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Nanophysics
Outline
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
http://www.ujf-grenoble.fr/PHY/intra/Formations/M1/Physique/UEs/2-5fois3ECTS/PHY425i/Planducours/
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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 mva0pa0p2 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 4k L13 3 F Metal box (N electrons) Fermi energy EN=nmaxN=π=⎜ ⎟ 2kFL F π π 326 2 h 1/ 3 2 p nN1/ 3 E=k=2e F10max 2 πF FLevel spacing: 1D =kF= =(6)⎜ ⎟ 1.610 2m hLV2 2 πh Δ =E(n)E(n1)=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 electronsscreening
2 ⎛ ⎞ e L E=expc 4πεLλ 0s
Decreases exponentially with box size
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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
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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
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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.)
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nonuniversal 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 ωDk2 Lω hD9 ε=hω= 2 Lε
F
LOW ENERGY
Phase coherence Energy length relaxation length Ballistic regime 2 h 2 2 ε(kk) k F 2m khv 2F F εh(kk)hvδk= k F F m L ε
M1-nanosciences
Nano-capacitance
d
L
2 εL 0 C= =εL 0 d
Lithography today
-20 1nmC=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
245 μ=9,274110J/Tesla=610eV/T=0.67K/T 0
1eV=11400K 4 1Tesla=10 Gauss
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