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PH575 Spring 2014 Lecture #13 Free electron theory: Sutton Ch. 7 pp 132 -> 144; Kittel Ch. 6.
E k( ) =
2k2
2m
D E( ) = V
2π 22m2
"#$
%&'
3/2
E1/2
Assumption: electrons metal do not interact with each other, not with the potential due to the ions EXCEPT that the ion potential confines them to the region of space occupied by the material. We will see that this assumption leads to a spherical Fermi surface and explains the properties of simple metals. It is remarkable that the theory works as well as it does, and it improves the classical Drude model that is still used with success.
Paul Drude (1863-1906)
Free electron bands & LCAO (tight binding) bands are qualitatively similar, with important exception of gaps. (Sutton, Fig. 7.6, Ge e.g.) So free electron model works well only for metals.
http://physics.njit.edu/~savrasov/Projects/Teaching/Phys%20485%20SPRING%202004/Lecture21Topics.htm
FE bands (Na) LCAO band (Na)
Metal, L vacuum
Energy
PE
vacuum
0
-V0
"Jellium" - free electrons a uniform positive BG
L
H ψ k = Ek ψ k
−2
2md 2
dx2+d 2
dy2+d 2
dz2#
$%
&
'( ψ k = Ek ψ k
0
+PBC
ψ kr( ) = 1
Veik ⋅r
Similar to Bloch form
−2
2md 2
dx2+d 2
dy2+d 2
dz2"
#$
%
&'ψ k (
r ) = Ekψ k (r )
ψ kr( ) = 1
Veik ⋅r
r = xx + yy + zz; r = r = x2 + y2 + z2k = kx x + ky y + kz z; k =
k = kx
2 + ky2 + kz
2
Ek( ) = ?; kx = ? ky = ? kz = ?
E k( ) =
2k2
2m
kx = nx2πL; ky = ny
2πL; kz = nz
2πL
E0 kx
ky
E k( ) =
2k2
2m
kx = nx2πL; ky = ny
2πL; kz = nz
2πL
E0 kx
ky
Constant energy surfaces are spheres in k space Each state occupies 8π3/L3 in k-space When all N valence electrons have filled up k-states, we reach the Fermi Energy
EF =
2kF2
2mIs the factor of 2 from spin (spin => 2 electrons/state) in the right place?
N 8π3
V12≡4πkF
3
3
Fermi parameters (see Kittel Ch.6, Table I):
EF =
2kF2
2m
kF = 3π 2 n
# e/vol
"#$
%&'
1/3
TF =EF
kB
vF =
kFm
Sodium: n = 2.65 × 1022 e-/cm3
kF = 0.92 × 108 cm-1:
TF = 37,500 K: temp. above which e obey classical statistics (can ignore quantum effects)
EF = 3.2 eV: kinetic energy of Fermi level e
vF = 1.1 × 108 cm s-1 : 1% c
Density of states (number of states per unit energy at a particular energy): We discussed this in a previous lecture (but then without spin).
S E( ) = 4πk3
32V8π 3 =
V3π 2
2m2
!
"#
$
%&3/2
E3/2
D E( ) = dSdE
=V2π 2
2m2
!
"#
$
%&3/2
E1/2
Important result: density of states for free electrons varies as E1/2; also electrons near VBM and CBM in semiconductors
3-D Density of states for free electron:
S counts the states. D E( ) = dSdE
⇒ dS = D(E)dE
dS = D(k)× annular vol of k-space
=2V2π( )3 × 4πk2dk
D E( )dE = 2V2π( )3
4πk2dk
D E( ) = 8πk2V
2π( )3dkdE
D E( ) = V2π 2
2m2
"#$
%&'
3/2
E1/2
E0 + dEE0
kx
ky
E =
2k2
2m; dEdk
=2km
Example: Find total energy at T = 0 (homework) Will use this later for electronic specific heat.
ETot = ED E( )dE0
EF
∫
E0 + dEE0
kx
ky
ETot = E f E( )D E( )dE0
∞
∫
= E 1e E−EF( )/kBT +1
D E( )dE0
∞
∫
Total energy at T ≠ 0
Free electron bands: square lattice
kx (ky= 0)
E E =2k +G
2
2m
Real spacea1 = a 1,0( )a2 = a 0,1( )
Reciprocalg1 = 2π
a 0,1( )g2 = 2π
a 1,0( )
G = 2π
a 0,0( )G = 2π
a ±1,±1( )G = 2π
a ±1,0( )G = 2π
a 0,2( )G = 2π
a 0,1( )G = 2π
a ±2,0( )
http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/illustr/band_bcc_free.gif
Free electron bands: bcc lattice
E =2k +G
2
2m
From Tsymbal (UNL)
Free electron bands: fcc lattice Example Copper
E =2k +G
2
2m
<- Free electron bands
<- Calculated with pseudopotential methods. Note d-bands!
