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**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)]. 1

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**Electronic properties of materials**

resistivity 24 orders of magnitude between copper and rubber! Copyright Stuart Lindsay 2008

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**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

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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: m* = effective mass A typical Fermi energy is 200 times the thermal energy.

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**Drude free-electron model**

Conduction electrons are a gas of free, non-interacting particles, exchanging energy only by scattering events. mean time between collisions drift velocity From Newton’s law: Ohms Law: conductivity

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**ρ(Cu)= 1.6 micro Ohm·cm =2.7·10-14 s (correct)**

But: The electron mean free path in copper is ≈ 40nm.

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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. The density of states between k and k+dk in a volume V is:

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**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 For lithium (a, lattice constant=3.49Å): n kF vF EF 4.6·1022cm-3 1.12Å-1 1.23·108cm/s 4.74eV

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**Transport in free electron metals**

At low applied electric fields, main source of excitation is thermal. Transport involves only a fraction of carriers: At 300K in Li: Copyright Stuart Lindsay 2008

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**Copyright Stuart Lindsay 2008**

Modified Drude Theory From Debye theory: Copyright Stuart Lindsay 2008

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**Electrons in crystals: Bloch’s theorem**

R: lattice translation vector 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. 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. 11

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**Single electrons in isolated atoms**

First Brillouin zone -k and k directions are equivalent (reduced zone scheme) Trial Bloch states for an 1D lattice: Interaction hamiltonian From I order Perturbation Theory: Electrons in the lattice Single electrons in isolated atoms 12

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**Copyright Stuart Lindsay 2008**

Band structure Nearest neighbors approximation (n=±1): On-site energy Hopping matrix element Copyright Stuart Lindsay 2008 13

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**Copyright Stuart Lindsay 2008**

Free electrons Electrons in a periodic potential Negative effective mass! Copyright Stuart Lindsay 2008

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group velocity At ka =±π the group velocity falls to zero: E=ε0. 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.

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**Extended and reduced Brillouin zones**

Extended Zone Reduced Zone Copyright Stuart Lindsay 2008

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**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. Semiconductor: 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) (Dielectric breakdown for large electric fields)

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**Why elements with 2 electrons/unit cell are most metals ?**

Partially filled Density of states (free electron model) A Fermi sphere of radius kF Cubic lattice in the reciprocal space A cube of side 2π/a 1° BZ completely filled, 2° BZ partially filled.

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**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.

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**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)

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**Probability of transition from m to k**

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: Probability of transition from m to k Density of states close to Ek

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**The Landauer Resistance**

Intensity of current per unit area 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:

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**N = allowed quantum states in the channel**

The factor 2 accounts for the two allowed spin states. For N=1: Landauer resistance: 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.

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**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.

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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: The resistance of a tunnel junction of gap L is: Φ = V0-E = workfunction of the metal

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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.

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**An exact conductance can be calculated from the Landauer-Buttiker equation:**

Tij = matrix elements that connect electronic states i on one side of the junction to states j on the other side.

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**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

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**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).

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**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: 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).

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Hanna-Tinkham theory Electric circuit model of the two junctions Coulomb blockade experiment. Normalized distribution of charges on the central particle Thermally activated hopping rates between the particle and the left or right electrodes,

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**I-V curves from nanoscale double junctions experiments**

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.

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**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.

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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): Red areas: SET is on at Coulomb steps Gate bias for level at Vsd = 0

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**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 , 291

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**Resonant tunneling Tunneling rate to the right**

Tunneling rate from the left central particle diameter = 2R Electrons incident from the left face a barrier of height V0 containing a localized state at energy E0.

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**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).

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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=4R B) 1 state Dashed lines: electronic transmission expected through the gap if no molecule is present between the two electrodes.

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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. e-h: large barriers weak coupling a-d: small barriers strong coupling

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**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.

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**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

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