1 Free Electron Fermi Gas
-
Upload
ngoccuong832003 -
Category
Documents
-
view
219 -
download
0
Transcript of 1 Free Electron Fermi Gas
-
8/8/2019 1 Free Electron Fermi Gas
1/45
Solid State Physics
for Illumination Engineering II
Lecture I
Prof. Shavkat Yuldashev
Dongguk University, September 2010
-
8/8/2019 1 Free Electron Fermi Gas
2/45
Free Electron Fermi Gas I
Electron Waves
In the classical picture, electrons are particles that follow Newton's laws of mechanics. They arecharacterized by their mass m0, their position = ( x , y , z ) , and their velocity However, this
intuitive picture is not sufficient for describing the behavior of electrons within solid crystals, where
it is more appropriate to consider electrons as waves. The wave-particle duality is one of the
fundamental features of quantum mechanics. Using complex numbers, the wave function for a free
electron can be written as
(1)
with the wave vector The wave vector is parallel to the electronmomentum
(2)
and it relates to the electron energy E as
with
-
8/8/2019 1 Free Electron Fermi Gas
3/45
In general, electron wave functions need to satisfy the Schrodinger equation
(3)
where the potential represents the periodic semiconductor crystal. This equation is often writtenas
(4)
with H called the Hamiltonian. The Schrodinger equation is for just one electron; all other electrons
and atomic nuclei are included in the potential . For the free electron, = 0 and the solution
to the Schrodinger equation is of the simple form given by Eq. (1.1).
The boundary conditions are n(0) = 0; n (L) = 0, as imposed by the infinite potential energy
barriers. They are satisfied if the wavefunction is sinelike with an integral numbern of half-
wavelengths between 0 and L:
(5)
where A is a constant. We see that (1.5) is the solution of (1.4), because
-
8/8/2019 1 Free Electron Fermi Gas
4/45
(6)
11.
According to the Pauli exclusion principle no two
electrons can have all their quantum numbers
identical. That is each orbital can be occupied by
at most one electron. This applies to electrons in
atoms, molecules, or solids.
-
8/8/2019 1 Free Electron Fermi Gas
5/45
-
8/8/2019 1 Free Electron Fermi Gas
6/45
The Fermi energy F is defined as the energy of the topmost filled level in the ground
state of the N electron system. By (1.6) with n = nF we have in one dimension:
(7)
The Fermi Dirac distribution gives the probability that the orbital at energy will be
occupied in an ideal electron gas in thermal equilibrium:
(8)
The quantity is the chemical potential, and we see that at absolute zero temperature the
chemical potential is equal to the Fermi energy, defined as the energy of the topmost filled
orbital at absolute zero.
-
8/8/2019 1 Free Electron Fermi Gas
7/45
The kinetic energy of the electron gas increases as the temperature is increased: some energy
levels are occupied which were vacant at absolute zero, and some levels are vacant which were
occupied at absolute zero temperature.
-
8/8/2019 1 Free Electron Fermi Gas
8/45
(9)
(10)
We now require the wavefunctions to be periodic in x, y, z with the period L. Thus
(11)
With the wavevector component k satisfy
(12)
-
8/8/2019 1 Free Electron Fermi Gas
9/45
Any component of kis of the form 2n/L , where n is a positive or negative integer.
The components of k are the quantum numbers of the problem, along with quantum number
ms for the spin direction. We confirm that these values of kx satisfy (1.11), for
(13)==
The energy
(1.14)
-
8/8/2019 1 Free Electron Fermi Gas
10/45
We see that there is one allowed wavevector that is, one distinct triplet of quantum
numbers kx , ky , kz - for the volume element (2/L)3 of kspace.
Thus in the sphere of volume 4kF3 /3 the total number of orbitals is
(15)
where the factor 2 on the left comes from the two allowed values ofms , the spin quantum number,
for each allowed value of k. Then
(16)
which depends only on the particle concentration.
Using (1.14)(17)
-
8/8/2019 1 Free Electron Fermi Gas
11/45
(19)
(20)
-
8/8/2019 1 Free Electron Fermi Gas
12/45
(21)
-
8/8/2019 1 Free Electron Fermi Gas
13/45
-
8/8/2019 1 Free Electron Fermi Gas
14/45
If N is the total number of electrons, only a fraction of the order of T / TF can be excited
thermally at temperature T, because only these lie within an energy range of the order of kBT
of the top of the energy distribution (Fig.5).
(22)
(23)
-
8/8/2019 1 Free Electron Fermi Gas
15/45
-
8/8/2019 1 Free Electron Fermi Gas
16/45
-
8/8/2019 1 Free Electron Fermi Gas
17/45
-
8/8/2019 1 Free Electron Fermi Gas
18/45
-
8/8/2019 1 Free Electron Fermi Gas
19/45
It is good approximation to evaluate the density of statesD () at F and take it outside of the integral:
-
8/8/2019 1 Free Electron Fermi Gas
20/45
-
8/8/2019 1 Free Electron Fermi Gas
21/45
-
8/8/2019 1 Free Electron Fermi Gas
22/45
-
8/8/2019 1 Free Electron Fermi Gas
23/45
-
8/8/2019 1 Free Electron Fermi Gas
24/45
-
8/8/2019 1 Free Electron Fermi Gas
25/45
-
8/8/2019 1 Free Electron Fermi Gas
26/45
-
8/8/2019 1 Free Electron Fermi Gas
27/45
-
8/8/2019 1 Free Electron Fermi Gas
28/45
-
8/8/2019 1 Free Electron Fermi Gas
29/45
-
8/8/2019 1 Free Electron Fermi Gas
30/45
-
8/8/2019 1 Free Electron Fermi Gas
31/45
-
8/8/2019 1 Free Electron Fermi Gas
32/45
-
8/8/2019 1 Free Electron Fermi Gas
33/45
-
8/8/2019 1 Free Electron Fermi Gas
34/45
Solution:
-
8/8/2019 1 Free Electron Fermi Gas
35/45
Solution:
-
8/8/2019 1 Free Electron Fermi Gas
36/45
Solution:
-
8/8/2019 1 Free Electron Fermi Gas
37/45
Solution:
-
8/8/2019 1 Free Electron Fermi Gas
38/45
The number of moles per cm3 is so that the concentration is
atoms cm3 . The mass of an atom of He3 is
Thus
Solution:
-
8/8/2019 1 Free Electron Fermi Gas
39/45
Solution:
-
8/8/2019 1 Free Electron Fermi Gas
40/45
-
8/8/2019 1 Free Electron Fermi Gas
41/45
Solution:
-
8/8/2019 1 Free Electron Fermi Gas
42/45
-
8/8/2019 1 Free Electron Fermi Gas
43/45
Solution:
-
8/8/2019 1 Free Electron Fermi Gas
44/45
Solution:
-
8/8/2019 1 Free Electron Fermi Gas
45/45
Rsq (ohms) = 10-9 c2 Rsq(gaussian) (30)(137)ohms 4.1k.
Solution: