siena, settembre 2005 (i) 1 theory of electron transport in semiconductor materials and structures...
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Siena, settembre 2005 (I) 1
Theory of electron transportin semiconductor materials and structures
Carlo Jacoboni
INFM-CNR National Research Center on Nano-Structures and Bio-Systems Laboratory S3
Università di Modena e Reggio Emilia, Italy
Tel. [email protected]
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CONTENTS1. Un po’ di storia2. Le origini della meccanica quantistica3. La fisica quantistica ed effetti quantistici4. Cristalli, stati di Bloch, bande di energia5. Dinamica pseudo-classica6. Metalli, isolanti e semiconduttori.7. Semiconduttori intrinseci ed estrinseci. Statistica degli elettroni
nei semiconduttori8. Modello semplice del trasporto in semiconduttori9. Funzione di distribuzione ed equazione di Boltzmann10. Scattering elettronici11. Strutture a semiconduttore – Giunzioni, diodi e mosfet12. Quantum wells, wires, dots e super-reticoli 13. Strutture mesoscopiche (Landauer)
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Some history Some history
1630 – Galileo, Dialogo sopra i due massimi sistemi del mondo1638 – Galileo, Discorsi e dimostrazioni matematiche intorno a due nuove scienze1687 – Newton, Principia1788 – Lagrange equations 1792 – Wedgwood: all bodies in the oven become incandescent at the same temp.1814 – Fraunhofer discovers the black lines in the solar spectrum1843 – Hamilton equations1855 – Maxwell equations1859 – Kirchhoff explains the black lines in the solar sp. and founds spectroscopy1879 – Stefan law: Etot=T4
1887 – Hertz confirms experimentally the propagation of the e.m. waves1887 – Michelson-Moreley experiment1896 – Mme Curie discovers radioactivity1897 – Thomson discovers the electron (cathodic rays)1900 – Reilaigh-Jeans law1900 – Planck law, photon quantization
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Siena, settembre 2005 (I) 4
Some Some history (2)history (2)1902 – Measurements of the photoelectric effect – Lénard
1905 – Golden year of Einstein1911 – Rutherford experimets1912 – X-ray diffraction from crystals (von Laue)1913 – Bohr atom1916 – Bohr-Sommerfeld quantization conditions1923 – de Broglie hypothesis1926 – Schroediger and Heisenberg equations1927 – Electron diffraction from crystals – Davisson e Germer (1925)1928 – Dirac equation
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Siena, settembre 2005 (I) 5
Luce: onde Luce: onde elettromagnetiche o elettromagnetiche o fotoni ?fotoni ?
Radiazione termica
Effetto fotoelettrico
Interferenza
Plack:
Einstein:
// hkcp
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Siena, settembre 2005 (I) 6
Interference
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Siena, settembre 2005 (I) 7
Luce: onde Luce: onde elettromagnetiche o elettromagnetiche o fotoni ?fotoni ?
Radiazione termica
Effetto fotoelettrico
Interferenza a singolo fotone
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Siena, settembre 2005 (I) 8
Single-photon Single-photon interferenceinterference
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Siena, settembre 2005 (I) 9
Atomo di Bohr: perché Atomo di Bohr: perché non collassa ?non collassa ?
nL
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Siena, settembre 2005 (I) 10
Electrons: Electrons: particles or particles or waves ?waves ?
rn 2
/hp
nL Bohr!!
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Siena, settembre 2005 (I) 11
Electron Electron interferenceinterference
Electrons are in superposition of states. The measurement …Theory predicts PROBABILITY
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Schroedinger equationSchroedinger equationE= ; p = k
(x,t) = A ei(kx-t)
For given energy and momentum: plane wave
Dx2(x) = -k2 (x)
Dt(x) = -i (x) m
k
m
p
22
222
Schroedinger equation
2
22
2 xmti
V
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Siena, settembre 2005 (I) 13
Schroedinger equation-Forma operatorialeSchroedinger equation-Forma operatoriale
Vxmt
i2
22
2
A f = g ; (AB) f = A (B f) ; (A+B) f = Af+ (B f)
Vm
pH
mm
p
pip
222
2222
222
Ht
i
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Siena, settembre 2005 (I) 14
Time-independent Schroedinger equationTime-independent Schroedinger equation
H
Hii
er
Ht
i
ti
)(
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Siena, settembre 2005 (I) 15
Wave Wave packetspackets
(x,t) = A(k) ei(kx-t) dk
v
x p ½
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Siena, settembre 2005 (I) 16
Energy eigenstates Energy eigenstates States with well defined energyStates with well defined energy
Ground state
First excited state
Second excited state
Infinitepotentialwell
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Siena, settembre 2005 (I) 17
Energy eigenstates Energy eigenstates States with well defined energyStates with well defined energy
Localized state
Extended state
Finitepotentialwell
Resonant state
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Siena, settembre 2005 (I) 18
Quantum effects: Total reflectionQuantum effects: Total reflection
V(x)
v
v
Vo
< Vo
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Siena, settembre 2005 (I) 19
Quantum effects: Quantum effects: Partial reflectionPartial reflection
V(x)
v
v
> Vo
v
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Siena, settembre 2005 (I) 20
Quantum effects: Quantum effects: TunnelTunnel
V(x)
v
v
<Vo
v
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Siena, settembre 2005 (I) 21
Pauli exclusion Pauli exclusion principleprinciple
Coulomb potential well Periodic table
Identical particles
Electron spin 1/2
Atoms