. For spherical energy surfaces, appropriate for holes and electrons in III-V semiconductors, m = m e or m = m h. Si is an indirect band gap semiconductor and the electron energy surfaces are prolate spheroids 18 oriented along the < 100 > axes; these six pockets of electrons are shown in Fig. 44. The electron energy in the conduction band for the minimum along the k x axis is given E e (k) = E C + h2 2 m L (k x k min ) 2 + h2 2 m T (k 2 x+k 2 y), (5.19) where k min is the position of the energy minimum along 18 Pointy like a zeppelin because m L > m T, not squashed like oblate spheroids. FIG. 44 Surfaces of constant energy for Si. the k x axis. The cyclotron effective mass is anisotropic (two masses m L and m T ) and depends on the angle θ of the magnetic field relative to the longitudinal axis of the energy surface 1 m 2 = sin2 θ + cos2 θ. (5.20) m T m L m 2 T For θ = 0, m = m T, while for θ = π/2, m = m L m T. Since there are six equivalent energy surfaces in Si, for any general direction of magnetic field there are three different cyclotron effective masses from the electrons. Equation 5.20 is derived assuming that the magnetic field lies in the x-z plane and is given by B = B(sin θ, 0, cos θ). The equation of motion is obtained from the Lorentz force h dk = ev B, (5.21) dt where the velocity is derived from v = 1 h ke, and is given by v = h(k x /m T, k y /m T, (k z k min )/m L ). Equation 5.21 gives the following three equations and dk x dt = eb m T k y cos θ, (5.22) dk y dt = eb m L (k z k min ) sin θ + eb m T k x cos θ, (5.23) dk z dt = eb m T k y sin θ. (5.24) These three equations can be used to derive Eq Figure 45 shows the absorption for silicon when θ = 30, from which the two cyclotron masses, m L and m T, can be determined.

24 FIG. 45 Cyclotron resonance signal from silicon at 4 K using ν = Hz radiation. The magnetic field lies in the (100) plane and is at θ = 30 to the [001] axis. Note 10 koe = 1 T. 24

25 25 VI. SOLID STATE ELECTRONICS A. Inhomogeneous semiconductors Semiconductors form the basis 19 for most electronic devices. A key feature in real devices is that the doping is inhomogeneous - the simplest example of this is the p-n junction diode. In an inhomogeneous semiconductor, there is an extra mechanism of charge transport due to the thermal diffusion of charges in a carrier concentration gradient. Initially when p-type and n-type semiconductors are placed in contact the excess electrons in the n-type side can lower their energy by flowing into the empty states in the p-type material, thereby charging it negatively. The charging produces an electric field that will tend to oppose the diffusion of further electrons - this electric field is responsible for the energy levels on either side of the junction being displaced relative to one another, Eventually a dynamic equilibrium will be reached - this occurs when the E F levels are coincident - a process called the principle of detailed balance. There will be a similar process happening in the valence band, that is, there is a diffusion of holes from the p-type material into the n-type. We will concentrate on the electrons, but will remember to add the hole effects. B. Diffusion currents Charged particles move in the gradient of a potential V according to the equation J = n q µ V = n q µe, where E = V is the electric field. Particles also diffuse from high to low concentrations. For a concentration gradient along the x-axis the diffusion current J diff (along the x-axis) is given by J diff = qd n, (6.1) where D is the diffusion constant. current density J is Hence the total J = n q µ E q D n, (6.2) where E is the electric field (and for electrons we put q = e). In a semiconductor the electron density is given by Eq. 3.3 n = N C exp[ (E C E F )/k B T ], where in an inhomogeneous semiconductor both E C (x) (known as band-bending) and E F (x) can depend on position. Hence 19 Hook and Hall, Chapter 6 n x = n E F k B T x n E C k B T x. In an isolated semiconductor in equilibrium, the Fermi level must be constant, therefore E F / x = 0. Then Eq. 6.2 in 1D becomes { J = nq µe D } E C (6.3) k B T x But the local or internal electric field E seen by the electrons is given by the force from the position dependent energy E C (x) qe = E C x. In an isolated inhomogeneous semiconductor, the total current density J = 0, as the diffusion current produces a charge imbalance which gives an electric field and current which