Electronic properties of solids:
Free Electron Fermi Gas
2
Outline
• Overview - role of electrons in solids
⚫ Drude model
• Free electron model (Sommerfeld theory)
– Pauli Exclusion Principle, Fermi Statistics
– Energy levels in 1 and 3 dimensions
• Similarities, differences from vibration waves
• Density of states, heat capacity
• Fermi surface
3
Role of Electrons in Solids
• Electrons are responsible for binding of crystals: they are the
‘glue’ that hold the nuclei together
Van der Waals - electronic polarizability
Ionic - electron transfer
Covalent - electron bonds
Metallic
• Electrons are responsible for important properties:
– Electrical conductivity in metals
(But why are some solids insulators?)
– Magnetism
– Optical properties
– . . . .
4
Starting Point for Understanding
Electrons in Solids
• Nature of a metal:
Electrons can become “free of the nuclei” and move between
nuclei since we observe electrical conductivity.
• Electron Gas
Simplest possible model for a metal - electrons are completely
“free of the nuclei”
-- nuclei are replaced by a smooth background
-- “Electrons in a box”
5
Electron Gas - History
• Electron Gas model predates quantum mechanics
• Electrons Discovered in 1897 – J. J. Thomson
• Drude-Lorentz Model: Electrons - classical particles
free to move in a box
6
Drude Model (1900)
Three assumptions:
• Electrons have a scattering time τ. The probability of
scattering within a time interval dt is dt/τ.
• Once a scattering event occurs, we assume the electron returns
to momentum p = 0.
• In between scattering events, the electrons, which are charge
−e particles, respond to the externally applied electric field E
and magnetic field B.
• In 1900 Paul Drude realized that he could apply Boltzmann’s
kinetic theory of gases to understanding electron motion
within metals.
7
Drude-Lorentz Model (1900-1905)
( ) (1 )( ( ) ) 0dt dt
p t dt p t Fdt
+ = − + +
( )( ) (1 )( ( ) )
d p t dtp t dt p t Fdt
dt + = − +
( ) ( )d p t p tF
dt = −
keeping linear terms (dt)
under an electric field( ) ( )d p t p t
eEdt
= − −
for a steady state 0mv
eE
+ =
8
Drude-Lorentz Model (1900-1905)
2e n
j env E Em
= − = =
e Ev
m
= −
under electric and magnetic fields
( ) ( )( )
d p t p te E v B
dt = − + −
for a steady state
0 ( )j B m
eE jn en
= − − +
E j=
9
Drude-Lorentz Model (1900-1905)
2
2
2
0
0
0 0
m B
ne ne
B mE j
ne ne
m
ne
= −
When the magnetic field is along the z direction,
Hall resistivity
for a steady state E j=0 ( )j B m
eE jn en
= − − +
10
Edwin Hall experiment (1879)
11
Drude-Lorentz Model (1900-1905)
2
2
2
0
0
0 0
m B
ne ne
B mE j
ne ne
m
ne
= −
Hall resistivity
Hall coefficient 1yx
HR
B ne
= = − from Drude model
When the magnetic field is along the z direction,
We could measure the density of electrons.
On the other hand, if the density of electrons is known, we could
measure the magnetic field.
Hall sensor
12
Drude-Lorentz Model (1900-1905)
The Oxford Solid State Physics by Simon (2017)
Hall coefficient1yx
HR
B ne
= = − from Drude model
13
Drude-Lorentz Model (1900-1905)
2e n
j env E Em
= − = =
Heat currentdT
J Kdx
= −
1 1
3 3v
K Cvl nc v v= =3
2v B
c k= 2 3B
k Tv
m=
2
831.11 10
2
BkK
T e
− =
WΩ/K2
the ratio of thermal conductivity to electric conductivity:
14
Drude-Lorentz Model (1900-1905)2
831.11 10
2
BkK
T e
− =
WΩ/K2
The Oxford Solid State Physics by Simon (2017)
15
Wiedemann–Franz law (1853)
The ratio of the electronic contribution of the thermal
conductivity (K) to the electrical conductivity (σ) of a metal is
constant at the same temperature.
Ludivg Lorenz (1872): The ratio is proportional to T.
8, 2 10
KcT c
−=
The simple model seems work!
16
Peltier effect
Peltier effect: Running electrical current through a material
also transports heat.
qj j=
31( )
3 2
B
q
k Tj nv= j env= −
2
Bk T
e = −
44.3 10
2
Bk
ST e
−= = − = − V/K
Seebeck coefficient
17
Seebeck coefficient S
44.3 10S
−= − V/KThe Oxford Solid State Physics by Simon (2017)
100 times smaller…
The actual specific heat per particle is much lower.
18
Quantum Mechanics
• 1911: Bohr Model for H atom
• 1923: Wave Nature of Particles, Louie de Broglie
• 1924-26: Development of Quantum Mechanics -
Schrödinger equation
• 1924: Bose-Einstein Statistics for identical particles
(phonons, ...)
• 1925-26: Pauli Exclusion Principle, Fermi-Dirac
statistics (electrons, ...)
• 1925: Spin of the electron (spin = 1/2), G. E. Uhlenbeck
and S. Goudsmit
19
Schrödinger Equation : 1-d line
• Suppose particles can move freely on a line with position x,
0 < x < L
• Schrödinger Eq. in 1-d with V = 00 L
2 2
2 ( ) ( )
2
dx x
m dx − =
( )
1
2
2 2 2* 2
0
2sin( ), , 1, 2, 3, ...
( ) ( ) 1, ( )2 2
L
n
x kx k n nL L
k nx x dx
m m L
= = =
= = =
Normalization
Boundary
condition• Solution with (x) = 0 at x = 0, L
20
Electrons on a line
• For electrons in a box, the energy is just the kinetic energy which is quantized because the waves must fit into the box
2 2
, , 1, 2, 3, ...2
kk n n
m L
= = =
Approaches continuum
as L becomes large
k
E
21
Electrons in 3-d
• Schrödinger Eq. in 3D with V = 0
• Solution:
2 2 2 22 2
2 2 2( ) ( ),
2
d d dr r
m dx dy dz − = = + +
2 2
( ) ,
2= ,
2
i k r i k rr e ike
k mk
m
= =
=
ˆ( ) 2, 1 (Same for , )
2Thus = ( , , )
xi k Lik r Lx i k r
x x y z
x y z
e e e k n k kL
k n n nL
+ = = =
22
Electrons in 3-d• From previous slide, where we have imposed periodic
boundary conditions, the energy can be written as
2 2 22 2 2
( )2 2
x y z
kk k k
m m = = + +
23
Density of States (DOS)
Solid-State Physics by Ibachand Lueth (2009)
24
DOS in 3-d
• Key point - exactly the same as for vibration waves - the values of
kx, ky, kz are equally spaced : ∆kx= 2/L, etc.
• Thus the volume in k space per state is (2/L)3 and the number of
states N with |k| < k0 is
• Density of states per unit energy is ( )dN dN dk
D EdE dk dE
= =
3 3
30
02
4( )
2 3 6
kL VN N k
= =
2 2 2
2
k dE kE
m dk m= → =
2
3/2
2 2 2 2 22( ) ( )
2 2 2
Vk m Vmk V mD E E
k = = =
25
DOS in 3-d
• Consider electrons with spin ½ , there are two allowed values of
the spin quantum number for each allowed value of k. Therefore
the density of states is
3 / 2
2 2
2( ) ( )
2
V mD E E E
=
D(E)
E
Filled
EF
Empty
At 0 K
26
What is special about electrons?
