Handout 3
Free Electron Gas in 2D and 1D
In this lecture you will learn:
Free electron gas in two dimensions and in one dimension
Density of States in k-space and in energy in lower dimensions
ECE 407 Spring 2009 Farhan Rana Cornell University
Electron Gases in 2D
In several physical systems electron are confined to move in just 2
dimensions
STM
micrograph
Examples, discussed in detail later in the course, are shown below:
Semiconductor Quantum Wells:
GaAs
Graphene:
InGaAs
quantum well
(1-10 nm)
GaAs
Semiconductor quantum
wells can be composed of
pretty much any
semiconductor from the
groups II, III, IV, V, and VI of
the periodic table
TEM
micrograph
Graphene is a single atomic layer
of carbon atoms arranged in a
honeycomb lattice
ECE 407 Spring 2009 Farhan Rana Cornell University
Electron Gases in 1D
In several physical systems electron are confined to move in just 1 dimension
Examples, discussed in detail later in the course, are shown below:
Semiconductor Quantum
Wires (or Nanowires):
Semiconductor Quantum
Point Contacts
(Electrostatic Gating):
GaAs
metal
Carbon Nanotubes
(Rolled Graphene
Sheets):
metal
InGaAs
Quantum well
InGaAs
Nanowire
GaAs
GaAs
ECE 407 Spring 2009 Farhan Rana Cornell University
Electrons in 2D Metals: The Free Electron Model
The quantum state of an electron is described by the time-independent
Schrodinger equation:
2 2
r V r r E r
2m
Consider a large metal sheet of area A= Lx Ly :
Use the Sommerfeld model:
The electrons inside the sheet are confined in a
two-dimensional infinite potential well with zero
potential inside the sheet and infinite potential
outside the sheet
V r 0
V r
A Lx Ly
Ly
Lx
for r inside the sheet
for r outside the sheet
The electron states inside the sheet are given
by the Schrodinger equation
free electrons
(experience no
potential when inside
the sheet)
ECE 407 Spring 2009 Farhan Rana Cornell University
Born Von Karman Periodic Boundary Conditions in 2D
Solve:
2 2
r E r
2m
Use periodic boundary conditions:
Solution is:
These imply that each
edge of the sheet is
folded and joined to
the opposite edge
x Lx , y , z x , y , z
x , y Ly , z x , y , z
1 i k . r
e
A Lx Ly
x
Ly
Lx
1 i k x x k y y
e
A
The boundary conditions dictate that the allowed values of kx , and ky are such
that:
e i k x Lx 1
e
i k y Ly
kx n
2
Lx
n = 0, 1, 2, 3,.
ky m
2
Ly
m = 0, 1, 2, 3,.
ECE 407 Spring 2009 Farhan Rana Cornell University
Born Von Karman Periodic Boundary Conditions in 2D
Labeling Scheme:
All electron states and energies can be labeled by the corresponding k-vector
k r
1 i k . r
e
A
2k 2
Ek
2m
Normalization: The wavefunction is properly normalized:
Orthogonality:
2
2
d r k r 1
Wavefunctions of two different states are orthogonal:
2
d r
i k k ' . r
r k r d 2r e
k ' , k
A
k* '
Momentum Eigenstates:
Another advantage of using the plane-wave energy eigenstates (as opposed to the
sine energy eigenstates) is that the plane-wave states are also momentum
eigenstates
Momentum operator: p
p r r k r
Velocity:
Velocity of eigenstates is:
k 1
vk
k E k
m
ECE 407 Spring 2009 Farhan Rana Cornell University
States in 2D k-Space
2
Lx
ky
k-space Visualization:
The allowed quantum states states can be
visualized as a 2D grid of points in the entire
k-space
kx n
2
Lx
ky m
2
Ly
kx
2
Ly
n, m = 0, 1, 2, 3, .
Density of Grid Points in k-space:
Looking at the figure, in k-space there is only one grid point in every small
area of size:
Lx
There are
2 2
Ly
2 2
Very important
result
grid points per unit area of k-space
ECE 407 Spring 2009 Farhan Rana Cornell University
The Electron Gas in 2D at Zero Temperature - I
Suppose we have N electrons in the sheet.
Then how do we start filling the allowed quantum states?
N
y
Suppose T~0K and we are interested in a filling scheme
that gives the lowest total energy.
The energy of a quantum state is:
2 k x2 k y2
2k 2
Ek
2m
2m
A Lx Ly
x
Ly
Lx
ky
Strategy:
Each grid-point can be occupied by two electrons
(spin up and spin down)
Start filling up the grid-points (with two electrons
each) in circular regions of increasing radii until
you have a total of N electrons
kx
kF
When we are done, all filled (i.e. occupied)
quantum states correspond to grid-points that are
inside a circular region of radius kF
ECE 407 Spring 2009 Farhan Rana Cornell University
The Electron Gas in 2D at Zero Temperature - II
ky
Each grid-point can be occupied by two electrons (spin
up and spin down)
kF
All filled quantum states correspond to grid-points that
are inside a circular region of radius kF
Area of the circular region =
kx
kF2
A
Number of grid-points in the circular region =
Number of quantum states (including
spin) in the circular region =
Fermi circle
kF2
kF2
A 2
kF
2
But the above must equal the total number N of electrons inside the box:
A 2
kF
2
n electron density
kF 2 n 2
N kF2
A 2
Units of the electron
density n are #/cm2
ECE 407 Spring 2009 Farhan Rana Cornell University
The Electron Gas in 2D at Zero Temperature - III
ky
All quantum states inside the Fermi circle are filled (i.e.
occupied by electrons)
All quantum states outside the Fermi circle are empty
kF
Fermi Momentum:
The largest momentum of the electrons is: kF
This is called the Fermi momentum
Fermi momentum can be found if one knows the electron
1
density:
kx
Fermi circle
kF 2 n 2
Fermi Energy:
2kF2
The largest energy of the electrons is:
This is called the Fermi energy EF :
Also:
EF
2 n
m
2m 2 2
kF
EF
2m
or
EF
Fermi Velocity:
kF
The largest velocity of the electrons is called the Fermi velocity vF : v F
ECE 407 Spring 2009 Farhan Rana Cornell University
The Electron Gas in 2D at Non-Zero Temperature - I
ky
A
Recall that there are
space
2 2
So in area dk x dk y
grid points is:
dk x
dk y
grid points per unit area of k-
kx
of k-space the number of
dk x dk y
d 2k
The summation over all grid points in k-space can be replaced by an area integral
all k
Therefore:
d 2k
2 2
d 2k
N 2 f k 2 A
f k
2
2
all k
f k is the occupation probability of a quantum state
ECE 407 Spring 2009 Farhan Rana Cornell University
The Electron Gas in 2D at Non-Zero Temperature - II
The probability f k that the quantum state of wavevector k is occupied by an
electron is given by the Fermi-Dirac distribution function:
f k
E k Ef K T
1 e
Therefore:
N 2 A
d 2k
2 2
Where:
2 k x2 k y2
2k 2
Ek
2m
2m
d 2k
1
f k 2 A
2
E
k
2 1 e Ef KT
Density of States:
The k-space integral is cumbersome. We need to convert into a simpler form an
energy space integral using the following steps:
d 2k 2 k dk
and
2k 2
2k
dk
dE
m
2m
Therefore:
2 A
d 2k
dk
dE
ECE 407 Spring 2009 Farhan Rana Cornell University
The Electron Gas in 2D at Non-Zero Temperature - III
N 2 A
Where:
d 2k
2 1 e E k Ef KT
g2D E
A dE g2D E
0
E E f KT
1 e
Density of states function is constant
(independent of energy) in 2D
g2D(E) has units: # / Joule-cm2
ky
The product g(E) dE represents the number of
quantum states available in the energy interval
between E and (E+dE) per cm2 of the metal
Suppose E corresponds to the inner circle
from the relation:
kx
2k 2
2m
And suppose (E+dE) corresponds to the outer
circle, then g2D(E) dE corresponds to twice the
number of the grid points between the two
circles
ECE 407 Spring 2009 Farhan Rana Cornell University
The Electron Gas in 2D at Non-Zero Temperature - IV
N A dE g2D E
0
1
A dE g2D E f E Ef
E E f KT
1 e
Where: g2D E
f E Ef
The expression for N can be visualized as the
integration over the product of the two functions:
Check: Suppose T=0K:
Ef
Ef
f E Ef
T = 0K
Ef
EF
Compare with the previous result at T=0K:
N A dE g2D E f E Ef A dE g2D E
1
0
g2D E
Ef
Ef
At T=0K (and only at T=0K) the Fermi level
Ef is the same as the Fermi energy EF
ECE 407 Spring 2009 Farhan Rana Cornell University
The Electron Gas in 2D at Non-Zero Temperature - V
For T 0K:
Since the carrier density is known, and does not change with temperature, the