balances the diffusion current. Hence from Eq. 6.3 D = k BT q µ; (6.4) this is the Einstein relation for diffusion. Typical values for Ge at room temperature are µ e = 0.39 m 2 /Vs D e = m 2 /s µ h = 0.19 m 2 /Vs D h = m 2 /s. In an electric field and a concentration gradient the electron and hole current densities will be { J e = eµ e ne + k } BT n and e x { J h = eµ h pe k } (6.5) BT p. e x Note that the signs of the electron and hole diffusion currents are different, as each diffuses down a concentration gradient but have opposite charges. For a semiconductor with an external applied voltage, the Fermi level will depend on position and the total current density is then J = nqd E F k B T x = nµ E F x = neµe ext, (6.6) where E ext is the electric field from the external voltage. Hence the effective conductivity is given by σ = neµ as before. But in the semiconductor the current can be driven by the internal electric field or by carrier diffusion. This applies to a semiconductor in equilibrium. C. Diffusion length An important parameter is the diffusion length L diff. Consider a region of semiconductor where excess carriers (say electrons) are injected (by photoionisation or in a p-n junction) to give a non-equilibrium carrier concentration n(x) = n(x) n 0, where n 0 is the equilibrium

26 26 concentration. These will diffuse away until they recombine with holes in a recombination time τ r. For diffusion along the x-axis J diff = qd n x, (6.7) where the diffusion current decreases as electrons recombine, to return the concentration back to the equilibrium value n 0. In the region between x x+dx the equation of continuity for the excess charge density is J diff (x + dx) = J diff (x) q n(x)dx τ r rate out rate in recombination rate which can written as J diff x = q n(x) τ r charge/unit area/sec Hence which has a solution qd 2 n(x) x 2 = q n(x) τ r, n(x) = n(0) exp( x/l diff ), (6.8) where L diff = Dτ r = µk B T τ r /e is the diffusion length. For τ r = 15 µs and for electrons in Ge at room temperature, L diff = 0.1 mm. The diffusion current for electrons, Eq. 6.7, can then be written as J diff,e = ed n x = ed e L e n(x) D. The p-n junction in equilibrium = ed e L e n(0)e x/l diff. (6.9) A p-n junction is an example of an inhomogeneous semiconductor. Electrons leaving the n-type region leave behind static positively charged positive ions, and similarly on the p- type side there will be a region of negatively charged acceptor ions. Therefore in the junction region there will be two depletion layers containing fixed equal and opposite charges - these can be considered to behave like a charged capacitor. If the function is at x = 0, the depletion layer extends to l p and +l n into the p and n-type materials respectively. With no applied external voltage, the carriers move so that the Fermi level is constant across the junction. In the depletion region, the current from the electric field balances the diffusion current at all points, FIG. 46 Comparison of band profiles for a p-n junction which is (a) unbiased; (b) forward based; and (c) reverse biased. as in Section VI.B, with no net current flow in a dynamic equilibrium. Band bending occurs. Define E i as the Fermi level in an intrinsic semiconductor. Then in the n-type region, from Eq. 3.3 we have n = N C exp[ (E C E F )/k B T ] = n i exp[(e F E i )/k B T ] = n i exp(e Φ n /k B T ) = N D, where N D is the donor concentration and Φ n is measured in volts. Hence ( ) ND ln. Φ n = E F E i e = k BT e A similar expression is obtained for the p-type region, therefore the total internal potential Φ of the junction is Φ = Φ n + Φ p = k BT e { ln ( ND n i ) + ln n i ( NA n i )}. (6.10) The internal potential is less than the energy gap; in Si, E gap = 1.12 ev, for N A = m 3, N D = m 3, n i = m 3, and Φ = 0.67 ev. The ionised impurity charge concentrations (immobile) in the depletion layer can be taken as N D and N A over

