Electrons in Solids - UES005

7/27/2019 Electrons in Solids - UES005

1/22


2/22


3/22


4/22


5/22


6/22


7/22


8/22


9/22


10/22

6 Free Elec tron Fermi Gas 137

The quantity y is the chemical potential (TP, Chapter 5) , and we seethat at absolute zero the chemical potential is eqml to the Fermi energy, de-fined as the energy of the topmost filled orbital at absolute zero.

The high energy tail of the distrihi~tions that part for which 6 - y 9 k,T;here the exponential term is dominant in the denominator of ( 5 ) , sothat f ( e ) - exp[(p -


11/22

On substituting (9 ) in (6) we have the energy ek of the orbital withwavevector k:

The magnitude k of the wavevector is related to the wavelength h by k = 2?rlh.The linear momentum p may be represented in quantum mechanics by

the operator p = -ifiV, whence for the orbital (9)

so that the plane wave $k is an eigenfunction of the linear momentum with theeigenvalue fik.The particle velocity in the orbital k is given by v =fiklm.

In the ground state of a system of N free electrons, the occupied orbitalsmay be represented as points inside a sphere in k space. The energy at the sur-face of the sphere is the Fermi energy; the wavevectors at the Fermi surfacehave a magnitude k, such that (Fig. 4):

From (10)we see that there is one allowed wavevector-that is, one dis-tinct triplet of quantum numbers k,, k,,, k,-for the volume element ( 2 7 r / ~ ) ~fk space. Thus in the sphere of volume 4?rk23 the total number of orbitals is

where the factor 2 on the left comes from the two allowed values of the spinquantum number for each allowed value of k. Then (15) gives


12/22

Energy, +Using (14) and ( l6 ),

Figure 5 Density of single-particle states as a func-tion of energy, for a free electron gas in three dimen-sions. The dashed curve represents the densityf ( E , T ) D ( E ) f filled orbitals at a finite temperature,but such that k,T is small in comparison with E Theshaded area represents the filled orbitals at absolutezero. The average energy is increased when the tem-perature is increased from 0 o T, for electrons arethermally excited from region 1 o region 2.

This relates the Fermi energy to the electron concentration NN. The electronvelocity vFat the Fermi surface is


13/22


14/22


15/22


16/22


17/22


18/22

E = (hk)2/2m


19/22

In a solid the electron encounters atoms periodically after distance a (nearest

neighbour distance). So the E-k space becomes periodic with 2p/a .


20/22


21/22

Since the nearest neighbour distance is different in different directions so the E-k diagrams

need to be constructed for all the 3 directions and the band-gap is defined as difference

b/w the top of the top-most filled band (valence band) and bottom of the next band

(conduction band) in 3-D.


22/22

Effective mass:

m* =1

2

2

2

The measured mass of the

electron from transport properties

of the solid is not same as the free

electron mass. This is due to

presence of bands.

Electrons in Solids - UES005

Documents

Transcript of Electrons in Solids - UES005