Fermi level at temperature T is found from the expression
n dE g2D E
0
1 e E E f KT
Ef
1 e K T
K
T
log
In general, the Fermi level Ef is a function of temperature and decreases from EF as
the temperature increases. The exact relationship can be found by inverting the
above equation and recalling that:
EF
to get:
EF
Ef T KT loge KT 1
ECE 407 Spring 2009 Farhan Rana Cornell University
Total Energy of the 2D Electron Gas
The total energy U of the electron gas can be written as:
d 2k
U 2 f k E k 2 A
f k Ek
2
all k
Convert the k-space integral to energy integral:U A dE g2D E f E Ef E
0
The energy density u is:
u
U
dE g2D E f E Ef E
A 0
Suppose T=0K:
EF
2 2
u dE g2D E E
Since:
We have: u
EF2
EF
1
n EF
2
ECE 407 Spring 2009 Farhan Rana Cornell University
2D Electron Gas in an Applied Electric Field - I
ky
f k
e
k t k
E
e
E
ky
E E x x
kx
kx
Electron distribution in k-space
when E-field is zero
Electron distribution is shifted in
k-space when E-field is not zero
Distribution function: f k
Distribution function: f k
e
E
Since the wavevector of each electron is shifted by the same amount in the
presence of the E-field, the net effect in k-space is that the entire electron
distribution is shifted as shown
Ly
Lx
ECE 407 Spring 2009 Farhan Rana Cornell University
2D Electron Gas in an Applied Electric Field - II
Current density (units: A/cm)
d 2k e
J 2 e
f k
Ev k
2 2
e
E
ky
Do a shift in the integration variable:
d k
2
e
f k v k
2
e
k
d 2k
J 2 e
f k
m
2 2
e 2
d 2k
J
f k E
2
2
m
2
2
ne
J
E E
m
J 2 e
kx
Electron distribution is shifted in
k-space when E-field is not zero
Distribution function: f k
e
E
Where:
n e 2
m
electron density = n (units: #/cm2)
Same as the Drude result - but
units are different. Units of are
Siemens in 2D
ECE 407 Spring 2009 Farhan Rana Cornell University
Electrons in 1D Metals: The Free Electron Model
The quantum state of an electron is described by the time-independent
Schrodinger equation:
2 2
x V x x E x
2 m x 2
Consider a large metal wire of length L :
Use the Sommerfeld model:
The electrons inside the wire are confined in a
one-dimensional infinite potential well with zero
potential inside the wire and infinite potential
outside the wire
V x 0
V x
for x inside the wire
for x outside the wire
free electrons
(experience no
potential when inside
the wire)
The electron states inside the wire are given by
the Schrodinger equation
ECE 407 Spring 2009 Farhan Rana Cornell University
Born Von Karman Periodic Boundary Conditions in 1D
Solve:
2 2
x E x
2 m x 2
Use periodic boundary conditions:
x L, y , z x , y , z
Solution is:
L
These imply that each
facet of the sheet is
folded and joined to
the opposite facet
1 i k x x
e
L
The boundary conditions dictate that the allowed values of kx are such that:
e i k x L 1
kx n
2
L
n = 0, 1, 2, 3,.
ECE 407 Spring 2009 Farhan Rana Cornell University
10
States in 1D k-Space
k-space Visualization:
The allowed quantum states states can be
visualized as a 1D grid of points in the entire
k-space
kx n
2
L
2
L
kx
n = 0, 1, 2, 3, .
Density of Grid Points in k-space:
Looking at the figure, in k-space there is only one grid point in every small
length of size:
L
There are
L
2
Very important
result
grid points per unit length of k-space
ECE 407 Spring 2009 Farhan Rana Cornell University
The Electron Gas in 1D at Zero Temperature - I
Each grid-point can be occupied by two electrons (spin
up and spin down)
All filled quantum states correspond to grid-points that
are within a distance kF from the origin
Length of the region = 2kF
Number of grid-points in the region =
kF
kF
kx
Fermi points
L
2k F
2
Number of quantum states (including
L
2
2k F
spin) in the region =
But the above must equal the total number N of electrons in the wire:
NL
2k F
n electron density
kF
N 2k F
Units of the electron
density n are #/cm
2
ECE 407 Spring 2009 Farhan Rana Cornell University
11
The Electron Gas in 1D at Zero Temperature - II
All quantum states between the Fermi points are filled (i.e.
