Guessi, Luiz Henrique Bugatti
Efeitos das correlações inter-átomos adsorvidos na densidade de estados do grafeno / Luiz Henrique Bugatti Guessi. - Rio Claro, 2016
83 f. : il., figs., gráfs.
Dissertação (mestrado) - Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas
Orientador: Antonio Carlos Ferreira Seridonio
1. Física. 2. Transporte quântico. 3. Grafeno. 4. Estrutura multiníveis. 5. Batimentos quânticos. 6. Scanning Tunneling Microscope. I. Título.
530 G936e
❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡✱ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s ❡ ❛ ◆♦ss❛ ❙❡♥❤♦r❛✱ ♣♦✐s s❡♠ ❡❧❡s ♥❡♥❤✉♠❛ ❞❛s ♠✐♥❤❛s ❝♦♥q✉✐st❛s s❡r✐❛♠ ♣♦ssí✈❡✐s❀
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❆♥t♦♥✐♦ ❈❛r❧♦s ❋❡rr❡✐r❛ ❙❡r✐❞♦♥✐♦✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛✱ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s✱ ♣❡❧❛s ♦♣♦rt✉♥✐❞❛❞❡s ❡ ❛ t♦❞♦s ❛s ❞✐s❝✉ssõ❡s ❢ís✐❝❛s q✉❡ t✐✈❡♠♦s ❞❡s❞❡ ❛ ✐♥✐❝✐❛çã♦ ❝✐❡♥tí✜❝❛✳ Pr✐♥❝✐♣❛❧♠❡♥t❡✱ ♣♦r ♥✉♥❝❛ t❡r ♠❡❞✐❞♦ ❡s❢♦rç♦s ♣❛r❛ ❝♦♠♣❧❡♠❡♥t❛r ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳
❆♦s ♠❡✉s ♣❛✐s✱ ▲✉✐③ ❈❛r❧♦s ●✉❡ss✐ ❡ ▼❛r✐❛ ❞❡ ▲♦✉r❞❡s ❇✉❣❛tt✐ ●✉❡ss✐✱ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦ q✉❡ ♠❡ ❞❡r❛♠ ❞✉r❛♥t❡ t♦❞♦s ❡ss❡s ❛♥♦s ❡ ♣❡❧❛ ❝♦♥✜❛♥ç❛ q✉❡ s❡♠♣r❡ ❞❡♣♦s✐t❛r❛♠ ❡♠ ♠✐♠❀
❆ ♠✐♥❤❛ ♥❛♠♦r❛❞❛ ❙✐❜❡❧✐ ❆♣✳ ❞❡ ❙♦✉③❛ ❈r✉③✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❝♦♥✜❛♥ç❛ ❡ ♣♦r s❡♠♣r❡ ❡st❛r ♣r❡s❡♥t❡ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦s ♠❛✐s ❞✐❢í❝❡✐s❀
❊ ♣♦r ✜♠✱ ❛♦s ❝♦❧❛❜♦r❛❞♦r❡s ❑✳ ❑r✐st✐♥ss♦♥✱ ▼✳ ❞❡ ❙♦✉③❛✱ ❋✳ ▼✳ ❙♦✉③❛✱ ❘✳ ❙✳ ▼❛❝❤❛❞♦ ❡ ■✳ ❆✳ ❙❤❡❧②❦❤ q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❞✐r❡t❛♠❡♥t❡ ♣❛r❛ ❛ ❡❧❛❜♦r❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳
❖❜r✐❣❛❞♦ ❛ t♦❞♦s✦ ▲✉✐③ ❍❡♥r✐q✉❡ ❇✉❣❛tt✐ ●✉❡ss✐
❘❡s✉♠♦
❋♦✐ ❞✐s❝✉t✐❞♦ t❡♦r✐❝❛♠❡♥t❡ ❛ ❉❡♥s✐❞❛❞❡ ▲♦❝❛❧ ❞❡ ❊st❛❞♦s ✭▲❉❖❙✮ ❞❡ ✉♠❛ ❢♦❧❤❛ ❞❡ ❣r❛❢❡♥♦ ❤♦s♣❡❞❛♥❞♦ ❞✉❛s ✐♠♣✉r❡③❛s ❞✐st❛♥t❡s ❧♦❝❛❧✐③❛❞❛s ♥♦ ❝❡♥tr♦ ❞❛ ❝é❧✉❧❛ ❤❡①❛❣♦♥❛❧✳ ❆♦ ❛❝♦♣❧❛r ❧❛t❡r❛❧♠❡♥t❡ ❛ ♣♦♥t❛ ❞♦ ▼✐❝r♦❝ó♣✐♦ ❞❡ ❱❛rr❡❞✉r❛ ♣♦r ❚✉✲ ♥❡❧❛♠❡♥t♦ ✭❙❚▼✮ s♦❜r❡ ♦ át♦♠♦ ❞❡ ❝❛r❜♦♥♦✱ ❞♦✐s ♥♦✈♦s ♥♦tá✈❡✐s ❡❢❡✐t♦s ❢♦r❛♠ ❞❡t❡❝t❛❞♦s✿ ✐✮ ✉♠❛ ❡str✉t✉r❛ ❞❡ ♠✉❧t✐♥í✈❡✐s ♥❛ ▲❉❖❙ ❡ ✐✐✮ ♣❛❞rõ❡s ❞❡ ❜❛t✐♠❡♥t♦s ♥❛ ▲❉❖❙ ✐♥❞✉③✐❞❛✳ ❚❛♠❜é♠ ❢♦r❛♠ ♠♦str❛❞♦s q✉❡ ❛♠❜♦s ♦s ❢❡♥ô♠❡♥♦s ♦❝♦rr❡♠ ♣ró①✐♠♦s ❛♦s ♣♦♥t♦s ❞❡ ❉✐r❛❝ ❡ sã♦ ❛❧t❛♠❡♥t❡ ❛♥✐s♦tró♣✐❝♦s✳ ❆❧é♠ ❞✐ss♦✱ ❢♦r❛♠ ♣r♦♣♦st♦s ❡①♣❡r✐♠❡♥t♦s ❞❡ ❝♦♥❞✉tâ♥❝✐❛ ❡♠♣r❡❣❛♥❞♦ ♦ ❙❚▼ ❝♦♠♦ ✉♠❛ s♦♥❞❛ ♣❛r❛ ❛ ♦❜s❡r✈❛çã♦ ❞❡ t❛✐s ♠❛♥✐❢❡st❛çõ❡s ❡①ót✐❝❛s ♥❛ ▲❉❖❙ ❞♦ ❣r❛❢❡♥♦ ✐♥❞✉③✐❞❛ ♣❡❧❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❛s ✐♠♣✉r❡③❛s✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ●r❛❢❡♥♦✱ ❡str✉t✉r❛ ♠✉❧t✐♥í✈❡✐s✱ ♣❛❞rõ❡s ❞❡ ❜❛t✐♠❡♥t♦s✱ ❙❚▼ ✲ ❙❝❛♥♥✐♥❣ ❚✉♥♥❡❧✐♥❣ ▼✐❝r♦s❝♦♣❡
❆❜str❛❝t
❲❡ ❞✐s❝✉ss t❤❡♦r❡t✐❝❛❧❧② t❤❡ ▲♦❝❛❧ ❉❡♥s✐t② ♦❢ ❙t❛t❡s ✭▲❉❖❙✮ ♦❢ ❛ ❣r❛♣❤❡♥❡ s❤❡❡t ❤♦st✐♥❣ t✇♦ ❞✐st❛♥t ❛❞❛t♦♠s ❧♦❝❛t❡❞ ❛t t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❤❡①❛❣♦♥❛❧ ❝❡❧❧s✳ ❇② ♣✉tt✐♥❣ ❧❛t❡r❛❧❧② ❛ ❙❝❛♥♥✐♥❣ ❚✉♥♥❡❧✐♥❣ ▼✐❝r♦s❝♦♣❡ ✭❙❚▼✮ t✐♣ ♦✈❡r ❛ ❝❛r❜♦♥ ❛t♦♠✱ t✇♦ r❡♠❛r❦❛❜❧❡ ♥♦✈❡❧ ❡✛❡❝ts ❝❛♥ ❜❡ ❞❡t❡❝t❡❞✿ ✐✮ ❛ ♠✉❧t✐❧❡✈❡❧ str✉❝t✉r❡ ✐♥ t❤❡ ▲❉❖❙ ❛♥❞ ✐✐✮ ❜❡❛t✐♥❣ ♣❛tt❡r♥s ✐♥ t❤❡ ✐♥❞✉❝❡❞ ▲❉❖❙✳ ❲❡ s❤♦✇ t❤❛t ❜♦t❤ ♣❤❡♥♦✲ ♠❡♥❛ ♦❝❝✉r ♥❡❛r❜② t❤❡ ❉✐r❛❝ ♣♦✐♥ts ❛♥❞ ❛r❡ ❤✐❣❤❧② ❛♥✐s♦tr♦♣✐❝✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ♣r♦♣♦s❡ ❝♦♥❞✉❝t❛♥❝❡ ❡①♣❡r✐♠❡♥ts ❡♠♣❧♦②✐♥❣ ❙❚▼ ❛s ❛ ♣r♦❜❡ ❢♦r t❤❡ ♦❜s❡r✈❛t✐♦♥ ♦❢ s✉❝❤ ❡①♦t✐❝ ♠❛♥✐❢❡st❛t✐♦♥s ✐♥ t❤❡ ▲❉❖❙ ♦❢ ❣r❛♣❤❡♥❡ ✐♥❞✉❝❡❞ ❜② ✐♥t❡r✲❛❞❛t♦♠s ❝♦rr❡❧❛t✐♦♥s✳
❑❡②✇♦r❞s✿ ●r❛♣❤❡♥❡✱ ♠✉❧t✐❧❡✈❡❧ str✉❝t✉r❡✱ ❜❡❛t✐♥❣ ♣❛tt❡r♥s✱ ❙❚▼ ✲ ❙❝❛♥♥✐♥❣ ❚✉♥♥❡❧✐♥❣ ▼✐❝r♦s❝♦♣❡✳
❙✉♠ár✐♦
▲✐st❛ ❞❡ ❋✐❣✉r❛s ✈✐
✶ ■♥tr♦❞✉çã♦ ✶
✷ Pr♦♣r✐❡❞❛❞❡s ❡❧❡trô♥✐❝❛s ❡ ❡str✉t✉r❛✐s ❞♦ ●r❛❢❡♥♦ ✹
✷✳✶ ❈á❧❝✉❧♦ t✐❣❤t✲❜✐♥❞✐♥❣ ❞♦ ●r❛❢❡♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✷ ❉✐s♣❡rsã♦ ❧✐♥❡❛r ❞♦ ●r❛❢❡♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✸ ❖ ♠♦❞❡❧♦ ❞❡ ❆♥❞❡rs♦♥ ✶✹
✸✳✶ ❖ ❍❛♠✐❧t♦♥✐❛♥♦ ❞❡ ❆♥❞❡rs♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✷ ❖ ❡❢❡✐t♦ ❑♦♥❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✸ ❖ ❡❢❡✐t♦ ❋❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✹ ▼♦❞❡❧♦ t❡ór✐❝♦ ✷✸
✹✳✶ ❖ ❍❛♠✐❧t♦♥✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✷ ❖ ❝á❧❝✉❧♦ ❞❛s ❋✉♥çõ❡s ❞❡ ●r❡❡♥ ❡ ❞❛ ▲❉❖❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✺ ❆✉t♦✲❡♥❡r❣✐❛s✱ ♣❛râ♠❡tr♦ ❞❡ ❋❛♥♦ ❡ ♦s❝✐❧❛çõ❡s ❞❡ ❋r✐❡❞❡❧ ✸✵
✺✳✶ ❆✉t♦✲❡♥❡r❣✐❛s ❞✐r❡t❛s ❡ ❝r✉③❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✺✳✶✳✶ ❈á❧❝✉❧♦ ❞❛s ❛✉t♦✲❡♥❡r❣✐❛s ❞✐r❡t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✺✳✶✳✷ ❆s ❛✉t♦✲❡♥❡r❣✐❛s ❝r✉③❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✺✳✶✳✸ P❛râ♠❡tr♦ ❞❡ ❋❛♥♦ ❡ ♦s❝✐❧❛çõ❡s ❞❡ ❋r✐❡❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✻ ❘❡s✉❧t❛❞♦s ❡ ❞✐s❝✉ssõ❡s ✹✵
✼ ❈♦♥❝❧✉sõ❡s ✹✼
❘❡❢❡rê♥❝✐❛s ✹✾
❙❯▼➪❘■❖
❆ ❖ ♠ét♦❞♦ ❞❛ ❊q✉❛çã♦ ❞❡ ▼♦✈✐♠❡♥t♦ ✭❊❖▼✮ ✺✷
❇ P✉❜❧✐❝❛çõ❡s ✺✹
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ✭❛✮ ❉✉❛s ✐♠♣✉r❡③❛s r♦t✉❧❛❞❛s ♣♦r 1 ❡ 2 ❛❞s♦r✈✐❞❛s ♥♦ ❝❡♥tr♦ ❞❛ ❝é❧✉❧❛ ❤❡①❛❣♦♥❛❧ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ❧❛t❡r❛❧ d ❛♦ ❧♦♥❣♦ ❞❛s ❞✐r❡çõ❡s
③✐❣③❛❣ ❡ ❛r♠❝❤❛✐r✳ ❆s ✐♠♣✉r❡③❛s s♦♠❜r❡❛❞❛s r❡♣r❡s❡♥t❛♠ ❛ s❡♣❛✲ r❛çã♦ ❡♥tr❡ ❛s ✐♠♣✉r❡③❛s 1 ❡ 2✳ ◆♦s ♣❛✐♥é✐s ✭❜✮ ❡ ✭❝✮✱ ❛ ▲❉❖❙ ❞♦ ❣r❛❢❡♥♦ ♥♦ sít✐♦ rs ✭s = A, B✮ é s♦♥❞❛❞❛ ♣❡❧❛ ♣♦♥t❛ ❞♦ ❙❚▼ ♥❛s
❞✐r❡çõ❡s ③✐❣③❛❣ ❡ ❛r♠❝❤❛✐r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✷✳✶ ❆ r❡❞❡ ❤❡①❛❣♦♥❛❧ ❞♦ ❣r❛❢❡♥♦✳ ❆s ❡s❢❡r❛s ❡♠ ❛③✉❧ ❡ ✈❡r♠❡❧❤♦ ❧♦❝❛❧✐✲
③❛❞❛s ♥♦s ✈ért✐❝❡s ❞♦s ❤❡①á❣♦♥♦s r❡♣r❡s❡♥t❛♠ ♦s át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ❞❛s s✉❜✲r❡❞❡s A ❡ B ❡ ❛s ❧✐♥❤❛s ❡♠ ♣r❡t♦ ✐♥❞✐❝❛♠ ❛s ❧✐❣❛çõ❡s q✉í✲
♠✐❝❛s σ ❞♦s ♦r❜✐t❛✐s sp2✳ a
1 ❡ a2 ❞❡✜♥❡♠ ♦s ✈❡t♦r❡s ♣r✐♠ár✐♦s ❡ ♦ ❧♦s❛♥❣♦ ❡♠ ❝✐♥③❛ ❝❧❛r♦ ❞❡♠❛r❝❛ ❛ ❝é❧✉❧❛ ✉♥✐tár✐❛ ❞❛ r❡❞❡✳ ❖s ✈❡t♦r❡s
x1 =a0( √
3
2 ,0)❡ x2 =a0( 2+√3
2 ,0)✐♥❞✐❝❛♠ ❛ ♣♦s✐çã♦ ❞❡ ❝❛❞❛ át♦♠♦ ❞❡ ❝❛r❜♦♥♦ ❞❡♥tr♦ ❞❛ ❝é❧✉❧❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s t✐❣❤t✲❜✐♥❞✐♥❣ ❞♦ ❣r❛❢❡♥♦✳ ✭❛✮ ❘❡❧❛çã♦ ❞❡ ❞✐s✲
♣❡rsã♦ ❞❛ ❡♥❡r❣✐❛ ❞❛❞❛ ♣❡❧❛ ❊q✳✭✷✳✶✺✮ ❡♠ ❢✉♥çã♦ ❞❡ kx ❡ ky✳ ✭❜✮
❘❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ♥♦ ❧✐♠✐t❡ ❞❡ ❜❛✐①❛s ❡♥❡r❣✐❛s ❡♠ t♦r♥♦ ❞♦s ♣♦♥✲ t♦s ❞❡ ❉✐r❛❝✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✸ ❖ ❤❡①á❣♦♥♦ ❡①t❡r♥♦ ✐❧✉str❛ ❛ r❡❞❡ r❡❝í♣r♦❝❛ ❞♦ ❣r❛❢❡♥♦✳ b1 ❡b2 sã♦
♦s ✈❡t♦r❡s ♣r✐♠ár✐♦s q✉❡ ❞❡❧✐♠✐t❛♠ ❛ ♣♦s✐çã♦ ❞♦s ♣♦♥t♦s ❞❛ r❡❞❡✳ ❖ ❤❡①á❣♦♥♦ s♦♠❜r❡❛❞♦ ✭❝✐♥③❛ ❝❧❛r♦✮ ❞❡s❝r❡✈❡ ❛ ♣r✐♠❡✐r❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥ ❡ ❡♠ ❝❛❞❛ ✈ért✐❝❡ ❞♦ ❤❡①á❣♦♥♦ ❡stã♦ ❧♦❝❛❧✐③❛❞♦s ♦s ♣♦♥t♦s ❞❡ ❉✐r❛❝ K❡ K✬✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✸✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ ✐♠♣✉r❡③❛ ❛❞s♦r✈✐❞❛ ♥❛ s✉♣❡r❢í❝✐❡ ♠❡tá❧✐❝❛✳ ❆s ❡s❢❡r❛s ❞♦✉r❛❞❛ ❡ ❝✐♥③❛s r❡♣r❡s❡♥t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ✐♠♣✉✲ r❡③❛ ❡ ❛ s✉♣❡r❢í❝✐❡ ♠❡tá❧✐❝❛✳ ❖ ♣❛râ♠❡tr♦ V ♠♦❞✉❧❛ ❛ ✐♥t❡♥s✐❞❛❞❡ ❞❡ ❤✐❜r✐❞✐③❛çã♦ ❡♥tr❡ ❛ ✐♠♣✉r❡③❛ ❡ ♦ ♠❡t❛❧ ❤♦s♣❡❞❡✐r♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙
✸✳✷ ❛✮ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❆♥❞❡rs♦♥ ❞❡ ✉♠❛ ✐♠♣✉r❡③❛ ♥♦ ❡s✲ ♣❛ç♦ ❞❛s ❡♥❡r❣✐❛s ❡♠ ✉♠❛ ❜❛♥❞❛ ❞❡ ❝♦♥❞✉çã♦ ❞❡ ❧❛r❣✉r❛2D s❡♠✐✲
♣r❡❡♥❝❤✐❞❛ ❝♦♠ s❡✉ ♥í✈❡❧ ❞❡ ❋❡r♠✐ ε❋✱ ❧♦❝❛❧✐③❛❞♦ ❡①❛t❛♠❡♥t❡ ♥♦
❝❡♥tr♦ ❞❛ ❜❛♥❞❛✳ ❖s ♥í✈❡✐s ❞❡ ❡♥❡r❣✐❛s ❞✐s❝r❡t♦s ❞❛ ✐♠♣✉r❡③❛εdσ ❡ εdσ+Usã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧❛s ❧✐♥❤❛s ❡♠ ♣r❡t♦✳ ❜✮ ❉❡♥s✐❞❛❞❡ ❞❡ ❡s✲ t❛❞♦s ❞❛ ✐♠♣✉r❡③❛ ❛❞s♦r✈✐❞❛ ♥♦ ♠❡t❛❧ ❤♦s♣❡❞❡✐r♦✳ ❆s r❡ss♦♥â♥❝✐❛s ❡♠ t♦r♥♦ ❞❡Edσ ❡ Edσ+U sã♦ ❝❤❛♠❛❞❛s ❞❡ ♣✐❝♦s ❞❡ ❍✉❜❜❛r❞✳ ✳ ✳ ✳ ✶✼ ✸✳✸ ✭❛✮ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞❡ ✉♠ ♠❡t❛❧ ❤♦s♣❡❞❡✐r♦ s❡♠✐ ♣r❡❡♥❝❤✐❞❛ ❞❡
❧❛r❣✉r❛2D✱ ♦♥❞❡ D é ❛ s❡♠✐✲❧❛r❣✉r❛ ❞❛ ❜❛♥❞❛✳ ❆s s❡t❛s ❡♠ ✈❡r❞❡
❛♣♦♥t❛♥❞♦ ♣❛r❛ ❝✐♠❛ ❡ ♣❛r❛ ❜❛✐①♦ r❡♣r❡s❡♥t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ♦r✐❡♥t❛çã♦ ❞♦s ❡❧étr♦♥s ❞❡s❡♠♣❛r❡❧❤❛❞♦s ♥♦ ♥í✈❡❧ εdσ ❛❝♦♣❧❛❞♦s
❞❡ ❢♦r♠❛ ❛♥t✐❢❡rr♦♠❛❣♥ét✐❝♦s ❝♦♠ ♦s ❡❧étr♦♥s ♣ró①✐♠♦s ❞❡ ε❋✳ ✭❜✮
❉❡♥s✐❞❛❞❡ ❞❡ ❡st❛❞♦s ❞❛ ✐♠♣✉r❡③❛ ❞❡♥tr♦ ❞♦ r❡❣✐♠❡ ❑♦♥❞♦✳ ❯♠❛ r❡ss♦♥â♥❝✐❛ ❡str❡✐t❛ ❡ ❛❣✉❞❛ ❡♠ t♦r♥♦ ❞♦ ♥í✈❡❧ ❞❡ ❋❡r♠✐ ❝♦♠ s❡♠✐✲ ❧❛r❣✉r❛ΓK ❡♠❡r❣❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✸✳✹ ❘❡♣r❡s❡♥t❛çã♦ ❡s♣❛❝✐❛❧ ❞♦ ♠♦❞❡❧♦ ❞❡ ❆♥❞❡rs♦♥ ❞❡ ✉♠❛ ✐♠♣✉r❡③❛ ♥❛ ♣r❡s❡♥ç❛ ❞♦ ❙❚▼ ✭❙❝❛♥♥✐♥❣ ❚✉♥♥❡❧✐♥❣ ▼✐❝r♦s❝♦♣❡✮✳ ❆s ❧✐♥❤❛s tr❛❝❡❥❛❞❛s ❞❡♠♦♥str❛♠ ♦s ♣♦ssí✈❡✐s ❝❛♠✐♥❤♦s ❞❡ t✉♥❡❧❛♠❡♥t♦ ❡♥tr❡ ❛ ♣♦♥t❛ ❡ ♦ ♠❡t❛❧ ❤♦s♣❡❞❡✐r♦✳ ❖s í♥❞✐❝❡s tc ❡ td ♠♦❞✉❧❛♠✱ r❡s♣❡❝✲
t✐✈❛♠❡♥t❡✱ ❛ ❛♠♣❧✐t✉❞❡ ❞❡ t✉♥❡❧❛♠❡♥t♦ ❞❛ ♣♦♥t❛ ♣❛r❛ ♦ ♠❡t❛❧ ❡ ❞❛ ♣♦♥t❛ ♣❛r❛ ❛ ✐♠♣✉r❡③❛✳ ❊ss❡ ❞♦✐s ♣♦ssí✈❡✐s ❝❛♠✐♥❤♦ ❞❡ t✉♥❡✲ ❧❛♠❡♥t♦ ♦r✐❣✐♥❛♠ ✉♠ ❡❢❡✐t♦ ❞❡ ✐♥t❡r❢❡rê♥❝✐❛ ❝❤❛♠❛❞♦ ✐♥t❡r❢❡rê♥❝✐❛ ❋❛♥♦ ❡①trí♥s❡❝♦✳ ❆s ❧✐♥❤❛s tr❛❝❡❥❛❞❛s ♥♦ ♠❡t❛❧ r❡♣r❡s❡♥t❛♠ ♦s ❡❧é✲ tr♦♥s ❧✐✈r❡s ✈✐❛❥❛♥❞♦ ♣♦r ❡❧❡✳ ❚♦❞❛✈✐❛✱ ❡ss❡s ♣♦❞❡♠ t✉♥❡❧❛r ❞♦ ♠❡t❛❧ ♣❛r❛ ❛ ✐♠♣✉r❡③❛ ❡ ❞❛ ✐♠♣✉r❡③❛ ♣❛r❛ ♦ ♠❡t❛❧✳ ❊ss❡ ♣r♦❝❡ss♦ ❞❡ t✉♥❡✲ ❧❛♠❡♥t♦ t❛♠❜é♠ ✐rá ❝❛r❛❝t❡r✐③❛r ✉♠ ❡❢❡✐t♦ ❞❡ ✐♥t❡r❢ê♥❝✐❛ ❝❤❛♠❛❞❛ ❞❡ ❋❛♥♦ ✐♥trí♥s❡❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✺✳✶ ❖s ♣❛✐♥é✐s ✭❛✮ ❡ ✭❜✮ r❡♣r❡s❡♥t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ❝♦♥t♦r♥♦s
♣❡r❝♦rr✐❞♦s ♣❡❧❛s ✐♥t❡❣r❛✐s ❞❛ ❊q✳ ✺✳✶✶ ❞❡♣❡♥❞❡♥t❡s ❞❛s ❢✉♥çõ❡s
H0(1) ❡ H0(2)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✻✳✶ ▲❉❖❙(rA) ❡♠ ❢✉♥çã♦ ❞❛ ❡♥❡r❣✐❛ ♣❛r❛ ❛ ❞✐r❡çã♦ ❛r♠❝❤❛✐r✱ ♥❛ s✉❜✲
r❡❞❡ A✳ ❆ r❡t❛ tr❛s❡❥❛❞❛ ❡♠ ✈❡r❞❡ ❡ ❛ ❝✉r✈❛ só❧✐❞❛ ✈❡r♠❡❧❤❛ r❡♣r❡✲
s❡♥t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡st❛❞♦s ❞♦ ❣r❛❢❡♥♦ ♣✉r♦ ❡ ♥❛ ♣r❡s❡♥ç❛ ❞❛s ✐♠♣✉r❡③❛s✳ ❆s ✐♠♣✉r❡③❛s ❛❞s♦r✈✐❞❛s ♥♦ ❣r❛❢❡♥♦ ♦r✐❣✐♥❛♠ ✉♠❛ ❡str✉t✉r❛ ♠✉❧t✐♥í✈❡✐s ❞❡s❝r✐t❛ ♣❡❧❛s ♦s❝✐❧❛çõ❡s ♣r❡✲ s❡♥t❡s ♥❛s ❡①tr❡♠✐❞❛❞❡s ❞❛s r❡t❛s✳ ❊♠ t♦r♥♦ ❞❡ E∼ −0.03D ✭♥ã♦
♠♦str❛❞❛✮ ❡E∼0.03D ✭✈❡❥❛ ♦ ✐♥s❡t✮✱ ♦❜s❡r✈❛✲s❡ ❞✉❛s r❡ss♦♥â♥❝✐❛s
❋❛♥♦ ❞❡✈✐❞♦ ❛ ♣r❡s❡♥ç❛ ❞❛s ✐♠♣✉r❡③❛s ❡ sã♦ ❝❤❛♠❛❞♦s ❞❡ ♣✐❝♦s ❞❡ ❍✉❜❜❛r❞✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙
✻✳✷ ▲❉❖❙(rA)❡♠ ❢✉♥çã♦ ❞❛ ❡♥❡r❣✐❛ ♣❛r❛ ❛ ❞✐r❡çã♦ ❛r♠❝❤❛✐r✳ ❆s ❝✉r✈❛s
❡♠ ✈❡r❞❡ ❡ ✈❡r♠❡❧❤♦ r❡♣r❡s❡♥t❛♠✱ ♥❛ s✉❛ ❞❡✈✐❞❛ ♦r❞❡r✱ ❛ ▲❉❖❙(rA)
❞♦ ❣r❛❢❡♥♦ ♣✉r♦ ❡ ❝♦♠ ✐♠♣✉r❡③❛s✳ ❆s ♦s❝✐❧❛çõ❡s ♣r❡s❡♥t❡s ♥❛ ❝✉r✈❛ ❡♠ ✈❡r♠❡❧❤♦ r❡♣r❡s❡♥t❛♠ ❛ ❡♠❡r❣ê♥❝✐❛ ❞❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ ♠✉❧t✐✲ ♥í✈❡✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✻✳✸ P❛❞rõ❡s ❞❡ ❜❛t✐♠❡♥t♦s ♥❛ ▲❉❖❙ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ♣♦s✐❝✐♦♥❛♠❡♥t♦
❞❛s ✐♠♣✉r❡③❛s ❧♦❝❛❧✐③❛❞❛s ♥❛ ❞✐r❡çã♦ ❛r♠❝❤❛✐r ♣❛r❛ ❛s s✉❜✲r❡❞❡s ❆ ✭♣❛✐♥❡❧ ✭❛✮✮ ❡ ❇ ✭♣❛✐♥❡❧ ✭❜✮✮✳ ❖s ♣✐❝♦s ❞❡ ❍✉❜❜❛r❞ sã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧❛s r❡ss♦♥â♥❝✐❛s ❋❛♥♦ ❧♦❝❛❧✐③❛❞♦s ❡♠ t♦r♥♦ ❞❡Edσ ❡Edσ+U✳ ✳ ✳ ✳ ✹✹ ✻✳✹ ✭❛✮ ∆▲❉❖❙ ❡♠ ❢✉♥çã♦ ❞❛ ❡♥❡r❣✐❛ ♣❛r❛ ❛s ✐♠♣✉r❡③❛s ❧♦❝❛❧✐③❛❞❛s ♥❛
s✉❜✲r❡❞❡ A ❞❛ ❣❡♦♠❡tr✐❛ ③✐❣③❛❣ ✭♣❛✐♥❡❧ ✭❛✮✮ ❡ B ✭♣❛✐♥❡❧ ✭❜✮✮✳ ◆♦t❡
q✉❡ ❡♠❜♦r❛ ❛ ❡str✉t✉r❛ ❞❡ ♠✉❧t✐♥í✈❡✐s é ♦❜s❡r✈❛❞❛ ❝❧❛r❛♠❡♥t❡✱ ♦s ❜❛t✐♠❡♥t♦s ❡stã♦ ❛✉s❡♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
Cap´ıtulo
1
■♥tr♦❞✉çã♦
❖ ❣r❛❢❡♥♦ é ✉♠ s✐st❡♠❛ ❜✐❞✐♠❡♥s✐♦♥❛❧ ✭✷❉✮ ❢♦r♠❛❞♦ ♣♦r át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ♦r❣❛✲ ♥✐③❛❞♦s ❡♠ ✉♠❛ ❡str✉t✉r❛ ❤❡①❛❣♦♥❛❧✱ s❡♠❡❧❤❛♥t❡ ❛ ✉♠ ❢❛✈♦ ❞❡ ♠❡❧ ✭✶✱ ✷✱ ✸✮✳ ❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ ♥♦tá✈❡❧ ❞❡ss❡ s✐st❡♠❛ é ❛ ❡①✐stê♥❝✐❛ ❞❡ ❝♦♥❡s ❞❡ ❉✐r❛❝ ♥❛ ♣r✐♠❡✐r❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥ ❡♠ s✉❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s✱ s✐♠✐❧❛r ❛♦s q✉❡ ❛♣❛r❡❝❡♠ ♥❛ ❞✐s♣❡r✲ sã♦ ❞❡ ❢ér♠✐♦♥s r❡❧❛t✐✈íst✐❝♦s s❡♠ ♠❛ss❛✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ s✐st❡♠❛s ❜❛s❡❛❞♦s ❡♠ ❣r❛❢❡♥♦ ♣r♦♠♦✈❡♠ ❝♦♥❞✐çõ❡s ❛❞❡q✉❛❞❛s ♣❛r❛ ❡♠✉❧❛r ❢❡♥ô♠❡♥♦s r❡❧❛t✐✈íst✐❝♦s ♥♦ ❞♦♠í♥✐♦ ❞❛ ❋ís✐❝❛ ❞❛ ▼❛tér✐❛ ❈♦♥❞❡♥s❛❞❛✳ ❈✉r✐♦s❛♠❡♥t❡✱ ♦ ❛♣❛r❡❝✐♠❡♥t♦ ❞❡ ❢ér♠✐♦♥s ❞❡ ❉✐r❛❝ q✉❛s❡✲r❡❧❛t✐✈íst✐❝♦s s❡♠ ♠❛ss❛ t❛♠❜é♠ ❢♦r❛♠ ♦❜s❡r✈❛❞♦s ❡♠ ❝♦♥❞✉t♦r❡s ♠♦❧❡❝✉❧❛r❡s ✭✹✮ ❡ ❡♠ ✐s♦❧❛♥t❡s t♦♣♦❧ó❣✐❝♦s ✭✺✮✳ ❊①♣❡r✐♠❡♥t♦s r❡❝❡♥t❡s ❡ tr❛❜❛❧❤♦s t❡ór✐❝♦s ❞❡♠♦♥str❛r❛♠ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❝♦♥tr♦❧❡ ❡❢❡t✐✈♦ ❞❡ ❛❞s♦rçã♦ ❞❡ ✐♠♣✉r❡③❛s ♠❛❣♥ét✐❝❛s ✐♥❞✐✈✐❞✉❛✐s✱ ♦s ❝❤❛♠❛❞♦s ❛❞❛t♦♠s✱ ❡♠ ✉♠❛ ú♥✐❝❛ ❢♦❧❤❛ ❞❡ ❣r❛❢❡♥♦ ✭✻✱ ✼✱ ✽✮✳ P❛r❛ ❡①♣❧♦r❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s ❞❡ss❛s ✐♠♣✉r❡③❛s✱ ❜❡♠ ❝♦♠♦ s❡✉s ❡❢❡✐t♦s s♦❜r❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❤♦s♣❡❞❡✐r♦✱ ♦ ▼✐❝r♦s❝ó♣✐♦ ❞❡ ❱❛rr❡✲ ❞✉r❛ ♣♦r ❚✉♥❡❧❛♠❡♥t♦ ❞❡ ❡❧étr♦♥s ✭❙❚▼✮ t❡♠ s✐❞♦ r❡❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❛ ❢❡rr❛♠❡♥t❛ ❡①♣❡r✐♠❡♥t❛❧ ♠❛✐s ❡✜❝✐❡♥t❡ ✭✾✮✳ ❖ ❙❚▼ ❝♦♥s✐st❡ ❡♠ ✉♠❛ ♣♦♥t❛ ♠❡tá❧✐❝❛ ❝❛♣❛③ ❞❡ ❞❡t❡❝t❛r ❛ ❞❡♥s✐❞❛❞❡ ❧♦❝❛❧ ❞❡ ❡st❛❞♦s ✭▲❉❖❙✮ ✈✐❛ ♠❡❞✐❞❛s ❞❡ ❝♦♥❞✉tâ♥❝✐❛ ❞✐✲ ❢❡r❡♥❝✐❛❧✳
◆♦t❛✈❡❧♠❡♥t❡✱ ❛ ♣♦♥t❛ ❞❡t❡❝t❛ ✉♠ ❢❛s❝✐♥❛♥t❡ ❢❡♥ô♠❡♥♦ ❡♥✈♦❧✈❡♥❞♦ ❡s♣❛❧❤❛✲ ♠❡♥t♦ ❡❧❡trô♥✐❝♦ ♣❡❧❛s ✐♠♣✉r❡③❛s✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦s❝✐❧❛çõ❡s ❞❡ ❋r✐❡❞❡❧✱ q✉❡ ❛♣❛✲ r❡❝❡♠ ♥♦ s✐♥❛❧ ❞❛ ❝♦♥❞✉tâ♥❝✐❛ ❝♦♠♦ ♣❛❞rõ❡s ❞❡ ♦s❝✐❧❛çõ❡s ❛♠♦rt❡❝✐❞❛s q✉❛♥❞♦ ❛ ♣♦s✐çã♦ ❞❛ ♣♦♥t❛ é ✈❛r✐❛❞❛ ✭✶✵✱ ✶✶✮✳ ❆s ♣r♦♣r✐❡❞❛❞❡s ♠❛❣♥ét✐❝❛s ❞❛s ✐♠♣✉r❡③❛s
(b) Lateral view: Zigzag (a) Top view:
(c) Front view: Armchair
B
adatom 1
STM tip
1 2
d
m=1
A
r
adatom 2
B A
B
r
1 2
m=1
B
m=1 m=2
m
=
1
m
=
2
Z
ig
z
a
g
Armchair
x e y e
1 2
2
s=A s=B
p=1
p=0 p=0
p=1
d
A
s
=
A
s
=
B
❋✐❣✉r❛ ✶✳✶✿ ✭❛✮ ❉✉❛s ✐♠♣✉r❡③❛s r♦t✉❧❛❞❛s ♣♦r 1 ❡ 2 ❛❞s♦r✈✐❞❛s ♥♦ ❝❡♥tr♦ ❞❛
❝é❧✉❧❛ ❤❡①❛❣♦♥❛❧ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ❧❛t❡r❛❧ d ❛♦ ❧♦♥❣♦ ❞❛s ❞✐r❡çõ❡s ③✐❣③❛❣ ❡ ❛r♠❝❤❛✐r✳ ❆s ✐♠♣✉r❡③❛s s♦♠❜r❡❛❞❛s r❡♣r❡s❡♥t❛♠ ❛ s❡♣❛r❛çã♦ ❡♥tr❡ ❛s ✐♠♣✉r❡③❛s 1 ❡ 2✳ ◆♦s
♣❛✐♥é✐s ✭❜✮ ❡ ✭❝✮✱ ❛ ▲❉❖❙ ❞♦ ❣r❛❢❡♥♦ ♥♦ sít✐♦rs ✭s=A, B✮ é s♦♥❞❛❞❛ ♣❡❧❛ ♣♦♥t❛ ❞♦ ❙❚▼ ♥❛s ❞✐r❡çõ❡s ③✐❣③❛❣ ❡ ❛r♠❝❤❛✐r✳
❡♠ ❣r❛❢❡♥♦ ❢♦r❛♠ ❛❜♦r❞❛❞❛s t❡♦r✐❝❛♠❡♥t❡ ❞❡♥tr♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ✉♠❛ ú♥✐❝❛ ✐♠♣✉✲ r❡③❛✱ ❞❡s❝r✐t❛ ♣❡❧♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❞❡ ❆♥❞❡rs♦♥ ✭✶✷✮ ♣❛r❛ ❞♦✐s ❝♦♥tr❛st❛♥t❡s ❧✐♠✐t❡s tér♠✐❝♦s ✭TK r❡❢❡r❡✲s❡ ❛ t❡♠♣❡r❛t✉r❛ ❞❡ ❑♦♥❞♦✮✿ ✐✮ T ≫TK✱ ♦♥❞❡ ❛ ❛♣r♦①✐♠❛çã♦
❞❡ ❝❛♠♣♦ ♠é❞✐♦ ❞❡ ❍❛rtr❡❡✲❋♦❝❦ é ❛♣❧✐❝á✈❡❧ ✭✶✸✱ ✶✹✮✱ ❡ ✐✐✮ T ≪ TK✱ ♦ r❡❣✐♠❡
❣♦✈❡r♥❛❞♦ ♣❡❧❛ ❢♦r♠❛çã♦ ❞❛ ♥✉✈❡♠ ❑♦♥❞♦✱ ♥♦ q✉❛❧ ♦s ❡❢❡✐t♦s ❞❡ ❢♦rt❡ ❝♦rr❡❧❛çã♦ t♦r♥❛♠✲s❡ ❝r✉❝✐❛✐s ✭✶✺✱ ✶✻✱ ✶✼✮✳ ◆♦ ú❧t✐♠♦ r❡❣✐♠❡✱ ❛❞✐❝✐♦♥❛♥❞♦ ✉♠❛ s❡❣✉♥❞❛ ✐♠✲ ♣✉r❡③❛ ♥♦ ❤♦s♣❡❞❡✐r♦✱ ✉♠ ✐♥t❡r❡ss❛♥t❡ ❡❢❡✐t♦ ❡♠❡r❣❡✿ ♦ ❡❢❡✐t♦ ❞❡ ❛❝♦♣❧❛♠❡♥t♦ ❞❡ tr♦❝❛ ❡♥tr❡ s♣✐♥s ❧♦❝❛❧✐③❛❞♦s ❢❛③ ✈❛r✐❛r ❛ ♦r✐❡♥t❛çã♦ ❞♦s ♠❡s♠♦s ❞❡✈✐❞♦ ❛ ❞✐stâ♥❝✐❛ ❞❡ s❡♣❛r❛çã♦ ❡♥tr❡ ❛s ✐♠♣✉r❡③❛s✳ ■ss♦ ♦❝♦rr❡ ♣♦rq✉❡ ❛ ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛ ❡♥tr❡ ♦s s♣✐♥s ❧♦❝❛❧✐③❛❞♦s é ♠❡❞✐❛❞❛ ♣❡❧♦s ❡❧étr♦♥s ❞❡ ❝♦♥❞✉çã♦ s✉❜♠❡t✐❞♦s ❛s ♦s❝✐❧❛✲ çõ❡s ❞❡ ❋r✐❡❞❡❧✳ ❚❛❧ ♠❡❝❛♥✐s♠♦ ❢♦r♠❛ ❛ ❜❛s❡ ❞❛ ✐♥t❡r❛çã♦ ❘❑❑❨ ✭❘✉❞❡r♠❛♥✲ ❑✐tt❡❧✲❑❛s✉②❛✲❨♦s✐❞❛✮✱ q✉❡ ♥♦ ❝❛s♦ ❞♦ ❣r❛❢❡♥♦ t♦r♥❛✲s❡ ❛❧t❛♠❡♥t❡ ❛♥✐s♦tró♣✐❝♦
✭✶✽✱ ✶✾✱ ✷✵✱ ✷✶✮✳
◆❡st❡ tr❛❜❛❧❤♦✱ ❡♠♣r❡❣❛♥❞♦ ❞✉❛s ✐♠♣✉r❡③❛s ❞❡ ❆♥❞❡rs♦♥✱ ❢♦✐ ♣r❡✈✐st❛ ❛ ❢♦r✲ ♠❛çã♦ ❞❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ ♠✉❧t✐♥í✈❡✐s ♥❛ ❞❡♥s✐❞❛❞❡ ❧♦❝❛❧ ❞❡ ❡st❛❞♦s ✭▲❉❖❙✮ ❞♦ ❣r❛❢❡♥♦ ❡ ❜❛t✐♠❡♥t♦s ✐♥❞✉③✐❞♦s ♥❛ ▲❉❖❙ ♥❛s ✈✐③✐♥❤❛♥ç❛s ❞♦s ♣♦♥t♦s ❞❡ ❉✐✲ r❛❝✱ ❝♦♠♦ r❡s✉❧t❛❞♦ ❞❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❛s ✐♠♣✉r❡③❛s ♠❡❞✐❛❞❛ ♣❡❧♦s ❡❧étr♦♥s ❞❡ ❝♦♥❞✉çã♦✳ P❛r❛ ❣❛r❛♥t✐r t♦t❛❧ ❛✉sê♥❝✐❛ ❞♦ ❡❢❡✐t♦ ❞❡ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ♦s s♣✐♥s ♣r♦✈❡✲ ♥✐❡♥t❡s ❞❛ ❜❧✐♥❞❛❣❡♠ ❛♥t✐✲❢❡rr♦♠❛❣♥ét✐❝❛ ❞♦ ❡❢❡✐t♦ ❑♦♥❞♦✱ ❢♦✐ ❝♦♥s✐❞❡r❛❞♦ ♦ ❧✐♠✐t❡
T ≫TK✳ ◆❡ss❡ r❡❣✐♠❡✱ ♣♦❞❡✲s❡ s❡❣✉r❛♠❡♥t❡ ❛✜r♠❛r q✉❡ ❛♣❡♥❛s ❛s ✢✉t✉❛çõ❡s ❞❡
❝❛r❣❛ ❡♥tr❡ ❛s ✐♠♣✉r❡③❛s ❞✐st❛♥t❡s ♥♦ ❣r❛❢❡♥♦ sã♦ r❡❧❡✈❛♥t❡s✱ ✈❡❥❛ ❋✐❣✳ ✻✳✶✭❛✮✳ ❚❛✐s ✢✉t✉❛çõ❡s ♣♦❞❡♠ s❡r ❞❡t❡❝t❛❞❛s ❝♦♠ ❛ ♣♦♥t❛ ❞♦ ❙❚▼ ❧♦❝❛❧✐③❛❞❛ s♦❜r❡ ✉♠ sít✐♦ ❞❛ s✉❜✲r❡❞❡ ❆ ♦✉ ❇ ✭✈❡❥❛ ❋✐❣s✳✻✳✶✭❜✮ ❡ ✭❝✮✮ q✉❡ r❡s✉❧t❛♠ ❡♠ ♣❛❞rõ❡s ❞❡ ❜❛t✐♠❡♥t♦s ✐♥❞✉③✐❞♦s ♥❛ ▲❉❖❙✱ ♦s q✉❛✐s s❡rã♦ ❞✐s❝✉t✐❞♦s ❛❜❛✐①♦✳ ❉❡✈✐❞♦ à ♥❛t✉r❡③❛ ❞✐s❝r❡t❛ ❞❛ r❡❞❡ ❞♦ ❣r❛❢❡♥♦✱ ❢♦r❛♠ ❞❡✜♥✐❞♦s í♥❞✐❝❡s q✉❡ ❞❡s❝r❡✈❡♠ ♦s s❡❣✉✐♥t❡s ❝♦♠♣r✐♠❡♥✲ t♦s ❝❛r❛❝t❡ríst✐❝♦s✿ m♣❛r❛ ❛ s❡♣❛r❛çã♦ ❡♥tr❡ ❛s ✐♠♣✉r❡③❛s ❡p❞❡s✐❣♥❛ ❛ ♣♦s✐çã♦ ❞❛
♣♦♥t❛ ❞♦ ❙❚▼ ✭✈❡❥❛ ❋✐❣✳ ✻✳✶✭❛✮✮✳ ❋♦✐ ♦❜s❡r✈❛❞♦ q✉❡ ♣❛r❛ ♦❜t❡r ✉♠❛ ❡str✉t✉r❛ ❞❡ ♠✉❧t✐♥í✈❡✐s ❡ ♣❛❞rõ❡s ❞❡ ❜❛t✐♠❡♥t♦s ❞✐st✐♥t♦s✱ ❛ r❡str✐çã♦m= 2p♣❛r❛p≫1❞❡✈❡ s❡r s❛t✐s❢❡✐t❛✳ ❆❧é♠ ❞✐ss♦✱ ❢♦✐ ♦❜s❡r✈❛❞♦ q✉❡ ♦s ♣❛❞rõ❡s ❞❡ ❜❛t✐♠❡♥t♦s sã♦ ❛❧t❛✲ ♠❡♥t❡ ❛♥✐s♦tró♣✐❝♦s✱ t❡♥❞♦ ❝♦♠♣♦rt❛♠❡♥t♦s ❞✐st✐♥t♦s ❛♦ ❧♦♥❣♦ ❞❛s ❞✐r❡çõ❡s ③✐❣③❛❣ ❡ ❛r♠❝❤❛✐r✳ ❖s r❡s✉❧t❛❞♦s ❛♣♦♥t❛♠ q✉❡ ❛ ▲❉❖❙ é s❡♥sí✈❡❧ ♣❛r❛ ✐♠♣✉r❡③❛s s❡♣❛✲ r❛❞❛s ♣♦r ❣r❛♥❞❡s ❞✐stâ♥❝✐❛s✱ r❡✈❡❧❛♥❞♦ q✉❡ ♦ ❣r❛❢❡♥♦ é ✉♠ ❤♦s♣❡❞❡✐r♦ ❛❞❡q✉❛❞♦ ♣❛r❛ ❛ ♦❜s❡r✈❛çã♦ ❞❡ ❡❢❡✐t♦ ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ✐♠♣✉r❡③❛s✳
❊st❡ tr❛❜❛❧❤♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ◆♦ ❈❛♣ít✉❧♦ ✷ ❛s ♣r♦♣r✐❡✲ ❞❛❞❡s ❡str✉t✉r❛✐s ❡ ❡❧❡trô♥✐❝❛s ❞♦ ❣r❛❢❡♥♦ ✈✐❛ ❝á❧❝✉❧♦ t✐❣❤t✲❜✐♥❞✐♥❣ sã♦ ❞✐s❝✉t✐❞❛s✳ ◆♦ ❈❛♣ít✉❧♦ ✸ sã♦ ❛♣r❡s❡♥t❛❞♦s ♦ ♠♦❞❡❧♦ ❞❡ ❆♥❞❡rs♦♥✱ ♦ ❡❢❡✐t♦ ❑♦♥❞♦ ❡ ♦ ❡❢❡✐t♦ ❋❛♥♦✳ ❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ♠♦❞❡❧♦ t❡ór✐❝♦✱ ❞♦ ❝á❧❝✉❧♦ ❞❛s ❢✉♥çõ❡s ❞❡ ●r❡❡♥ ✈✐❛ ❛♣r♦①✐♠❛çã♦ ❍✉❜❜❛r❞ ■ ❡ ❞❛ ❞❡❞✉çã♦ ❞❛ ❡①♣r❡ssã♦ ❛♥❛❧ít✐❝❛ ❞❛ ▲❉❖❙ sã♦ ❢❡✐t❛s ♥♦ ❈❛♣ít✉❧♦ ✹✳ ❖ ❝á❧❝✉❧♦ ❞❛s ❛✉t♦✲❡♥❡r❣✐❛s sã♦ ❞❡t❛❧❤❛❞♦s ♥♦ ❈❛♣ít✉❧♦ ✺✳ P♦r ♠❡✐♦ ❞♦s ❝á❧❝✉❧♦s r❡❛❧✐③❛❞♦s ♥♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s✱ ❛ ❢♦r♠❛çã♦ ❞❛ ❡str✉t✉r❛ ♠✉❧t✐✲ ♥í✈❡✐s ❡ ❞♦s ♣❛❞rõ❡s ❞❡ ❜❛t✐♠❡♥t♦s q✉â♥t✐❝♦s sã♦ ❞✐s❝✉t✐❞♦s ♥♦ ❈❛♣ít✉❧♦ ✻✳ P♦r ú❧t✐♠♦✱ ❛s ❝♦♥❝❧✉sõ❡s sã♦ ❛♣r❡s❡♥t❛❞❛s ♥♦ ❈❛♣ít✉❧♦ ✼✳ ◆♦s ❆♣ê♥❞✐❝❡s ❡♥❝♦♥tr❛♠✲ s❡ ✉♠❛ ❡①♣❧✐❝❛çã♦ ❞❡t❛❧❤❛❞❛ ❞♦ ♠ét♦❞♦ ❞❛ ❊q✉❛çã♦ ❞♦ ▼♦✈✐♠❡♥t♦ ❡ ♦s tr❛❜❛❧❤♦s ♣✉❜❧✐❝❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦ ♠❡str❛❞♦✳
Cap´ıtulo
2
Pr♦♣r✐❡❞❛❞❡s ❡❧❡trô♥✐❝❛s ❡ ❡str✉t✉r❛✐s ❞♦
●r❛❢❡♥♦
◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❛❜♦r❞❛❞❛s ❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❡str✉t✉r❛✐s ❡ ❡❧❡✲ trô♥✐❝❛s ❞♦ ❣r❛❢❡♥♦✳ ■♥✐❝✐❛❧♠❡♥t❡ ❛ ❢♦r♠❛çã♦ ❞❛ r❡❞❡ ❤❡①❛❣♦♥❛❧ ❡ s❡✉s ❛s♣❡❝t♦s ❣❡♦♠étr✐❝♦s s❡rã♦ ❞✐s❝✉t✐❞♦s✳ P♦st❡r✐♦r♠❡♥t❡✱ ♦ ❝á❧❝✉❧♦ ❞❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❡ ♦ ❧✐♠✐t❡ ❞❡ ❞✐s♣❡rsã♦ ❧✐♥❡❛r ❞♦ ❣r❛❢❡♥♦ ♣❛r❛ ❜❛✐①❛s ❡♥❡r❣✐❛s ❡♠ t♦r♥♦ ❞♦s ♣♦♥t♦s ❞❡ ❉✐r❛❝ s❡rã♦ ❡st✉❞❛❞♦s ❞❡♥tr♦ ❞❛ ❛♣r♦①✐♠❛çã♦ t✐❣❤t✲❜✐♥❞✐♥❣✳
✷✳✶ ❈á❧❝✉❧♦ t✐❣❤t✲❜✐♥❞✐♥❣ ❞♦ ●r❛❢❡♥♦
❖ ❣r❛❢❡♥♦ é ✉♠❛ ❡str✉t✉r❛ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❢♦r♠❛❞❛ ♣♦r át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ♦r❣❛♥✐③❛❞♦s ❡♠ ✉♠❛ r❡❞❡ ❤❡①❛❣♦♥❛❧✳ ❊ss❛ r❡❞❡ ❞♦ t✐♣♦ ❢❛✈♦ ❞❡ ♠❡❧ é ♦r✐❣✐♥❛❞❛ ❣r❛ç❛s ❛♦s ♦r❜✐t❛✐s ❤í❜r✐❞♦ssp2 q✉❡ sã♦ ❞❡s❝r✐t♦s ♣❡❧❛ t❡♦r✐❛ ❞❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ♦r❜✐t❛✐s ❛tô♠✐❝♦s ✭▲❈❆❖ ✕ ▲✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❛t♦♠✐❝ ♦r❜✐t❛❧s✮ ❡ ♦r✐❣✐♥❛❞♦s ♣❡❧❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❡♥tr❡ ♦s ♦r❜✐t❛✐s2s✱ 2px ❡ 2py q✉❡ ❢♦r♠❛♠ ✉♠❛ ❣❡♦♠❡tr✐❛
tr✐❛♥❣✉❧❛r ♣❧❛♥❛r ♥♦ ♣❧❛♥♦ ①②✳ ❚♦❞❛ ❛ ❡str✉t✉r❛ ❣❡♦♠étr✐❝❛ ❡ ❡st❛❜✐❧✐❞❛❞❡ ❞❛ r❡❞❡ ❞♦ ❣r❛❢❡♥♦ sã♦ ❝❛r❛❝t❡r✐③❛❞❛s ♣❡❧❛s ❧✐❣❛çõ❡s q✉í♠✐❝❛s σ ❡♥tr❡ ♦s ♦r❜✐t❛✐s sp2✳ ❏á ♦ ♦r❜✐t❛❧ 2pz ❢❛③ ❧✐❣❛çõ❡s q✉í♠✐❝❛s ❞♦ t✐♣♦ π ❝♦♠ s❡✉s ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s✳ ❊ss❡s
♦r❜✐t❛✐s sã♦ ♦s r❡s♣♦♥sá✈❡✐s ♣❡❧❛ ❝♦♥❞✉çã♦ ❡❧❡trô♥✐❝❛ ♥♦ ❣r❛❢❡♥♦✱ ✉♠❛ ✈❡③ q✉❡ ❛s ❧✐❣❛çõ❡s q✉í♠✐❝❛s ❞♦ t✐♣♦π sã♦ ♠❛✐s ❢r❛❝❛s ❡ ♣❡r♠✐t❡♠ q✉❡ ♦s ❡❧étr♦♥s t✉♥❡❧❡♠ ❞❡
✉♠ ❝❛r❜♦♥♦ ♣❛r❛ ♦✉tr♦✳ ❆ss✐♠✱ t♦❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧❡trô♥✐❝❛s ❞♦ ❣r❛❢❡♥♦ sã♦
✷✳✶ ❈á❧❝✉❧♦ t✐❣❤t✲❜✐♥❞✐♥❣ ❞♦ ●r❛❢❡♥♦
♠✉✐t♦ ❜❡♠ ❞❡s❝r✐t❛s ❝♦♥s✐❞❡r❛♥❞♦ ❛♣❡♥❛s ♦ ♦r❜✐t❛❧ 2pz ♥♦ ❝á❧❝✉❧♦ t✐❣❤t✲❜✐♥❞✐♥❣
✭✸✮✳
❆ r❡❞❡ ❞❡ ❣r❛❢❡♥♦ é ✐❧✉str❛❞❛ ♥❛ ❋✐❣✳ ✭✷✳✶✮✳ ❙✉❛ ❣❡♦♠❡tr✐❛ ❤❡①❛❣♦♥❛❧ é ❢♦r♠❛❞❛ ♣♦r ❞✉❛s s✉❜✲r❡❞❡s A ❡ B✱ ❞❡✜♥✐❞❛s ♣❡❧❛s ❝♦r❡s ✈❡r♠❡❧❤❛ ❡ ❛③✉❧✳ P❛r❛ q✉❡ ♦
❣r❛❢❡♥♦ ❢♦r♠❡ ✉♠❛ r❡❞❡ ❞❡ ❇r❛✈❛✐s✱ ❞❡✜♥❡✲s❡ ✉♠❛ ❝é❧✉❧❛ ✉♥✐tár✐❛ ❞❡ ♠♦❞♦ q✉❡ ❡st❛ ❡♥❣❧♦❜❡ ❞♦✐s át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ❞❡ s✉❜✲r❡❞❡s ❞✐❢❡r❡♥t❡s✳ ❯♠❛ ♣♦ssí✈❡❧ ❣❡♦♠❡tr✐❛ é ♦ ❧♦s❛♥❣♦ ❞❡✜♥✐❞♦ ♥❛ ❋✐❣✳ ✭✷✳✶✮✳ ❖s ✈❡t♦r❡s ♣r✐♠ár✐♦s ❞❛ r❡❞❡ sã♦ ❞❛❞♦s ♣♦r
a1 = √
3a0(12, √
3
2 ) ❡ a2 = √
3a0(−12, √
3
2 )✱ ♦♥❞❡ a0 é ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ❞❡ ♠ó❞✉❧♦ 1,42 ˚❆✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ r❡❞❡ é ❞❡s❝r✐t❛ ❛♣❡♥❛s ♣❡❧♦ ♠♦✈✐♠❡♥t♦ ❞❡ tr❛♥s❧❛çã♦ ❞❛ ❝é❧✉❧❛ ✉♥✐tár✐❛✱ ♦❜t✐❞♦ ♣❡❧❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ss❡s ✈❡t♦r❡s✳
^x
^y
x
1
x
2
a
1
a
2
❋✐❣✉r❛ ✷✳✶✿ ❆ r❡❞❡ ❤❡①❛❣♦♥❛❧ ❞♦ ❣r❛❢❡♥♦✳ ❆s ❡s❢❡r❛s ❡♠ ❛③✉❧ ❡ ✈❡r♠❡❧❤♦ ❧♦❝❛❧✐③❛❞❛s ♥♦s ✈ért✐❝❡s ❞♦s ❤❡①á❣♦♥♦s r❡♣r❡s❡♥t❛♠ ♦s át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ❞❛s s✉❜✲r❡❞❡sA❡B ❡
❛s ❧✐♥❤❛s ❡♠ ♣r❡t♦ ✐♥❞✐❝❛♠ ❛s ❧✐❣❛çõ❡s q✉í♠✐❝❛sσ ❞♦s ♦r❜✐t❛✐ssp2✳ a1 ❡ a2 ❞❡✜♥❡♠ ♦s ✈❡t♦r❡s ♣r✐♠ár✐♦s ❡ ♦ ❧♦s❛♥❣♦ ❡♠ ❝✐♥③❛ ❝❧❛r♦ ❞❡♠❛r❝❛ ❛ ❝é❧✉❧❛ ✉♥✐tár✐❛ ❞❛ r❡❞❡✳ ❖s ✈❡t♦r❡s x1 = a0(√3
2 ,0)❡ x2 =a0(2+ √
3
2 ,0)✐♥❞✐❝❛♠ ❛ ♣♦s✐çã♦ ❞❡ ❝❛❞❛ át♦♠♦ ❞❡ ❝❛r❜♦♥♦ ❞❡♥tr♦ ❞❛ ❝é❧✉❧❛✳
❆♣ós ❞✐s❝✉t✐r ❛s ♣r♦♣r✐❡❞❛❞❡s ❣❡♦♠étr✐❝❛s ❞❛ r❡❞❡✱ ♦ ♣r✐♠❡✐r♦ ♣❛ss♦ ♣❛r❛ ❞❡s✲ ❝r❡✈❡r ❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞♦ ❣r❛❢❡♥♦ ✈✐❛ ❛♣r♦①✐♠❛çã♦ t✐❣❤t✲❜✐♥❞✐♥❣ é ❞❡✜♥✐r ♦
✷✳✶ ❈á❧❝✉❧♦ t✐❣❤t✲❜✐♥❞✐♥❣ ❞♦ ●r❛❢❡♥♦
❍❛♠✐❧t♦♥✐❛♥♦ ❞❡ ✉♠ ❡❧étr♦♥ ♥❛ ♣r❡s❡♥ç❛ ❞♦ ♣♦t❡♥❝✐❛❧ ❞❛ r❡❞❡✳ ❆ss✐♠
H = P
2
2♠e
+ X
❘ ➍ ●
[❱❛t(x−x1−R) +❱❛t(x−x2−R)], ✭✷✳✶✮
♦♥❞❡ P ❡ ♠esã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ ♠♦♠❡♥t♦ ❡ ❛ ♠❛ss❛ ❞♦ ❡❧étr♦♥✱ ❱❛t(x−x1−R) ❡ ❱❛t(x−x2 −R) sã♦ ♦s ♣♦t❡♥❝✐❛✐s ❞❡ ❝❛❞❛ át♦♠♦ ❞❡ ❝❛r❜♦♥♦ ♥♦ sít✐♦ R ❡ G é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ✈❡t♦r❡s q✉❡ ❞❡s❝r❡✈❡♠ ❛ r❡❞❡✳ ❆❧é♠ ❞✐ss♦✱ ♦ ❍❛♠✐❧t♦♥✐❛♥♦ ♣♦❞❡ s❡r r❡❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
H = P
2
2♠e
+❱❛t(x−x1) +
+ ❱❛t(x−x2) +
X
R6=0
[❱❛t(x−x1−R) +❱❛t(x−x2−R)], ✭✷✳✷✮
♦♥❞❡ ♦s ❞♦✐s ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡s❝r❡✈❡♠ ♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❞❡ ✉♠ ú♥✐❝♦ át♦♠♦ ❞❡ ❝❛r❜♦♥♦ ❡ é r♦t✉❧❛❞♦ ♣♦r H❛t✳ ❆ ú❧t✐♠❛ ♣❛rt❡ ❞♦ ❍❛♠✐❧t♦♥✐❛♥♦ é ❞❡✜♥✐❞❛ ♣♦r H❝r = △❯✱ ✐st♦ é✱ ♣❡❧♦ t❡r♠♦ q✉❡ ❝♦♥té♠ t♦❞❛s ❛s ✐♥❢♦r♠❛çõ❡s ❞♦ ♣♦t❡♥❝✐❛❧ ♣❡r✐ó❞✐❝♦ ❞❛ r❡❞❡✳
❉❡✈❡ ✜❝❛r ❝❧❛r♦ q✉❡ ❛ ❛♣r♦①✐♠❛çã♦ t✐❣❤t✲❜✐♥❞✐♥❣ só é ✈á❧✐❞❛ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡
H❛t ≫H❝r, ✭✷✳✸✮
♣♦✐s ❛ ❤✐♣ót❡s❡ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❛♣r♦①✐♠❛çã♦ é q✉❡ ❛ s♦❜r❡♣♦s✐çã♦ ❞❛s ❢✉♥çõ❡s ❞❡ ♦♥❞❛ ❡♥tr❡ ♦s át♦♠♦s ✈✐③✐♥❤♦s s❡❥❛♠ ♣❡q✉❡♥❛s✱ ❞❡ ♠♦❞♦ q✉❡ ❛♣❡♥❛s ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s s❡❥❛♠ ❧❡✈❛❞❛s ❡♠ ❝♦♥s✐❞❡r❛çã♦✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ ❡♥❡r❣✐❛ ❞♦ ❡❧étr♦♥ ♥❛s ♣r♦①✐♠✐❞❛❞❡s ❞♦ át♦♠♦ ❞❡ ❝❛r❜♦♥♦ s❡rá ❜❡♠ ❞❡✜♥✐❞❛ ♣♦r H❛t ❡ ❛ ❡♥❡r❣✐❛ ❡①tr❛ ❞❛❞❛ ♣♦r H❝r s❡rá s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛✱ ♣❡r♠✐t✐♥❞♦ q✉❡ ❡❧❛ s❡❥❛ tr❛t❛❞❛ ❞❡ ❢♦r♠❛ ♣❡rt✉r❜❛t✐✈❛ ✭✷✷✮✳ ❖ ❣r❛❢❡♥♦ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ❞❡♥tr♦ ❞❡ss❛ ❛♣r♦①✐♠❛çã♦ ♣♦rq✉❡ ♦s ♦r❜✐t❛✐s 2pz sã♦ ❧♦❝❛❧✐③❛❞♦s ❡ ♦s ú♥✐❝♦s r❡s♣♦♥sá✈❡✐s ♣❡❧♦
tr❛♥s♣♦rt❡ ❡❧❡trô♥✐❝♦ ♥❛ r❡❞❡✳
P❛r❛ ❞❡s❝r❡✈❡r ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❞♦ ❡❧étr♦♥ ♥❛ r❡❞❡✱ s❡rá ❝♦♥s✐❞❡r❛❞♦ ♦ s❡❣✉✐♥t❡
✷✳✶ ❈á❧❝✉❧♦ t✐❣❤t✲❜✐♥❞✐♥❣ ❞♦ ●r❛❢❡♥♦
❆♥s❛t③✿
Ψk =
X
R➍ ●
åi(k·R)Φ(x
−R), ✭✷✳✹✮
♦♥❞❡ Φ(x−R) é ❛ s♦♠❛ ❞♦s ♦r❜✐t❛✐s ❛tô♠✐❝♦s 2pz ❞♦ ❝❛r❜♦♥♦✳ ❆ ❢✉♥çã♦ Φ(x) é ❝❛r❛❝t❡r✐③❛❞❛ ♣❡❧❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ♦r❜✐t❛✐s Φ1(x) ❡ Φ2(x)✱ ♦♥❞❡ Φ1,2(x) sã♦ ❛s ❢✉♥çõ❡s ❞♦s ♦r❜✐t❛✐s ❛tô♠✐❝♦s ❞♦s ❝❛r❜♦♥♦s 1 ❡ 2✳ ❊ss❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ Ψk s❛t✐s❢❛③ ♣❡r❢❡✐t❛♠❡♥t❡ ♦ t❡♦r❡♠❛ ❞❡ ❇❧♦❝❤ ❡ é ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ ❞❡ ❲❛♥♥✐❡r✳
■st♦ é✱ ❡ss❛ ❢✉♥çã♦ ❞❡s❝r❡✈❡ ♣❡r❢❡✐t❛♠❡♥t❡ ❛ ♣❡r✐♦❞✐❝✐❞❛❞❡ ❞❛ r❡❞❡ ❞❡ ♠♦❞♦ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ |Φ(x)|2 =|Φ(x−R)|2 s❡❥❛ s❛t✐s❢❡✐t❛✱ ♣♦✐s ❡❧❛s s❡ ❞✐❢❡r❡♠ ❛♣❡♥❛s ♣♦r
✉♠❛ ❢❛s❡ ❣❧♦❜❛❧ ✭✷✷✮✳
❆♥t❡s ❞❡ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤r♦❞✐♥❣❡r ❞♦ s✐st❡♠❛✱ s❡rá ❛♥❛❧✐s❛❞♦ ❝♦♠♦ ♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❞❛ ❊q✳ ✭✷✳✷✮ ❛t✉❛ s❡♣❛r❛❞❛♠❡♥t❡ ❡♠ ❝❛❞❛ ♦r❜✐t❛❧ ❛tô♠✐❝♦✳ ▲♦❣♦✱
H❛t
1(2)Φ1(2)(x) +△❯1(2)Φ1(2)(x) = (ǫ1(2)+△❯1(2))Φ1(2)(x), ✭✷✳✺✮
♦♥❞❡△❯1(2) =PR6=0[❱❛t(x−x1−R) +❱❛t(x−x2−R)] +❱❛t(x−x2(1))r❡♣r❡✲
s❡♥t❛ ♦ ♣♦t❡♥❝✐❛❧ ❞❛ r❡❞❡ ❡H❛t
1(2) =
P2
2♠e +❱❛t(x−x1(2))❞❡✜♥❡ ♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❞♦
át♦♠♦ ❞❡ ❝❛r❜♦♥♦ 1(2)✳ ❈♦♠♦ ❛ r❡❞❡ é ❢♦r♠❛❞❛ ❛♣❡♥❛s ♣♦r át♦♠♦s ❞❡ ❝❛r❜♦♥♦✱ s❡✉s ❛✉t♦✈❛❧♦r❡sǫ1(2)sã♦ ✐❣✉❛✐s✳ ❈♦♠ ♦ ✐♥t✉✐t♦ ❞❡ s✐♠♣❧✐✜❝❛r ♦s ❝á❧❝✉❧♦s✱ ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ é ❞❡✜♥✐❞❛ ♣❡❧♦s ❛✉t♦✈❛❧♦r❡s ǫ1(2)✳ ▲♦❣♦✱
HΦ1(2)(x) = △❯1(2)Φ1(2)(x). ✭✷✳✻✮
P♦r ♠❡✐♦ ❞♦s ❛✉t♦✈❛❧♦r❡s ♦❜t✐❞♦s ♥❛ ❊q✳ ✭✷✳✻✮✱ ♦ s❡❣✉✐♥t❡ ❍❛♠✐❧t♦♥✐❛♥♦ t✐❣❤t✲ ❜✐♥❞✐♥❣ ❢♦✐ ❞❡✜♥✐❞♦
HΨk(x) = ❊(k)Ψk(x), ✭✷✳✼✮
q✉❡ r❡s✉❧t❛ ❡♠
❊(k)hΦj|Ψki =hΦj| △❯j|Ψki. ✭✷✳✽✮
✷✳✶ ❈á❧❝✉❧♦ t✐❣❤t✲❜✐♥❞✐♥❣ ❞♦ ●r❛❢❡♥♦
❚r❛❜❛❧❤❛♥❞♦ ✐♥✐❝✐❛❧♠❡♥t❡ ❝♦♠ hΦ1|Ψki ❡hΦ2|Ψki✱ ❢♦r❛♠ ♦❜t✐❞♦s
hΦ1|Ψki = b1hΦ1(x)|Φ1(x)i+b2hΦ1(x)|Φ2(x)i + å−i(k·a1)(b
1hΦ1(x)|Φ1(x−a1)i+b2hΦ1(x)|Φ2(x−a1)i) + å−i(k·a2)(b
1hΦ1(x)|Φ1(x−a2)i+b2hΦ1(x)|Φ2(x−a2)i) = b1hΦ1(x)|Φ1(x)i+b2hΦ1(x)|Φ2(x)i
+ å−i(k·a1)(b
1hΦ1(x)|Φ1(x−a1)i+b2hΦ1(x)|Φ2(x−a1)i) + å−i(k·a2)(b
1hΦ1(x)|Φ1(x−a2)i+b2hΦ1(x)|Φ2(x−a2)i) = b1 +b2γ0 1 +å−i(k·a1)+å−i(k·a2)
✭✷✳✾✮
❡
hΦ2|Ψki = b1hΦ2(x)|Φ1(x)i+b2hΦ2(x)|Φ2(x)i + åi(k·a1)(b
1hΦ2(x)|Φ1(x−a1)i+b2hΦ2(x)|Φ2(x−a1)i) + åi(k·a2)(b
1hΦ2(x)|Φ1(x−a2)i+b2hΦ2(x)|Φ2(x−a2)i) = b1hΦ2(x)|Φ1(x)i+b2hΦ2(x)|Φ2(x)i
+ åi(k·a1)(b
1hΦ2(x)|Φ1(x−a1)i+b2hΦ2(x)|Φ2(x−a1)i) + åi(k·a2)(b
1hΦ2(x)|Φ1(x−a2)i+b2hΦ2(x)|Φ2(x−a2)i) = b2+b1γ0 1 +åi(k·a1)+åi(k·a2)
, ✭✷✳✶✵✮
♦♥❞❡γ0 =
´
Φ∗
1(x)Φ2(x−R)dx=
´
Φ∗
2(x)Φ1(x−R)dx✳ ❖s t❡r♠♦s q✉❡ ❞❡s❝r❡✈❡♠ ❛ s♦❜r❡♣♦s✐çã♦ ❞❛s ❢✉♥çõ❡s ❞❡ ♦♥❞❛ ❝♦♠ ♦s s❡❣✉♥❞♦s ✈✐③✐♥❤♦s sã♦ ✐❣✉❛✐s ❛ ③❡r♦✳ ❊ss❛s ✐❣✉❛❧❞❛❞❡s sã♦ ✈á❧✐❞❛s✱ ♣♦✐s ❛s ❢✉♥çõ❡s ❞❡ ♦♥❞❛ sã♦ ❜❡♠ ❧♦❝❛❧✐③❛❞❛s✳ ◆♦✈❛♠❡♥t❡✱ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ s✐♠♣❧✐✜❝❛r ♦s ❝á❧❝✉❧♦s✱ ❢♦✐ ❝♦♥s✐❞❡r❛❞♦γ0 = 0✱ ❞❛❞♦ q✉❡γ0 ≪1✳
❖ ♣ró①✐♠♦ ♣❛ss♦ é ❝❛❧❝✉❧❛r hΦj| △❯j|Ψki, q✉❡ r❡s✉❧t❛ ❡♠
hΦ1| △❯1|Ψki = b1hΦ1(x)| △❯1|Φ1(x)i+b2hΦ1(x)| △❯1|Φ2(x)i
+ X
R6=0
åi(k·R)b
1hΦ1(x)| △❯1|Φ1(x−R)i
+ X
R6=0
åi(k·R)b
2hΦ1(x)| △❯1|Φ2(x−R)i
= b2γ1 1 +å−i(k·a1)+å−i(k·a2)
, ✭✷✳✶✶✮
✷✳✶ ❈á❧❝✉❧♦ t✐❣❤t✲❜✐♥❞✐♥❣ ❞♦ ●r❛❢❡♥♦
❡
hΦ2| △❯2|Ψki = b1hΦ2(x)| △❯2|Φ1(x)i+b2hΦ2(x)| △❯2|Φ2(x)i
+ X
R6=0
åi(k·R)b
1hΦ2(x)| △❯2|Φ1(x−R)i
+ X
R6=0
åi(k·R)b
2hΦ2(x)| △❯2|Φ2(x−R)i
= b1γ1 1 +åi(k·a1)+åi(k·a2)
, ✭✷✳✶✷✮
♦♥❞❡ γ1 = hΦ1(x)| △❯1|Φ2(x)i = hΦ2(x)| △❯2|Φ1(x)i ❡ é ❝❤❛♠❛❞♦ ❞❡ t❡r♠♦ ❞❡ ❤♦♣♣✐♥❣✳ ❊ss❡ ♣❛râ♠❡tr♦ ❞❡✜♥❡ ❛ ❛♠♣❧✐t✉❞❡ ❞❡ t✉♥❡❧❛♠❡♥t♦ ❡♥tr❡ ❡❧étr♦♥s ❞❡ át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ✈✐③✐♥❤♦s✳ ◆♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s ♦ t❡r♠♦ ❞❡ ❤♦♣♣✐♥❣ γ1 s❡rá r♦t✉❧❛❞♦ ♣♦r t✳ ❖s ✈❛❧♦r❡s ❡s♣❡r❛❞♦s ❞♦s ♣♦t❡♥❝✐❛✐s hΦ1,2(x)| △❯1,2|Φ1,2(x)i
❡ P
R6=0hΦ1,2(x)| △❯1,2|Φ1,2(x−R)i sã♦ ♥✉❧♦s✱ ♣♦✐s ❛♣❡♥❛s ❛s s♦❜r❡♣♦s✐çõ❡s ❞❡ ♦♥❞❛ ❧♦❝❛❧ ❡ ❡♥tr❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s ❡stã♦ s❡♥❞♦ ❝♦♥s✐❞❡r❛❞❛s ♥♦ ❝á❧❝✉❧♦✳
❙✉❜st✐t✉✐♥❞♦ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥❛s ❊q✳ ✭✷✳✾✮✱ ✭✷✳✶✵✮✱ ✭✷✳✶✶✮ ❡ ✭✷✳✶✷✮ ♥❛ ❡q✉❛çã♦ ❊q✳ ✭✷✳✽✮✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r❡s é ❢♦r♠✉❧❛❞♦ ♥❛ ♥♦t❛çã♦ ♠❛tr✐❝✐❛❧ ♣❡❧❛ ❡q✉❛çã♦
❊(k) 0
0 ❊(k)
!
b1
b2
!
= 0 γ1α(k)
γ1α∗(k) 0
!
b1
b2
!
, ✭✷✳✶✸✮
♦♥❞❡α(k) = 1 +å−i(k·a1)+å−i(k·a2) ❡α∗(k)é ♦ s❡✉ ❝♦♠♣❧❡①♦ ❝♦♥❥✉❣❛❞♦✳ P♦rt❛♥t♦✱
♦s ❛✉t♦✈❛❧♦r❡s sã♦ ♦❜t✐❞♦s ♣❡❧♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ ❞❛ ❊q✳ ✭✷✳✶✸✮✱ q✉❡ r❡s✉❧t❛ ❡♠
❊(k) =±γ1|α(k)|, ✭✷✳✶✹✮
♦✉ ❛✐♥❞❛
❊(k) = ±γ1
v u u
t1 + 4 cos(akx
2 ) cos
√
3aky
2
!
+ 4 cos2
akx
2
, ✭✷✳✶✺✮
♦♥❞❡ a =√3a0✳ ❆ ❋✐❣✳ ✭✷✳✷✮ ✭❛✮ ✐❧✉str❛ ❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞♦ ❣r❛❢❡♥♦ ♦❜t✐❞❛
✷✳✷ ❉✐s♣❡rsã♦ ❧✐♥❡❛r ❞♦ ●r❛❢❡♥♦
♣❡❧♦ ❝á❧❝✉❧♦ t✐❣❤t✲❜✐♥❞✐♥❣✳ ❆s ❜❛♥❞❛s ❞❡ ✈❛❧ê♥❝✐❛ ❡ ❝♦♥❞✉çã♦ sã♦ r❡♣r❡s❡♥t❛❞❛s ♣❡❧❛s ❝✉r✈❛s ✐♥❢❡r✐♦r ❡ s✉♣❡r✐♦r ❛♦ ♥í✈❡❧ ❞❡ ❡♥❡r❣✐❛ ③❡r♦ q✉❡ r❡♣r❡s❡♥t❛ ♦ ♥í✈❡❧ ❞❡ ❋❡r♠✐✳ ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡ ❛s ❜❛♥❞❛s ♥ã♦ s❡ t♦❝❛♠ ❞❡✈✐❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❣❛♣ ♣♦♥t✉❛❧ ❡①❛t❛♠❡♥t❡ ❡♠ ③❡r♦✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✳ ✭✷✳✷✮ ✭❜✮✳ ❊ss❡s ♣♦♥t♦s sã♦ ❝❤❛♠❛❞♦s ❞❡ ♣♦♥t♦s ❞❡ ❉✐r❛❝ K ❡ K✬✱ ❡ ❡♠ t♦r♥♦ ❞❡❧❡s✱ ♥♦ ❧✐♠✐t❡ ❞❡
❜❛✐①❛s ❡♥❡r❣✐❛s✱ ♦ ❣r❛❢❡♥♦ ♣♦ss✉✐ ✉♠❛ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❧✐♥❡❛r ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❢ér♠✐♦♥s r❡❧❛t✐✈íst✐❝♦s s❡♠ ♠❛ss❛ ✭✸✮✳ ◆♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s K ❡ K✬ s❡rã♦
r♦t✉❧❛❞♦s ♣♦r K±✳ ▼❛✐s ❞❡t❛❧❤❡s s❡rã♦ ❞✐s❝✉t✐❞♦s ♥❛ ♣ró①✐♠❛ s❡çã♦✳
a) b)
❋✐❣✉r❛ ✷✳✷✿ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s t✐❣❤t✲❜✐♥❞✐♥❣ ❞♦ ❣r❛❢❡♥♦✳ ✭❛✮ ❘❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞❛ ❡♥❡r❣✐❛ ❞❛❞❛ ♣❡❧❛ ❊q✳✭✷✳✶✺✮ ❡♠ ❢✉♥çã♦ ❞❡ kx ❡ ky✳ ✭❜✮ ❘❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ♥♦
❧✐♠✐t❡ ❞❡ ❜❛✐①❛s ❡♥❡r❣✐❛s ❡♠ t♦r♥♦ ❞♦s ♣♦♥t♦s ❞❡ ❉✐r❛❝✳
✷✳✷ ❉✐s♣❡rsã♦ ❧✐♥❡❛r ❞♦ ●r❛❢❡♥♦
❆ r❡❞❡ r❡❝í♣r♦❝❛ ❞♦ ❣r❛❢❡♥♦ é ✐❧✉str❛❞❛ ♥❛ ❋✐❣✳ ✭✷✳✸✮ ❡ ❢♦r♠❛❞❛ ♣❡❧♦s ✈❡t♦r❡s ♣r✐♠ár✐♦sb1 = 32aπ0(
√
3,1)❡b2 = 32aπ0(− √
3,1)✳ ◆♦t❡ q✉❡ ❛ r❡❞❡ t❛♠❜é♠ ♣♦ss✉✐ ✉♠ ❢♦r♠❛t♦ ❤❡①❛❣♦♥❛❧✳ ❖s ♣♦♥t♦s ❡♠ ♣r❡t♦ ♥♦s ✈ért✐❝❡s ❡ ♥♦ ❝❡♥tr♦ ❞♦ ❤❡①á❣♦♥♦ sã♦ ♦s ♣♦♥t♦s ❞❛ r❡❞❡ r❡❝í♣r♦❝❛ ❡ ♦ ❤❡①á❣♦♥♦ ♠❡♥♦r ✭❝✐♥③❛ ❝❧❛r♦✮ ❞❡❧✐♠✐t❛ ❛ ♣r✐♠❡✐r❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥✳ ❊♠ ❝❛❞❛ ✈ért✐❝❡ ❞❛ ♣r✐♠❡✐r❛ ③♦♥❛ ❡stã♦ ❧♦❝❛❧✐③❛❞♦s ♦s ♣♦♥t♦s ❞❡
✷✳✷ ❉✐s♣❡rsã♦ ❧✐♥❡❛r ❞♦ ●r❛❢❡♥♦
❉✐r❛❝K❡K✬✱ t♦❞❛✈✐❛✱ ❛♣❡♥❛s ✉♠ t❡rç♦ ❞❡ ❝❛❞❛ ✉♠ ❞❡❧❡s ❡stá ❞❡♥tr♦ ❞❛ ♣r✐♠❡✐r❛
③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥✳ ❉❡ss❛ ❢♦r♠❛✱ t♦❞❛ ❛ ❢ís✐❝❛ ❞❡ ❜❛✐①❛s ❡♥❡r❣✐❛s ❞♦ ❣r❛❢❡♥♦ ❞❡♥tr♦ ❞❛ ♣r✐♠❡✐r❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥ é ❞❡s❝r✐t❛ ♣♦r ❞♦✐s ❝♦♥❡s ❞❡ ❉✐r❛❝ ❛ss♦❝✐❛❞♦s ❛s s✉❜✲r❡❞❡s A ❡ B✳
Primeira zona de Brillouin
Pontos da rede recíproca
^
x
^
y
K b
1
b
2
a
1
a
2
❋✐❣✉r❛ ✷✳✸✿ ❖ ❤❡①á❣♦♥♦ ❡①t❡r♥♦ ✐❧✉str❛ ❛ r❡❞❡ r❡❝í♣r♦❝❛ ❞♦ ❣r❛❢❡♥♦✳ b1 ❡ b2 sã♦ ♦s ✈❡t♦r❡s ♣r✐♠ár✐♦s q✉❡ ❞❡❧✐♠✐t❛♠ ❛ ♣♦s✐çã♦ ❞♦s ♣♦♥t♦s ❞❛ r❡❞❡✳ ❖ ❤❡①á❣♦♥♦ s♦♠❜r❡❛❞♦ ✭❝✐♥③❛ ❝❧❛r♦✮ ❞❡s❝r❡✈❡ ❛ ♣r✐♠❡✐r❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥ ❡ ❡♠ ❝❛❞❛ ✈ért✐❝❡ ❞♦ ❤❡①á❣♦♥♦ ❡stã♦ ❧♦❝❛❧✐③❛❞♦s ♦s ♣♦♥t♦s ❞❡ ❉✐r❛❝ K❡K✬✳
❆ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❧✐♥❡❛r ❞♦ ❣r❛❢❡♥♦ é ♦❜t✐❞❛ ♣❡❧❛ ❡①♣❛♥sã♦ ❡♠ sér✐❡ ❞❡ ❚❛②❧♦r ❞❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ ❣r❛❢❡♥♦ ❡♠ t♦r♥♦ ❞♦s ♣♦♥t♦sK❡K✬✳ ❈♦♠♦ ❛ ❊q✳ ✭✷✳✶✹✮
é ♣r♦♣♦r❝✐♦♥❛❧ ❛ α(k)✱ ❡①♣❛♥❞✐r ❛ ❡♥❡r❣✐❛ t♦t❛❧ s❡rá ❡q✉✐✈❛❧❡♥t❡ ❛ ❡①♣❛♥❞✐r
α(K+q) ≈ α(K) +q· ∇kα(k)|k=K, ✭✷✳✶✻✮ ♦♥❞❡ q ≪ |K|✳ ◆❡ss❡ ❝❛s♦✱ ♦ ♣♦♥t♦ ❞❡ ❉✐r❛❝ ❡s❝♦❧❤✐❞♦ ♣❛r❛ r❡❛❧✐③❛r ❛ ❡①♣❛♥sã♦
✷✳✷ ❉✐s♣❡rsã♦ ❧✐♥❡❛r ❞♦ ●r❛❢❡♥♦
❡♠ sér✐❡ ❞❡ ❚❛②❧♦r ❢♦✐
K= 4π 3√3a0
(1,0), ✭✷✳✶✼✮
q✉❡ é ✐❧✉str❛❞♦ ♥❛ ❋✐❣✳ ✭✷✳✸✮✳ ❖s ❞❡♠❛✐s ♣♦♥t♦s ❞❡ ❉✐r❛❝ sã♦ ♦❜t✐❞♦s ♣❡❧❛ r♦t❛çã♦ ❞♦ ✈❡t♦r K✳ ❊♥tã♦✱ ♦ ♣r✐♠❡✐r♦ ♣❛ss♦ ♣❛r❛ ♦❜t❡r ❊(K+q) é ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❡
α(k)❡♠ t♦r♥♦ ❞♦ ♣♦♥t♦ K✱ s❡♥❞♦
α(K) = 1 +å−i(K·a1)+å−i(K·a2). ✭✷✳✶✽✮ ❖s ♣r♦❞✉t♦s ❡s❝❛❧❛r❡s ❛❝✐♠❛ ❡♥tr❡ ♦s ✈❡t♦r❡s K✱ a1 ❡ a2 r❡s✉❧t❛♠ ❡♠
K·a1 = 2
3π ✭✷✳✶✾✮
❡
K·a2 = −2
3π. ✭✷✳✷✵✮
❊♥tã♦✱ s✉❜st✐t✉✐♥❞♦ ❛s ❊qs✳ ✭✷✳✶✾✮ ❡ ✭✷✳✷✵✮ ♥❛ ❡①♣r❡ssã♦ ✭✷✳✶✽✮✱ ♦ ✈❛❧♦r ❞❡ α(K) s❡rá
α(K) = 1 +å−i(23π)+åi( 2
3π)= 0. ✭✷✳✷✶✮
❖ ✈❛❧♦r ❞❡ ❊(K) = 0 ❥á ❡r❛ ❡s♣❡r❛❞♦✱ ♣♦✐s ❡①❛t❛♠❡♥t❡ ♥♦s ♣♦♥t♦s ❞❡ ❉✐r❛❝ ❛ ❡♥❡r❣✐❛ é ♥✉❧❛✳ ■ss♦ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞♦ ♥❛ ❋✐❣✳ ✭✷✳✷✮✳
P❛r❛ ♦❜t❡r ♦ ✈❛❧♦r ❞♦ s❡❣✉♥❞♦ t❡r♠♦ ❞❛ ❡①♣❛♥sã♦✱ ✐♥✐❝✐❛❧♠❡♥t❡ é ♥❡❝❡ssár✐♦ ❞❡t❡r♠✐♥❛r ♦s ✈❛❧♦r❡s ❞❡ ∇kα(k) ♥♦ ♣♦♥t♦ k=K✳ ❆ss✐♠✱ ♦ ❣r❛❞✐❡♥t❡ r❡s✉❧t❛ ❡♠
∇kjα(k)|k=K =−i(a1jå−
i(K·a1)+a
2jå−i(K·a2))ˆej, ✭✷✳✷✷✮
♦♥❞❡ j = ①✱② ❡ r❡♣r❡s❡♥t❛ ❛ ♦r✐❡♥t❛çã♦ q✉❡ ♦ ♦♣❡r❛❞♦r ∇ ❛t✉❛✳ ❊♥tã♦✱ s✉❜st✐✲
t✉✐♥❞♦ ♦ r❡s✉❧t❛❞♦ ❞❛ ❊q✳ ✭✷✳✷✷✮ ♥♦ s❡❣✉♥❞♦ t❡r♠♦ ❞❛ ❊q✳ ✭✷✳✶✻✮ ❡ ❝♦♠ ❛❧❣✉♠❛s
✷✳✷ ❉✐s♣❡rsã♦ ❧✐♥❡❛r ❞♦ ●r❛❢❡♥♦
♠❛♥✐♣✉❧❛çõ❡s ❛❧❣é❜r✐❝❛s✱ é ♦❜t✐❞♦
q· ∇kα(k)|k=K =
3
2a0(qx+iqy)❡ −i(π
2). ✭✷✳✷✸✮
▲♦❣♦✱ ♦ ✈❛❧♦r ❞♦ ♠ó❞✉❧♦ ❞❡ α(K+q) s❡rá
|α(K+q)| ≈ 3
2a0|q| ✭✷✳✷✹✮
❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱
❊(K+q)≈ ±3
2a0γ1|q|=±~v❋|q|, ✭✷✳✷✺✮
♦♥❞❡ v❋ = 32a0γ1~ é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❋❡r♠✐ ❡ é ❛♣r♦①✐♠❛❞❛♠❡♥t❡
c
300✱ s❡♥❞♦ c ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③✳ ▲♦❣♦✱ ❡ss❛ ❡q✉❛çã♦ ❞❡♠♦♥str❛ q✉❡ ♦ ❣r❛❢❡♥♦ ♣♦ss✉✐ ✉♠❛ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❧✐♥❡❛r ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❢ér♠✐♦♥s r❡❧❛t✐✈íst✐❝♦s s❡♠ ♠❛ss❛ ❡ ♣r♦♠♦✈❡ ❝♦♥❞✐çõ❡s ❛♣r♦♣r✐❛❞❛s ♣❛r❛ ❡♠✉❧❛r ❢❡♥ô♠❡♥♦s r❡❧❛t✐✈íst✐❝♦s ♥♦ ❞♦♠í♥✐♦ ❞❛ ❋ís✐❝❛ ❞❛ ▼❛tér✐❛ ❈♦♥❞❡♥s❛❞❛ ✭✸✮✳
Cap´ıtulo
3
❖ ♠♦❞❡❧♦ ❞❡ ❆♥❞❡rs♦♥
❖ ♠♦❞❡❧♦ ❞❡ ❆♥❞❡rs♦♥ ❢♦✐ ❡❧❛❜♦r❛❞♦ ❡♠ ✶✾✻✶ ♣♦r P✳ ❲✳ ❆♥❞❡rs♦♥ ✭✶✷✮✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡s❝r❡✈❡r t❡♦r✐❝❛♠❡♥t❡ ❛ ❢♦r♠❛çã♦ ❞❡ ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s ❧♦❝❛❧✐③❛❞♦s ❡♠ ♠❡t❛✐s ❞❡ tr❛♥s✐çã♦ ❞❡✈✐❞♦ à ♣r❡s❡♥ç❛ ❞❡ ✐♠♣✉r❡③❛s✳ ❖ ♠♦❞❡❧♦ ❞❡s❝r❡✈❡ ❛s ✐♠♣✉r❡③❛s ❞❡ ❢♦r♠❛ s✐♠♣❧❡s✱ ❞❡ ♠♦❞♦ q✉❡ ❡ss❛s s❡❥❛♠ r❡♣r❡s❡♥t❛❞❛s ♣♦r ✉♠ ú♥✐❝♦ ♥í✈❡❧ ❞✐s❝r❡t♦ ❞❡ ❡♥❡r❣✐❛✳ ❖ ❢❛t♦ ❞❡ss❡ ♠♦❞❡❧♦ ❞❡s❝r❡✈❡r r❡s✉❧t❛❞♦s t❡ór✐❝♦s ♠✉✐t♦ ♣ró①✐♠♦s ❛♦s r❡s✉❧t❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❢❛③ ❝♦♠ q✉❡ ❡❧❡ s❡❥❛ ❛❝❡✐t♦ ❡ ✉t✐❧✐③❛❞♦ ❛té ♦s ❞✐❛s ❞❡ ❤♦❥❡ ♣❡❧❛ ❝♦♠✉♥✐❞❛❞❡ ❝✐❡♥tí✜❝❛✳ ❉❡ss❡ ♠♦❞♦✱ ♦ ♠♦❞❡❧♦ ❞❡ ❆♥❞❡rs♦♥ s❡rá ✉t✐❧✐③❛❞♦ ♥♦ tr❛❜❛❧❤♦ ♣❛r❛ ❞❡s❝r❡✈❡r ❛s ✐♠♣✉r❡③❛s ♠❛❣♥ét✐❝❛s ♥♦ ❣r❛❢❡♥♦✳ P♦r s✐♠♣❧✐❝✐❞❛❞❡✱ ♥❡st❡ ❝❛♣ít✉❧♦ s❡rá ❞✐s❝✉t✐❞♦ ♦ ♠♦❞❡❧♦ ❞❡ ❆♥❞❡rs♦♥ ♣❛r❛ ✉♠❛ ✐♠♣✉r❡③❛ ❛❞s♦r✈✐❞❛ ❡♠ ✉♠❛ s✉♣❡r❢í❝✐❡ ♠❡tá❧✐❝❛✳ ❯♠❛ ❜r❡✈❡ ❡①♣❧❛♥❛çã♦ ❞♦ ❡❢❡✐t♦ ❑♦♥❞♦ ❡✱ ♣♦st❡r✐♦r♠❡♥t❡✱ ❞♦ ❡❢❡✐t♦ ❋❛♥♦ s❡rã♦ r❡❛❧✐③❛❞❛s ❛ ❧✉③ ❞❡ss❡ ♠♦❞❡❧♦✳
✸✳✶ ❖ ❍❛♠✐❧t♦♥✐❛♥♦ ❞❡ ❆♥❞❡rs♦♥
❖ s✐st❡♠❛ ❢ís✐❝♦ ✐❧✉str❛❞♦ ♥❛ ❋✐❣✳ ✭✸✳✶✮ ♣♦❞❡ s❡r ♠♦❞❡❧❛❞♦ t❡♦r✐❝❛♠❡♥t❡ ♣❡❧♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❞❡ ❆♥❞❡rs♦♥ ❞❡ ✉♠❛ ✐♠♣✉r❡③❛✳ ❊ss❡ ❍❛♠✐❧t♦♥✐❛♥♦ é ❞❡s❝r✐t♦ ❡♠ s❡❣✉♥❞❛ q✉❛♥t✐③❛çã♦ ♥♦ ❡s♣❛ç♦ ❞♦s ♠♦♠❡♥t♦s ♣❡❧❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦✿
H❙■❆▼=X
kσ
εkσc†kσckσ +X
σ
εdσd†σdσ +Ud↑†d↑d†↓d↓+ 1
√
N
X
kσ
[Vc†kσdσ+V∗d†σckσ]. ✭✸✳✶✮ ❖ ♣r✐♠❡✐r♦ t❡r♠♦ ❞♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❞❡s❝r❡✈❡ ♦ ♠❡t❛❧ ❤♦s♣❡❞❡✐r♦ ❝♦♠♦ ✉♠ ❣ás
✸✳✶ ❖ ❍❛♠✐❧t♦♥✐❛♥♦ ❞❡ ❆♥❞❡rs♦♥
❋✐❣✉r❛ ✸✳✶✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ ✐♠♣✉r❡③❛ ❛❞s♦r✈✐❞❛ ♥❛ s✉♣❡r❢í❝✐❡ ♠❡tá❧✐❝❛✳ ❆s ❡s❢❡r❛s ❞♦✉r❛❞❛ ❡ ❝✐♥③❛s r❡♣r❡s❡♥t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ✐♠♣✉r❡③❛ ❡ ❛ s✉♣❡r❢í❝✐❡ ♠❡tá❧✐❝❛✳ ❖ ♣❛râ♠❡tr♦ V ♠♦❞✉❧❛ ❛ ✐♥t❡♥s✐❞❛❞❡ ❞❡ ❤✐❜r✐❞✐③❛çã♦ ❡♥tr❡ ❛ ✐♠♣✉r❡③❛ ❡
♦ ♠❡t❛❧ ❤♦s♣❡❞❡✐r♦✳
❞❡ ❡❧étr♦♥s ♥ã♦ ✐♥t❡r❛❣❡♥t❡✳ ▲♦❣♦✱ ♦ ♦♣❡r❛❞♦r c†kσ (ckσ)❝r✐❛ ✭❛♥✐q✉✐❧❛✮ ❡❧étr♦♥s ♥❛ ❜❛♥❞❛ ❝♦♠ ✉♠❛ ❞❛❞❛ ❡♥❡r❣✐❛ εkσ✱ ♦♥❞❡ k é ♦ ♥ú♠❡r♦ ❞❡ ♦♥❞❛ ❡ σ é ♦ s♣✐♥ ❞♦
❡❧étr♦♥✳
❖ s❡❣✉♥❞♦ t❡r♠♦ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛ ❊q✳ ✭✸✳✶✮ ❝❛r❛❝t❡r✐③❛ ❛ ✐♠♣✉r❡③❛ ❛❞s♦r✈✐❞❛ ♥♦ ♠❡t❛❧ ❤♦s♣❡❞❡✐r♦✱ ♦♥❞❡ d†jσ ✭djσ✮ ❝r✐❛ ✭❛♥✐q✉✐❧❛✮ ❡❧étr♦♥s ❝♦♠ ✉♠ ❞❛❞♦ s♣✐♥ σ
♥♦ ❡st❛❞♦ Edσ✳ ❊ss❡ ♥í✈❡❧ ❞❡ ❡♥❡r❣✐❛ ❞✐s❝r❡t♦ r❡♣r❡s❡♥t❛ ♦s ♦r❜✐t❛✐s ❛tô♠✐❝♦s d ❡ f✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣❛r❛ ♠❡t❛✐s ❞❡ tr❛♥s✐çã♦ ❡ t❡rr❛s r❛r❛s✳
❖ t❡r❝❡✐r♦ t❡r♠♦ ❞♦ ❍❛♠✐❧t♦♥✐❛♥♦ é ♦ r❡s♣♦♥sá✈❡❧ ♣♦r ❞❡s❝r❡✈❡r ❛ ✐♥t❡r❛çã♦ ❈♦✉❧♦♠❜✐❛♥❛ ❧♦❝❛❧U♥♦ sít✐♦ ❞❛ ✐♠♣✉r❡③❛✳ ❚❛❧ ✐♥t❡r❛çã♦ ❢❛✈♦r❡❝❡ ❛ ❢♦r♠❛çã♦ ❞❡ ✉♠ ❡st❛❞♦ ♠❛❣♥ét✐❝♦ ❧♦❝❛❧✐③❛❞♦ ❞❡♥tr♦ ❞♦ ♠♦❞❡❧♦✱ ♣♦✐s ❡❧❛ ✐♥✐❜❡ ❛ ❞✉♣❧❛ ♦❝✉♣❛çã♦ ❞❛ ✐♠♣✉r❡③❛✳ ➱ ✐♠♣♦rt❛♥t❡ ❡♥❢❛t✐③❛r q✉❡ ❛ ❢♦r♠❛çã♦ ❞♦ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦ ❞❡♣❡♥❞❡ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞❛ ♣♦s✐çã♦ ❡♥❡r❣ét✐❝❛ ❞♦ ❡st❛❞♦ Edσ ❡♠ r❡❧❛çã♦ ❛♦ ♥í✈❡❧ ❞❡ ❋❡r♠✐ ❡ ❞❛ ✐♥t❡♥s✐❞❛❞❡ ❞❡ ❤✐❜r✐❞✐③❛çã♦✳ ❚❛❧ ❞✐s❝✉ssã♦ s❡rá r❡❛❧✐③❛❞❛ ♥♦ ❞❡❝♦rr❡r ❞❡st❛ s❡çã♦✳ ▲♦❣♦✱ ❛ ✐♥t❡r❛çã♦ ❈♦✉❧♦♠❜✐♥❛ ♣♦❞❡ s❡r ❞❡s❝r✐t❛ ♣❡❧❛ ✐♥t❡❣r❛❧
U=
ˆ
dr1dr2|φd(f)(r1)|2
e2
|r1−r2||φd(f)(r2)|
2, ✭✸✳✷✮
❝♦♠ φd(f) s❡♥❞♦ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❞♦ ♦r❜✐t❛❧ ❛tô♠✐❝♦ d (f) ❡ ♦s ✈❡t♦r❡s r1 ❡ r2 ❞❡❧✐♠✐t❛♠ ❛s ♣♦s✐çõ❡s ❞♦s ❡❧étr♦♥s ♥♦ sít✐♦ ❞❛ ✐♠♣✉r❡③❛ ✭✷✸✮✳
❖ ú❧t✐♠♦ t❡r♠♦ ❞❛ ❊q✳ ✭✸✳✶✮ é ♦ r❡s♣♦♥sá✈❡❧ ♣♦r ❞❡s❝r❡✈❡r ❛ ❤✐❜r✐❞✐③❛çã♦
✸✳✶ ❖ ❍❛♠✐❧t♦♥✐❛♥♦ ❞❡ ❆♥❞❡rs♦♥
❡♥tr❡ ♦s ❡❧étr♦♥s ❞❛ ❜❛♥❞❛ ❞❡ ❝♦♥❞✉çã♦ ❡ ♦ ❡st❛❞♦ Edσ✳ ❖s ♣❛râ♠❡tr♦s V ❡ V∗ r❡♣r❡s❡♥t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ❤✐❜r✐❞✐③❛çã♦ ❞❛ ✐♠♣✉r❡③❛ ❝♦♠ ♦ ♠❡t❛❧ ❤♦s♣❡❞❡✐r♦ ❡ s❡✉ ❝♦♠♣❧❡①♦ ❝♦♥❥✉❣❛❞♦ ❡N é ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❡st❛❞♦s ❞♦ ❤♦s♣❡❞❡✐r♦✳
◆♦ ♠♦❞❡❧♦ ❞❡ ❆♥❞❡rs♦♥✱ ❛ s♦❜r❡♣♦s✐çã♦ ❞♦s ❡st❛❞♦s ❞❛ ❜❛♥❞❛ ❝♦♠ ♦ ❡st❛❞♦ ❞✐s❝r❡t♦ é ❞❛❞♦ ❡♠ t❡r♠♦s ❞♦ ♣❛râ♠❡tr♦ Γ✱ ❝❤❛♠❛❞♦ ❞❡ ♣❛râ♠❡tr♦ ❞❡ ❆♥❞❡rs♦♥✱ ♦♥❞❡ Γ =π|V|2ρ0 ❡ ρ0 é ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡st❛❞♦s ❞♦ ♠❡t❛❧ ❤♦s♣❡❞❡✐r♦✳ ▲♦❣♦✱ ❡ss❡ ♣❛râ♠❡tr♦ ❞❡s❝r❡✈❡ ❛ s❡♠✐✲❧❛r❣✉r❛ ❞♦ ❛❧❛r❣❛♠❡♥t♦ ❞♦ ❡st❛❞♦ Edσ✱ ✈❡❥❛ ❋✐❣✳ ✸✳✷ ✭❜✮✳ ❆✐♥❞❛✱ ♦ ♣❛râ♠❡tr♦ ❞❡ ❆♥❞❡rs♦♥ é ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ t❡♠♣♦ ❞❡ ✈✐❞❛ ❞♦ ❡❧étr♦♥ ♥♦ ❡st❛❞♦ ❧♦❝❛❧✐③❛❞♦ ❡ t❛❧ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ é ♦❜t✐❞❛ ♣❡❧❛ r❡❣r❛ ❞❡ ❖✉r♦ ❞❡ ❋❡r♠✐ ❞❛❞❛ ♣♦r
1
τ = 2π
|V|2ρ0
~ =
2Γ
~ , ✭✸✳✸✮
♦♥❞❡ τ é ♦ t❡♠♣♦ ❞❡ ✈✐❞❛ ❞♦ ❡❧étr♦♥ ♥♦ ❡st❛❞♦ Edσ ✭✷✸✮✳
❈♦♠♦ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞♦ ♥❛ ❞✐s❝✉ssã♦ ❛❝✐♠❛✱ t♦❞❛ ❛ ❢ís✐❝❛ ❞❡s❝r✐t❛ ♣❡❧♦ ♠♦❞❡❧♦ ❞❡ ❆♥❞❡rs♦♥ s❡rá ❞❛❞❛ ❡♠ t❡r♠♦s ❞♦s ♣❛râ♠❡tr♦s εdσ✱ U ❡ Γ✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛
T ≫T❑✱ ♦s s❡❣✉✐♥t❡s r❡❣✐♠❡s ♣♦❞❡♠ s❡r ❡①♣❧♦r❛❞♦s ♥♦ ♠♦❞❡❧♦✿
✶✳ ◗✉❛♥❞♦U≫εdσ ≫Γ✱ ✐st♦ é✱ ❛ ✐♠♣✉r❡③❛ ❡stá ❢r❛❝❛♠❡♥t❡ ❛❝♦♣❧❛❞❛ ❛♦ ♠❡t❛❧ ❤♦s♣❡❞❡✐r♦✱ ❛ t❛①❛ ❞❡ tr♦❝❛ ❡♥tr❡ ❡❧étr♦♥s ❞❛ ❜❛♥❞❛ ❡ ❞♦ ❡st❛❞♦ ❧♦❝❛❧✐③❛❞♦ sã♦ ❜❛✐①❛s ♦ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ ♦❝♦rr❛ ❛ ❢♦r♠❛çã♦ ❞❡ ✉♠ ❡st❛❞♦ ♠❛❣♥ét✐❝♦ ❧♦❝❛❧✐③❛❞♦ q✉❛♥❞♦ εdσ ❡ εdσ +U ❡stã♦ ❛❜❛✐①♦ ❡ ❛❝✐♠❛ ❞♦ ♥í✈❡❧ ❞❡ ❋❡r♠✐✳ ❚♦❞❛✈✐❛✱ s❡ ❛♠❜♦s ♦s ❡st❛❞♦s ❡st✐✈❡r❡♠ ❛❝✐♠❛ ♦✉ ❛❜❛✐①♦ ❞❡ ε❋ ❛ ✐♠♣✉r❡③❛
s❡ ❡♥❝♦♥tr❛rá ✈❛③✐❛ ♦✉ ❞✉♣❧❛♠❡♥t❡ ♦❝✉♣❛❞❛ ✭✷✸✮✳
✷✳ P❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ U ≫ Γ ≫ εdσ✱ ❛ ✐♠♣✉r❡③❛ s❡ ❡♥❝♦♥tr❛ ❢♦rt❡♠❡♥t❡
❛❝♦♣❧❛❞❛ ❛♦ ❤♦s♣❡❞❡✐r♦✳ ❉❡✈✐❞♦ ❛ ✐♥t❡♥s✐❞❛❞❡ ❞♦ ❛❝♦♣❧❛♠❡♥t♦✱ ♦s ❡st❛❞♦s ❞❛ ✐♠♣✉r❡③❛ ❡ ❞♦ ❝♦♥tí♥✉♦ s❡ s♦❜r❡♣õ❡♠ ❛♦ ♣♦♥t♦ ❞❡ ❛❧❛r❣❛r ♦s ❡st❛❞♦s εdσ
❡ εdσ +U✳ ▲♦❣♦✱ ♦ ♥ú♠❡r♦ ❞❡ ❡❧étr♦♥s ❝♦♥t✐❞♦s ♥❛ ✐♠♣✉r❡③❛ s❡rá ❞❛❞♦ ❡♠ t❡r♠♦s ❞❡ ✉♠❛ ♦❝✉♣❛çã♦ ♠é❞✐❛ ❞❡ ❡❧étr♦♥s ❝♦♠ s♣✐♥s ✉♣ ❡ ❞♦✇♥✳ ❊♥tr❡t❛♥t♦✱ ❡ss❡ ❛❧❛r❣❛♠❡♥t♦ ❛♠♣❧✐✜❝❛ ❛ t❛①❛ ❞❡ tr♦❝❛ ❡♥tr❡ ♦s ❡❧étr♦♥s ❞❡ ❝♦♥❞✉çã♦ ❡ ♦ ❡st❛❞♦ εdσ✱ s✉♣r✐♠✐♥❞♦ ❛ss✐♠ ❛ ❢♦r♠❛çã♦ ❞❡ ✉♠ ❡st❛❞♦ ♠❛❣♥ét✐❝♦✱ ♠❡s♠♦
q✉❛♥❞♦ ❛♣❡♥❛s ♦ ❡st❛❞♦ εdσ é ♦❝✉♣❛❞♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♥❡ss❡ r❡❣✐♠❡ ❝❤❛♠❛❞♦
❞❡ ✈❛❧ê♥❝✐❛ ✐♥t❡r♠❡❞✐ár✐❛✱ só ♦❝♦rr❡rá ❛ ❢♦r♠❛çã♦ ❞❡ ✉♠ ❡st❛❞♦ ♠❛❣♥ét✐❝♦ q✉❛♥❞♦ ❛ ♦❝✉♣❛çã♦ ♠é❞✐❛ ❡♥tr❡ ❡❧étr♦♥s ❞❡ s♣✐♥s ✉♣ ❡ ❞♦✇♥ ❢♦r❡♠ ❞✐❢❡r❡♥t❡s✳