• Fermions - obey Pauli exclusion principle
• Electrons have spin s = 1/2 - two electrons (spin up and spin down)
can occupy each state
• Kinetic energy = p2/2m = (ħ2/2m) k2
• If we know the number of electrons per unit volume Nelec/V, the
lowest allowed energy state is for the lowest Nelec/2 states to be
filled with 2 electrons each, and all the (infinite) number of other
states to be empty.
27
What is special about electrons?
• All states are filled up to the Fermi momentum kF and Fermi
energy EF = (ħ2/2m) kF2 given by
3
3
4
(2 ) 3 2F
V Nk
=
21/33
( )F
Nk
V
=
2 22/33
( )2
F
NE
m V
=
28
Fermi-Dirac distribution
• At finite temperatures, electrons are not all in the lowest energy
states
• Fermi-Dirac distribution gives the probability that a state at
energy will be occupied in an ideal electron gas in thermal
equilibrium
where is the chemical potential. At T = 0 K, = εF.
B
1( )
exp[( ) / ] 1f
k T
=
− +
f ()
E
1
½
f ()
E
1
½
T = 0 K T ≠ 0 K
29Introduction to Solid State Physics by Kittel (2005)
30Introduction to Solid State Physics by Kittel (2005)
31Introduction to Solid State Physics by Kittel (2005)
( )dN
D E EdE
=
3 3/2
3
2 4
(2 ) 3F
VN k E
=
3D
( )dN
D EdE
=
2
2
2
(2 )F
VN k E
=
2D
independent of E
1( )
dND E
dE E=
1/22
(2 )F
LN k E
=
1D
32
Typical values for electrons
• Conclusion: For typical metals the Fermi energy (or the
Fermi temperature) is much greater than ordinary
temperatures.
Solid-State Physics by Ibachand Lueth (2009)
33Introduction to Solid State Physics by Kittel (2005)
34
Heat Capacity for Electrons
• Just as for phonons the definition of heat capacity is C = dU/dT
where U = total internal energy
~ kBT
f ()
E
1
½
• For T << TF = EF / kB it is easy to see that roughly
U ~ U0 + Nelec (T/ TF) kBT so that
C = dU/dT ~ Nelec kB (T/TF)
35
• Quantitative evaluation: F
0
0 0
( ) ( ) ( ) ( ) ( )e e
U T U T U d D f d D
= − = −
0
( ) 1 ( ) ( ) ( ) ( )F
F
f D d d D f
= − − +
0 0
= ( ) 1 ( ) ( ) 1 ( ) ( )
+ ( ) ( ) ( ) ( ) ( )
F F
F F
F F
F F
f D d f D d
D f d f D d
− − − −
− +
0
0 0
= ( ) ( ) 1 ( ) + ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
F
F
F F
F
F F
F F F
d D f d D f
D d f D d f D d
− − −
− + +
36
• Quantitative evaluation: F
0
0 0
( ) ( ) ( ) ( ) ( )e e
U T U T U d D f d D
= − = −
0
0 0
= ( ) ( ) 1 ( ) + ( ) ( ) ( )
( ) ( ( ) ( ) ( ) ( ) )
F
F
F F
F
F F
F F
d D f d D f
D d f D d f D d
− − −
− + +
Energy cost of
empty state below F
Energy cost of occupied
state above F
N N
0
= ( ) ( ) 1 ( ) + ( ) ( ) ( ) F
F
F Fd D f d D f
− − −
37
F
FF F
