Chapter 5. Carrier Modeling

Transcription

1 Chapter 5 Carrier Modeling References: Bohr's Atomic Model Nucleus at the center: - protons/neutrons. Electrons orbit around the nucleus. Orbits- restricted to certain discrete (quantized) values of radii. Orbits correspond to energy. Energy of electron quantized.

2 Bohr's Atomic Model Silicon Atom

3 Electrons transfer to outer orbits by absorbing energy. Electrons must release energy to come to inner orbits. Bohr Model - not adequate to fully describe the atom. Quantum mechanics offers a better description of the atom. Offers a basic understanding of electron behavior - in terms of quantum numbers.

4 Properties of materials generally determined by the behavior of electrons in its atoms. 1925: Schrodinger developed an equation that describes the wave motion of the electron. Each solution described by 3 quantum numbers n, l, and m. Pauli later showed that a fourth quantum number, the spin number, is required to fully describe an electron. Widely accepted view on electron behavior today takes four quantum numbers to completely describe the state and motion of any electron orbiting an atom's nucleus. The farther away a sub-shell is from the nucleus, the higher is the energy content of its electrons.

5 Quantum Numbers Four numbers used to describe the state and motion of any electron in an atom. 1. Principal Quantum Number, n The main energy shell that the electron belongs to. Assigned an integer = 1, 2, 3, 4,. 2. Orbital Angular Momentum Number, l Determines which energy subshell or orbital within the main energy shell the electron occupies. Has values ranging from 0 to n-1. Assigned a letter with the convention: 's' for l = 0; 'p' for l = 1; 'd' for l = 2; and 'f' for l = 3, etc. 3. Magnetic Number, m: Specifies the orientation of the angular momentum of the electron. Assigned integer values ranging from -l to +l. 4. Spin Number, s: Specifies the spin of the electron on its axis. Assigned values: + ½ and ½.

6 First ten orbitals and corresponding quantum numbers of a hydrogen atom Eg: For l =1 (p-orbital), there are three possible values for m: -1, 0, and +1. Each value can accommodate up to two electrons. A 'p' orbital can have a total of 6 electrons. An 's' orbital can only have a maximum of 2 electrons, a 'd' orbital can have 10 electrons. The two electrons occupying every m value can be distinguished using the spin number - specifies the direction in which the electron is spinning on its axis. This spin number can only have a value of either -1/2 or +1/2.

7 Pauli's Exclusion Principle - No two electrons can occupy the same quantum state simultaneously. -No two electrons in an atom can have exactly the same set of quantum numbers. -Forbids any two electrons to occupy the same quantum state. Pauli's Exclusion Principle applies to fermions - particles with a half-integer spin E.g: electrons, protons, neutrons, and muons. Not applicable to 'bosons', - particles with integer spins.

8 Energy Bands Recall: Isolated atom, electrons can only occupy certain discrete energy levels, can not have same quantum numbers. When a large number of atoms are brought together to form a crystal, the permissible energy levels split into a band of energies so that the crowded electrons will still have unique quantum numbers. Energy Bands - sets of closely-spaced energy levels that arise when atoms are brought together to form a crystal. Splitting of an energy level into an energy band

9 Energy Bands Versus Interatomic Distance Isolated atoms have discrete electron energy levels. If these atoms are brought closer, the initial quantized energy levels will eventually split into a large number of closely spaced, discrete energy levels. This large number of closely spaced energy levels is known as an energy band. Changes that accompany: 1) the permissible energy levels become a function of interatomic distance 2) Discrete energy levels become energy bands, i.e., split into a large number of closely spaced energy levels. 3) The valence electrons are no longer localized to a particular atom.

