Electrons in Nanostructures
• The
electronic properties of bulk materials are dominated by
electron scattering.
• Electrons travel at a drift velocity depending on the applied
voltage (Ohm’s law).
• The scattering events that contribute to resistance occur with
mean free paths that are typically tens of nm in many metals.
(1 every 40 nm in Cu).
If the size of the structure is of the same scale as the mean
free path of an electron, Ohm’s law may not apply, giving rise
to quantum effects [quantum confinement in 1D (nanowires)
and 0D (quantum dots)].
Electronic properties of materials
A
L
R

L
A
resistivity
24 orders of magnitude
between copper and rubber!
Copyright Stuart Lindsay 2008
The Fermi Liquid
• Individual electrons interact strongly
• Excited carriers behave like “free electrons” owing to screening
by “correlation hole”
Fermi energy = energy of
the highest occupied state at
zero temperature.
= chemical potential at T=0
Mobile electrons are produced from thermal fluctuations that
promote electrons from below the Fermi energy to above it.
Carriers are not produced alone, but in correlated pairs
(electron/hole), so that Coulomb interactions may be ignored.
These correlated pairs can be treated as quasi-particles in
stationary states of the system!
The energy of the quasi-particle can be written as:
2k 2
E
2m *
m* = effective mass
A typical Fermi energy is 200 times the thermal energy.
Drude free-electron model
Conduction electrons are a gas of free, non-interacting particles,
exchanging energy only by scattering events.

mean time between collisions
From Newton’s law:
m
Ohms Law:
v

 eE
J  E  ne v
conductivity
v
drift velocity

1

ne 
2

ρ(Cu)= 1.6 micro Ohm·cm
m
=2.7·10-14 s
But:
3kT
v 
m
l
v

 1nm
The electron mean free path in copper is ≈ 40nm.
(correct)
Sommerfeld Model
Conduction electrons are described as a free quantum gas.
Two electrons (spin-up and spin down) can occupy each of the dn
states per unit wave vector.
2k 2
E
2m *
The density of states between k and k+dk in a volume V is:
Vk 2 dk
dn 
2 2
Putting 2 electrons (spin up/spin down) into each of the dn
states at T=0:
kF
k F3
Vk 2 dk
N  2 

V
2
2
2
3
0
2
2
F
 k
EF 
2m
k F
vF 
m
N
k F3
n  2
V 3
Electron density = valence electrons per unit cell / unit cell volume
For lithium (a, lattice constant=3.49Å):
n
kF
vF
EF
4.6·1022cm-3
1.12Å-1
1.23·108cm/s
4.74eV
Transport in free electron metals
At low applied electric fields, main source of excitation is thermal.
Transport involves only a
fraction of carriers:
k BT
f ( n) 
EF
At 300K in Li:
0.025
f(n)
 0.005
4.74
Copyright Stuart Lindsay 2008
Modified Drude Theory
2
ne
 k BT
1
 

m EF
 k BT 
3

CV  nk B 
2
 EF 
From Debye theory:
CV  T 3
Copyright Stuart Lindsay 2008
Electrons in crystals: Bloch’s theorem
U (r )  U (r  R )
R: lattice translation vector
 n ,k r  R   exp ik  R  n ,k (r )
Bloch’s theorem
All the properties of an infinite crystal can be described in terms
of the basic symmetries and properties of a unit cell of the lattice.
k max
2

a
Crystal momentum
Wavelenghts less than a lattice constant are not physically
meaningful. Measurable quantities must always have the
periodicity of the lattice. For an infinite crystal: k=0.


a
 k 
First Brillouin zone

-k and k directions are equivalent
(reduced zone scheme)
a
Trial Bloch states for an 1D lattice:
 T   exp ikna s (r  na)
n
From I order Perturbation Theory:
Interaction
hamiltonian
Ek  Es   T U op  T
Electrons
in the lattice
Single electrons in
isolated atoms
Band structure
Nearest neighbors approximation (n=±1):
Ek  Es   s U op  s   s U op  s ( r  a ) expika 
  s U op  s ( r  a ) exp ika
Es   s U op  s   0
On-site energy
 s U op  s (r  a)  
Hopping matrix element
Copyright Stuart Lindsay 2008
E k   0  2 cos ka
Free electrons
2k 2
E
*
2m
Electrons in a
periodic potential
dE (k )
0
dk
Negative
effective mass!
2

