❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s
❉❡♣❛rt❛♠❡t♦ ❞❡ ❋ís✐❝❛
❊❢❡✐t♦s ❞❡ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❞❡♣❡♥❞❡♥t❡ ❞❛ ♣♦s✐çã♦ ❡♠
♠♦♥♦❝❛♠❛❞❛s ❡ ❜✐❝❛♠❛❞❛s ❞❡ ❣r❛❢❡♥♦
■♦❧❛♥❞❛ ▼❛r✐❛♥♦ ❚❛✈❛r❡s
■♦❧❛♥❞❛ ▼❛r✐❛♥♦ ❚❛✈❛r❡s
❊❢❡✐t♦s ❞❡ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❞❡♣❡♥❞❡♥t❡ ❞❛ ♣♦s✐çã♦ ❡♠
♠♦♥♦❝❛♠❛❞❛s ❡ ❜✐❝❛♠❛❞❛s ❞❡ ❣r❛❢❡♥♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈✉rs♦ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡✲ ❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❋ís✐❝❛✳
❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ❘❛✐♠✉♥❞♦ ◆♦❣✉❡✐r❛ ❞❛ ❈♦st❛ ❋✐❧❤♦
Dados Internacionais de Catalogação na Publicação Universidade Federal do Ceará
Biblioteca do Curso de Física
T23e Tavares, Iolanda Mariano
Efeitos de campo magnético dependente da posição em monocamadas e bicamadas de grafeno / Iolanda Mariano Tavares. – Fortaleza, 2014.
60 f.: il. color. enc.; 30 cm.
Dissertação (Mestrado em Física) – Universidade Federal do Ceará, Centro de Ciências, Departamento de Física, Programa de Pós-Graduação em Física, Mestrado Acadêmico em Física, Fortaleza, 2014.
Orientação: Prof. Dr. Raimundo Nogueira da Costa Filho. Área de concentração: Física da Matéria Condensada.
1. Grafeno. 2. Níveis de Landau. 3. Estrutura eletrônica do grafeno. I. Costa Filho, Raimundo Nogueira da. II. Título.
iii
iv
❆●❘❆❉❊❈■▼❊◆❚❖❙
❙ã♦ ✐♥ú♠❡r❛s ♣❡ss♦❛s às q✉❛✐s ❡✉ s♦✉ ✐♠❡♥s❛♠❡♥t❡ ❣r❛t❛✱ ♠❛s✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r tê✲❧❛s ❝♦❧♦❝❛❞♦ ♥♦ ♠❡✉ ❝❛♠✐♥❤♦✱ ♠❡ ♣r♦♣♦r❝✐♦♥❛♥❞♦ ✉♠❛ ót✐♠❛ ❡①♣❡r✐ê♥❝✐❛ ❞❡ ✈✐❞❛✳ ❆ ❝♦♠❡ç❛r ♣❡❧❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♠✐♥❤❛ ♠ã❡ ■✈❛♥✐ ▼❛r✐❛♥♦✱ ♠❡✉ ♣❛✐ ❘✐❝❛r❞♦ ❚❛✈❛r❡s ❡ ♠✐♥❤❛ ❞♦❝❡ ✐r♠ã ■r✐s✱ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦ só ❢♦✐ ♣♦ssí✈❡❧ ❝♦♠ ♦ ❛♣♦✐♦ ❝♦♥st❛♥t❡ ❞❡❧❡s✳ ❉❡✈♦ ♠✉✐t♦ à ❝r✐❛çã♦ q✉❡ t✐✈❡ ❞♦s ♠❡✉s ♣❛✐s✱ ❝♦♠ ❝❡rt❡③❛ sã♦ ❛s ♠❛✐♦r❡s r❡❢❡rê♥❝✐❛s q✉❡ ❡✉ ❜✉s❝♦ s❡❣✉✐r✳ ❙♦✉ ♠✉✐t♦ ❣r❛t❛ ♣❡❧♦ ❝❛r✐♥❤♦ ❡ ❛t❡♥çã♦ q✉❡ ♠❡ ❞❡❞✐❝❛♠✱ ❡s♣❡r♦ s❡♠♣r❡ ❝♦rr❡s♣♦♥❞❡r à ❛❧t✉r❛✳
❆♦ Pr♦❢❡ss♦r ❘❛✐♠✉♥❞♦ ◆♦❣✉❡✐r❛✱ ♠❛♥✐❢❡st♦ ♠✐♥❤❛ ❛❞♠✐r❛çã♦ ❡ r❡s♣❡✐t♦ ♣❡❧❛ ♦r✐❡♥t❛çã♦ q✉❡ r❡❝❡❜✐ ♥❛ ❢♦r♠✉❧❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ q✉❡ ❝♦♥tr✐❜✉✐✉ ♣❛r❛ ♦ ♠❡✉ ❛♠❛❞✉r❡❝✲ ✐♠❡♥t♦ ❡♥q✉❛♥t♦ ♣❡ss♦❛ ❡ ♣❡sq✉✐s❛❞♦r❛✳ ❙❡✉ ♣r♦✜ss✐♦♥❛❧✐s♠♦ ❡ ❞❡❞✐❝❛çã♦ sã♦ r❡❢❡rê♥❝✐❛s q✉❡ ✐r❡✐ ❧❡✈❛r ❛♦ ❧♦♥❣♦ ❞❛ ♠✐♥❤❛ ❝❛rr❡✐r❛ ❛❝❛❞ê♠✐❝❛✳
❙♦✉ ❣r❛t❛ ❛ t♦❞♦s ♣r♦❢❡ss♦r❡s q✉❡ t✐✈❡ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❘❡❣✐♦♥❛❧ ❞♦ ❈❛r✐r✐ ✲ ❯❘❈❆✱ q✉❡ ✜③❡r❛♠ ❝r❡s❝❡r ❡♠ ♠✐♠ ❛ s❡❞❡ ♣♦r ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ♠❡ ♣r♦♣♦r❝✐♦♥❛r❛♠ ♦ ✐♥❣r❡ss♦ ♥♦ ❝✉rs♦ ❞❡ ♠❡str❛❞♦✱ ❝♦♠ ❞❡st❛q✉❡ ♣❛r❛ ♦s ♣r♦❢❡ss♦r❡s ❆❧❡①❛♥❞r❡ ▼❛❣♥♦ ✲ ♠❡✉ ♣r✐♠❡✐r♦ ♦r✐❡♥t❛❞♦r ❞❡ ✐♥✐❝✐❛çã♦ ❝✐❡♥tí✜❝❛ ✲ ❋r❛♥❝✐s❝♦ ❆✉❣✉st♦✱ ❊❞✉❛r❞♦ ❙♦✉③❛ ❡ ❏♦ã♦ ❍❡r♠í♥✐♦ ✭❯❋❈❆✮✱ ♣❡❧❛ ❡♥♦r♠❡ ❛❥✉❞❛ ❞✉r❛♥t❡ ❛ s❡❧❡çã♦ ❞♦ ♠❡str❛❞♦✳ ❚❛♠❜é♠ ❛♦s ♣r♦✲ ❢❡ss♦r❡s ❞❛ ♣ós✲❣r❛❞✉❛çã♦ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ✲ ❯❋❈✱ ♠❡✉ ♠✉✐t♦ ♦❜r✐❣❛❞❛✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s ❞❡ ❧♦♥❣❛ ❞❛t❛✱ ❞❛ é♣♦❝❛ ❞❡ ❣r❛❞✉❛çã♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ❆♥❛ ■③❛❜❡❧❛ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ❡①❝❡❧❡♥t❡s ❝♦♥✈❡rs❛s✳ ❚❛♠❜é♠ ❛♦s ❛♠✐❣♦s q✉❡ ✜③ ❞✉r❛♥t❡ ♦ ♠❡str❛❞♦✱ ♠❡✉s ❝♦❧❡❣❛s ❞❡ ❛♣❛rt❛♠❡♥t♦ ❏♦sé ■❧❤❛♥♦ ❡ ❉❛♠✐❛♥❛ ❱✐❡✐r❛✱ q✉❡ ❛♠❡♥✐③❛r❛♠ ❛ s❛✉❞❛❞❡ q✉❡ s❡♥t✐ ❡st❛♥❞♦ ✜s✐❝❛♠❡♥t❡ ❧♦♥❣❡ ❞❛ ♠✐♥❤❛ ❢❛♠í❧✐❛ ❞✉r❛♥t❡ ❡ss❡s ú❧t✐♠♦s ❞♦✐s ❛♥♦s✳ ❉♦s ❛♠✐❣♦s ♠❛✐s r❡❝❡♥t❡s✱ t❡♥❤♦ ♠✉✐t♦ ❛ ❛❣r❛❞❡❝❡r ❛♦ ▼❛r❝✉s ❱✐♥í❝✐✉s✱ s❡♠ ♦ q✉❛❧ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♠♣♦s✐çã♦ ❞❡st❡ tr❛❜❛❧❤♦ s❡r✐❛ ♠✉✐t♦ ♠❛✐s ❞✐❢í❝✐❧✱ ♥✉♥❝❛ ✈♦✉ ❡sq✉❡❝❡r s❡✉ ❛♣♦✐♦ ❡♠♦❝✐♦♥❛❧ ❡ ✐♥t❡❧❡❝t✉❛❧ ❞✉r❛♥t❡ ♦s ♠♦♠❡♥t♦s ❜♦♥s ❡ r✉✐♥s✳
P♦r ✜♠✱ ♠❛s ♥ã♦ ♠❡♥♦s ✐♠♣♦rt❛♥t❡✱ r❡❣✐str♦ ♠❡✉ ❛❣r❛❞❡❝✐♠❡♥t♦ à ❝♦♦r❞❡♥❛çã♦ ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❞❛ ❯❋❈✳ ❊ ❛♦ ❈♦♥s❡❧❤♦ ◆❛❝✐♦♥❛❧ ❞❡ ❉❡s❡♥✲ ✈♦❧✈✐♠❡♥t♦ ❈✐❡♥tí✜❝♦ ❡ ❚❡❝♥♦❧ó❣✐❝♦ ✭❈◆Pq✮✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
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✧❬✳✳✳❪ ❖ q✉❡ q✉❡r♦ ❞✐③❡r é q✉❡ ♥ã♦ ❤á s♦♠❡♥t❡ ❛ ❜❡❧❡③❛ ♥❡ss❛ ❞✐♠❡♥sã♦✱ ❡♠ ✉♠ ❝❡♥tí♠❡tr♦❀ ❡①✐st❡ ❜❡❧❡③❛ t❛♠✲ ❜é♠ ♥❛s ♠❡♥♦r❡s ❞✐♠❡♥sõ❡s✱ ♥❛ ❡str✉t✉r❛ ✐♥t❡r♥❛✱ ❡ ♥♦ s❡✉ ♣r♦❝❡ss♦✧✳
✭❘✐❝❤❛r❞ ❋❡②♥♠❛♥ ✲ ❖❞❡ ❛ ✉♠❛ ✢♦r✮✳
✧❬✳✳✳❪ ❊♠ t♦r♥♦ ❞❡ t❡✉s ♣❛ss♦s✱ ❛ ♣❛✐s❛❣❡♠ q✉❡ t❡ ❛❜r✐❣❛ s❡rá s❡♠♣r❡ ❡♠ t✉❛ ❛♣r❡❝✐❛çã♦ ❛q✉✐❧♦ q✉❡ ♣❡♥✲ s❛s ❞❡❧❛✱ ♣♦rq✉❡ ❝♦♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛ q✉❡ ❛♣❧✐❝❛r❡s à ◆❛t✉r❡③❛✱ ♦❜r❛ ✈✐✈❛ ❞❡ ❉❡✉s✱ ❛ ◆❛t✉r❡③❛ ✐❣✉❛❧♠❡♥t❡ t❡ ♠❡❞✐rá✧✳
vi
❘❊❙❯▼❖
◆♦s ú❧t✐♠♦s ❛♥♦s ♦ ❣r❛❢❡♥♦ t❡♠ s❡ ♠♦str❛❞♦ ✉♠ ✐♥t❡r❡ss❛♥t❡ ♦❜❥❡t♦ ❞❡ ✐♥✈❡st✐❣❛çã♦ ♣❛r❛ ❝✐❡♥t✐st❛s ❛♦ r❡❞♦r ❞♦ ♠✉♥❞♦✱ t❛♥t♦ ♣❡❧❛s s✉❛s ❝❛r❛❝t❡ríst✐❝❛s ♣❡❝✉❧✐❛r❡s ❜❡♠ ❝♦♠♦ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧❡trô♥✐❝❛s✳ ▼✉✐t❛s ♣❡sq✉✐s❛s ❥á ❢♦r❛♠ r❡❛❧✐③❛❞❛s ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❞❡s✈❡♥❞❛r ❡ss❡ ❛✐♥❞❛ r❡❝é♠ ❞❡s❝♦❜❡rt♦ ❛❧ótr♦♣♦ ❞♦ ❝❛r❜♦♥♦✱ ❥á t❡♥❞♦ r❡♥❞✐❞♦ ✉♠ ♣rê♠✐♦ ◆♦❜❡❧ ❡♠ ✷✵✶✵✳ ❆q✉✐ s❡rã♦ ❛❜♦r❞❛❞❛s ❛❧❣✉♠❛s ❞❡ss❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ ❣r❛❢❡♥♦ ❢♦❝❛♥❞♦ ❡♠ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡str✉t✉r❛✐s ❞❡ r❡❞❡ ❡ t❛♠❜é♠ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧❡trô♥✐❝❛s✳ ❙❡rá ❡st✉❞❛❞❛ ❛ ✐♥t❡r❛çã♦ ❞♦s ❢ér♠✐♦♥s ❞❡ ❉✐r❛❝✱ ♣❛rtí❝✉❧❛s r❡❧❛t✐✈íst✐❝❛s s❡♠ ♠❛ss❛✱ s♦❜ ❛ ❛çã♦ ❞❡ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦ ❡♠ ✉♠❛ ❡ ❞✉❛s ❝❛♠❛❞❛s ❞❡ ❣r❛❢❡♥♦✱ ❝♦♠ ❜❛s❡ ♥❛ t❡♦r✐❛ ❞❡ ❉✐r❛❝ ♣❛r❛ ❜❛✐①❛s ❡♥❡r❣✐❛s ❡ ♥♦ ♠♦❞❡❧♦ ❞❡ ❛♣r♦①✐♠❛çã♦tight −binding✱ ❝♦♠♣r♦✈❛♥❞♦ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ♦s ❞♦✐s ♠♦❞❡❧♦s✳ ❊s❝r❡✈❡♥❞♦ ♦ ❍❛♠✐❧t♦♥✐❛♥♦ ♣❛r❛ ✉♠❛ ❢♦❧❤❛ ❞❡ ❣r❛❢❡♥♦ ❝♦♥✜❣✉r❛✲s❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r✱ s❡♥❞♦ ♣♦ssí✈❡❧ ♦❜t❡r ❛ ❞✐s♣❡rsã♦ ❞❛s ❜❛♥❞❛s ❞❡ ❡♥❡r❣✐❛s ❞♦ ♠❛t❡r✐❛❧ ❡ ♦s ❝❤❛♠❛❞♦s ♥í✈❡✐s ❞❡ ▲❛♥❞❛✉✱ ❡st❡s ú❧t✐♠♦s ♦r✐❣✐♥❛❞♦s ♣❡❧❛ ♣r❡s❡ç❛ ❞❡ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ ❞❡ ✐♥t❡r❛çã♦✳ ❊♠ s❡❣✉✐❞❛ ❡ss❡s ♠❡s♠♦s ♣❛ss♦s s❡rã♦ r❡❛❧✐③❛❞♦s ♣❛r❛ ♦ s✐st❡♠❛ ❞❡ ❜✐❝❛♠❛❞❛✳ ❊♠ ❛❞✐çã♦✱ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛q✉✐ ♦s ❡❢❡✐t♦s ❞❡ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❞❡♣❡♥❞❡♥t❡ ❞❛ ♣♦s✐çã♦✱ ❛ ✜♠ ❞❡ s❡r❡♠ ❡s❝r✐t♦s ♦s ♥í✈❡✐s ❞❡ ▲❛♥❞❛✉✳ ❊st❡ tr❛❜❛❧❤♦ ✐rá ❞❡♠♦♥str❛r ❛ss✐♠✱ q✉❡ ✉♠ ❞♦s ❢❛t♦r❡s q✉❡ ❢❛③❡♠ ♦ ❣r❛❢❡♥♦ tã♦ ❛tr❛t✐✈♦ ♣❛r❛ ♣❡sq✉✐s❛ é ❥✉st❛♠❡♥t❡ ❛ ✐♥t❡r❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s ♥❡st❡ ♠❛t❡r✐❛❧✳
vii
❆❇❙❚❘❆❈❚
■♥ t❤❡ ♣❛st ❢❡✇ ②❡❛rs ❣r❛♣❤❡♥❡ ❤❛s ❜❡❡♥ s❤♦✇♥ t♦ ❜❡ ❛♥ ✐♥t❡r❡st✐♥❣ ♦❜❥❡❝t ♦❢ ✐♥✈❡st✐❣❛t✐♦♥ t♦ s❝✐❡♥t✐sts ❛r♦✉♥❞ t❤❡ ✇♦r❧❞✱ ❞✉❡ t♦ ✐ts ♣❡❝✉❧✐❛r ❝❤❛r❛❝t❡r✐st✐❝s ❛s ✇❡❧❧ ❛s ✐ts ❡❧❡❝tr♦♥✐❝s ♣r♦♣❡rt✐❡s✳ ▼✉❝❤ r❡s❡❛r❝❤ ❤❛✈❡ ❜❡❡♥ ❞♦♥❡ ✐♥ ♦r❞❡r t♦ ✉♥✈❡✐❧ t❤✐s st✐❧❧ ♥❡✇❧② ❞✐s❝♦✈❡r❡❞ ❝❛r❜♦♥ ❛❧❧♦tr♦♣❡✱ ❤❛✈✐♥❣ ❛❧r❡❛❞② r❡♥❞❡r❡❞ ❛ ◆♦❜❡❧ ♣r✐③❡ ✐♥ ✷✵✶✵✳ ❍❡r❡ ✐t✬❧❧ ❜❡ ❞✐s❝✉ss❡❞ s♦♠❡ ♦❢ t❤❡s❡ ❣r❛♣❤❡♥❡ ❝❤❛r❛❝t❡r✐st✐❝s ❢♦❝✉s✐♥❣ ♦♥ t❤❡ str✉❝t✉r❛❧ ❧❛tt✐❝❡ ♣r♦♣❡rt✐❡s ❛♥❞ ❛❧s♦ ♦♥ t❤❡ ❡❧❡❝tr♦♥✐❝s ♣r♦♣❡rt✐❡s✳ ❚❤❡ ✐♥t❡r❛❝t✐♦♥ ♦❢ ❉✐r❛❝ ❢❡r♠✐♦♥s✱ ♠❛ss❧❡ss r❡❧❛t✐✈✐st✐❝ ♣❛rt✐❝❧❡s✱ ✉♥❞❡r ❛♥ ❡①t❡r♥❛❧ ❤♦♠♦❣❡♥❡♦✉s ♠❛❣♥❡t✐❝ ✜❡❧❞ ✐♥ ♦♥❡ ❛♥❞ t✇♦ ❧❛②❡rs ♦❢ ❣r❛♣❤❡♥❡ ✇✐❧❧ ❜❡ st✉❞✐❡❞✱ ❜❛s❡❞ ♦♥ t❤❡ ❧♦✇✲❡♥❡r❣② ❉✐r❛❝ t❤❡♦r② ❛♥❞ t❤❡ t✐❣❤t✲❜✐♥❞✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥ ♠♦❞❡❧✱ st❛t✐♥❣ ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ♠♦❞❡❧s✳ ❇② ✇r✐t✐♥❣ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ❢♦r ❛ s✐♥❣❧❡ s❤❡❡t ♦❢ ❣r❛♣❤❡♥❡✱ ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠ ✐s s❡t ✉♣✱ ❛♥❞ t❤❡ ❡♥❡r❣② ❜❛♥❞ ❞✐s♣❡rs✐♦♥ ♦❢ t❤❡ ♠❛t❡r✐❛❧ ❛♥❞ t❤❡ s♦ ❝❛❧❧❡❞ ▲❛♥❞❛✉ ❧❡✈❡❧s ❛r❡ ❢♦✉♥❞✱ t❤❡ ❧❛tt❡r✱ ♦r✐❣✐♥❛t❡❞ ❜② t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ♠❛❣♥❡t✐❝ ✜❡❧❞ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♣❧❛♥❡✳ ◆❡①t✱ t❤❡s❡ s❛♠❡ st❡♣s ✇❡r❡ ♣❡r❢♦r♠❡❞ t♦ t❤❡ ❜✐❧❛②❡r s✐st❡♠✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ❡✛❡❝ts ♦❢ ❛ ♣♦s✐t✐♦♥ ❞❡♣❡♥❞❡♥t ♠❛❣♥❡t✐❝ ✜❡❧❞ ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞ ❤❡r❡✱ s♦ t❤❛t t❤❡ ▲❛♥❞❛✉ ❧❡✈❡❧s ❝❛♥ ❜❡ ✇r✐tt❡♥✳ ❚❤✐s ✇♦r❦ ✇✐❧❧ ❞❡♠♦♥str❛t❡ t❤❛t ♦♥❡ ♦❢ t❤❡ ❢❛❝t♦rs t❤❛t ♠❛❦❡ ❣r❛♣❤❡♥❡ s♦ ❛ttr❛❝t✐✈❡ t♦ r❡s❡❛r❝❤ ✐s ♣r❡❝✐s❡❧② t❤❡ ♣❛rt✐❝❧❡ ✐♥t❡r❛❝t✐♦♥ ✐♥ t❤✐s ♠❛t❡r✐❛❧✳
viii
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❖ ❣r❛❢❡♥♦ ✭❛✮ ♣♦❞❡ s❡r ❡♠♣✐❧❤❛❞♦ ❢♦r♠❛♥❞♦ ✸❉✲❣r❛✜t❡ ✭❜✮✱ ♣♦❞❡ s❡r ❡♥✲ r♦❧❛❞♦ ❢♦r♠❛♥❞♦ ✶❉✲♥❛♥♦t✉❜♦ ✭❝✮ ♦✉ ❡♥✈♦❧t♦✱ ❢♦r♠❛♥❞♦ ✵❉✲❢✉❧❡r❡♥♦ ✭❞✮ ❬✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ✭❛✮❊sq✉❡♠❛ ❞♦ ❡❢❡✐t♦ ❍❛❧❧ q✉â♥t✐❝♦ ❡♠ ✉♠ s❡♠✐❝♦♥❞✉t♦r ❜✐❞✐♠❡♥s✐♦♥❛❧
❝♦♥✈❡♥❝✐♦♥❛❧✱ ✭❜✮ ♥❛ ❜✐❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦ ❡ ✭❝✮ ♥❛ ♠♦♥♦❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦✳ ❖s ♣✐❝♦s ❧❛r❛♥❥❛s ✭❡❧étr♦♥s✮ ❡ ❛③✉✐s ✭❜✉r❛❝♦s✮ ♠♦str❛♠ ❛s s❡q✉ê♥❝✐❛s ❞❡ ♥í✈❡✐s ❞❡ ▲❛♥❞❛✉ ❡♠ ❢✉♥çã♦ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ♣♦rt❛❞♦r❡s n ❡ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ B✳ ❊♠ ✈❡r♠❡❧❤♦ ❡stã♦ ❛s ♠❡❞✐❞❛s ❞❡ σxy✳ ❆ ❝♦♥st❛♥t❡ e r❡♣r❡✲
s❡♥t❛ ❛ ❝❛r❣❛ ❞♦ ❡❧étr♦♥✱ h é ❛ ❝♦♥st❛♥t❡ ❞❡ P❧❛♥❝❦ ❡g✱ ❛ ❞❡❣❡♥❡r❡s❝ê♥❝✐❛ ❞❡ ✈❛❧❡ ❡ ❞❡ s♣✐♥ ❬✻❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ❋♦t♦❣r❛✜❛ ❞❡ ✉♠❛ ③♦♥❛ ❝♦♠ ✉♠❛ ❡ ❞✉❛s ❝❛♠❛❞❛s ❞❡ ❣r❛❢❡♥♦ ✉t✐❧✐③❛❞❛s
♥✉♠ ❡♥s❛✐♦ ❞❡ tr❛♥s♠✐tâ♥✐❝❛ ♦♥❞❡ s❡ ✈❡r✐✜❝❛ ❛ ❛❜s♦rçã♦ ❞❡ 2,3%❞❛ ❧✉③ ❬✶✷❪✳ ✹
✶✳✹ ❊sq✉❡♠❛ ❞❡ ✉♠ ❞✐s♣♦s✐t✐✈♦ ❞❡ ❣r❛❢❡♥♦ ♥❛ ❢♦r♠❛ ❞❡ ✉♠❛ ♣♦♥t❡ ❍❛❧❧ ❝♦♥✲ str✉í❞♦ ❡♠ ✉♠ s✉❜str❛t♦ Si❝♦❜❡rt♦ ♣♦r ✉♠❛ ❝❛♠❛❞❛ ❞❡ SiO2 ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✻ ✶✳✺ ✭❛✮ ❆rr❛♥❥♦ ❤❡①❛❣♦♥❛❧ ❞❡ át♦♠♦s ❞❡ ❝❛r❜♦♥♦ q✉❡ ❢♦r♠❛♠ ✉♠❛ ❢♦❧❤❛ ❞❡
❣r❛❢❡♥♦✳ ❊st❛ ❡str✉t✉r❛ é ✉♠❛ r❡❞❡ ❞❡ ❇r❛✈❛✐s ❝♦♠ ❞♦✐s át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ♣♦r ❝é❧✉❧❛ ✉♥✐tár✐❛ ✭❡♠ ❝✐♥③❛✮✱ ❡ s❡✉s ✈❡t♦r❡s ♣r✐♠✐t✐✈♦s sã♦ ❛1 ❡ ❛2✳ ❊♠ ✭❜✮ ❡stá r❡♣r❡s❡♥t❛❞❛ ❛ r❡❞❡ r❡❝í♣r♦❝❛ ❞❛ r❡❞❡ ♠♦str❛❞❛ ❡♠ ✭❛✮✳ ❆1 ❡ ❆2 sã♦ ♦s ✈❡t♦r❡s ♣r✐♠✐t✐✈♦s ❞❛ r❡❞❡ r❡❝í♣r♦❝❛✳ ❆ ár❡❛ ❡♠ ❝✐♥③❛ é ❛ ♣r✐♠❡✐r❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥✱ ♦s ♣♦♥t♦s ❑ ❡ ❑✬ sã♦ ♦s ♣♦♥t♦s ♦♥❞❡ ❛s ❜❛♥❞❛s ❞❡ ✈❛❧ê♥❝✐❛ ❡ ❞❡ ❝♦♥❞✉çã♦ s❡ t♦❝❛♠ ❬✶✾❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✻ ❋♦r♠❛çã♦ ❞♦s ♦r❜✐t❛✐ssp2✱ ❛tr❛✈és ❞❛ ❝♦♠❜✐♥❛çã♦ ❞♦ ♦r❜✐t❛❧ s ❝♦♠ ♦r❜✐t❛✐s
px ❡py✱ ❡♥q✉❛♥t♦ ♦ ♦r❜✐t❛❧ pz ♣❡r♠❛♥❡❝❡ ✐♥✈❛r✐❛♥t❡ ❬✷✵❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ix
✷✳✸ ◆í✈❡✐s ❞❡ ▲❛♥❞❛✉ En ❡♠ ❢✉♥çã♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❇✱ ❛s ❝✉r✈❛s r❡♣r❡✲
s❡♥t❛♠ ❛ ✈❛r✐❛çã♦ ♥❂✵✱✳✳✳✱✹✳ ❊st❡s sã♦ r❡s✉❧t❛❞♦s ♣❛r❛ ❛ ❊q✳✷✳✷✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✹ ◆í✈❡✐s ❞❡ ▲❛♥❞❛✉ En ❡♠ ❢✉♥çã♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ B0✱ ❛s ❝✉r✈❛s r❡♣r❡✲
s❡♥t❛♠ ❛ ✈❛r✐❛çã♦ ♥❂✵✱✳✳✳✱✹✳ ❊st❡s sã♦ r❡s✉❧t❛❞♦s ♣❛r❛ ❛ ❊q✳✷✳✹✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✶ ❊str✉t✉r❛ ❝r✐st❛❧✐♥❛ ❞♦ ❣r❛❢❡♥♦ ❝♦♠ ❞✉❛s s✉❜r❡❞❡s ❆ ❡ ❇✳ ❖s ✈❡t♦r❡s ♣r✐♠✐✲
t✐✈♦s q✉❡ ❢♦r♠❛♠ ❛ r❡❞❡ sã♦ ❛1 ❡ ❛2✳ ❊♥q✉❛♥t♦ ❘i ✭❝♦♠ ✐❂✶✱✷✱✸✮ ❧♦❝❛❧✐③❛♠
♦s ✈✐③✐♥❤♦s ♠❛✐s ♣ró①✐♠♦s ❬✷✸❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✷ ❉✐s♣❡rsã♦ ❞❛s ❜❛♥❞❛s ❞❡ ❡♥❡r❣✐❛s ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❡ ❊q✳ ✸✳✶✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✸ ▼♦♥♦❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦ ✐♠❡rs❛ ❡♠ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ✉♥✐❢♦r♠❡ B =
B0z✳ ❖♥❞❡ ✉s❛♠♦sˆ B0 = 0,5πT✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✹ ▼♦♥♦❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦ ✐♠❡rs❛ ❡♠ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ✉♥✐❢♦r♠❡ B =
B0z✳ ❆q✉✐ t❡♠♦sˆ kx = 0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✺ ▼♦♥♦❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦ ✐♠❡rs❛ ❡♠ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ♣❡r✐ó❞✐❝♦ B =
B0cos(kBx)ˆz✳ ❆q✉✐ ✉s❛♠♦s kx = 0✱ ❡B0 =πT✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✸✳✻ ▼♦♥♦❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦ ✐♠❡rs❛ ❡♠ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ♣❡r✐ó❞✐❝♦ B =
B0cos(kBx)ˆz✳ ❆q✉✐ ✉s❛♠♦s ky = 0✱ ❡B0 = 1,5πT✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✶ ■❧✉str❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❛ ❡str✉t✉r❛ ❞❡ ❞✉❛s ❢♦❧❤❛s ❞❡ ❣r❛❢❡♥♦✳ ❖ s❛❧t♦
❞✐r❡t♦ ❡♥tr❡ ♦s ♣♦♥t♦s ✈❡r♠❡❧❤♦s ❞❛ ❢♦❧❤❛ ❞❡ ❜❛✐①♦ ❡ ❞❡ ❝✐♠❛ é r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ ♣❛râ♠❡tr♦ γ1✳ ❖ s❛❧t♦ ❞❡ ❜❛✐①❛ ❡♥❡r❣✐❛ ❡♥tr❡ ♦s ♣♦♥t♦s ❛③✉✐s ❡stá r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ ♣❛râ♠❡tr♦ γ3✳ ❘❡♣r❡s❡♥t❛♥❞♦ ♦ s❛❧t♦ ❡♥tr❡ ♦s ♣♦♥t♦s ✈❡r♠❡❧❤♦s ❡ ♦s ❛③✉✐s t❡♠✲s❡ γ4 ❬✷✼❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✷ ❱✐sã♦ ❞❡ ❝✐♠❛ ❞❛ ❡str✉t✉r❛ ❝r✐st❛❧✐♥❛ ❞❡ ❞✉❛s ❢♦❧❤❛s ❞❡ ❣r❛❢❡♥♦✳ ➪t♦♠♦s
A1 ❡ B1 ❞❛ ❢♦❧❤❛ ❞❡ ❜❛✐①♦ sã♦ ♠♦str❛❞♦s ❡♠ ♣r❡t♦ ❡ ❛③✉❧✳ ◆❛ ❢♦❧❤❛ ❞❡ ❝✐♠❛ A2 ❡ B2 sã♦ ♦s ♣♦♥t♦s ❛③✉✐s ❡ ✈❡r♠❡❧❤♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ ♣❛rt❡ s♦♠❜r❡❛❞❛ ✐♥❞✐❝❛ ❛ ❝é❧✉❧❛ ✉♥✐tár✐❛ ❝♦♥✈❡♥❝✐♦♥❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✸ ❇❛♥❞❛s ❞❡ ❜❛✐①❛ ❡♥❡r❣✐❛ ❞❛ ❜✐❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦ ❝r❡s❝❡♥❞♦ ❛ ♣❛rt✐r ❞♦
♦r❜✐t❛❧2pz ♣❧♦t❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦ ❡✐①♦kx ♥♦ ❡s♣❛ç♦ r❡❝í♣r♦❝♦✱ ✐♥t❡rs❡❝t❛♥❞♦
♦s ♣♦♥t♦s ❑✬ ✭K−✮ ❡ ❑ ✭K+✮✱ Γ ❝♦rr❡s♣♦♥❞❡ ❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥✳ ❊♠
❞❡st❛q✉❡ t❡♠♦s ❛s ❜❛♥❞❛s ♥❛ ♣r♦①✐♠✐❞❛❞❡ ❞♦ ♣♦♥t♦ ❑ ❬✷✻❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ❆✳✶ ❖s ✈❡t♦r❡s ♣r✐♠✐t✐✈♦s~a1 ❡~a2✱ ♦ ✈❡t♦r ❡s♣❛❝✐❛❧ ~x ❝♦♥❡❝t❛♥❞♦ ♦s ❝❡♥tr♦s ❞❡
❞♦✐s ❤❡①á❣♦♥♦s ❡ ♦s ✈❡t♦r❡s ✉♥✐tár✐♦s ~eA ❡ ~eB ❞✐st✐♥❣✉✐♥❞♦ ❛ s✉❜r❡❞❡ ❆
x
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✶✳✶ ❈♦♥❞✉t✐✈✐❞❛❞❡s tér♠✐❝❛s ❞♦ ❣r❛❢❡♥♦✱ ♥❛♥♦t✉❜♦ ❞❡ ❝❛r❜♦♥♦✱ ❞✐❛♠❛♥t❡ ❡ ❝♦❜r❡✳ ✺ ✹✳✷ ❱❛❧♦r❡s ❞♦s ♣❛râ♠❡tr♦s t✐❣❤t✲❜✐♥❞✐♥❣ ❞❡t❡r♠✐♥❛❞♦s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ♣❛r❛
♦ ❣r❛✜t❡ ❡ ❜✐❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✸ ❊♥❡r❣✐❛ ❞❡ ❢♦r♠❛çã♦ ♣❛r❛ ♦s ❞✐✈❡rs♦s ❡♠♣✐❧❤❛♠❡♥t♦s ❞❛ ❜✐❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦
xi
❙✉♠ár✐♦
❆❣r❛❞❡❝✐♠❡♥t♦s ✈
❘❡s✉♠♦ ✈✐✐
❆❜str❛❝t ✈✐✐✐
▲✐st❛ ❞❡ ❋✐❣✉r❛s ①
▲✐st❛ ❞❡ ❚❛❜❡❧❛s ①✐
✶ ■♥tr♦❞✉çã♦ ✶
✶✳✶ ●r❛❢❡♥♦✿ ❛s♣❡❝t♦s ❣❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ❉✐s♣♦s✐t✐✈♦s ❞❡ ❣r❛❢❡♥♦ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❆♣r❡s❡♥t❛çã♦ ❞❛ t❡♦r✐❛ ❞❡ ❉✐r❛❝ ♣❛r❛ ❜❛✐①❛s ❡♥❡r❣✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❡str✉t✉r❛✐s ❞❛ r❡❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹ ❉❡❧✐♥❡❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✷ ▼♦♥♦❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦ s♦❜ ❡❢❡✐t♦ ❞❡ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ✶✹ ✷✳✶ ▼♦❞❡❧♦ ❝♦♥tí♥✉♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳✶ ❈❛❧✐❜r❡ ❞❡ ▲❛♥❞❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✶✳✷ ❈❛❧✐❜r❡ s✐♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✶✳✸ ❈❛♠♣♦ ❞❡♣❡♥❞❡♥t❡ ❞❛ ♣♦s✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✶✳✹ ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✸ ❆♣r♦①✐♠❛çã♦ t✐❣❤t✲❜✐♥❞✐♥❣ ✷✸
❙❯▼➪❘■❖ xii
✹ ❇✐❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦ ✸✵
✹✳✶ ❈♦♥s✐❞❡r❛çõ❡s ❣❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✹✳✷ ▼♦❞❡❧♦ t✐❣❤t✲❜✐♥❞✐♥❣ ♣❛r❛ ❜✐❝❛♠❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✷✳✶ ❊♠♣✐❧❤❛♠❡♥t♦ ❆❆ ❞❛ ❜✐❝❛♠❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✺ ❈♦♥❝❧✉sõ❡s ❡ ♣❡rs♣❡❝t✐✈❛s ✹✵
❆♣ê♥❞✐❝❡ ✲ ❈♦♦r❞❡♥❛❞❛s ❞♦s ♣♦♥t♦s ❞❡ ❉✐r❛❝ ✹✷
1
✶✳ ■♥tr♦❞✉çã♦
❯♠ ❞♦s ❡❧❡♠❡♥t♦s ♠❛✐s ❛❜✉♥❞❛♥t❡s ♥♦ ✉♥✐✈❡rs♦ é ♦ át♦♠♦ ❞❡ ❝❛r❜♦♥♦ C✱ é t❛♠❜é♠ ❡♥❝♦♥tr❛❞♦ ❡♠ t♦❞❛s ❛s ❢♦r♠❛s ❝♦♥❤❡❝✐❞❛s ❞❡ ✈✐❞❛✳ ❈♦♥s✐❞❡r❛♥❞♦ s✐st❡♠❛s à ❜❛s❡ ❞❡ ❝❛r❜♦♥♦ ❡♥❝♦♥tr❛♠♦s ✈ár✐❛s ❡str✉t✉r❛s q✉❡ ❧✐❣❛♠ s❡✉s át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ❞❡ ❞✐❢❡r❡♥t❡s ♠❛♥❡✐r❛s✱ ❝❤❛♠❛♠♦s ❡❧❛s ❞❡ ❛❧ótr♦♣♦s✳ ❖s ❛❧ótr♦♣♦s ❞♦ ❝❛r❜♦♥♦ r❡❧❛t✐✈❛♠❡♥t❡ ❝♦♥❤❡❝✐❞♦s sã♦ ❞✐❛♠❛♥t❡ ❡ ❣r❛✜t❡✳ ❆♠❜♦s sã♦ ❢♦r♠❛❞♦s ♣♦r át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ♠❛s tê♠ ♣r♦♣r✐❡❞❛❞❡s ❞✐❢❡r❡♥t❡s✳ ❖ ❞✐❛♠❛♥t❡ é ✉♠ ♠❛t❡r✐❛❧ ❞✐❡❧étr✐❝♦ ❡①tr❡♠❛♠❡♥t❡ ❞✉r♦✱ ❝♦♥s✐st❡ ❞❡ q✉❛tr♦ ♦r❜✐t❛✐s ❤í❜r✐❞♦ssp3✱ ✐st♦ é✱ t♦❞♦s ♦s q✉❛tr♦ ❡❧étr♦♥s ❢♦r❛ ❞❛ ❝❛♠❛❞❛ ❞❡ ✈❛❧ê♥❝✐❛ ♣❡r♠✐t❡♠ ❧✐❣❛çõ❡s ❝♦✈❛❧❡♥t❡s ❛ ♦✉tr♦s q✉❛tr♦ át♦♠♦s ❞❡ ❝❛r❜♦♥♦✳ ❏á ♦ ❣r❛✜t❡ é ✉♠ ❞♦s ♠❛t❡r✐❛✐s ♠❛✐s s♦✜st✐❝❛❞♦s ❡ ❝r✐st❛❧✐③❛✲s❡ ❡♠ ✉♠ s✐st❡♠❛ ❤❡①❛❣♦♥❛❧✱ ❝♦♥s✐st❡ ❞❡ três ♦r❜✐t❛✐s ❤í❜r✐❞♦ssp2 ❡ é ✉♠ ❝♦♥❞✉t♦r ❡❧étr✐❝♦✳ P♦❞❡♠♦s ♥♦t❛r q✉❡ ❛♠❜♦s ♦s ❛❧ótr♦♣♦s tê♠ ♣r♦♣r✐❡❞❛❞❡s q✉❛s❡ ♦♣♦st❛s✳
❆q✉✐✱ ✐r❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ❣r❛✜t❡ ❡♠ ♠❛✐s ❞❡t❛❧❤❡✳ ❈❛❞❛ át♦♠♦ ❞❡ ❝❛r❜♦♥♦ ❞❡ss❡ ❛❧ótr♦♣♦ ✉s❛ ❛♣❡♥❛s três ❞♦s q✉❛tr♦ ❡❧étr♦♥s ❢♦r❛ ❞❛ ❝❛♠❛❞❛ ❞❡ ✈❛❧ê♥❝✐❛ ❡♠ ❧✐❣❛çõ❡s ❝♦✈❛✲ ❧❡♥t❡s ❛ três ♦✉tr♦s át♦♠♦s ❞❡ ❝❛r❜♦♥♦✳ P♦rt❛♥t♦✱ ❡❧❡s s❡ ❛rr❛♥❥❛♠ ♥✉♠ ♣❧❛♥♦✱ ❢♦r♠❛♥❞♦ ✉♠❛ ❝❛♠❛❞❛ ❢♦rt❡♠❡♥t❡ ❧✐❣❛❞❛ ❝♦♠ ❡str✉t✉r❛ ❤❡①❛❣♦♥❛❧✳ ❆♦ ❝♦♥trár✐♦ ❞♦ ❞✐❛♠❛♥t❡ ♥ós ✈❡♠♦s q✉❡ ❝❛❞❛ át♦♠♦ ❞❡ ❝❛r❜♦♥♦ ♥♦ ❣r❛✜t❡ t❡♠ ✉♠ ❡❧étr♦♥ ❞❡s❧♦❝❛❞♦✳ ❊st❡ é ❧✐✈r❡ ♣❛r❛ s❡ ♠♦✈❡r ❛tr❛✈és ❞♦ ♣❧❛♥♦✳ P♦r ❡ss❛ r❛③ã♦ ♦ ❣r❛✜t❡ ❝♦♥❞✉③ ❡❧❡tr✐❝✐❞❛❞❡ ❛♣❡♥❛s ❛♦ ❧♦♥❣♦ ❞❡ át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ♥♦ ♣❧❛♥♦✳
✶✳✶ ●r❛❢❡♥♦✿ ❛s♣❡❝t♦s ❣❡r❛✐s 2
❋✐❣✉r❛ ✶✳✶✿ ❖ ❣r❛❢❡♥♦ ✭❛✮ ♣♦❞❡ s❡r ❡♠♣✐❧❤❛❞♦ ❢♦r♠❛♥❞♦ ✸❉✲❣r❛✜t❡ ✭❜✮✱ ♣♦❞❡ s❡r ❡♥r♦❧❛❞♦ ❢♦r♠❛♥❞♦ ✶❉✲♥❛♥♦t✉❜♦ ✭❝✮ ♦✉ ❡♥✈♦❧t♦✱ ❢♦r♠❛♥❞♦ ✵❉✲❢✉❧❡r❡♥♦ ✭❞✮ ❬✶❪✳
❊♠ r❡s✉♠♦✱ ♦ ❣r❛❢❡♥♦ é ♠❛✐s ❞✉r♦ q✉❡ ♦ ❞✐❛♠❛♥t❡✱ ♣♦ré♠ ❡①tr❡♠❛♠❡♥t❡ ✢❡①í✈❡❧ ❡ ✉♠ ❝♦♥❞✉t♦r ❞❡ ❡❧❡tr✐❝✐❞❛❞❡ ♠✉✐t♦ ♠❡❧❤♦r q✉❡ ♦✉tr♦s ♠❛t❡r✐❛✐s✳ ❆❜❛✐①♦ ❝♦♥t✐♥✉❛r❡♠♦s ❛ ❢❛❧❛r ❞❛s ✐♥t❡r❡ss❛♥t❡s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ ❣r❛❢❡♥♦✱ ❝♦♠ ❞❡st❛q✉❡ ♣❛r❛ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧❡trô♥✐❝❛s✳
✶✳✶ ●r❛❢❡♥♦✿ ❛s♣❡❝t♦s ❣❡r❛✐s
❖ ❣r❛❢❡♥♦ é ✉♠❛ ❢♦r♠❛ ❛❧♦tró♣✐❝❛ ❞♦ ❝❛r❜♦♥♦ ❝♦♥st✐t✉í❞❛ ❞❡ ✉♠❛ ♠♦♥♦❝❛♠❛❞❛ ❞❡ át♦♠♦s ❞♦ ❡❧❡♠❡♥t♦ ❧✐❣❛❞♦s ❡♠ ✉♠ ❛rr❛♥❥♦ ❤❡①❛❣♦♥❛❧✳ ❉❡ ❢❛t♦✱ ♦ ❣r❛❢❡♥♦ ♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ✉♠❛ ❞❛s ✐♥ú♠❡r❛s ❝❛♠❛❞❛s q✉❡ ❝♦♥st✐t✉❡♠ ❛ ❣r❛✜t❡✳ ❯♠ ❝r✐st❛❧ ❞❡ ❣r❛✜t❡ ❝♦♠ 1mm ❞❡ ❡s♣❡ss✉r❛ ❝♦♥s✐st❡ ❞❡ três ♠✐❧❤õ❡s ❞❡ ❝❛♠❛❞❛s ❞❡ ❣r❛❢❡♥♦✱ s♦❜r❡♣♦st❛s✱ ✉♥✐❞❛s ♣♦r ✐♥t❡r❛çõ❡s ✐♥t❡r♠♦❧❡❝✉❧❛r❡s✳ ❖ Prê♠✐♦ ◆♦❜❡❧ ❞❡ ❋ís✐❝❛ ❡♠ ✷✵✶✵ ❢♦✐ ❞❛❞♦ ❛♦s ❝✐❡♥t✐st❛s ❆♥❞r❡ ●❡✐♠ ❡ ❑♦♥st❛♥t✐♥ ◆♦✈♦s❡❧♦✈✶✱ ♣❡❧♦ s✉❝❡ss♦ ♦❜t✐❞♦ ♥❛ ♣r♦❞✉çã♦✱
✐s♦❧❛♠❡♥t♦✱ ✐❞❡♥t✐✜❝❛çã♦ ❡ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦ ❣r❛❢❡♥♦✳
◆♦ s❡✉ ❛rt✐❣♦✷✱ ♦s ❞♦✐s r❡✈❡❧❛r❛♠ q✉❡ t❛❧ ❝❛♠❛❞❛ ♣♦❞✐❛ s❡r ✐s♦❧❛❞❛ ❡♠ q✉❛♥t✐❞❛❞❡
s✉✜❝✐❡♥t❡ ♣❛r❛ ✈❡r✐✜❝❛r s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❡r❛ ❡stá✈❡❧✳ ❊♠ ✉♠❛ ❞❛s ❡t❛♣❛s ❞♦ tr❛❜❛❧❤♦ ❡❧❡s ✉t✐❧✐③❛r❛♠ ✜t❛ ❛❞❡s✐✈❛ ♣❛r❛ r❡♠♦✈❡r ❝❛♠❛❞❛s ❞❡ ❣r❛❢❡♥♦ ❞❡ ❝r✐st❛✐s ❞❡ ❣r❛✜t❡✳ ❙❡✲ ❣✉♥❞♦ ♦ ❝♦♠✉♥✐❝❛❞♦ ❞❛ ❆❝❛❞❡♠✐❛ ❘❡❛❧ ❙✉❡❝❛ ❞❡ ❈✐ê♥❝✐❛s✱ q✉❡ ❝♦♥❝❡❞❡✉ ♦ ♣rê♠✐♦✱ ♦ ❣r❛❢❡♥♦ é ✧♦ ♣r✐♠❡✐r♦ ♠❛t❡r✐❛❧ ❝r✐st❛❧✐♥♦ ✈❡r❞❛❞❡✐r❛♠❡♥t❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❡ é r❡♣r❡s❡♥t❛✲ t✐✈♦ ❞❡ t♦❞❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ♠❛t❡r✐❛s ✷❉✱ q✉❡ ✐♥❝❧✉✐✱ ♣♦r ❡①❡♠♣❧♦✱ ♠♦♥♦❝❛♠❛❞❛s ❞❡ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ✭BN✮ ❡ ❞❡ ❞✐ss✉❧❢❡t♦ ❞❡ ♠♦❧✐❜❞ê♥✐♦ ✭M oS2✮✱ ❛♠❜♦s ♣r♦❞✉③✐❞♦s ❛♣ós ✷✵✵✹✧✳
✶✳✶ ●r❛❢❡♥♦✿ ❛s♣❡❝t♦s ❣❡r❛✐s 3
❖ ♠❛t❡r✐❛❧ é ♣r❛t✐❝❛♠❡♥t❡ tr❛♥s♣❛r❡♥t❡✱ ❞❡✐①❛♥❞♦ ♣❛ss❛r 97,7% ❞❛ ❧✉③ ✈✐sí✈❡❧✳ ➱
♠❛✐s r❡s✐st❡♥t❡ q✉❡ ✉♠❛ ❛♠♦str❛ ❞❡ ❛ç♦ ❞❡ ♠❡s♠❛ ❞✐♠❡♥sã♦ ❡ ❡st✐❝❛ ❛té20%s❡♠ r♦♠♣❡r✳
❯♠❛ r❡❞❡ ❤✐♣♦tét✐❝❛ ❞❡1m2 ❞❡ ❣r❛❢❡♥♦ t❡r✐❛ ❛♣❡♥❛s 0,77mg ❡✱ ❡st❡♥❞✐❞❛✱ s✉♣♦rt❛r✐❛ ✉♠ ♦❜❥❡t♦ ❞❡4kg✳ ❖ ❣r❛❢❡♥♦ ❝♦♥❞✉③ ❝❛❧♦r ❡ ❝♦rr❡♥t❡ ❡❧étr✐❝❛ ♠❡❧❤♦r q✉❡ ♦ ❝♦❜r❡✱ ♦ q✉❡ ♣♦❞❡
s❡r ❛tr✐❜✉í❞♦ ❛♦s ❡❧étr♦♥s ❞❡s❧♦❝❛❧✐③❛❞♦s✳
❆ ❝♦♥q✉✐st❛ ❡①♣❡r✐♠❡♥t❛❧ ❞♦ ❣r✉♣♦ ❞❡ ●❡✐♠ ♠♦t✐✈♦✉ ♦✉tr❛s ✐♠♣♦rt❛♥t❡s ♣❡sq✉✐s❛s✱ t❛♥t♦ ❡①♣❡r✐♠❡♥t❛✐s ❝♦♠♦ t❡ór✐❝❛s✱ ❡♠ ✈✐st❛ ❞❛s ✐♥t❡r❡ss❛♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❡❧❡trô♥✐❝❛s✳ ❖s r❡s✉❧t❛❞♦s t❡ór✐❝♦s ♠♦str❛❞♦s ♣♦r ❲❛❧❧❛❝❡ ❬✸❪ ♣r❡✈✐❛♠ ♦ ❣r❛❢❡♥♦ ❝♦♠♦ ✉♠ ✐♠♣♦rt❛♥t❡ ❝❡♥ár✐♦ ❞❡ ❞✐s❝✉ssã♦ ♥ã♦ só ♣❛r❛ ✉♠ r❡✈♦❧✉❝✐♦♥ár✐♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ t❡❝♥♦❧ó❣✐❝♦✱ ♠❛s t❛♠✲ ❜é♠ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ s✐t✉❛çõ❡s ❢ís✐❝❛s ✐♥t❡r❡ss❛♥t❡s ❝♦♠♦ ❛s ❛❝❤❛❞❛s ♥❛ ❡❧❡tr♦❞✐♥â♠✐❝❛ q✉â♥t✐❝❛ ✭❊❉◗✮✳ ❉❡ ❢❛t♦✱ ❲❛❧❧❛❝❡✱ ♥✉♠ ❡st✉❞♦ ❢❡✐t♦ ♣❛r❛ ❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧❡trô♥✐❝❛s ❞♦ ❣r❛✜t❡ ❬✸❪✱ ♠♦str♦✉ ❝♦♠♦ ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ s❡♠✐♠❡tá❧✐❝♦ ♥ã♦ ✉s✉❛❧ ❡r❛ ✈✐st♦ ♥❛ ❡str✉✲ t✉r❛ ❡❧❡trô♥✐❝❛ ❞♦ ❣r❛❢❡♥♦✳ ❆ ♣❛rt✐❝✉❧❛r ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞♦ ❣r❛❢❡♥♦ ♠♦str❛✈❛ ❛❧é♠✱ ♣❛r❛ ❜❛✐①❛s ❡♥❡r❣✐❛s✱ ✉♠❛ ❞✐s♣❡rsã♦ ❞❛ ❡♥❡r❣✐❛ q✉❡ ❡r❛ ❧✐♥❡❛r ♥♦ ✈❡t♦r ❞❡ ♦♥❞❛✱ ❞✐❢❡r❡♥t❡ ❞♦ ✉s✉❛❧♠❡♥t❡ ❡♥❝♦♥tr❛❞♦ ♣❛r❛ s❡♠✐❝♦♥❞✉t♦r❡s✳ ❊st❛ ♣❛rt✐❝✉❧❛r r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦✱ ❝♦♥✲ s✐❞❡r❛❞❛ ✉♠ ❞♦s ❛s♣❡❝t♦s ♠❛✐s ✐♥t❡r❡ss❛♥t❡s ♥♦ ❣r❛❢❡♥♦✱ ✐♥❞✐❝❛ q✉❡ ♣❛r❛ ❜❛✐①❛s ❡♥❡r❣✐❛s ❛s ❡①❝✐t❛çõ❡s sã♦ ❞♦ t✐♣♦ ❡♥❝♦♥tr❛❞♦ ♣❛r❛ ❢ér♠✐♦♥s ❞❡ ❉✐r❛❝ ♥❛ ❊❉◗✱ ❞✐❢❡r❡♥❝✐❛❞❛ ✉♥✐✲ ❝❛♠❡♥t❡ ♣❡❧❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛s q✉❛s✐♣❛rtí❝✉❧❛s✱ q✉❡ ♥♦ ❝❛s♦ ❞♦ ❣r❛❢❡♥♦ é ✐❣✉❛❧ à ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❋❡r♠✐υF❂ ❝✴✸✵✵ ❛♦ ✐♥✈és ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③c✳
✶✳✶ ●r❛❢❡♥♦✿ ❛s♣❡❝t♦s ❣❡r❛✐s 4
❋✐❣✉r❛ ✶✳✷✿ ✭❛✮❊sq✉❡♠❛ ❞♦ ❡❢❡✐t♦ ❍❛❧❧ q✉â♥t✐❝♦ ❡♠ ✉♠ s❡♠✐❝♦♥❞✉t♦r ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝♦♥✲ ✈❡♥❝✐♦♥❛❧✱ ✭❜✮ ♥❛ ❜✐❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦ ❡ ✭❝✮ ♥❛ ♠♦♥♦❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦✳ ❖s ♣✐❝♦s ❧❛r❛♥❥❛s ✭❡❧étr♦♥s✮ ❡ ❛③✉✐s ✭❜✉r❛❝♦s✮ ♠♦str❛♠ ❛s s❡q✉ê♥❝✐❛s ❞❡ ♥í✈❡✐s ❞❡ ▲❛♥❞❛✉ ❡♠ ❢✉♥çã♦ ❞❛ ❞❡♥✲ s✐❞❛❞❡ ❞❡ ♣♦rt❛❞♦r❡sn❡ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦B✳ ❊♠ ✈❡r♠❡❧❤♦ ❡stã♦ ❛s ♠❡❞✐❞❛s ❞❡σxy✳ ❆
❝♦♥st❛♥t❡er❡♣r❡s❡♥t❛ ❛ ❝❛r❣❛ ❞♦ ❡❧étr♦♥✱hé ❛ ❝♦♥st❛♥t❡ ❞❡ P❧❛♥❝❦ ❡g✱ ❛ ❞❡❣❡♥❡r❡s❝ê♥❝✐❛ ❞❡ ✈❛❧❡ ❡ ❞❡ s♣✐♥ ❬✻❪✳
❈♦♠♦ ❝✐t❛♠♦s✱ ♦ ❣r❛❢❡♥♦ ❛♣r❡s❡♥t❛ ❛✐♥❞❛ ♣r♦♣r✐❡❞❛❞❡s ó♣t✐❝❛s ✐♥t❡r❡ss❛♥t❡s✱ ❞❡✲ ✈✐❞♦ ❛ s✉❛ ❛❜s♦rçã♦ ❞❡ ❧✉③✳ ❊st✉❞♦s ❢❡✐t♦s ❛tr❛✈és ❞❡ ❡♥s❛✐♦s ❞❡ r❡✢❡❝tâ♥❝✐❛ ❡ tr❛♥s♠✐tâ♥✲ ❝✐❛ ❬✶✷❪ ❝♦♠♣r♦✈❛♠ q✉❡ ♦ ❛✉♠❡♥t♦ ❞❛ ❛❜s♦rçã♦ ❞❛ ❧✉③ ♥♦ ❣r❛❢❡♥♦ s❡ ❞á ♣r♦♣♦r❝✐♦♥❛❧♠❡♥t❡ ❝♦♠ ♦ ♥ú♠❡r♦ ❞❡ ❝❛♠❛❞❛s ❞❡st❡✳ ■ss♦ ♣♦t❡♥❝✐❛❧✐③❛ s✉❛ ✉t✐❧✐③❛çã♦ ❡♠ ❛♣❧✐❝❛çõ❡s ❡❧❡tr♦✲ ó♣t✐❝❛s✳
❋✐❣✉r❛ ✶✳✸✿ ❋♦t♦❣r❛✜❛ ❞❡ ✉♠❛ ③♦♥❛ ❝♦♠ ✉♠❛ ❡ ❞✉❛s ❝❛♠❛❞❛s ❞❡ ❣r❛❢❡♥♦ ✉t✐❧✐③❛❞❛s ♥✉♠ ❡♥s❛✐♦ ❞❡ tr❛♥s♠✐tâ♥✐❝❛ ♦♥❞❡ s❡ ✈❡r✐✜❝❛ ❛ ❛❜s♦rçã♦ ❞❡2,3% ❞❛ ❧✉③ ❬✶✷❪✳
❆ ❝♦♥❞✉t✐✈✐❞❛❞❡ tér♠✐❝❛ ♥♦ ❣r❛❢❡♥♦ é ❞❡♥♦♠✐♥❛❞❛ ♣❡❧❛s ✈✐❜r❛çõ❡s ♥❛ r❡❞❡✱ ❛ ❝♦♥❞✉çã♦ tér♠✐❝❛ é ❡❢❡t✉❛❞❛ ♣♦r ❢ô♥♦♥s✸✳ ❊st✉❞♦s r❡❛❧✐③❛❞♦s ❝♦♠ ♥❛♥♦t✉❜♦s ❞❡ ❝❛r❜♦♥♦
✸◆❛ ❢ís✐❝❛ ❞❡ ♠❛tér✐❛ ❝♦♥❞❡♥s❛❞❛ é ✉♠❛ q✉❛s✐♣❛rtí❝✉❧❛ q✉❡ ❞❡s✐❣♥❛ ✉♠ q✉❛♥t✉♠ ❞❡ ✈✐❜r❛çã♦ ❡♠ ✉♠
✶✳✷ ❉✐s♣♦s✐t✐✈♦s ❞❡ ❣r❛❢❡♥♦ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s 5
❞❡♠♦♥str❛r❛♠ q✉❡ ❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ tér♠✐❝❛ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛✳ ❖s ✈❛❧♦r❡s ♦❜t✐❞♦s ♣❛r❛ ♦ ❣r❛❢❡♥♦ ❬✶✸❪ sã♦ ❡①tr❡♠❛♠❡♥t❡ ❡❧❡✈❛❞♦s✱ ✉❧tr❛♣❛s✲ s❛♥❞♦ ♦s ❞❡ ❞✐❛♠❛♥t❡ ❝✈❞✹ ❡ ❛✐♥❞❛ ❧✐❣❡✐r❛♠❡♥t❡ s✉♣❡r✐♦r❡s ❛♦s ❞❡ ♥❛♥♦t✉❜♦ ❞❡ ❝❛r❜♦♥♦✱
t❛♥t♦ ♣❛r❛ ♠✉✐❧t✐❝❛♠❛❞❛ ✭▼❲✮ ❝♦♠♦ ♣❛r❛ ♠♦♥♦❝❛♠❛❞❛ ✭❙❲✮✳ ◆❛ ❚❛❜❡❧❛ ✶✳✶ sã♦ ❝♦♠✲ ♣❛r❛❞♦s ♦s ✈❛❧♦r❡s ❞❡ ❝♦♥❞✉t✐✈✐❞❛❞❡ tér♠✐❝❛ ❞♦ ❣r❛❢❡♥♦✱ ♥❛♥♦t✉❜♦ ❞❡ ❝❛r❜♦♥♦ ✭❈◆❚✮✱ ❞✐❛♠❛♥t❡ ❡ ❝♦❜r❡ à t❡♠♣❡r❛t✉r❛ ❛♠❜✐❡♥t❡✿
▼❛t❡r✐❛❧ ❑✭❲✴♠❑✮ ❬✶✹❪ ●r❛❢❡♥♦ ✹✽✹✵✲✺✸✵✵ ▼❲ ✲ ❈◆❚ ❃✸✵✵✵
❙❲ ✲ ❈◆❚ ✸✺✵✵ ❉✐❛♠❛♥t❡ ❈❱❉ ✷✵✵✵
❈♦❜r❡ ✹✵✵
❚❛❜❡❧❛ ✶✳✶✿ ❈♦♥❞✉t✐✈✐❞❛❞❡s tér♠✐❝❛s ❞♦ ❣r❛❢❡♥♦✱ ♥❛♥♦t✉❜♦ ❞❡ ❝❛r❜♦♥♦✱ ❞✐❛♠❛♥t❡ ❡ ❝♦❜r❡✳ ▼✉✐t♦s ❡st✉❞♦s t❡ór✐❝♦s tê♠ s✐❞♦ ❢❡✐t♦s s♦❜r❡ ♥❛♥♦✜t❛s ❞❡ ❣r❛❢❡♥♦ q✉❡ ❛♣♦♥t❛♠ ♣r♦♣r✐❡❞❛❞❡s ♠❛❣♥ét✐❝❛s ✐♥t❡r❡ss❛♥t❡s ❡ ❛❧❣✉♥s ❞❛❞♦s ♦❜t✐❞♦s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ❞❡♠♦♥✲ str❛r❛♠ q✉❡ ♥❛♥♦✜t❛s ❞❡ ❣r❛❢❡♥♦ ❡①✐❜❡♠ ♣r♦♣r✐❡❞❛❞❡s ❞❡ tr❛♥s♣♦rt❡ ♠❛❣♥ét✐❝♦ ♥♦tá✈❡✐s ❬✶✺✱ ✶✻❪✳ ❊♠ ❣❡r❛❧✱ ❛ ❡str✉t✉r❛ r❡ss♦♥❛♥t❡ ❞❛ tr❛♥s♠✐ssã♦ é s✐❣♥✐✜❝❛♥t❡♠❡♥t❡ ♠❛✐s ♥♦tá✈❡❧ ♣❛r❛ ❡❧étr♦♥s ❝♦♠ ❡s♣❡❝tr♦ ❧✐♥❡❛r ✭❉✐r❛❝✮ ❞♦ q✉❡ ♣❛r❛ ♦s ✉s✉❛✐s ❡❧étr♦♥s ❝♦♠ ❡s♣❡❝tr♦ ♣❛r❛❜ó❧✐❝♦✳
❖ ❣r❛❢❡♥♦ t❡♠ ❛♣❧✐❝❛çõ❡s ♣r♦♠✐ss♦r❛s✳ ➱ ✉♠ ❝♦♥❞✉t♦r tr❛♥s♣❛r❡♥t❡✱ ✢❡①í✈❡❧ ❡ ♠❡❝❛♥✐❝❛♠❡♥t❡ r❡s✐st❡♥t❡✱ q✉❡ ♣♦❞❡r✐❛ s❡r ✉s❛❞♦ ❡♠ t❡❧❛s ✉❧tr❛✜♥❛s✱ ✢❡①í✈❡✐s ❡ s❡♥sí✈❡✐s ❛♦ t♦q✉❡ ♣❛r❛ t✈✱ ❝♦♠♣✉t❛❞♦r❡s✱ ❝❡❧✉❧❛r❡s ❡ ❧✐✈r♦s ❞✐❣✐t❛✐s✳ ❚r❛♥s✐st♦r❡s ❞❡ ❣r❛❢❡♥♦ s❡r✐❛♠ ♠❛✐s rá♣✐❞♦s q✉❡ ♦s ❞❡ s✐❧í❝✐♦ ❡ ❝❤✐♣s ❝♦♠ ♠❛✐♦r ❝❛♣❛❝✐❞❛❞❡ ❞❡ ♣r♦❝❡ss❛♠❡♥t♦ ♣♦❞❡r✐❛♠ s❡r ❢❛❜r✐❝❛❞♦s✳
P♦r ❞✉❛s ✈❡③❡s ❛ ♣r♦❞✉çã♦ ❡ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ✉♠ ♥♦✈♦ ❛❧ótr♦♣♦ ❞♦ ❝❛r❜♦♥♦ r❡♥✲ ❞❡✉ ♦ ◆♦❜❡❧✳ ❊♠ ✶✾✾✻✱ ♦ ♣rê♠✐♦ ❞❡ ◗✉í♠✐❝❛ ❢♦✐ ❝♦♥❢❡r✐❞♦ ❛ três ❝✐❡♥t✐st❛s ♣❡❧❛ ❞❡s❝♦❜❡rt❛ ❡ ♣❡❧♦ ❡st✉❞♦ ❞♦s ❢✉❧❡r❡♥♦s✳
❆ s❡❣✉✐r ❢❛❧❛r❡♠♦s ♠❛✐s ❞❛ ♣r♦❞✉çã♦ ❞❡ ❞✐s♣♦s✐t✐✈♦s à ❜❛s❡ ❞❡ss❡ ❛❧ótr♦♣♦ ❞♦ ❝❛r❜♦♥♦✳ ❊♠ s❡❣✉✐❞❛ s❡rá ❢❡✐t❛ ✉♠❛ ✐♥tr♦❞✉çã♦ ❛♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ❉✐r❛❝ ♣❛r❛ ♣❛rtí❝✉❧❛s r❡❧❛t✐✈íst✐❝❛s s❡♠ ♠❛ss❛✱ q✉❡ sã♦ ♦s ❢ér♠✐♦♥s✱ ♠♦str❛♥❞♦ ❝♦♠♦ s❡ ❝♦♥✜❣✉r❛ ❛ ❡str✉t✉r❛ ❞❛ r❡❞❡ ❤❡①❛❣♦♥❛❧ ❞♦ ❣r❛❢❡♥♦✳
✶✳✷ ❉✐s♣♦s✐t✐✈♦s ❞❡ ❣r❛❢❡♥♦ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s
P❛r❛ ✐♥✈❡st✐❣❛r ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ❛s ✐♥t❡r❡ss❛♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❡❧étrô♥✐❝❛s ❞♦
✶✳✷ ❉✐s♣♦s✐t✐✈♦s ❞❡ ❣r❛❢❡♥♦ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s 6
❣r❛❢❡♥♦ é ♥❡❝❡ssár✐♦ ❝♦♥str✉✐r ✉♠ ❞✐s♣♦s✐t✐✈♦ ❞❡ ❣r❛❢❡♥♦✳ ❊♠ ❣❡r❛❧ ❡st❡s ❞✐s♣♦s✐t✐✈♦s sã♦ ❢❡✐t♦s ❞❡♣♦s✐t❛♥❞♦✲s❡ ✢♦❝♦s ❞❡ ❣r❛❢❡♥♦ s♦❜r❡ ✉♠ s✉❜str❛t♦ ❞❡ s✐❧í❝✐♦ ❛❧t❛♠❡♥t❡ ❞♦♣❛❞♦ ❝♦❜❡rt♦ ♣♦r ✉♠❛ ❝❛♠❛❞❛ ❞❡ ❡s♣❡ss✉r❛ ❜❡♠ ❞❡✜♥✐❞❛ ❞❡ ó①✐❞♦ ❞❡ s✐❧í❝✐♦✳
❆ ❡str✉t✉r❛ ❡♠ ♠✉❧t✐❝❛♠❛❞❛s ❢♦r♠❛❞❛ ♣❡❧♦ s✐❧í❝✐♦ ❛❧t❛♠❡♥t❡ ❞♦♣❛❞♦✱ q✉❡ ❢✉♥✲ ❝✐♦♥❛ ❞❡ ❢♦r♠❛ s❡♠❡❧❤❛♥t❡ ❛ ✉♠ ❝♦♥t❛t♦ ♠❡tá❧✐❝♦✱ ♦ ❞✐ó①✐❞♦ ❞❡ s✐❧í❝✐♦✱ q✉❡ é ✐s♦❧❛♥t❡✱ ❡ ♦ ❣r❛❢❡♥♦✱ q✉❡ é ✉♠ s❡♠✐❝♦♥❞✉t♦r ❞❡gap ♥✉❧♦✱ t❡♠ ✉♠ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡ ❡❢❡✐t♦ ❞❡ ❝❛♠♣♦ q✉❡ ♣❡r♠✐t❡ ❝♦♥tr♦❧❛r ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❝❛r❣❛ ♥♦ ❣r❛❢❡♥♦ ❛tr❛✈és ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠❛ ❞✐❢❡✲ r❡♥ç❛ ❞❡ ♣♦t❡♥❝✐❛❧ ❡♥tr❡ ♦ ❣r❛❢❡♥♦ ❡ ♦ s✐❧í❝✐♦✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠ ❡st❡ t✐♣♦ ❞❡ s✉❜str❛t♦ é ♣♦ssí✈❡❧ ✈❡r ✢♦❝♦s ❞❡ ❣r❛❢❡♥♦ s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ✉t✐❧✐③❛♥❞♦✲s❡ ❛♣❡♥❛s ✉♠ ♠✐❝r♦s❝ó♣✐♦ ó♣t✐❝♦✳ ◗✉❛♥❞♦ ❛ ❝❛♠❛❞❛ ❞❡ ❞✐ó①✐❞♦ ❞❡ s✐❧í❝✐♦ t❡♠ ❡s♣❡ss✉r❛ ❛♣r♦♣r✐❛❞❛ ❬✶✼❪✱ ♦❝♦rr❡ ✐♥✲ t❡r❢❡rê♥❝✐❛ ❡♥tr❡ ❛ ❧✉③ r❡✢❡t✐❞❛ ♥❛s ❞✐❢❡r❡♥t❡s ❝❛♠❛❞❛s ❣❡r❛♥❞♦ ✉♠ ❜♦♠ ❝♦♥tr❛st❡ ó♣t✐❝♦ ❡♥tr❡ ❛s r❡❣✐õ❡s ❝♦♠ ❣r❛❢❡♥♦ ❡ s❡♠ ❣r❛❢❡♥♦ ♥❛ s✉♣❡r❢í❝✐❡✳
❖s ❞✐s♣♦s✐t✐✈♦s ❞❡ ❣r❛❢❡♥♦ ♠❛✐s ❜ás✐❝♦s✱ q✉❡ s❡r✈❡♠ ♣❛r❛ ♠❡❞✐r s✉❛ r❡s✐stê♥❝✐❛✱ sã♦ ❢❡✐t♦s ✉t✐❧✐③❛♥❞♦ té❝♥✐❝❛s ❞❡ ❧✐t♦❣r❛✜❛✱ ❞❡s❡♥❤❛♥❞♦ ♦ tr❛♥síst♦r ♥♦ ❣r❛❢❡♥♦✳ ❆ ♣r✐♥❝í✲ ♣✐♦ ♣r❡❝✐s❛♠♦s ❛♣❡♥❛s s❛❜❡r q✉❡ é ♣♦ssí✈❡❧ ❝♦♥t❛❝t❛r ❡❧❡tr✐❝❛♠❡♥t❡ ♦ ❣r❛❢❡♥♦ ❡ ❝♦rr♦ê✲❧♦ ♥❛ ❢♦r♠❛ ❞❡ ✉♠❛ ♣♦♥t❡ ❍❛❧❧✱ q✉❡ ♣❡r♠✐t❡ r❡❛❧✐③❛r ♠❡❞✐❞❛s✱ ♥❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ q✉❛tr♦ t❡r♠✐♥❛✐s✱ ❞❛ r❡s✐stê♥❝✐❛ ❞♦ ❣r❛❢❡♥♦✳
❋✐❣✉r❛ ✶✳✹✿ ❊sq✉❡♠❛ ❞❡ ✉♠ ❞✐s♣♦s✐t✐✈♦ ❞❡ ❣r❛❢❡♥♦ ♥❛ ❢♦r♠❛ ❞❡ ✉♠❛ ♣♦♥t❡ ❍❛❧❧ ❝♦♥str✉í❞♦ ❡♠ ✉♠ s✉❜str❛t♦Si ❝♦❜❡rt♦ ♣♦r ✉♠❛ ❝❛♠❛❞❛ ❞❡SiO2 ❬✶✽❪✳
◆♦s ♣r✐♠❡✐r♦s ❡st✉❞♦s ❞❡ ❣r❛❢❡♥♦ ❞✐s♣♦s✐t✐✈♦s ♠✉✐t♦ s✐♠♣❧❡s ♥❛ ❢♦r♠❛ ❞❡ ♣♦♥t❡ ❍❛❧❧ ❢♦r❛♠ ✉s❛❞♦s✳ ❍♦❥❡✱ ❥á ❢♦r❛♠ ❢❡✐t♦s ❞✐s♣♦s✐t✐✈♦s ❞❡ ❣r❛❢❡♥♦ s✉s♣❡♥s♦✱ ❡♠ ❢♦r♠❛ ❞❡ ♥❛♥♦✜t❛s ❡ ❡♠ ❢♦r♠❛ ❞❡ ♣♦♥t♦s q✉â♥t✐❝♦s✱ ❞❡♥tr❡ ♦✉tr♦s✳ ■ss♦ ♠♦str❛ q✉❡ ❛ ♣❡sq✉✐s❛ ❞❡ tr❛♥s♣♦rt❡ ❡❧étr✐❝♦ ❡♠ ❣r❛❢❡♥♦ ❛✈❛♥ç♦✉ ❜❛st❛♥t❡ ❡ ❛✐♥❞❛ ❛♣r❡s❡♥t❛ ✈ár✐❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❡ r♦t❛s ❞❡ ✐♥✈❡st✐❣❛çã♦✳
✶✳✸ ❆♣r❡s❡♥t❛çã♦ ❞❛ t❡♦r✐❛ ❞❡ ❉✐r❛❝ ♣❛r❛ ❜❛✐①❛s ❡♥❡r❣✐❛s 7
♦ ❝♦♥s✉♠♦ ❡ ❛ ♣♦❧✉✐çã♦ ❛♦ ♠❡✐♦ ❛♠❜✐❡♥t❡✳ ❏❡sús ❞❡ ❧❛ ❋✉❡♥t❡✱ ❈❊❖ ❞❛ Graphenea✱ ✉♠❛ ❞❛s ♣r✐♥❝✐♣❛✐s ♣r♦❞✉t♦r❛s ♠✉♥❞✐❛✐s ❞❡ ❣r❛❢❡♥♦ ❧❛♠✐♥❛❞♦ ❛✜r♠❛ q✉❡ ✧♦ ❣r❛❢❡♥♦ ❥á é ✉t✐❧✐③❛❞♦ ♣❛r❛ ❢❛❜r✐❝❛r ❡❧❡tr♦❞♦s ❞❡ ❜❛t❡r✐❛s✱ t❡❧❛s tát❡✐s✱ ❝é❧✉❧❛s ❢♦t♦✈♦❧t❛✐❝❛s ✭s♦❧❛r❡s✮✱ ❞✐s♣♦s✐t✐✈♦s ❞❡ ❡❧❡trô♥✐❝❛ ❞✐❣✐t❛❧ ❡ ❛♥❛❧ó❣✐❝❛ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛✱ ❡ ❝♦♠♣♦st♦s ❛✈❛♥ç❛❞♦s ♣❛r❛ ❛ ✐♥❞ústr✐❛ ❛❡r♦♥á✉t✐❝❛ ♦✉ ♣❛r❛ ❛ ❛❧t❛ ❝♦♠♣❡t✐çã♦ ❞❡ ✈❡❧❛✧✺✳
❚❛❧ é ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛❞❛ ❛ ❡st❡ t✐♣♦ ❞❡ ❝❛r❜♦♥♦✱ q✉❡ ❛ ❈♦♠✐ssã♦ ❊✉r♦♣❡✐❛ ❞❡s✲ t✐♥❛rá ❯❙✩ ✶✱✸ ❜✐❧❤ã♦ ♣❛r❛ ❛♣♦✐❛r ♥♦s ♣ró①✐♠♦s ❞❡③ ❛♥♦s ♣r♦❥❡t♦s ❡✉r♦♣❡✉s ♣✐♦♥❡✐r♦s ♣❛r❛ ♣❡sq✉✐s❛r ❛s ❛♣❧✐❝❛çõ❡s ❞♦ ❣r❛❢❡♥♦✳ ❆ ❧✐st❛ ❞❡ ❛♣❧✐❝❛çõ❡s t❡❝♥♦❧ó❣✐❝❛s ❞♦ ❣r❛❢❡♥♦✱ q✉❡ s❡❣✉♥❞♦ s❡✉s ♣ró♣r✐♦s ❞❡s❝♦❜r✐❞♦r❡s sã♦ ✧t❛♥t❛s q✉❡ ♥ã♦ ♣♦❞❡♠ s❡r ❡♥✉♠❡r❛❞❛s✧✱ ❛✉♠❡♥t❛♠ s❡♠ ❝❡ss❛r✱ ♠ês ❛ ♠ês✳
❆❜❛✐①♦ s❡❣✉❡ ✉♠❛ ✐♥tr♦❞✉çã♦ à ❡❧❡tr♦❞✐♥â♠✐❝❛ q✉â♥t✐❝❛ ❞♦ ❣r❛❢❡♥♦✱ ❝♦♠ ❡♥❢♦q✉❡ ❡♠ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡str✉t✉r❛✐s ❞❡ r❡❞❡✳
✶✳✸ ❆♣r❡s❡♥t❛çã♦ ❞❛ t❡♦r✐❛ ❞❡ ❉✐r❛❝ ♣❛r❛ ❜❛✐①❛s ❡♥❡r✲
❣✐❛s
P❛✉❧ ❆❞r✐❡♥ ▼❛✉r✐❝❡ ❉✐r❛❝ ✭❇r✐st♦❧✱ ✽ ❞❡ ❆❣♦st♦ ❞❡ ✶✾✵✷ ✲ ❚❛❧❧❛❤❛ss❡❡✱ ✷✵ ❞❡ ❖✉t✉❜r♦ ❞❡ ✶✾✽✹✮ ❢♦✐ ✉♠ ❢ís✐❝♦ t❡ór✐❝♦ ❜r✐tâ♥✐❝♦✳ ❊st✉❞♦✉ ❡♥❣❡♥❤❛r✐❛ ❡❧étr✐❝❛ ♥❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❞❡ ❇r✐st♦❧✱ ❝♦♠♣❧❡t❛♥❞♦ ♦ ❝✉rs♦ ❡♠ ✶✾✷✶✳ ❋❡③ ❝♦♥tr✐❜✉✐çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ❡ ❊❧❡tr♦❞✐♥â♠✐❝❛ ◗✉â♥t✐❝❛✳ ❋♦✐ ♣r♦❢❡ss♦r ❧✉✲ ❝❛s✐❛♥♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❈❛♠❜r✐❞❣❡ ❡ ♣❛ss♦✉ ♦s ú❧t✐♠♦s ❞❡③ ❛♥♦s ❞❛ s✉❛ ✈✐❞❛ ♥❛ ❋❧♦r✐❞❛ ❙t❛t❡ ❯♥✐✈❡rs✐t②✳ ❊♥tr❡ ♦✉tr❛s ❞❡s❝♦❜❡rt❛s✱ ❢♦r♠✉❧♦✉ ❛ ❊q✉❛çã♦ ❞❡ ❉✐r❛❝✱ q✉❡ ❞❡s❝r❡✈❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s ❢ér♠✐♦♥s ❡ q✉❡ ♦ ❧❡✈♦✉ à ♣r❡✈✐sã♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❛ ❛♥t✐♠❛tér✐❛✳
❊♠ s✉❛ t❡s❡✱ ❞❡❢❡♥❞✐❞❛ ❡♠ ✶✾✷✻✱ ❞❡s❡♥✈♦❧✈❡✉ ✉♠❛ ✈❡rsã♦ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ✐♥❝♦r♣♦r❛♥❞♦ ❛ ♠❡❝â♥✐❝❛ ♠❛tr✐❝✐❛❧ ❞❡ ❲❡r♥❡r ❍❡✐s❡♥❜❡r❣ ❝♦♠ ❛ ♠❡❝â♥✐❝❛ ♦♥❞✉❧❛tór✐❛ ❞❡ ❊r✇✐♥ ❙❝❤rö❞✐♥❣❡r ♥✉♠ ú♥✐❝♦ ❢♦r♠❛❧✐s♠♦ ♠❛t❡♠át✐❝♦✳ ❊♠ ✶✾✷✽✱ ❞❡s❡♥✈♦❧✈❡✉ ❛ ❝❤❛♠❛❞❛ ❊q✉❛çã♦ ❞❡ ❉✐r❛❝ q✉❡ ♦ ❧❡✈♦✉ ❛ ♣r❡✈❡r ❛ ❡①✐stê♥❝✐❛ ❞♦ ♣ós✐tr♦♥✱ ❛ ❛♥t✐♣❛rtí❝✉❧❛ ❞♦ ❡❧étr♦♥✱ q✉❡ ❢♦✐ ♦❜s❡r✈❛❞♦ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ❡♠ ✶✾✸✷ ♣♦r ❈❛r❧ ❉❛✈✐❞ ❆♥❞❡rs♦♥✳
◆❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ é ✉♠❛ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ r❡❧❛t✐✈íst✐❝❛ q✉❡ ❞❡s❝r❡✈❡ ❝♦♠ s✉❝❡ss♦ ♣❛rtí❝✉❧❛s ❡❧❡♠❡♥t❛r❡s ❞❡ s♣✐♥ ✲1
2 ✱ ❝♦♠♦ ♦ ❡❧étr♦♥✳ ❆♥t❡r✐♦r♠❡♥t❡✱ ❛ ❡q✉❛çã♦ ❞❡ ❑❧❡✐♥✲●♦r❞♦♥ ✭✉♠❛ ❡q✉❛çã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ♥❛s ❞❡r✐✈❛❞❛s t❡♠♣♦r❛✐s ❡ ❡s♣❛❝✐❛✐s✮ ❢♦✐ ♣r♦♣♦st❛ ♣❛r❛ ❛ ♠❡s♠❛ ❢✉♥çã♦✱ ♠❛s ❛♣r❡s❡♥t♦✉ sér✐♦s ♣r♦❜❧❡♠❛s ♥❛ ❞❡✜♥✐çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳
❉✐r❛❝ t❡♥t♦✉ ❡♥❝♦♥tr❛r ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❝♦♠ ✉♠❛ ❞❡♥s✐❞❛❞❡ ❞❡
✺❋♦♥t❡ ❊❋❊ ✲ ❆❣❡♥❝✐❛ ❊❋❊✱ ❞✐s♣♦♥í✈❡❧ ❡♠ ❁❤tt♣✿✴✴t❡❝♥♦❧♦❣✐❛✳t❡rr❛✳❝♦♠✳❜r✴❡❧❡tr♦♥✐❝♦s✴❣r❛❢❡♥♦✲❞❡✲
✶✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❡str✉t✉r❛✐s ❞❛ r❡❞❡ 8
♣r♦❜❛❜✐❧✐❞❛❞❡ ♣♦s✐t✐✈❛✳ ❆ ❡q✉❛çã♦ q✉❡ ❢♦r♠✉❧♦✉ é ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ♣♦rt❛♥t♦ ❡❧✐♠✐♥♦✉ ❡ss❡ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛✳ ❊ ♠♦str♦✉ q✉❡ ♦ s♣✐♥ ♣♦❞❡r✐❛ s❡r ❞❡❞✉③✐❞♦ ❢❛❝✐❧♠❡♥t❡ ❞❛ ❡q✉❛çã♦✱ ❛♦ ✐♥✈és ❞❡ ♣♦st✉❧❛❞♦✳ ❈♦♥t✉❞♦✱ ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ ♥ã♦ é ♣❡r❢❡✐t❛♠❡♥t❡ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛ t❡♦r✐❛ ❞❛ r❡❧❛t✐✈✐❞❛❞❡✱ ♣♦✐s ♥ã♦ ♣r❡✈ê ❛ ❝r✐❛çã♦ ❡ ❞❡str✉✐çã♦ ❞❡ ♣❛rtí❝✉❧❛s✱ ❛❧❣♦ q✉❡ ❛♣❡♥❛s ✉♠❛ t❡♦r✐❛ q✉â♥t✐❝❛ ❞❡ ❝❛♠♣♦s ♣♦❞❡r✐❛ tr❛t❛r✳
P❛✉❧ ❉✐r❛❝ ❡❧❛❜♦r♦✉ s✉❛ ❡q✉❛çã♦ ❝♦♠♦ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ r❡❧❛t✐✈íst✐❝❛ ❞❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❞❡s❝r❡✈❡r ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠ ❡❧étr♦♥ ♥❛ ♠❡❝â♥✐❝❛ q✉â♥✲ t✐❝❛✳ ❖r✐❣✐♥❛❧♠❡♥t❡ ❢♦✐ ❝♦♥❝❡❜✐❞❛ ♣❛r❛ s❡r ❛♣❧✐❝❛❞❛ ❛ ❡❧étr♦♥s ♠❛ss✐✈♦s s❡ ♠♦✈❡♥❞♦ ♥♦ ❡s♣❛ç♦ tr✐❞✐♠❡♥s✐♦♥❛❧ ✭✸❉✮ ❡ ❞❡♣♦✐s ❣❡r♦✉ ✉♠❛ ❡❧❡tr♦❞✐♥â♠✐❝❛ q✉â♥t✐❝❛✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛ ❛ ♦✉tr❛s s✐t✉❛çõ❡s ❛❧é♠ ❞❡ ❡❧étr♦♥s ♠❛ss✐✈♦s ❡♠ ✸❉✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❡♠ ♦✉tr❛s ❞✐♠❡♥sõ❡s ❡ ♦✉tr♦s t✐♣♦s ❞❡ ♣❛rtí❝✉❧❛s ✲ s❡♠ ♠❛ss❛ ♦✉ s❡♠ ❝❛r❣❛✳ ❘❡❝❡❜❡✉ ❡♠ ✶✾✸✸✱ ❥✉♥t♦ ❝♦♠ ❊r✇✐♥ ❙❝❤rö❞✐♥❣❡r✱ ♦ ◆♦❜❡❧ ❞❡ ❋ís✐❝❛ ❝♦♠ ♦ tr❛❜❛❧❤♦ ❚❤❡♦r② ♦❢ ❊❧❡❝tr♦♥s ❛♥❞ P♦s✐tr♦♥s✳
◆❛ ❢ís✐❝❛ ❞❛ ♠❛tér✐❛ ❝♦♥❞❡♥s❛❞❛✱ ♦s ❢ér♠✐♦♥s ❞❡ ❉✐r❛❝ ✭❛ss✐♠ ❝❤❛♠❛❞❛s ❛s ♣❛rtí❝✉✲ ❧❛s q✉❡ ♦❜❡❞❡❝❡♠ ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝✮✱ ❛♣❛r❡❝❡♠ ❝♦♠♦ q✉❛s✐♣❛rtí❝✉❧❛s ❡♠ ❛❧❣✉♠❛s ❡s✲ tr✉t✉r❛s ❞❡ ❜❛♥❞❛s ❛ ❜❛✐①❛s ❡♥❡r❣✐❛s✳ ❖ ❡①❡♠♣❧♦ ♠❛✐s ❢❛♠♦s♦ é ❝♦♠ ❝❡rt❡③❛ ♦ ❣r❛❢❡♥♦✱ ♠❛s ♥ã♦ é ♦ ú♥✐❝♦✳ ❈♦♠♦ ✈❡r❡♠♦s ♠❛✐s ❛ ❢r❡♥t❡✱ ♦ ❣r❛❢❡♥♦ ❝♦♥té♠ ❢ér♠✐♦♥s ❞❡ ❉✐r❛❝ s❡♠ ♠❛ss❛ ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s✳
✶✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❡str✉t✉r❛✐s ❞❛ r❡❞❡
❖ ❣r❛❢❡♥♦ é ✉♠❛ s✐♠♣❧❡s ❡str✉t✉r❛ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❞❡ át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ❛❣r✉♣❛❞♦s ❡♠ r❡❞❡s ❤❡①❛❣♦♥❛✐s ♦✉ t✐♣♦ ❝♦❧♠é✐❛ ❞❡ ❛❜❡❧❤❛✳ ➱ ✈✐st❛ ❝♦♠♦ ✉♠❛ r❡❞❡ ❞❡ ❇r❛✈❛✐s ❛tr❛✈és ❞❡ ✉♠❛ r❡❞❡ tr✐❛♥❣✉❧❛r ❝♦♠ ❞♦✐s át♦♠♦s ❆ ❡ ❇ ♥❛ ❜❛s❡ ❝♦♠♦ ❛♣❛r❡❝❡ ✐❧✉str❛❞♦ ♥❛ ❋✐❣✉r❛ ✶✳✺✿
✶✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❡str✉t✉r❛✐s ❞❛ r❡❞❡ 9
❋✐❣✉r❛ ✶✳✺✿ ✭❛✮ ❆rr❛♥❥♦ ❤❡①❛❣♦♥❛❧ ❞❡ át♦♠♦s ❞❡ ❝❛r❜♦♥♦ q✉❡ ❢♦r♠❛♠ ✉♠❛ ❢♦❧❤❛ ❞❡ ❣r❛❢❡♥♦✳ ❊st❛ ❡str✉t✉r❛ é ✉♠❛ r❡❞❡ ❞❡ ❇r❛✈❛✐s ❝♦♠ ❞♦✐s át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ♣♦r ❝é❧✉❧❛ ✉♥✐tár✐❛ ✭❡♠ ❝✐♥③❛✮✱ ❡ s❡✉s ✈❡t♦r❡s ♣r✐♠✐t✐✈♦s sã♦ ❛1 ❡ ❛2✳ ❊♠ ✭❜✮ ❡stá r❡♣r❡s❡♥t❛❞❛ ❛ r❡❞❡ r❡❝í♣r♦❝❛ ❞❛ r❡❞❡ ♠♦str❛❞❛ ❡♠ ✭❛✮✳ ❆1 ❡ ❆2 sã♦ ♦s ✈❡t♦r❡s ♣r✐♠✐t✐✈♦s ❞❛ r❡❞❡ r❡❝í♣r♦❝❛✳ ❆ ár❡❛ ❡♠ ❝✐♥③❛ é ❛ ♣r✐♠❡✐r❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥✱ ♦s ♣♦♥t♦s ❑ ❡ ❑✬ sã♦ ♦s ♣♦♥t♦s ♦♥❞❡ ❛s ❜❛♥❞❛s ❞❡ ✈❛❧ê♥❝✐❛ ❡ ❞❡ ❝♦♥❞✉çã♦ s❡ t♦❝❛♠ ❬✶✾❪✳
s❡ ❞❡✜♥❡♠ ❞✉❛s s✉❜r❡❞❡s✻ ❆ ❡ ❇✳ ❆s ❧✐❣❛çõ❡s ❞♦s át♦♠♦s ❞❛ s✉❜r❡❞❡ ❆ ❛♦s ❞❛ s✉❜r❡❞❡
❇ sã♦ ❢♦r♠❛❞❛s ♣♦r três ❧✐❣❛çõ❡s ❝♦✈❛❧❡♥t❡s ❡♥tr❡ ♦r❜✐t❛✐s sp2✱ ♦s q✉❛✐s sã♦ ♦r✐❣✐♥❛❞♦s ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ ✉♠❛ ❤✐❜r✐❞✐③❛çã♦ ❞♦ ♦r❜✐t❛❧s ♠❛✐s ❡①t❡r♥♦ ❝♦♠ ