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Eq Schrodinger e Simmetria

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Eq Schrodinger e Simmetria

  1. 1. Simmetria traslazionale Elettroni non-interagenti in un potenziale periodico tale che L’Hamiltoniana per un elettrone singolo in tale potenziale: La soluzione è difficile perchè U(r) è complicato, tuttavia la simmetria ci viene in aiuto.
  2. 2. Simmetria traslazionale
  3. 3. Simmetria traslazionale Gli elettroni si comportano come onde piane di ampiezza modulata dal reticolo
  4. 4. Hamiltoniana efficace
  5. 5. Conteggio dei
  6. 6. Densità degli stati
  7. 7. Singolarità di Van Hove Presenza di divergenze nella densità degli stati d  /dk=0 singolarità in D
  8. 8. Singolarità di Van Hove Nei casi 2-D e 3-D
  9. 9. Singolarità di Van Hove
  10. 10. Singolarità di Van Hove 2-D 3-D Le singolarità di Van Hove si manifestano nelle densità di stati elettronici e vibrazionali e possono essere messe parzialmente in evidenza dagli esperimenti.
  11. 11. Teorema di Bloch: analisi di Fourier Riscrivendo l’eq. di Schroedinger nello spazio di Fourier è possibile semplificarne la complessità e trovarne semplici soluzioni numeriche. Ipotesi: qualunque funzione periodica sul reticolo cristallino R presenta componenti di Fourier non nulle solo per vettori del reticolo reciproco K
  12. 12. Teorema di Bloch: analisi di Fourier
  13. 13. Teorema di Bloch: analisi di Fourier Una autofunzione  dell’Hamiltoniana non è necessariamente periodica nel reticolo, però può sempre essere scritta come sovrapposizione di onde piane che lo sono:
  14. 14. Teorema di Bloch: analisi di Fourier Date le nuove forme trovate per U(r) e  (r) riscriviamo l’eq. di Schroedinger:
  15. 15. Teorema di Bloch: analisi di Fourier
  16. 16. Modello di Kronig-Penney Nel 1931 Kronig e Penney trovarono una soluzione non approssimata, evidenziando l’esistenza di bande di energia.
  17. 17. Esercitazione: sol. numerica modello K-G
  18. 18. Risultati: sol. numerica modello K-G
  19. 19. Simmetria rotazionale La teoria della rappresentazione dei gruppi descrive le conseguenze di queste simmetrie sulle soluzioni dell’eq. di Schroedinger. Benchè utile la riduzione dello “sforzo computazionale” può variare tra un fattore 3 e 24, cioè nettamente inferiore alla semplificazione introdotta dal teorema di Bloch ( ~ 10 23 ). La vera importanza risiede nella possibilità di determinare delle regole di selezione che in funzione della simmetria degli stati elettronici predicono la probabilità di transizione tra gli stati, determinata dall’interazione con agenti esterni (p. es. radiazione). Più specificatamente distinguiamo transizioni permesse e proibite .
  20. 20. Simmetria rotazionale Prima zona di Brillouin per un reticolo diretto fcc . Il reticolo reciproco è un bcc con spaziatura 4  /a.
  21. 21. Simmetria rotazionale Prima zona di Brillouin per un reticolo diretto bcc . Il reticolo reciproco è un fcc con spaziatura 4  /a.
  22. 22. Simmetria rotazionale Supponiamo che per un certo cristallo esista il gruppo di simmetria G. Le due funzioni possono essere rese ortogonali ed avere il medesimo autovalore dell’energia: Si divide la BZ in zone irriducibili e si risolve l’eq. di Schroedinger solo in queste. Questa conclusione però si basa sull’ipotesi che i due vettori d’onda k e Gk siano “distinguibili”. Ipotesi talvolta errata se differiscono per un vettore di reticolo reciproco K: se la “nuova” funzione d’onda è un multiplo dell’originale nessun problema, ma se è diversa (linearmente indipendente) allora siamo in presenza di una degenerazione degli stati (stati diversi, stessa energia).

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