E (k )
1

m  2

k 2
Copyright Stuart Lindsay 2008
d 1 dE( k )
vg 

dk  dk
group velocity
At ka =±π the group velocity falls to zero: E=ε0.
2

 2a
k
Bragg diffraction condition
The combination of forward and backward (reflected) wave results
in a standing wave: the electron does not propagate at these values
of k.
The flattening of the function E(k) causes an increase in the density
of states near ka=±π.
Extended and reduced Brillouin zones
Extended Zone
Reduced Zone
Copyright Stuart Lindsay 2008
Band structure and electronic properties
• Metals : EF lies inside an allowed band (1 electron/unit cell)
• Insulators : The Fermi level lies at the top of a band (full band).
Large gap between bands.
(Dielectric breakdown
2U  k B T
for large electric fields)
• Semiconductor: Full band (valence band).
2U  k B T
- Dope with free electrons in the conduction band (e.g., P, As):
m*>0 (n type donors, negative carriers)
- Dope to take electrons from valence band (e.g., B, Ga):
m*<0, “positive” carriers (holes) (p type donors)
Why elements with 2 electrons/unit cell are most metals ?
Partially filled
Cubic lattice in the
reciprocal space
A cube of side 2π/a
Density of states
(free electron model)
A Fermi sphere of
radius kF
1° BZ completely
filled, 2° BZ
partially filled.
Electrons in a quantum point contact
Filled states
A bias V, applied across the two electrodes, will shift the Fermi
levels of one relative to the other by an amount eV.
The net current will be proportional to the number of states in this
energy range.
At what size quantum effects dominate?
Upper limit:
the size of the nanostructure approaches the electron mean free
path for scattering (tens to hundreds of nm at room temperature).
Lower limit:
only one mode of transmission available in the channel, i.e. Fermi
wavelength in diameter (2π/kf).
For lithium ≈ 6Å (atomic dimensions)
Fermi Golden Rule can be applied also to the case of electrons that
tunnel from one bulk electrode to another by means of a small
connecting constriction.
From Perturbation Theory:
2
P(m, k ) 
 m Hˆ   k

Probability of transition
from m to k
2
 (Ek )
Density of states close to Ek
The Landauer Resistance
i  ne v
Intensity of current per unit area
1 dE
 v   vg 
 dk
dn
dn dk
n
eV 
eV
dE
dk dE
group velocity
no. of states in the energy range dE
per unit energy (eV)
In 1D the distance between allowed k points is 2π/L.
Per unit length:
dn
1

dk 2
1 1
n
eV
2 vg
1 1
2e 2
i  nev g  2 N

 eV  ev g  N
V
2 v g
h
N = allowed quantum states in the channel
The factor 2 accounts for the two allowed spin states.
For N=1:
2e 2
G0 
 77.5S
h
Landauer resistance:
1
RL 
 12.9k
G0
The Landauer resistance is independent of the material lying
between the source and the sink of electrons.
It is a fundamental constant associated to quantum transport.
Landauer resistance is NOT a resistance in the ohmic sense:
no power is dissipated in the quantum channel!
It reflects how the probability of transmission changes as the
applied voltage is changed.
If the restriction is smaller than the scattering length of the
electrons, it cannot be described as a resistance in the ‘Ohm’s
law” sense. Dissipation requires scattering.
This occurs in bulk electrodes, but not in the nanochannel!
This is the reason why the high current densities in the STM (109
A/m2) do not damage the sample.
If the source and sink of electrons are connected by N channels
(N different electronic wavefunctions can occupy the gap), the
resistance of the gap is Rg is:
1
1 h
Rg  RL 
N
N 2e 2
The resistance of a tunnel junction of gap L is:



