M1 nanosciences
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Niveau: Supérieur, Master, Bac+4
M1-nanosciences 1 Nanophysics CHAPITER I. Orders of magnitude in nanophysique CHAPITER II. Conductance of nanowires and circuits CHAPITER III. Nanoelectronics, transistors, mosfets, one-electron devices CHAPITER IV. Nanomagnetism an spintronics CHAPITER V. Quantum computing CHAPITER VI. Molecular motors and wratchets Outline
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M1-nanosciences 1 Nanophysics CHAPITER I. Orders of magnitude in nanophysique CHAPITER II. Conductance of nanowires and circuits CHAPITER III. Nanoelectronics, transistors, mosfets, one-electron devices CHAPITER IV. Nanomagnetism an spintronics CHAPITER V. Quantum computing CHAPITER VI. Molecular motors and wratchets Outline
- zyx
- box size
- physical radius minimize
- quantum computing
- atomic scale
- coulomb potential
- nanophysique chapiter
- kinetic coulomb
Nanophysics
Outline
CHAPITER I.Orders of magnitude in nanophysique CHAPITER II.Conductance of nanowires and circuits CHAPITER III. Nanoelectronics, transistors, mosfets, oneelectron devices CHAPITER IV.Nanomagnetism an spintronics CHAPITER V.Quantum computing CHAPITER VI.Molecular motors and wratchets
http://www.ujf-grenoble.fr/PHY/intra/Formations/M1/Physique/UEs/2-5fois3ECTS/PHY425i/Planducours/
M1-nanosciences
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Chapiter 1: Orders of magnitudes for nanoscale objects
a) Atomic scale (1Å) 1Å=0.1 nm
-9 1 nanometer=1 nm=10 m
Bohr radius of the hydrogen atom
+Ze
-e r
2 |Ψ(r)|
r
2 2 r rp Ze E=H(p,r)= − 2m4πεr 0
kinetic
Coulomb attraction
Quantum uncertainty
ΔpΔr~h
h p ~Δp= Δr
2 2 hZe E= − 2 2m(Δr)4πε0Δr M1-nanosciences
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Atomic Bohr radius, angular momentum
2 2 hZe E= − 2 ( )r 2mΔr4πε0Δ
Physical radius minimize E
2 2 ∂E2hZe =0= − + 3 2 ∂Δr2m(Δr)4πε(Δr) 0
kinetic 1 α 2 (Δr) a 0
1Coulomb α Δrpotential
2 |Ψ(r)|
2 4πεh 0 a Δr=a= =0.5A=0.05nm0 0 2 mZe r Angular momentum L Ze hElectron velocity L = L mva0pa0p≈ 2 av p hZe Z 0 = = = = c mc mca4πεhc137.37 ≈h 0 0 8 C=3 10 m/sec M1-nanosciences 3
Energy-length
ψ
L
x
relation: quantum confinement
(x,y,z) k,k,k x y z
Asink xsinksink z x y z
2 m n ph2 2 2 k=,k=,k=E=(k+k+k) x y z x y z Energy L L L 2m 2 2 h3π Ground state energy (m=n=p=1) E= 0 2 2m L -19 E =1.8 V L=1nm0e10 J~1.13
L=100nm E =0.113 meV~1K 0 3 4⎛k L⎞13 3 F Metal box (N electrons) Fermi energy E↔N=nmaxN=π=⎜ ⎟ 2kFL F π π 3⎝2⎠6 2 h 1/ 3 2 p n⎛N⎞ 1/ 3 E=k=2e F10max 2 πF FLevel spacing: 1D =kF= =(6)≈⎜ ⎟ 1.610 2m hL⎝V⎠ 2 2 πh Δ =E(n)−E(n−1)=2n In a 1nm box, there is only 4 electrons (spin)2 max 2mL 2 πhv F = M1-nanosciencesL4
Coulomb interaction in a nano-box
2 electrons in a nanobox
2 e E= =1.4eV c 4πεL 0
Big box: many electrons⇒screening
2 ⎛ ⎞ e L E=exp⎜−⎟ c 4πεLλ 0⎝s⎠
Decreases exponentially with box size
M1-nanosciences
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Example: spectroscopy of a quantum box
eV/2
E+n) i2g
E n) i+1g
E n) i g
eU
eV/
A gate voltage allows to shift energy levels uniformly
Color map of tunnelling Current trough the box
Mendeleev table of quantum box as artificial atom
Hund rule holds for large boxes
M1-nanosciences
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E corr
M1-nanosciences
)E0) i
E i
Energy-length
relation:spectral rigidity
v F
impurities
Energy-time uncertainty E h
L
(x=0)
(L)
v F
L τ= v F
(0) expi)
2 h hvh F E== = 2π corr τL mL
7
Sensitivity to boundary condition Diffusive motion « Spectral rigidity » of the electron Random walk L h hD 2 Lv le E= = F τ=D=2 corr τL D3
Energy scales
Level spacing
2 2πh Δ = 2D mS 2 3 8πh Δ = 3D 3/ 2 (2m)Vε F
Level width
⎡ ⎤ hhD γ= ⎢,⎥ 2 τL ⎢ϕ γ⎥ ⎣ ⎦
LOW ENERGIES
Diffusive regime
E c
hv F l e
Correlation diffusive/ballistic (Thouless) (spectral rigidity) hD E=, c 2 L hv F E= c L HIGH ENERGIES
Universal regime (indep. of box shape, disorder, etc.)
M1-nanosciences
nonuniversal regime
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Elastic mean free path
l e
Diffusive regime
l e
Fermi wavelength
F
F
hv F l e
Lϕ
Lϕ
Box size
Lε
Energies/lengths scales
E c
HIGH ENERGY
LONG LENGTH
régime diffusif 2D ω≈Dk≈ 2 Lω hD9 ε=hω= 2 Lε
F
LOW ENERGY
Phase coherence Energy length relaxation length Ballistic regime 2 h 2 2 ε≈(k−k) k F 2m khv 2F F ε≈h(k−k)≈hvδk= k F F m L ε
M1-nanosciences
Nano-capacitance
d
L
2 εL 0 C= =εL 0 d
Lithography today
-20 1nm⇒C=10 F
2 e E= =2eV Coulomb 2C
L=10nm d=0.1nm
-17 C=10 F
2 e E= =2meV Coulomb 2C Large energies: major effect on nanoscale transistors
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Magnetic moments: Bohr magneton
Magnetic momentμ
a 0
v
Μ=current x surface
e e ea v ehe 2 2 0 μ= ⋅πa= ⋅πa= = =L 0 0 2πa τ02 2m2m v
Bohr magneton
−24−5 μ=9,2741⋅10J/Tesla=6⋅10eV/T=0.67K/T 0
1eV=11400K 4 1Tesla=10 Gauss
M1-nanosciences
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