27 27 FIG. 47 Left: (a) Net charge density ρ(x), and (b) free charge density, in a p-n junction. Right: Electric field across an equilibrium p-n junction. FIG. 48 Energy band diagram at zero bias showing the number of conduction band electrons able to cross from N P. the depletion layer widths d n and d p. Charge neutrality gives N D d n = N A d p = Q d. (6.11) These charges act like a capacitor C d with charge Q d = N D d n per unit area with a mean charge separation of (d n + d p )/2. The voltage across this capacitor is the internal voltage Hence Φ = Q d C d = e N D d n (d n + d p ) 2εε 0 = en Dd 2 n 2εε 0 + en Ad 2 p 2εε 0 = Φ n + Φ p. d 2 n = 2εε 0Φ n en D and d 2 p = 2εε 0Φ p en A. If N D = N A, then Φ n = Φ p = Φ/2, and the width of the depletion layer is 4εε0 Φ W = d n + d p =. (6.12) en D For a typical dopant level in Si, W 2µm. In the space charge region the Fermi level is far from the conduction and valence bands. Consequently the carrier concentrations are small ( n i ). E. A biased p-n junction Consider a p-n device, across which a voltage V is applied. Almost all the applied voltage will be dropped occur across the junction itself, as the bulk material has a very low resistance. In forward bias (V > 0) the p- type region is positive and the n-type region is negative; current then flows from p-type to n-type. The opposite occurs in reverse bias, V < 0, see Fig.46. For many situations, a p-n junction can be treated as an abrupt junction in which the energy bands change discontinuously at x = 0. The electron concentration at FIG. 49 Energy band diagram at forward bias, showing an increased number of conduction band electrons able to cross N P. x > 0 (in the n-type) is fixed at N D (majority carriers) so that the electron concentration at x = 0 (just in the p-type) from the electrons which diffuse across the junction is n p (x = 0 ) = N D exp[ e(φ V )/k B T ] = n 0 p exp(ev/k B T ), (6.13) where V is the bias voltage and n 0 p is the equilibrium electron concentration (the minority carrier concentration) in the p-type region. Hence there is injection of minority carriers (for V > 0, forward bias) into the p-type region. This produces a diffusion current into the p-region where recombination will occur. This leads to an excess electron density n(x) for x < 0 given by n(x) = n(0) exp( x /L e ), (6.14) where L e is the diffusion length for electrons and n(0) = n 0 p [exp(ev/k B T ) 1] with n 0 p = n 2 i /N A. The diffusion length is typically 50 µm, much greater than the depletion layer thickness - hence the abrupt junction approximation is normally satisfactory. The diffusion current of electrons is given by J diff = qd dn dx = ed e n(0) L e = ed e L e n 0 p [exp(ev/k B T ) 1], (6.15)

28 28 TABLE VII Carrier concentrations in a p-n junction (diffusion model) Zero bias p-type n-type Electron Minority carriers Majority carriers concentrations n 0 p = n 2 i /N A = N D e eφ/k BT n n = N D Hole Majority carriers Minority carriers concentrations p p = N A p 0 n = n 2 i /N D = N A e eφ/k BT Bias V p-type n-type Electron Diffusing minority carriers Majority carriers concentrations at x = 0 n p (0) = n 0 p e ev /k BT n n = N D Hole Majority carriers Diffusing minority carriers concentrations at x = 0 p p = N A p n(0) = p 0 n e ev /k BT where D e = µ e k B T/e is the diffusion constant. A similar expression will apply to the hole current which will add to the electron current. Hence the total current can be written as I = I 0 (exp(ev/k B T ) 1), (6.16) which is the famous Shockley s law (1949) which describes the I-V characteristics of a p-n junction diode. I 0 is the saturation current, which for a cross-section area A is ( ) De I 0 = ea = ea n 2 i n 0 P + D h p 0 n L e L h ( De + L e N A D h L h N D ). (6.17) I 0 is strongly temperature dependent because of the n 2 i factor. In a biased p-n junction the diffusion current is carried by the excess minority carriers in a diffusion length each side of the space charge region, known as the diffusion region. These carriers recombine with the majority carriers which are supplied by the voltage source and hence a majority carrier current flows in the bulk n- type and p-type material. For Ge, this simple theory works well. But in Si, the intrinsic carrier concentration n i is much smaller and there is also a contribution to the current from carrier generation or recombination in the space-charge region. In reverse bias, any electron-hole pairs which are thermally generated in the space-charge region are rapidly separated by the strong electric field there and give a generation current. In forward bias, the recombination of electron and holes in the space-charge region produces a recombination current. In real diodes the experimental Shockley s law may become I = I 0 (exp(ev/αk B T ) 1), (6.18) where the diode ideality factor α lies between 1 and 2. In real diodes the doping is not usually the same in both regions (a single-sided junction). A more heavily doped p-type region would give a p + n junction. The width of the depletion layer is determined by the doping level on the lower doped side. F. Applications of p-n junctions These junctions have many applications, a few of which are described below: 1. Varactor diode A reverse biased (V < 0) p-n junction acts as a capacitor C = εε 0 A/d, where A is the cross-sectional area and d is the width of the depletion layer. Equation 6.11 shows that d (Φ V ) 1/2. Hence C (Φ V ) 1/2, and the capacitance can be voltage controlled. A typical value might be 100 pf for an area of 1 mm Photodiodes (or p-i-n diodes) Optical fibre communications require fast optical detectors. A thin region of lightly doped (or intrinsic) Si can be placed between p-type and n-type material. A reverse bias then depletes the whole central region to give a thick (50 µm) depletion layer. Light from a fibre produces electron-hole pairs which are rapidly separated by the internal electric field. The transit time can be very fast (1 ns) so a rapid response is obtained. 3. Solar cells Solar cells, see Figs , are large area p-n junctions. Light absorbed in the depletion region give a lightdependent current source I L which is added to the diode

29 29 FIG. 52 A solar cell: (a) note the top contact design, and the penetration of photons to various depths; (b) the energy-band diagram for a solar cell. Only those electron-hole pairs which separate across the depletion layer contribute to the output current. FIG. 50 Example of a current-voltage characteristic for a silicon p-n junction. Reverse voltages and currents are given as negative values. FIG. 53 Solar cell: (a) equivalent circuit; (b) the currentvoltage characteristics show a downward shift as the light becomes stronger. characteristic I = I 0 [exp(ev/k B T ) 1] I L. (6.19) The short-circuit current is I L and the open-circuit voltage V oc (I = 0) is V oc = k BT e ( IL + I 0 ln I 0 ). (6.20) The maximum power out IV occurs for (IV )/ V = 0 and is 0.82I L V oc. FIG. 51 The large reverse bias on a p-i-n (p-v-n) photodiode produces depletion through the whole of the v layer. Photons absorbed there produce carriers that are swept away to the terminals. 4. Light-emitting diodes (LEDs) A p-n junction can be constructed so that a forwardbiased current produces light. The injected minority carriers recombine to emit light. Recombination can occur

30 30 Integrating it becomes εε 0 dv dx = ρ ax + constant = ρ a x ρ a d p. where the constant of integration is determined knowing that at the edge of the depletion layer the electric field is zero, dv/dx x= dp = 0. On the n-type side of the junction we have εε 0 dv dx = ρ dx + ρ d d n. Continuity of dv/dx at x = 0 gives ρ a d p = ρ d d n ; ρ a and ρ p have opposite signs and this latter equation can be rewritten as N A d p = N D d n. Integrating again gives εε 0 V0 V p dv = 0 d p ( ρ p x ρ a d p )dx = [ ρ ] 0 a 2 x2 ρ a d p x d p FIG. 54 Current-voltage characteristics of a solar cell under illumination. εε 0 (V 0 V p ) = ρ a d 2 p/2 ρ a d 2 p = ρ a d 2 p/2. Similar arguments on the n-type side gives εε 0 (V n V 0 ) = ρ d d 2 n/2. Therefore the potential across the junction is V = V p V n = 1 2εε 0 (ρ a d 2 p ρ d d 2 n) = e 2εε 0 (N A d 2 p+n D d 2 n) FIG. 55 Transition processes for electrons. The photon is shown leaving the initial state of the radiative transition. directly across the band-gap (for the direct gap semiconductors usually used) or via impurity levels. Some examples are: (i) GaAs diodes at 1.4 ev (band gap) in the infra-red. (ii) GaP doped with Zn and O which form a neutral impurity centre. This produces red light at 1.7 ev. (iii) GaP doped with N which gives colours from red to green as the N concentration varies. (iv) GaAs x P 1 x doped with N. This is a direct gap semiconductor with colours from red to green as x varies. There are also semiconductor diode lasers. V = e 2εε 0 N D d 2 n(1 + N D /N A ). (6.21) Considering the deletion layers to act like a charged capacitor, then the charge on one plate will be Q = en D d n and Eq can be rewritten as Therefore V = C = Q V = 1 2 V 1 2 Q2 2eεε 0 N A + N D N A N D. (6.22) [ ( )] 1 NA 2 N D 2eεε 0. (6.23) N A + N D G. Appendix: Calculation of capacitance of pn-junction The electrostatics of the depletion layer can be determined by solving Poisson s equation 2 V = ρ/εε 0 ; on the p-type side of the junction this can be written as d 2 V dx 2 = ρ εε 0.

