siena, settembre 2005 (i) 1 theory of electron transport in semiconductor materials and structures...

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]

I


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)


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


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


Luce: onde Luce: onde elettromagnetiche o elettromagnetiche o fotoni ?fotoni ?

Radiazione termica

Effetto fotoelettrico

Interferenza

Plack:

Einstein:

// hkcp


Interference


Luce: onde Luce: onde elettromagnetiche o elettromagnetiche o fotoni ?fotoni ?

Radiazione termica

Effetto fotoelettrico

Interferenza a singolo fotone


Single-photon Single-photon interferenceinterference


Atomo di Bohr: perché Atomo di Bohr: perché non collassa ?non collassa ?

nL


Electrons: Electrons: particles or particles or waves ?waves ?

rn 2

/hp

nL Bohr!!


Electron Electron interferenceinterference

Electrons are in superposition of states. The measurement …Theory predicts PROBABILITY

I


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


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


Time-independent Schroedinger equationTime-independent Schroedinger equation

H

Hii

er

Ht

i

ti

)(


Wave Wave packetspackets

(x,t) = A(k) ei(kx-t) dk

v

x p ½


Energy eigenstates Energy eigenstates States with well defined energyStates with well defined energy

Ground state

First excited state

Second excited state

Infinitepotentialwell


Energy eigenstates Energy eigenstates States with well defined energyStates with well defined energy

Localized state

Extended state

Finitepotentialwell

Resonant state


Quantum effects: Total reflectionQuantum effects: Total reflection

V(x)

v

v

Vo

< Vo


Quantum effects: Quantum effects: Partial reflectionPartial reflection

V(x)

v

v

> Vo

v


Quantum effects: Quantum effects: TunnelTunnel

V(x)

v

v

<Vo

v


Pauli exclusion Pauli exclusion principleprinciple

Coulomb potential well Periodic table

Identical particles

Electron spin 1/2

Atoms

siena, settembre 2005 (i) 1 theory of electron transport in semiconductor materials and structures...

Documents