electrons in solids - ues005
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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 -
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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
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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
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E = (hk)2/2m
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In a solid the electron encounters atoms periodically after distance a (nearest
neighbour distance). So the E-k space becomes periodic with 2p/a .
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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.
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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.