31 31 VII. GROWTH AND DOPING OF SEMICONDUCTORS A. Growth of bulk crystals (a) Pure Si is obtained 20 from mined quartz (SiO 2 ), which is melted at 1700 C in a carbon-arc furnace. This reduces the quartz to give molten metallurgical grade Si with many impurities, at the level > 100 ppm (parts per million). (b) The Si interacts with HCl to give trichlorosilane, SiHCl 3, leaving most impurities behind. This is reduced with H 2 gas to give polycrystalline Si with < 1 ppb (parts per billion) impurities. (c) Further purification is done from the melt. For a liquid and solid phase in contact in equilibrium, the ratio of the concentrations of a given impurity in the liquid (C L ) to that in the solid, (C S ) is the distribution coefficient k d = C S /C L, which is usually < 1, see Fig. 56. Hence the semiconductor can purified by solidification from the melt. In practice, a molten region is passed through an impure ingot (zone refining), see Fig. 57; this transfers the impurities to one end of the ingot. (d) Bulk semiconductors are normally grown from the melt, using the following: FIG. 56 Phase diagram for impurities with distribution coefficient k = C S/C L < 1. In the Czochralski technique, see Fig. 58, a single crystal is pulled from the melt, starting with an oriented seed crystal. In the Bridgman technique, see Fig. 59, the semiconductor is melted in a crucible and crystallisation is induced by lowering the temperature. In the floating zone technique, see Fig. 60, a polycrystalline rod is recrystallised as a single-crystal. B. Epitaxial growth In epitaxial growth, the semiconductor is grown on a singe-crystal oriented substrate by vapour-phase epitaxy (VPE) or chemical vapour deposition (CVD), liquid-phase epitaxy (LPE) or solid-phase epitaxy (SPE). These processes occur at lower temperatures than bulk melt growth and gives higher purity and perfection and greater compositional uniformity. In VPE, gases are passed over the substrate. Chemical or thermal reactions occur, catalysed by the substrate and the crystal grows, see Fig. 61. Typical reactions to produce Si are: SiCl H 2 Si(s) + 4 HCl Chemical reaction of adsorbed gases SiH 4 (silane) Si(s) + 2 H 2 Thermal decomposition on hot substrate 20 see Jenkins Chap. 2. FIG. 57 The distribution of solute in an ingot of material: (a) single zone pass; (b) the effect of multiple zone passes. A more recent technique is molecular beam epitaxy (MBE). This is a thermally controlled evaporation process in an ultra-high vacuum (UHV) apparatus, see Fig. 62. Compounds or elements are heated in crucibles (Knudsen cells) to generate neutral atomic or molecular beams directed onto a heated substrate. The beams travel without interaction (Knudsen or molecular flow) to the substrate. For GaAs, the Ga atoms have a sticking probability of 1. As 2 or As 4 molecules do not stick unless Ga is present, when they dissociate and react with the Ga adatoms to form GaAs. The crystal growth rate is very slow, about 1 atomic layer per second. But this enables precise layers and abrupt changes in composition to be incorporated. This has led to heterostructures and quantum wells (see Sec. VIII.C).

32 32 FIG. 59 The horizontal Bridgman technique for the growth of GaAs from the melt. FIG. 58 Schematic diagram of the Czochralski (vertical pulling) of single-crystal production. C. Diffusion doping of semiconductors The controlled doping of semiconductors is vital for the production of devices. These can be incorporated during the growth of the crystal (chemical doping). This can also be done by the diffusion of impurities into the surface of a semiconductor by thermal diffusion into the solid. First the dopant atoms are deposited on the surface, usually by chemical reaction from gases passed over the sample in an oven. For example, passing PH 3 and O 2 over Si gives FIG. 60 The floating-zone technique for the growth of single crystals. 4 PH O 2 2 P 2 O H 2 and 2 P 2 O Si 4 P + 5 SiO 2, which delivers P atoms to the Si surface. A drive-in step is used, which heats up the Si for a time t in, so that the dopants diffuse into the semiconductor. The depth to which the dopants diffuse is given by the diffusion length Dtin. D is the atomic diffusion coefficient, which is thermally activated D = D 0 exp( E/k B T ), (7.1) where E is the activation energy for the creation of the defect. A typical diffusion constant at 1200 C is FIG. 61 Cold-wall reaction vessel for pyrolytic growth of Si.

33 33 FIG. 62 Schematic diagram of an MBE chamber. FIG. 64 Sequence of operations in the production of a chip. E. Chip production A modern silicon fabrication plant uses a wide variety of techniques to make devices. A silicon wafer will be processed on which many integrated circuits or chips will be produced, see Fig. 64. FIG. 63 Schematic diagram of an ion implanter. D = m 2 s 1. Hence a drive-in time of 4000 s would give a diffusion depth of about 1 µm. Note that the dopant concentration will depend on the depth below the surface. D. Ion implantation Another technique is ion implantation (see Fig. 63) in which the semiconductor is bombarded with high-speed positively charged ions, such as B or Sb for silicon and Si, Se or S for GaAs. The energy of the beams is typically kev, though energies up to 2 MeV are used. The ions penetrate for a specific range up to several microns. Note that the ions can travel along relatively open channels in a given crystal structure. A specific dopant profile can be produced by using several implants with different energies. There are two problems with ion implantation. First, considerable structural damage is caused to the crystal. Second, many of the dopants will initially end up on interstitial sites where they will not be electrically active (for a dopant to work it must be on a substitutional site). Hence the sample must be annealed after implantation to restore the crystallinity of the semiconductor and provide enough thermal energy for the implanted ions to move onto lattice sites where they become electrically active.

34 34 VIII. HETEROSTRUCTURES AND NANOELECTRONICS We describe a few modern developments. A. Introduction to low-dimensional systems So far we have been concerned with essentially bulk properties of semiconductors. These properties are described by coefficients that are independent of the size and shape of the specimens. If one or more of the dimensions of the sample are reduced significantly then the sample is described as a low-dimensional system. 21 We can use modern lithographic techniques and molecular beam epitaxy (MBE) to produce systems which behave as films, lines and dots. Departures from bulk behaviour occur when the size of the sample becomes comparable with the wavelength of the relevant excitations in the solid. This phenomenon is called a quantum size effect. A less fundamental effect is when the specimen size is comparable with the mean free path of the excitations. This is known as a size effect (e.g., the size dependence of the thermal conductivity of a sapphire rod). 22 B. The two-dimensional electron gas (2DEG) Consider electrons confined to a film of thickness d by infinite potential barriers at z = 0 and z = d. Motion in the xy-plane is unconstrained. As in the 3D case we solve Schrodinger s equation for the band structure with boundary conditions ψ = 0 at z = 0 and z = d. The solution ψ(x, y, z) = e ikxx e ikyy sin(k z z) with k x = 2πp/L, k y = 2πq/L, and k z = nπ/d. L is the macroscopic dimension of the sample in the x and y directions. The energy levels are E = h2 (kx 2 + ky 2 + k 2m z) 2 = h2 e 2m e L 2 (p2 + q 2 ) + n2 h 2 8m e d 2 ; if d is small, the final term can be large. The lowest state with n = 2 (with p = q = 0) is higher in energy than n = 1 by 3 h 2 /8m e d 2, which exceeds k B T at room temperature for d < 60 Å. The motion of electrons in the z-direction is frozen, and they behave as free particles in two dimensions. 21 Hook and Hall, Chap Hook and Hall, Sec FIG. 65 The allowed wavevectors for 2D motion in the xyplane. The allowed states form a simple 2D lattice of spacing 2π/L in both the k x and k y directions. To calculate the 2D density of states the number of states in the ring between radii k k + dk must be determined. 1. Density of states (DOS) in 2D The 2D density of electron states in k-space is given by 2πk dk g(k) dk = (2π/L) 2 = L2 k dk ; (8.1) 2π and therefore g 2D (E) is given by g 2D (E) = m e π h 2 (8.2) which is independent of energy. If there are three 2D subbands that are occupied, the step-like density of states shown in Fig. 66 will be obtained. In practice quantum wells can be produced in semiconductor heterojunctions prepared by MBE or by using Metal oxide semiconductor (MOS) junctions. C. Quantum wells The advent of molecular beam epitaxy (MBE) has led to new types of devices. In heterostructures, alternate layers of GaAs and Al x Ga 1 x As are grown on a GaAs substrate with atomic precision, and because epitaxial growth is much slower than melt growth there is lower

35 35 (a) Al Ga As GaAs Al Ga As a 2 ψ 1 (b) E e1 ΔE C E C E(Al Ga As) g E(GaAs) g ΔE V E V FIG. 67 (a) A sample consisting of a thin layer of GaAs of width a sandwiched between layers of Al 0.3 Ga 0.7 As. (b) The electronic structure of a GaAs quantum well. E g(gaas) is the forbidden gap of bulk GaAs. The lowest confined level in the conduction band lies E e1 above the conduction band edge of bulk GaAs and its wavefunction is ψ 1. doped undoped FIG. 66 Density of states of electrons in a thin film. E 1, E 2 and E 3 are the bound state energies. The dashed line shows the energy dependence of the DOS of 3D electrons in a thick film, where g(e) E 1/2. disorder and higher purity. Layer widths can be controlled from a few nm up to µm. Conduction and valence band offsets, E C and E V, shown in Fig. 67 exist between the GaAs and Al x Ga 1 x As so that quantum wells are formed for both electrons and holes in the GaAs layers. The free carriers are now confined along the z-axis and form standing waves in the potential well; the electrons are still free to move in the xy-plane. For an infinite potential well, width a, the energies of the electron states are E(k) = E p + h2 k 2 2m = h2 p 2 8m a 2 + h2 k 2 2m, (8.3) where p is a quantum number giving the number of halfwavelengths of the electron standing wave in the well. Because of the different effective masses, different levels will be formed for heavy and light holes. The energy states for a given quantum number p are known as subbands AlGaAs E 1 E 2 GaAs E 1 E 2 FIG. 68 The subband structure at the interface in a GaAs active channel in a HEMT structure. E 1 and E 2 are the confined levels. The approximate positions of E 1 and E 2 are shown in the lower part of the figure, as well as the shape of the wavefunctions. In the upper part is the approximate potential, including contributions of the conduction band offset. D. High electron mobility transistors If a vertical electric field is superimposed on the band structure, then a narrow quantum well can be formed at an AlGaAs/GaAs interface. Electrons (or holes) in this well populate the lowest sub-band and form a twodimensional electron gas (2DEG). These electrons can come from donors in a doped layer which is separated by a spacer layer. This is called modulation doping and can give very high mobilities as the charged impurities are well removed from the carriers. Such structures are used in high electron mobility transistors, more commonly known as HEMTs. Mobilities of 10 m 2 /Vs can be achieved, compared to 0.06 m 2 /Vs in Si-MOSFETs,

36 36 which gives the possibility of much faster devices. 1. Nanostructures Given the 2DEG in a heterostructure the electrons can be further confined in the xy-plane by using electrodes (or gates) above the 2DEG. If the lateral dimensions of a device, L, can be reduced to be comparable with the de Broglie wavelength λ then electron interference effects can occur. It can also be less than the electron mean free path. Two types of scattering are distinguished. In elastic scattering the electron wave remains coherent (i.e., no random phase shift) after scattering whereas in inelastic scattering there is a random phase shift and the electron wave after scattering is incoherent with the wave before scattering, thus destroying any interference effects. We define an elastic mean free path l e and an inelastic mean free path l i. Several regimes are distinguished: Diffusive regime: L > l e. This is the normal situation considered so far. Kinetic equations can be used, using average velocities and currents. Quantum transport regime: l e < L < l i. Interference effects must be considered. Scattering at the boundaries may be important. Ballistic transport regime: L < l e < l i. The electron may pass through the device without being scattered. Many new effects then arise. Mesoscopic physics is the physics of systems or structures where average statistical properties cannot be used. The detailed response of these systems will depend on the specific structure of the individual device. For example, a single impurity atom may determine the response. If it moves, the explicit response will change. 2. Quantum dots and the single-electron transistor A very interesting example of a mesoscopic system is a quantum dot in which a small isolated region is weakly coupled to two reservoirs. The electrons in the dot form an almost isolated system with a series of energy levels E p where p is a quantum number. An integral number of electrons reside on the dot (typically 1000). Electrons can tunnel between the dot and the two reservoirs and hence the states in the dot will be occupied up to the Fermi level. The energy of the dot and hence the number of electrons can also be controlled by a gate electrode which is capacitatively coupled to the dot. As the gate potential is swept, the conductance (in the low bias voltage limit) through the dot exhibits a series of resonances as the energy levels of the electrons in the dot match the Fermi level E F in the reservoirs. At each resonance, one more electron is added to the dot. This is known as a Coulomb blockade as adding electrons helps or hinders the conductance through the dot. The quantum dot system can be used to make a single-electron transistor since the current through the device is controlled by the gate potential. 3. Low-dimensional physics A very important consequence of the production of heterostructures has been the study of the physics of the the two-dimensional electron gas (2DEG). At room temperature the properties of a a 2DEG are similar to 3D electrons. But at low temperatures, novel effects occur, such as the Quantum Hall Effect (QHE), discovered by von Klitzing in l980 (Nobel prize 1985). One-dimensional conductors have also been studied. A striking result is that the conductance G = I/V of a onedimensional ballistic conductor in is quantised in units of 2e 2 /h. E. Density of free electron states in 1D, 2D, and 3D The density of free electron states g(e) = dn/de in 1D, 2D and 3D are derived in this section. g(e) is the number of available states per unit energy per unit volume; this could also be called the density of available states per unit energy. Whatever the dimension d the density of states (DOS) can be written as g(e) = dn de = dn dk dk dk = g(k) de de. (8.4) The kinetic energy of free electrons is given by so so Eq. 8.4 becomes dk de = E = h2 k 2 2m ( ) 1 de = m dk h 2 k, g(e) = dn de = dn dk dk de = g(k) m h 2 k. (8.5) The number of states in a cubic box of side L is found by imposing periodic boundary conditions, for example, in the x-direction this condition is e ikx(x+l) = e ikxx. Therefore e ikxl = 1, which means that k x = m x 2π L, where m x is an integer. Similar boundary conditions apply in all three directions, so the wavevector is: k = (kx, k y, k z ) = 2π L (m x, m y, m z ), (8.6) where the quantum numbers m x, m y and m z are integers. Hence Eq. 8.6 tells us that the electron states are uniformly distributed in k-space, with spacing 2π L.

37 37 (a) 1. 3D DOS: g 3D (E) In 3D the available states between k k + dk are contained in a shell of radius k and thickness dk, see Fig. 69(a), which has a volume 4πk 2 dk. The electron states are uniformly distributed in k- space, with spacing 2π L. Therefore the number dn of energy states contained in the shell is (b) k y dn = 2 4πk2 dk (2π/L) 3, where the extra factor of 2 is due to the spin degeneracy of spin 1/2 electrons. The volume of the sample is L 3, so the number density is defined as n = N/V = N/L 3. Therefore the density of states per unit volume in k-space is dn dk = g(k) = 8πk2 (2π) 3 = k2 π 2 k f Using Eq. 8.5 the 3D density of free states is g 3D = k2 π 2 m h 2 k = mk π 2 h 2 = m π 2 h 2mE. (8.7) 3 k x 2. 2D DOS: g 2D(E) (c) k y In 2D the available states between k k + dk are contained in a ring of radius k and thickness dk with area 2πkdk, see Fig. 69(b). The electron states are uniformly distributed in k- space, with spacing 2π L ; therefore the number dn of energy states contained in the ring is dn = 2 2πkdk (2π/L) 2, -k f k f k x where the extra factor of 2 is due to the spin degeneracy of spin 1/2 electrons. The area of the sample is L 2, so the number density is defined as n = N/L 2. Therefore density of states per unit volume in k-space is dn dk = g(k) = 4πk (2π) 2 = k π Using Eq. 8.5 the 2D density of free states is FIG. 69 (a) Visualization of 3-dimensional k-space. The number of available states is determined by the number of points contained in the shell of radius k and thickness dk. (b) The ground state of 2DEG at T = 0 can be represented by a Fermi circle of radius k F in k-space, centred on k = 0. States satisfying k < k F are occupied, and those k > k F are unoccupied; the boundary between the two is called the Fermi surface. (c) A Fermi segment for a 1DEG. The confinement along the y direction increases the spacing k y between the adjacent states. When k y > k F, the condition L y < λ F is satisfied. The second subband is too high in energy, and only the first subband is populated. 3. 1D DOS: g 1D (E) g 2D = k π m h 2 k = m π h 2. (8.8) In 1D the available states between k k + dk are contained at the two points at ±k with thickness dk, see Fig. 69(k), which has length 2dk.