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Electrons in NanostructuresElectrons 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 materialsElectronic properties of materials
A
LR
Copyright Stuart Lindsay 2008
L
A
24 orders of magnitude between copper and rubber!
resistivity
• Individual electrons interact strongly
• Excited carriers behave like “free electrons” owing to screening by “correlation hole”
The Fermi Liquid
Fermi energyFermi energy = energy of the highest occupied state at zero temperature.= chemical potential at T=0
*
22
2m
kE
The energy of the quasi-particle can be written as:
m*m* = effective mass
A typical Fermi energy is 200 times the thermal energy.
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/holeelectron/hole), so that Coulomb interactions may be ignored.
These correlated pairs can be treated as quasi-particles in These correlated pairs can be treated as quasi-particles in stationary states of the system!stationary states of the system!
Drude free-electron modelDrude free-electron model
Ev
em
vne EJ
mean time between collisions
Ohms Law:
v drift velocity
From Newton’s law:
Conduction electrons are a gas of free, non-interacting particles, exchanging energy only by scattering events.
conductivity
m
ne
2
1
ρ(Cu)= 1.6 micro Ohm·cm =2.7·10-14 s (correct)
But:
nmv
l 1
m
kTv
3
The electron mean free path in copper is ≈ 40nm.
Sommerfeld ModelSommerfeld 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.
*
22
2m
kE
The density of states between k and k+dk in a volume V is:
2
2
2dkVk
dn
VkdkVk
N F
kF
2
3
02
2
322
m
kE F
F 2
22
m
kv F
F
n kF vF EF
4.6·1022cm-3 1.12Å-1 1.23·108cm/s 4.74eV
For lithium (a, lattice constant=3.49Å):
2
3
3Fk
V
Nn
Putting 2 electrons (spin up/spin down) into each of the dn states at T=0:
Electron density = valence electrons per unit cell / unit cell volume
Transport in free electron metalsTransport in free electron metals
F
B
E
Tknf )(
Copyright Stuart Lindsay 2008
Transport involves only a fraction of carriers:
At 300K in Li:
0050744
0250.
.
.)n(f
At low applied electric fields, main source of excitation is thermal.
Modified Drude TheoryModified Drude Theory
F
B
E
Tk
m
ne
21
F
BBV E
TknkC
2
3
From Debye theory:
Copyright Stuart Lindsay 2008
3TCV
Electrons in crystals: Bloch’s theoremElectrons in crystals: Bloch’s theorem
)()( R rUrU R: lattice translation vector
akmax
2
)( exp ,, rRkRr knkn i Bloch’s theoremBloch’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.
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.
aa
k
First Brillouin zone
-k and k directions are equivalent (reduced zone scheme)
n
sT narikna )(exp
Trial Bloch states for an 1D lattice:
From I order Perturbation Theory:
Top
Tsk UEE
Electrons in the lattice
Single electrons in isolated atoms
Interactionhamiltonian
Band structureBand structure
ikaexp)ar(U
ikaexp)ar(UUEE
sop
s
sop
ssop
ssk
Nearest neighbors approximation (n=±1):
0 sop
ss UE )( arU sop
s
On-site energy Hopping matrix element
Copyright Stuart Lindsay 2008
0)(
dk
kdE
Negative effective mass!
2
2
2
)(1
k
kEm
*
22
2m
kE
Free electronsElectrons in a periodic potential
Copyright Stuart Lindsay 2008
kaEk cos20
dk
)k(dE
dk
dvg
1
group velocity
At ka =±π the group velocity falls to zero: E=ε0.
ak
22
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 propertiesBand structure and electronic properties
• MetalsMetals : EF lies inside an allowed band (1 electron/unit cell)
• InsulatorsInsulators : The Fermi level lies at the top of a band (full band). Large gap between bands.
• SemiconductorSemiconductor: Full band (valence band).
- 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)
TkU B2
TkU B2
(Dielectric breakdown for large electric fields)
Why elements with 2 electrons/unit cell are most metals ?
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.
Partially filled
Electrons in a quantum point contactElectrons in a quantum point contact
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.
Filled states
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:
)(ˆ2),(
2
kkm EHkmP
Density of states close to Ek
Probability of transition from m to k
The Landauer ResistanceThe Landauer Resistance
vi ne Intensity of current per unit area
eVdE
dk
dk
dneV
dE
dnn
2
1
dk
dn
In 1D the distance between allowed k points is 2π/L.
Per unit length:
no. of states in the energy range dEper unit energy (eV)
dk
dEvv g
1 group velocity
eVv
ng
1
2
1
For N=1:S
h
eG 5.77
2 2
0
k.G
RL 9121
0
Landauer resistance:
N = allowed quantum states in the channelThe factor 2 accounts for the two allowed spin states.
Vh
eNeveV
vNnevi g
gg
221
2
12
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:
22
11
e
h
NR
NR Lg
The resistance of a tunnel junction of gap L is:
L.exp.L.expRR L 021912021
Φ = V0-E = workfunction of the metal
A. Propagation of quantum modes in a very narrow channel
Break junctionsBreak junctions
B. Landauer steps in the conductance of a gold break junction.
As the wire narrows down to dimensions of a few Fermi wavelengths, quantum jumps are observed in the current.
An exact conductance can be calculated from the Landauer-Buttiker equation:
222
j,iijT
h
eG
Tij = matrix elements that connect electronic states i on one side of the junction to states j on the other side.
The Coulomb BlockadeThe Coulomb Blockade
Bulk electrode
nm-sized conducting islandET can occur by hopping or
by resonant tunneling through the island
Single Electron TransistorSingle Electron TransistorA gate electrode applied to the island can alter its potential to overcame the blockade
Two possible Electron Transfer mechanisms:
Resonant tunnelingResonant tunneling: electron tunnels straight through the whole structure.
Hopping mechanismHopping 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 blockadeCoulomb blockade).
a
eV
08
When the applied bias exceeds the Coulomb blockade barrier current can flow again.
→ Coulomb staircaseCoulomb staircase
The voltage required to charge a spherical island of radius a is given by:
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 theoryHanna-Tinkham theory
Electric circuit model of the two junctions Coulomb blockade experiment.
n
n
n
n
)n(e)n(e)V(I 1122
)n( Normalized distribution of charges on the central particle
j 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 TransistorSingle Electron Transistor
Electrodes: n-GaASIsland: n-GaAs circular quantum dotInsulator: AlGaAs
A finite source-drain voltage (Vsd) opens a window of potential for tunneling via the quantum dot.
= 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.
White areas (Coulomb staircase):
0sddV
dI
Gate bias for level at Vsd = 0
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.
Red areas: SET is on
maxdV
dI
sd
at Coulomb steps
SET as a micromechanical sensorSET as a micromechanical sensor
R.G. Knobel and A.N. Cleland, Nature 2003 424, 291
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.
Resonant tunnelingResonant tunneling
central particlediameter = 2R
L Tunneling rate from the leftR
Tunneling rate to the right
Electrons incident from the left face a barrier of height V0 containing a localized state at energy E0.
220
2
)()(
4
RL
RL
EEh
eG
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).
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=4R
B) 1 state Dashed lines: electronic transmission expected through the gap if no molecule is present between the two electrodes.
In case of strong coupling betwwen the electrodes and the quantum dot, tunneling predominates and the whole system must be tretaed quantum mechanically.
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 barriersstrong coupling
e-h:large barriersweak coupling
Localization in disordered systemsLocalization in disordered systemsThe 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.
Peierl’s distortion: Peierl’s distortion: observed in linear conductive polymers
Localization in nanometric structuresLocalization in nanometric structures
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