occupied by electrons)
All quantum states outside the Fermi points are empty
Fermi Momentum:
The largest momentum of the electrons is: kF
This is called the Fermi momentum
Fermi momentum can be found if one knows the electron
density:
kF
kx
Fermi points
n
2
Fermi Energy:
2kF2
The largest energy of the electrons is:
2m 2 2
kF
EF
2m
This is called the Fermi energy EF :
EF
Also:
2 2 n2
8m
or
8m
EF
Fermi Velocity:
kF
The largest velocity of the electrons is called the Fermi velocity vF : v F
ECE 407 Spring 2009 Farhan Rana Cornell University
The Electron Gas in 1D at Non-Zero Temperature - I
Recall that there are L
2
space
grid points per unit length of k-
dk x
So in length dk x of k-space the number of
grid points is:
kx
L
dk x
2
The summation over all grid points in k-space can be replaced by an integral
all k
dk x
2
Therefore:
dk x
f k x
2
N 2 f k x 2 L
all k
f k x is the occupation probability of a quantum state
ECE 407 Spring 2009 Farhan Rana Cornell University
12
The Electron Gas in 1D at Non-Zero Temperature - II
The probability f k x that the quantum state of wavevector k x is occupied by an
electron is given by the Fermi-Dirac distribution function:
f k x
1
1 e E k x Ef K T
Therefore:
2k x2
Ek
2m
Where:
dk
dk x
1
x
f k x 2 L
E k x E f KT
2
2
1 e
N 2L
Density of States:
The k-space integral is cumbersome. We need to convert into a simpler form an
energy space integral using the following steps:
dk
dk x
2L2
2
0 2
and
2L
2k 2
2k
dE
dk
2m
m
Therefore:
dk x
2
2L
L dE
0
2m 1
E
ECE 407 Spring 2009 Farhan Rana Cornell University
The Electron Gas in 1D at Non-Zero Temperature - III
dk x
1
1
L dE g1D E
E
k
E
KT
x
f
1 e E E f KT
2 1 e
0
N 2L
Where:
g1D E
2m 1
E
Density of states function in 1D
g1D(E) has units: # / Joule-cm
The product g(E) dE represents the number of
quantum states available in the energy interval
between E and (E+dE) per cm of the metal
Suppose E corresponds to the inner points
from the relation:
2 2
k
2m
kx
And suppose (E+dE) corresponds to the outer
points, then g1D(E) dE corresponds to twice the
number of the grid points between the points
(adding contributions from both sides)
ECE 407 Spring 2009 Farhan Rana Cornell University
13
The Electron Gas in 1D at Non-Zero Temperature - IV
N L dE g1D E
0
1
L dE g1D E f E Ef
E E f KT
1 e
g1D E
2m 1
Where: g1D E
E
f E Ef
The expression for N can be visualized as the
integration over the product of the two functions:
Check: Suppose T=0K:
Ef
N L dE g1D E f E Ef L dE g1D E
f E Ef
1
8m
L
Ef
T = 0K
Ef
Ef
8m
Ef
Compare with the previous result at T=0K:
8m
EF
At T=0K (and only at T=0K) the Fermi level
Ef is the same as the Fermi energy EF
ECE 407 Spring 2009 Farhan Rana Cornell University
The Electron Gas in 1D at Non-Zero Temperature - V
For T 0K:
Since the carrier density is known, and does not change with temperature, the
Fermi level at temperature T is found from the expression
n dE g1D E
0
E E f KT
1 e
In general, the Fermi level Ef is a function of temperature and decreases from EF as
the temperature increases.
ECE 407 Spring 2009 Farhan Rana Cornell University
14
Total Energy of the 1D Electron Gas
The total energy U of the electron gas can be written as:
dk x
f k x E k x
2
U 2 f k x E k x 2 L
all k
Convert the k-space integral to energy integral: U L dE g1D E f E Ef E
0
U
dE g1D E f E Ef E
L 0
The energy density u is:u
Suppose T=0K:
EF
u dE g1D E E
0
8m
Since: n
We have:
32
8m E F
3
EF
1
n EF
3
ECE 407 Spring 2009 Farhan Rana Cornell University
1D Electron Gas in an Applied Electric Field - I
f k x
k x t k x
e
Ex
e
E
E E x x
kx
kx
Electron distribution in k-space
when E-field is zero
Distribution function: f k x
Electron distribution is shifted in
k-space when E-field is not zero
Distribution function: f k x
Ex
Since the wavevector of each electron is shifted by the same amount in the
presence of the E-field, the net effect in k-space is that the entire electron
distribution is shifted as shown
L
ECE 407 Spring 2009 Farhan Rana Cornell University
15
1D Electron Gas in an Applied Electric Field - II
Current (units: A)
dk x
e
I 2 e
f kx
E x v k x
e
Ex
E E x x
Do a shift in the integration variable:
kx
dk x
e
f k x v k x
Ex
2
e
k x
Ex
dk
x f k
I 2 e
x
m
2
I 2 e
dk
e 2
x f k E
2
x x
m
2
ne
I
E E
m
Electron distribution is shifted in
k-space when E-field is not zero
Distribution function: f k x
Ex
Where:
n e 2
m
electron density = n (units: #/cm)
Same as the Drude result - but
units are different. Units of are
Siemens-cm in 1D
ECE 407 Spring 2009 Farhan Rana Cornell University
ECE 407 Spring 2009 Farhan Rana Cornell University
16
