Update of solid state physics 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 2

Electrons in crystals

A free particle:

What happens in a crystal where electrons are subjected to a periodic potential created by the crystalline lattice?

Update of solid state physics 3

Electron in a periodic potential

and

Physical properties must be periodic

and

and

Update of solid state physics 4

The dependence is called the dispersion law, or energy spectrum

1st Brilluoin zone

Quasi-momentum is defined up to reciprocal lattice vector, G, obeying the equation

where g is an integer. One can express the reciprocal

lattice vector as where in 1D case

General form: Bloch function

where

Update of solid state physics 5

Energy bands in crystals

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 Schrdinger equation, just standing waves

Not all energy values are allowed there are gaps (forbidden values) in the energy spectrum

Update of solid state physics 6

Update of solid state physics 7

Tight-binding approach

Periodic potential is formed by atomic-like potential

Degenerate levels

Avoided level crossing Energy bands

Update of solid state physics 8

Parameter, a

Two-level system with splitting

Assume

Hamiltonian:

Energy levels:

Avoided crossing

Wave function:

Update of solid state physics 9

Concept of effective mass

/a /a

Update of solid state physics 10

Effective mass

Negative effective mass - holes

Quadratic spectrum

Update of solid state physics 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 me and mh, respectively.

Energy spectra of real materials can be rather complicated

Update of solid state physics 12

Surfaces of constant energy for silicon

Electrons Holes 1 Holes 2

Update of solid state physics 13

Energy bands of semiconductors

Ge Si GaAs

FCC

Update of solid state physics 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 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 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 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 18

Occupation probability and chemical potential

Electron obey Fermi-Dirac distribution:

Chemical potential is found from the normalization condition:

Update of solid state physics 19

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

Electron-like Hole-like

Update of solid state physics 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.

Ec is the edge of the conduction band. Like hydrogen atom!

Update of solid state physics 21

Rydberg Approx. 0.001

Effective Bohr radius:

Energy terms:

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 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 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 24

Ohms law:

Drude formula In magnetic field

friction Lorentz force

Update of solid state physics 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 26

- Hall coefficient

Update of solid state physics 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 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 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 30

Both acoustical and optical phonons can assist inter-valley 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 31

The sample contained a donor density of nv = 4.8x1019 m-3 and an acceptor density of nA = 2.1x1019 m-3.

Measured electron mobility in GaAs (circles) as function of temperature, including the theoretical contributions of relevant scattering mechanisms (full lines).

Minimal Update of Solid State PhysicsElectrons in crystalsElectron in a periodic potentialSlide Number 4Energy bands in crystalsSlide Number 6Tight-binding approachSlide Number 8Concept of effective massSlide Number 10Slide Number 11Slide Number 12Energy bands of semiconductorsOccupation of energy bandsDensity of statesSlide Number 16Slide Number 17Occupation probability and chemical potentialSlide Number 19DopingSlide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Scattering mechanismsSlide Number 28Slide Number 29Slide Number 30Slide Number 31