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Solid State Physics^ FREE ELECTRON MODEL
Lecture 14^ A.H. Harker Physics and Astronomy
UCL
The Free Electron Model
6.^
Basic Assumptions
In the free electron model, we assume that the valence electrons canbe treated as free, or at least moving in a region constant potential,and non-interacting. We’ll examine the assumption of a constant po-tential first, and try to justify the neglect of interactions later.
2
Constant Potential Imagine stripping the valence electrons from the atoms, and arrang-ing the resulting ion cores on the aatomic positions in the crystal. Resulting potential – periodic array of Coulombic attractions. From atomic theory, we are used to the idea that different electronicfunctions must be orthogonal to each other (remember we used thisidea in discussing the short-range repulsive part of interatomic po-tentials) i.e. if
ψ(c
r)^ is a core function and
ψ(v
r)^ is a valence function
∫^ ψ
(r)c ψ(rv
)dr^
Let’s see how orthogonality might be achieved for a slowly-varyingwave.
To achieve orthogonality: we need high spatial frequency (large
k ) components in the wave.
Large
k^ →
large energy.
5
So the extra energy caused by the orthogonality partly cancels theCoulomb potential. This can be formalised in
pseudopotential theory
The potential is weakened, and the constant potential assumption is a reasonable one. The netresult is that the effective potential seen by the electrons does nothave very strong dependence on position.
6
So finally we assume that the attractive potential of the ion cores canbe represented by a flat-bottomed potential. We go further, and assume that the potential is deep enough that wecan use a simple ‘particle-in-a-box’ model – the free electron model.
Orders of magnitude For a typical solid, the interatomic spacing is about
^2.^5
×^10
−^10
m.
Assume each atom is in a cube with that dimension, and releases onevalence electron, giving an electron density
Ne
/V^
≈^6
×^10
28 m
−^3.
Putting in the numbers, we find^ •^ E
≈F
9 ×
19 J = 6 eV
•^ TF
≈^70
,^ 000 K
-^ kF
≈^1
.^2 ×
−m
1 , comparable with the reciprocal lattice spacing
2.^5 ×
−m
We can also estimate the electron velocity at the Fermi energy:
v=F^
ℏkF m
≈^
1.^4 ×
−m s
which is fast, but not relativistic.
total energy of electrons =
∫^ E
F^ E g 0
(E) d
E^ =
3 Ne 5
E,F
so that the average energy per electron is
3 EF 5
.^
13
The Fermi surface In later sections we shall talk a good deal about the Fermi surface.This is an constant-energy surface in reciprocal space (k-space) withenergy corresponding to the Fermi energy. For the free electron gas,this is a sphere of radius
k. F
14
6.^
Some simple properties of the free electron gas
Thermionic emission If the work function
φ^ is small enough, if the material is heated the
electrons may acquire enough thermal energy to escape the metal. Asmall electric field is used to draw them away. The current
J^ =
BT
2 exp
(^ −
φ kTB
)^ ,
with a theoretical value
B^ =
emk
(^2) B= 1 (^2) πℏ
.^2 ×
A m
−^2 K
−^2.
Experimentally the exponential dependence is confirmed, with simi-lar values for
B.
Field emission A large applied field alters the potential outside the metal enough toallow electrons to
tunnel
out.
17
Very large fields are needed, but a sharp metal tip can give an imagewhich shows where the atoms are – fields vary across the atoms. More detail from newer scanning probe microscopes.
18
Photoemission A photon with energy greater than the work function can eject anelectron from the metal. 6.2.
X-ray emission (Auger spectroscopy) A high-energy electron incident on a metal may knock out an elec-tron from a core state (almost unchanged from the atomic state). Analectron from the band can fall into the empty state, emitting an x-ray.