0( ) ( ) ( ) ( ) ( ) ( )[1 ( )]
eU T d D f d D f
= − + − −
( )0
( , )( )
el e F
d df TC U d D
dT dT
= = −
Introduction to Solid State Physics by Kittel (2005)
38
( )0
( , )( )
el F F
df TC D d
dT
−
2 2
exp[( ) / ]( , )
(exp[( ) / ] 1)
F F
F
df
d
− −=
− +
Fx
−=
2 2
2( )
( 1)F
x
el B F x
eC k TD x dx
e
−
+ 100F
2 2
2( )
( 1)
x
el B F x
eC k TD x dx
e
−
+
F
FF F
0( ) ( ) ( ) ( ) ( ) ( )[1 ( )]
eU T d D f d D f
= − + − −
( )0
( , )( )
el e F
d df TC U d D
dT dT
= = −
39
2 2
2( )
( 1)
x
el B F x
eC k TD x dx
e
−
+
2
2 2
0
0
12 ( ) 4 ( )
1 1B F B Fx x
xk TD k TD x dx
e e
= − ++ +
2 2 1( )
( 1)( 1)B F x x
k TD x dxe e
−
−
=+ + even function
2 2
0
1[ ]
12 ( ) x
B F
dek TD x dx
dx
+= −
1
1 1
x
x x
e
e e
−
−=
+ + 1
( )1 ( )
x
x m
xm
ee
e
− −
−=
−= − = − −
− −
40
2
10
4 ( ) ( ) x m
el B F
m
C k TD x e dx
−
=
= − −
2
21 0
( 1)4 ( )
m
y
B F
m
k TD ye dym
−
=
−= −
2
21
( 1)4 ( )
m
B F
m
k TDm
=
−= −
41
2 2 2 2 2 21
( 1) 1 1 1 1 1 1( ...)1 2 3 4 5 6
m
m m
=
−= − − + − + − +
2 2 2 2 2 21
1 1 1 1 1 1 1... (2)
1 2 3 4 5 6m m
=
= + + + + + + =
2 2 2 2 2 2 21
1 1 1 1 1 1 1 (2)...
2 2 4 6 8 10 4m m
=
= + + + + + =
2 2 2 2 2 2
1 1 1 1 1 1 2... (2) (2)
1 2 3 4 5 6 2 − + − + − + = −
21
( 1) (2)
2
m
m m
=
−= −
42
2 FB F
BF
1,
2
el
TC N k T
kT
= =
2
21
( 1)4 ( )
m
el B F
m
C k TDm
=
−= −
3( )
2F
F
ND
=
2 214 ( )( (2)) 2 ( ) (2)
2B F B F
k TD k TD = − − =
22 ( ) (2)
B Fk TD =
2
22 ( )
6B F
k TD
=
43
2 FB F
BF
1,
2
el
TC N k T
kT
= =
Free electron model:
Drude model:
B
3
2el
C N k=
2 2 2 21 1
( ) ( )3 4 3 4
q B B
F F
T Tj Nk v k j
T e T
= = −
qj ST j=
21
( )3 4
B
F
TS k
e T
= −
2
Bk
e−
1
3q el
j U v=
j env= −
44
2e n
j env E Em
= − = =
21 1
3 3 2B
F
TK Cvl n k v v
T
= =
2 2B F
k Tv
m=
2 FB F
BF
1,
2
el
TC N k T
kT
= =
23
2
Bk Tn
m=
2 3B
k Tv
m=
22
3
Bk Tn
m
=
Drude model:
1 1 3
3 3 2B
K Cvl n k v v= =
45
Heat capacity
• Comparison of electrons in a metal with phonons
– Electrons dominate at low T in a metal
– Phonons dominate at high T because of reduction factor T/TF
Phonons approach
classical limit
C ~ 3 Natom kB
C
T
T 3 Electrons have
C ~ Nelec kB (T/TF)
T
46
Heat Transport
• Both electron and phonon can carry thermal energy
(Electrons dominate in metals).
• Similar to electric conduction, only the electrons near the
Fermi energy can contribute thermal current.
• Effects of electrons and phonons are comparable in poor
metals like alloys.
• Phonons dominate in non-metals.
47
Heat capacity
• Experimental results for metals
C/T = + A T 2 + ….
theo = (2/2) (Nelec/EF) kB2 is the free electron gas result.
Solid-State Physics by Ibach and Lueth (2009)
48
Effective mass of electrons
Introduction to Solid State Physics by Kittel (2005)
49
DOS of TM ions
Solid-State Physics by Ibach and Lueth (2009)
50
Heat capacity
• Experimental results for metals
C/T = + A T 2 + ….
It is most informative to find the ratio exp / theo where theo =
(2/2) (Nelec/EF) kB2 is the free electron gas result. Equivalently
since EF 1/m, we can consider the ratio exp / theo = mth / m,
where mth is a thermal effective mass for electrons in the metal.
51
Thermal effect mass
Causes of (free):
• Electron-ion interaction
• Electron-phonon interaction
• Electron-electron interaction
• Heavy fermion
• 4f or 5f electron wave-functions overlap with
neighboring ions
•
• CeAl3, CeCu2Si2, UPt3, UBe13
2 3th 10 -10 m
m
52
Interpretation of Ohm’s law:
Electrons act like a gas
• An electron is a particle - like a molecule.
• Electrons come to equilibrium by scattering like molecules
(electron scattering is due to defects, phonons, and electron-
electron scattering).
• Electrical conductivity occurs because the electrons are
charged and they move and equilibrate.
• What is different from usual molecules?
Electrons obey the Pauli exclusion principle. This limits the
allowed scattering which means that electrons act like a
weakly interacting gas.
53
The origin of resistivity
• Impurities - wrong atoms, missing atoms, extra atoms, …
– Proportional to concentration
(Dominated at low T)
Introduction to Solid State Physics by Kittel (2005)
• Lattice vibrations - atoms out of their ideal places
– Proportional to mean square displacement
(Dominated at room temperature)
54
The origin of resistivity
lattice impurity = +
T impurityn Matthiessen’s rule
Solid-State Physics by Ibachand Lueth (2009)
Impurity scattering
dominates at low T
55
The origin of resistivity
( ) (0)lattice impurity
T = −
The resistivity ratio of a
specimen is usually defined as
the ratio of its resistivity at
room temperature to its
residual resistivity.
Introduction to Solid State Physics by Kittel (2005)
A copper specimen with a
resistivity ratio of 1000
corresponding to an impurity
concentration of about 20 ppm.
56
The origin of resistivity
( ) (0)lattice impurity
T = −
For copper single crystal,
8 9(1.57 10 cm/s)(2 10 s)
Fl v −= =
In exceptionally pure specimens the resistivity ratio may be as
high as 106, whereas in some alloys (e.g., manganin) it is as low
as 1.1.
0.3cm=
57Introduction to Solid State Physics by Kittel (2005)
58
The Oxford Solid State Physics by Simon (2017)
Fermi surface
• The filling of the states is described by the Fermi surface –
the surface in k-space that separates filled from empty states
• For the electron gas this is a sphere of radius kF.
59
Why Drude model works
The Oxford Solid State Physics by Simon (2017)
• The shift of Fermi sea has finite averaged velocity, drift
velocity.
• The electron with higher kinetic energy would scatter back.
60
Summary
• Role of electrons in solids:
Determine binding
“electronic” properties (conductivity, …)
• The starting point for understanding electrons in solids is
completely different from that for nuclei
• Simplest model - Electron Gas
– Failure of classical mechanics (Drude model)
– Success of quantum mechanics (Sommerfeld theory)
– Pauli Exclusion Principle, Fermi Statistics
• Similarities, differences from vibration waves
• Density of States, Heat Capacity
61
Shortcomings of the free electron model
• The mean free path of electrons in metal is around 10 nm at
RT and even 1 mm at low temperatures. Why the electrons
would not be scattered by the charged atoms?
• How about insulators where there are no free electrons?
• There are many electrons in atoms. Why do core electrons
not contribute to anything?
• Why the Hall coefficients of some materials are with wrong
sign?
• In optical spectra of metal there are usually many features
and thus metal have their characteristic colors. How to
explain it via the free electron model?