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Electrons in metals
+ + + + + + + +
Ene
rgy
E
Spatial coordinate x
Nucleus with
localized core
electrons
Jellium model:
electrons shield potential to a large
extent
Electron “sees” effective smeared potential
Electron in a box
In one dimension:
In three dimensions:
)r(E)r()r(V)r(m
2
2
where
otherwise
Lz,y,xfor.constV
)z,y,x(V
00
222222
22 zyx kkkmm
kE
where zzyyxx nL
k,nL
k,nL
k
2222
2
8zyx nnn
mL
hE
and ,...,,n,n,n zyx 321
zksinyksinxksinL
)r( zyx
/ 232
Fixed boundary conditions:
+ + + + + + + +x
0 L
)Lx()x( 00
+
+
++
++
++
Periodic boundary conditions:
)z,y,x()Lz,Ly,Lx(
rki/
eL
)r(23
1
zzyyxx nL
k,nL
k,nL
k
222
and ,...,,,n,n,n zyx 3210
kx
m
kx
2
22
Ldk x
2
“free electron parabola”
density of states
Remember the concept of
dE
# of states
in ]dEE,E[
41111
)k(E 1 )k(E 2 E
))k(EE( 1
1. approach use the technique already applied for phonon density of states
k
))k(EE()E(D~
E
EE
E
dE)E(D~1
1
k
EE
E
dE))k(EE(1
1
where )E(D
~V
:)E(D1
Density of states per unit volume
Because I copy this part of the lecture from my solid state slides, I use E as the single particle energy.
In our stat. phys. lecture we labeled the single particle energy to distinguish it from the total energy of the N-particle system.
Please don’t be confused due to this inconsistency.
3
3k
Vd k
2
1/ Volume occupied by a state in k-space
k
))k(EE()E(D~
kx
ky
kz
L
2
L
2
L
2
Volume( )
VL
3322
Free electron gas: m
k
m
kE
22
2222
Independent from
and
dkkkd 23 4
Independent from
and
mEk 21
dE
E
mdk
2
1
dEE
mmE))k(EE()E(D
~V
)E(D2
124
2
1123
Em
)E(D//
3
2321
2
2
2
1
2
Each k-state can be occupied with 2 electrons of spin up/down
Em
)E(D/ 23
22
2
2
1
k2
dk
2. approach calculate the volume in k-space enclosed by the spheres
.constm
k)k(E
2
22and
.constdE)k(E
kx
ky
L
2
32
2
4
L/
dkkdk)k(D
~
# of states between spheres with k and k+dk :
dEE
mdk
2
1
22 2
mE
k
with)E(D
~V
)E(D1
2
2 spin states
Em
)E(D/ 23
22
2
2
1
E
D(E)
E’ E’+dE
D(E)dE =# of states in dE / Volume
The Fermi gas at T=0
E
f(E
,T=
0)
EF
1
E
D(E)
EF0
0
dE)T,E(f)E(Dn
Electron density
#of states in [E,E+dE]/volume
Fermi energy
depends on T
Probability that state is occupied
0
0
FE
dE)E(D dEEm FE/
0
0
23
22
2
2
1
3222
0 32
/
F nm
E T=0
00 5
3FEnU
0
00
FE
dE)E(DEUEnergy of the electron gas @ T=0:dEEE
m FE/
0
0
23
22
2
2
1
25023
22 5
22
2
1 /
F
/
Em
2300
23
22 5
121 /
FF
/
EEm
3222
0 32
/
F nm
E
there is an average energy of 0
5
3FE per electron without thermal stimulation
with electron density 322 1
10cm
n we obtain KT@eVTkeVE BF 30040
11240
Energy of the electron gas: ( )
21FE E
k
E kU
e
0
( )1FE E
EU D E dE
e
Specific Heat of a Degenerate Electron Gas
here: strong deviation from classical value
only a few electrons in the vicinity of EF can be scattered by thermal energy
into free states
Specific heat much smaller than expected from classical consideration
D(E)
Den
sity
of o
ccup
ied
stat
es
EEF
energy of
electron
state
0
dE)T,E(f)E(DEU
#states in [E,E+dE]
probability of occupation,
average occupation #
2kBT
Before we calculate U let us estimate:
These Tk)E(D
BF 222
1# of electrons
increase energy from TkE BF toTkE BF Tkn
E
TkTk)E(DU B
F
BBF 2
2Tk)E(DU BF Tk)E(DC BFel2 π
2
3
subsequent more precise calculation
Calculation of Cel from
0
dE)T,E(f)E(DEU
0
dET
f)E(DE
T
UC
Vel
22
1
TBkFEE
TBkFEE
B
F
e
e
Tk
EE
T
f
0
dET
f)E(DEE F
0
0 dET
f)E(DE
T
nE FFTrick:
Significant contributions only in the vicinity of EF
)E(DTkC FBel2
2
3
3
2
0
dET
f)E(DEEC Fel
with Tk
EE:x
B
F and dxTkdE B
E
D(E
)
EF
)E(D)E(D F
0
dET
fEE)E(DC FFel
21
x
x
e
e
T
x
T
f
TBk/FEx
x
FBel dxe
ex)E(DTkC
2
22
1
decreases rapidly to zero for x
dx
e
ex)E(DTkC
x
x
FBel 2
22
1
)E(DTkC FBel2
2
3
F
/
F Em
)E(D23
22
2
2
1
with 322
20 3
2
/
F nm
E
and
F
BBel E
TkknC
2
2
in comparison with Bclassical
el knC2
3
1 for relevant temperatures
W.H. Lien and N.E. Phillips, Phys. Rev. 133, A1370 (1964)
Heat capacity of a metal:
3ATTC
electronic contributionlattice contribution
@ T<<ӨD