Free electron bands & LCAO (tight binding) bands are qualitatively similar, with important exception of gaps. (Sutton, Fig. 7.6, Ge e.g.) So free electron model works well only for metals.
http://physics.njit.edu/~savrasov/Projects/Teaching/Phys%20485%20SPRING%202004/Lecture21Topics.htm
FE bands (Na - bcc) LCAO band (Na)
Free electron bands & LCAO (tight binding) bands are qualitatively similar, with important exception of gaps. (Sutton, Fig. 7.6, Ge e.g.) So free electron model works well only for metals.
FE bands (Na - bcc) LCAO band (Na)
There is a gap in the energy spectrum! This same result can be obtained by treating the periodic ion potential as a perturbation of the smooth potential (we won't do this, but it's discussed in Kittel, for example)
http://www.chembio.uoguelph.ca/educmat/chm753/properties/electronic/nearly_free_model.html
Free electrons (plane waves) don't interact with the lattice much until the wave vector becomes comparable with 1/a, then they are Bragg reflected as we discussed before (nearly free) and we have interference between a plane wave and its oppositely directed counterpart. These superpositions are standing waves with the same kinetic energy, but total energy is different
http://www.chembio.uoguelph.ca/educmat/chm753/properties/electronic/nearly_free_model.html
eikx − e− ikx
sin kx( )
eikx + e− ikx
cos kx( )
Lower energy state piles up electron density on the atoms (stronger Coulomb attraction to cores), while the higher energy state piles up electron density between the atoms. True plane waves have constant electron density at all positions (this energy is intermediate). Similar to the LCAO approach in this limit. Thus there is a gap in the energy spectrum!
http://www.chembio.uoguelph.ca/educmat/chm753/properties/electronic/nearly_free_model.html
eikx − e− ikx
sin kx( )
eikx + e− ikx
cos kx( )
Fermi Function • Probability that a state of energy E is occupied at temperature T. • f(µ) = 0.5 • Varies sharply over width of ≈kBT about EF. Step function T = 0.
0.5 1.0 1.5 2.0E�eV⇥0.2
0.4
0.6
0.8
1.0
f �E⇥f E( ) = 1
eE− µkBT +1
µ = 1eV T = 0.015 eV = ?K T = 0.3 eV = ?K
kBT
kBT
0.0 0.5 1.0 1.5 2.00.0
0.1
0.2
0.3
0.4
0.5
kBT
E
Electronic specific heat • Electrons are a repository of energy in metals and contribute to the specific heat. Dominant contribution at low T where phonons are frozen out.
• If electrons are a "classical gas" (no QM), equipartition theorem => electronic energy is Eelec = (3/2)kBT. Electronic specific heat is Celec = dEelec/dT= (3/2)kB
• Actual values in metals at low temperatures are less than 1% of this, and are NOT temperature independent.
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi2.html
0.5 1.0 1.5 2.0E�eV⇥
�1.0�0.5
0.5
1.0
1.5
2.0
many functions
Electronic specific heat Nearly free electrons are a repository of energy in metals and contribute to the specific heat. At low temperatures, when phonons (lattice vibrations) are frozen out, this contribution dominates.
D E( ) = V
2π 22m2
"#$
%&'
3/2
E1/2
ETot = E f E( )D E( )dE0
∞
∫
= E 1e E−µ( )/kBT +1
D E( )dE0
∞
∫
df E,T( )dT
kBT
Cv =dETot
dT= E
df E,T( )dT
D E( )dE0
∞
∫
Area under black curve is electronic specific heat. Very slight asymmetry is crucial! Easy enough to calculate numerically, but let's look for analytical insight. What is dominant contribution? Weakest?
0.5 1.0 1.5 2.0E�eV⇥
�1.0�0.5
0.5
1.0
1.5
2.0
many functions
EF = 1 eV kBT = 0.1 eV
Cv =dETot
dT= E
df E,T( )dT
D E( )dE0
∞
∫
D(E) varies slowly - remove from integral (but what value?
df E,T( )dT
= ?
Change variables: x = E − µ2kBT
Why can lower limit of integration be extended to -∞?
Helpful: x2
cosh2 x-∞
∞
∫ dx = π2
6
Cv =dETot
dT= E
df E,T( )dT
D E( )dE0
∞
∫
Cv =π 2
2D EF( )kB2T
Interpretation: Depends linearly on temperature - NOT independent Size depends only on density of states at Fermi surface - small for semimetals; large for so-called “heavy fermions” Also interpreted as each electron near Fermi surface contributing roughly kB to heat capacity:
Cv = D EF( )kBT × kB
Cv =dETot
dT= E
df E,T( )dT
D E( )dE0
∞
∫
Cv =π 2
2D EF( )kB2T
Cv,FE =π 2
2NkB
TTF
Much smaller than expected from classical theory:
Cv,class =32NkB