R  RL exp 1.02  L  12.9 exp 1.02  L
Φ = V0-E = workfunction of the metal

Break junctions
As the wire narrows down to dimensions of a few Fermi wavelengths,
quantum jumps are observed in the current.
A. Propagation of quantum
modes in a very narrow
channel
B. Landauer steps in the
conductance of a gold break
junction.
An exact conductance can be calculated from the Landauer-Buttiker
equation:
2e 2
G
h
 Tij
2
i,j
Tij = matrix elements that connect electronic states i on one side
of the junction to states j on the other side.
The Coulomb Blockade
Bulk electrode
nm-sized conducting island
ET can occur by hopping or
by resonant tunneling
through the island
Single Electron Transistor
A gate electrode applied to the
island can alter its potential to
overcame the blockade
Two possible Electron Transfer mechanisms:
Resonant tunneling: electron tunnels straight through the whole
structure.
Hopping mechanism: electron hops on the center particle and
then hops on the other electrode.
The hopping mechanism (negligible tunneling regime) is very
sensitive to the potential of the center particle because the
charging effect of a small particle for the transfer of one electron
can be quite significant.
If the charging energy is greater than the thermal energy
available, further hopping is inhibited (Coulomb blockade).
When the applied bias exceeds the Coulomb blockade barrier
current can flow again.
→ Coulomb staircase
The voltage required to charge a spherical island of radius a is
given by:
V 
e
80 a
Taking: ε=1 and ε0=8.85·10-12 F·m-1 one obtains:
ΔV = 0.7 V for a = 1nm
ΔV=7 mV for a = 100nm
Condition for Coulomb blockade is that the electron localizes on
the quantum dot (negligible tunneling).
• Hanna-Tinkham theory
Electric circuit model of
the two junctions Coulomb
blockade experiment.
n  

I ( V )  e  ( n )   
n  
( n )
j

2

2
  e  ( n )
n  
n  

1


1

Normalized distribution of charges on the central particle
Thermally activated hopping rates between the particle
and the left or right electrodes,
I-V curves from nanoscale
double junctions experiments.
dots: experimental points;
lines: Coulomb blockade
theory.
Q0 = residual floating charge
on the island
Hanna and Thinkham, Phys. Rev. B, 1991.
Single Electron Transistor
= an isolated metal particle
coupled by tunnel junctions to two
microscopic electrodes.
The isolated metal particle is
coupled with a gate electrode that
allows to control the potential of
the metal particle independently.
Electrodes: n-GaAS
Island: n-GaAs circular quantum dot
Insulator: AlGaAs
A finite source-drain voltage (Vsd)
opens a window of potential for
tunneling via the quantum dot.
A 3D-plot: dI/dVsd (z-axis) as a function of the source-drain
potential applied between the electrodes and the gate potential
applied to the quantum dot.
White areas (Coulomb staircase):
dI
0
dVsd
Red areas: SET is on
dI
 max
dVsd
at Coulomb
steps
Gate bias for level at Vsd = 0
SET as a micromechanical sensor
1μ
A sensor with single-electron sensitivity!
An electrochemical sensor based on the capacitive coupling of a
vibrating beam to the gate of a SET.
R.G. Knobel and A.N. Cleland, Nature 2003 424, 291
Resonant tunneling
L
Tunneling rate from the left
R
Tunneling rate
to the right
central particle
diameter = 2R
Electrons incident from the left face a barrier of height V0 containing a
localized state at energy E0.
L R
4e
G
h ( E  E0 ) 2  (L  R ) 2
2
Zero-bias
conductance
(2 counts both
spin channels)
G = 0.5 G0 when L=R and E = E0
At resonance (E=E0), the localized state is acting like a metallic
channel that connects the left and right electrodes.
If the tunneling rates are small enough, charge accumulation on the
localized state becomes significant, resulting in Coulomb blockade.
The Coulomb blockade requires that the tunneling resistance of the
contacts to the central particle exceeds twice the Landauer
resistance (i.e. h/e2).
In case of strong coupling betwwen the electrodes and the quantum
dot, tunneling predominates and the whole system must be tretaed
quantum mechanically.
Transmission vs. energy
for a tight-binding model
of a resonant tunneling
through
a
molecule
bound into a gap in a 1D
wire.
A) 2 states in the conduction band, L=R
C) 2 states, L=4R
B) 1 state
Dashed lines: electronic transmission expected through the gap
if no molecule is present between the two electrodes.
Time development of the charge density for a wave packet incident
from the left on a pair of barriers containing a localized resonant
state. Electron is modeled as a Gaussian wave packet launched
from the left.
a-d:
small barriers
strong coupling
e-h:
large barriers
weak coupling
Localization in disordered systems
The impact of disorder on electron transport becomes more
significant in nanometric systems.
• Temperature dependence of resistivity in nanostructures
Charge
density
distribution
calculated for electrons in random
potentials.
W/V = width of the potential
distribution in relation to the mean
value of the potential.
For W/V=8 the electrons are
almost completely localized.
Localization in nanometric structures
Peierl’s distortion: observed in linear conductive polymers
This distortion results in halving of the Brillouin zone in wave
vector space because the real space lattice is now doubled in size.
metallic (half-filled band) to insulator (full band) transition