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51
5 Electrons in Solids
Adiabatic approximation
(see Chapter 4) : no interaction
between electrons and moving nuclei.
Extension: nuclei + core electrons
→
atomic core
⇒
calculation of electronic states in approximation of rigid
atomic cores.
One electron approximation:
Schrödinger eq. for
> 10²³
interacting electrons in periodic potential of atomic cores!
Simplification: one electron in electrostatic potential of atomic
cores and all other electrons.
Other electrons largely screen potential of atomic nuclei
.
Electron electron interaction neglected:
important for magnetism
and superconductivity!
Fig. 5.1 Sketch of the potential for
one electron in a periodic lattice of
positive cores (+). The vacuum
level E
vac
is the level to which the
electon has to be promoted in order
for it to leave the crystal and to
escape to infinity. The simplest
approximation of this system is an
electron in a square potential well (
) with infinitely high walls at the
surface of the crystal.
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5.1 Free Electron Model
5.1.1
Free Electron Gas and Fermi
Statistics
Additional approximation:
interaction of electron with
atomic (cores is neglected).
⇒
Model of free electron gas in square wall potential
Metal = cube of side L
Infinite walls (actual work function typ. 5 eV)
Schrödinger eq. for a particle in an infinite square wall
potential:
−+
=
h
²
()
() ()
' ()
2
m
rV
r
r
E
r
∆
ψψ
ψ
Vxyz
xyz L
(,,)
,,
=
≤≤
∞
V = const. for 0
otherwise,
0
E = E‘  V
0
⇒
−=
h
²
2
m
rE
r
∆
in cube
(5.1)
Infinite walls
⇒
r
=
0
outside cube
(5.2)
(Fixed boundary conditions)
Normalization condition:
*
rr
d
r
box
∫
=
1
(5.3)
53
Solution of (5.1) with (5.3):
ψ
( )
sin
r
L
kx
ky
kz
xyz
=
2
3
2
(5.4)
with
()
E
k
mm
kkk
xyZ
==
+
+
hh
²²
²
22
222
(5.5)
(5.2)
⇒
(0,
y
,
z
) =
(L,
y
,
z
) = 0
0
≤≤
yz L
,
⇒
=
=
=
k
L
n
k
L
n
k
L
n
nnn
n
n
xx
yy
zz
i
i
π
,,
.
= 1,2,3.
.
= 0:
not square integrable
< 0: no linearly independent solution
(5.6)
L macroscopic
⇒
quasicontinuous descrete
k
values
ππ
La
<<
⇒
often
→
∫
∑
dk
ks
t
a
t
e
s
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Fig. 5.2
Spatial form of the first
three wavefunctions of a free
electron in a square well potential
of length
L
in the
x
direction. The
wavelengths corresponding to the
quantum numbers
n
x
=1,2,2,.
.. are
λ
=2
L
,
L
, 2
L
/3, .
..
55
Density of States
Fig. 5.3 a), b).
Prepresentation of the states of an infinite square well by means of a lattice of
allowed wave vector values in
k
space. Because of the two possible spin orientations, each
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This note was uploaded on 01/19/2010 for the course MATERIALS M504 taught by Professor Adelung during the Spring '02 term at Uni Kiel.
 Spring '02
 Adelung

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