10 Isolated carbon atoms contain six electrons occupying 1s, 2s and 2p orbital in pairs. Consider valence electrons occupying 2s and 2p orbitals. As lattice constant is reduced, there is an overlap of the electron wavefunctions occupying adjacent atoms. This leads to a splitting of the energy levels into bands containing 2N states (2s) and 6N states (2p), N= # atoms in crystal. Further reduction of the lattice constant causes the 2s and 2p energy bands to merge and split again into two bands containing 4N states each. At zero Kelvin, the lower band is completely filled with electrons and labeled as the valence band. The upper band is empty and labeled as the conduction band. Characteristics of Energy Bands Energy bands generally have: a width that increases as inter-atomic distance decreases and as the energy level becomes higher. energy levels whose number depends on the number of atoms involved and the number of discrete electron energy states in an isolated atom

11 Types of Energy Bands Permissible and Forbidden Energy Bands - Energy bands are either permissible or forbidden.- Permissible bands include: (a) Conduction Band energy band whose electrons can be used to conduct current. (b) Valence Band the highest energy band below the conduction band that is occupied. (c) Core Bands all bands below the valence band. Forbidden band: a band of non-permissible energy levels between the valence and conduction bands, the range of which is known as the Band Gap' (Eg) Classification of a solid: conductor, semiconductor, insulator, depending on bandgap. A conductor: valence band partially filled at 0 K. Insulator: valence band is completely filled at 0 K. Large bandgap. A semiconductor: valence band is completely filled at 0 K. Intermediate bandgap.

12 Energy Bands in Solids The energy bands in a solid determine whether the solid is a conductor, a semiconductor, or an insulator Electrons in a solid: found either in the conduction band or the valence band. Electrons in the conduction band - free to move. Used to conduct electricity. Electrons in valence bands - not mobile. Can not be used to carry an electrical charge. Electrons in valence bands can be supplied with external energy in order to excite them to jump into the conduction band. Conduction band is separated from the valence band by a band gap. Valence electron must gain energy that's large enough to overcome this band gap before it can transfer to the conduction band.

13 In a conductor, the valence and conduction bands overlap. Free valence electrons can easily move to the conduction band and participate in conduction, resulting in a very low electrical resistance. Insulator, the band gap is large. Transfer of valence electrons into the conduction band difficult. Insulator has high electrical resistance. Semiconductor, the band gap is not large. Valence electrons can be more easily induced to jump into the conduction band. Electrical conduction requires partially-filled valence bands or the presence of empty energy levels that excited electrons can transfer to. If energy band is completely filled, the electrons have no vacant energy levels to move up to. Unoccupied energy levels provide the destination electrons can move to upon gaining energy. This is why metals are good conductors - they have partially filled valence bands and highly mobile electrons that are not localized to any specific atom as charge carriers.

14 Possible energy band diagrams of a crystal. Shown are a) a half filled band, b) two overlapping bands, c) an almost full band separated by a small bandgap from an almost empty band and d) a full band and an empty band separated by a large bandgap. Electronic properties of a semiconductor are dominated by the highest partially empty band and the lowest partially filled band. It is often sufficient to only consider those bands.

15 Energy of most energetic electron in a metal at 0 K is known as the Fermi energy (E F ). Work function of a metal = the minimum energy required to remove electron from the Fermi level of the metal to vacuum level. Vacuum level = energy of a free electron (K.E. = 0) in vacuum. Energy of electron on metal surface affected by the optical, electrical, and mechanical properties of surface. Condition of surface affects work function. Changing surface of metal (e.g. charging, oxidation, contamination) affects work function. Work Function (Φ) Measured in ev - is the minimum amount of energy needed to remove an electron from the surface of a conductor, to a point just outside the conductor where the electron has zero kinetic energy.

16 This model is inadequate for semiconductors. Atoms in semiconductor crystals produce a periodic potential which affects electron behavior in the material. To find energy levels in semiconductors requires: Analysis of periodic potentials. Use of periodic wave functions, called Bloch functions. Due to the periodic potential, mass of electron is adjusted in to effective mass. Results: energy levels are grouped in bands, separated by energy band gaps.

17 Energy bands for diamond Vs lattice constant Energy band diagrams of common semiconductors The energy band diagrams of semiconductors are rather complex. The energy is usually plotted as a function of the wavenumber, k, along the main crystallographic directions in the crystal, since the band diagram depends on the direction in the crystal. The energy band diagrams contain multiple completely-filled and completely-empty bands. In addition, there are multiple partially-filled band.

18 Direct and Indirect Bandgap Direct bandgap: conduction band minimum and the valence band maximum occur at the same value for the wavenumber. Indirect bandgap: E V and E C occur at different values of wavenumber. Direct bandgap materials provide more efficient absorption and emission of light. Energy band diagrams Germanium Silicon Gallium Arsenide

19 Simplified Energy Band Diagram: only electrons in the highest almost-filled band (valence band) and the lowest almost-empty band (conduction band) dominate the behavior of the semiconductor.

20 Valence band edge, E v. Conduction band edge, E c. Vacuum level, E vacuum. Electron affinity, χ. Bandgap energy E g. Electron affinity, energy to remove electron from conduction band edge to vacuum level. Conductors, Insulators and Semiconductors

21 Conductors, Insulators and Semiconductors A: Occurs when atoms contain only one valence electron per atom, e.g. Cu, Au, Ag. B: Atoms with two valence electrons. Still highly conducting if the resulting filled band overlaps with an empty band. D: No conduction expected. Completely filled band is separated from the next higher empty band by a larger energy gap. E.g. insulators. C: Semiconductor. Completely filled band is close enough to the next higher empty band. Electrons can make it into the next higher band. This yields an almost full band below an almost empty band. Almost full band = valence band since occupied by valence electrons. Almost empty band will be called the conduction band, as electrons are free to move in this band and contribute to the conduction of the material.

22 Energy bandgap (E g ) Versus Temperature (T) E g of semiconductors tends to decrease as the temperature is increased. Increased thermal energy leads to increased interatomic spacing. This effect is quantified by the linear expansion coefficient of a material. Increased interatomic spacing reduces the average potential seen by the electrons in the material. In turn reduces the energy bandgap. Modulation of the interatomic distance e.g. by applying compressive (tensile) stress causes increase (decrease) of the bandgap. Energy bandgap (E g ) Versus Temperature (T) The temperature dependence of the energy bandgap, E g, has been experimentally determined yielding the following expression for E g as a function of the temperature, T: E g ( T ) = E g (0) 2 αt T + β where E g (0), α and β are fitting parameters. For Si: Eg(0) = ev, α = K 2, β = 636 Κ

23 Energy bandgap (E g ) Versus Temperature (T)

24 Energy bandgap (E G ) Versus Doping (N) Increase in doping concentration, N (density) cause bandgap to decrease. At high density, the wavefunctions of the electrons bound to the impurity atoms start to overlap. At cm -3, average distance between two impurity atoms is only 10 nm. Overlap of wavefunction forces formation of an energy band rather than a discreet level. For shallow impurity level, Band gap reduction is: 2 2 3q q N Eg ( N) = 16πε ε kt where N = doping density q = electronic charge ε s = dielectric constant of the semiconductor k = Boltzmann's constant T = temperature in Kelvin. For Si, this simplifies to: S S Bandgap shrinkage can typically be ignored for doping densities less than cm -3.

25 Doping dependence of the energy bandgap of GaAs (top/red curve), germanium (black curve) and silicon (bottom/blue curve). To find filled or empty bands: Find how many electrons can be placed in each band. Find how many electrons are available. Each band is formed due to the splitting of one or more atomic energy levels. Thus, minimum number of states in a band = 2 x number of atoms in the material. Factor of two since every energy level can contain two electrons with opposite spin.

26 Only the valence electrons assumed of interest. Core electrons - tightly bound to the atom, not allowed to freely move in the material.