Minimal Update of Solid State Physics

Transcription

1 Minimal Update of Solid State Physics It is expected that participants are acquainted with basics of solid state physics. Therefore here we will refresh only those aspects, which are absolutely necessary for understanding of the course. Update of solid state physics 1

2 A free particle: Electrons in crystals What happens in a crystal where electrons are subjected to a periodic potential created by the crystalline lattice? Update of solid state physics 2

3 Electron in a periodic potential and Physical properties must be periodic and and Update of solid state physics 3

4 General form: Bloch function where Quasi-momentum vector, G, obeying the equation is defined up to reciprocal lattice where g is an integer. One can express the reciprocal lattice vector as where in 1D case 1 st Brilluoin zone The dependence energy spectrum is called the dispersion law, or Update of solid state physics 4

5 Energy bands in crystals Not all energy values are allowed there are gaps (forbidden values) in the energy spectrum Physical origin Bragg reflections from periodic potential If the Bragg condition is met, i. e., in one dimension, there is no propagating solutions of the Schrödinger equation, just standing waves Update of solid state physics 5

6 Update of solid state physics 6

7 Tight-binding approach Periodic potential is formed by atomic-like potential Degenerate levels Avoided level crossing Energy bands Update of solid state physics 7

8 Avoided crossing Two-level system with splitting Assume Wave function: Parameter, a Hamiltonian: Energy levels: Update of solid state physics 8

9 Concept of effective mass π/a π/a Update of solid state physics 9

10 Quadratic spectrum Effective mass Negative effective mass - holes Update of solid state physics 10

11 For the states close to the band edges the spectrum is close to quadratic. For such states one can consider electrons and holes as quasiparticles with effective masses m e and m h, respectively. Energy spectra of real materials can be rather complicated Update of solid state physics 11

12 Surfaces of constant energy for silicon Electrons Holes 1 Holes 2 Update of solid state physics 12

13 Energy bands of semiconductors FCC Ge Si GaAs Update of solid state physics 13

14 Occupation of energy bands 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. Update of solid state physics 14

15 Density of states Two dimensional system, periodic boundary conditions Momentum is quantized in units of A quadratic lattice in k-space, each of them is g-fold degenerate (spin, valleys). Assume that, the limit of continuous spectrum. Number of states between k and k+dk: Update of solid state physics 15

16 Number of states per volume per the region k,k+dk Density of states -Number of states per volume per the region E,E+dE. Since Update of solid state physics 16

17 Electron density of states in the effective mass approximation as a function of energy, in one, two, and three dimensions Update of solid state physics 17

18 Occupation probability and chemical potential Electron obey Fermi-Dirac distribution: Chemical potential is found from the normalization condition: Update of solid state physics 18

19 Hole-like Electron-like Thermal smearing of the Fermi function (left), and the density of states for 3D case as well as the spherical carrier density, n(e), right Update of solid state physics 19

20 Doping Implanting suitable impurities, e. g., Si instead of Ga in GaAs: Only 3 electrons can participate the covalent bonds with adjacent As atoms. The remaining electron remains bound to attractive potential of Si like an atom in the medium with dielectric constant of GaAs. Concept of envelope functions like free electron with effective mass. E c is the edge of the conduction band. Like hydrogen atom! Update of solid state physics 20

21 Energy terms: Rydberg Approx Effective Bohr radius: Can be easily ionized into the conduction band! Acceptors: Si instead of As. Typical n-dopants for Si are Sb and P, p-dopants are B and Al, typical binding energy is about 50 mev; For GaAs Si (6 mev) and Be, Zn (30 mev), respectively. Update of solid state physics 21

22 While calculating the chemical potential all the carriers (both in the bands and in localized states) must be taken into account, as well as neutrality condition must be used: neutral ionized free Doping allows to engineer properties of semiconductors! Update of solid state physics 22

23 Diffusive transport Between scattering events electrons move like free particles with a given effective mass. In 1D case the relation between the final velocity and the effective free path, l, is then Assuming where is the drift velocity while is the typical velocity and introducing the collision time as we obtain in the linear approximation: Mobility Update of solid state physics 23

24 Ohm s law: In magnetic field Drude formula friction Lorentz force Update of solid state physics 24

25 Important quantity is the product of the cyclotron frequency, by the relaxation time, Generally, conductivity is a tensor, Resistivity tensor, Update of solid state physics 25

26 - Hall coefficient Update of solid state physics 26

27 Scattering mechanisms Bloch electrons have infinite conductance, finite resistance is due to carrier scattering. There are several scattering mechanisms: In pure crystals lattice vibrations (phonons). Electron-phonon scattering has different facets. Impurities, defects Neutral impurities break crystal symmetry, charged impurities create screened Coulomb field acting upon electrons. For scattering by charged impurities multiplied by a logarithmic correction Update of solid state physics 27

28 Lattice vibrations: acoustical and optical modes For 1D chain: According to quantum mechanics, lattice vibrations can be regarded as quasi-particles phonons with (quasi) momentum and energy The dependence is called the dispersion law, or energy spectrum of phonons. For 1D chain Update of solid state physics 28

29 More about electron-phonon scattering deformation potential scattering, for acoustic phonons piezoelectric scattering (in polar materials, like GaAs) Piezoelectric materials are extensively used for various applications: transducers, actuators, tunneling microscopy, etc Update of solid state physics 29

30 Both acoustical and optical phonons can assist intervalley transitions in materials with degenerate valleys Combination of impurity and phonon scattering determines the temperature dependence of the electron mobility, Mobility is a crucial characteristic of a material Update of solid state physics 30

31 Measured electron mobility in GaAs (circles) as function of temperature, including the theoretical contributions of relevant scattering mechanisms (full lines). The sample contained a donor density of n v = 4.8x10 19 m -3 and an acceptor density of n A = 2.1x10 19 m -3. Update of solid state physics 31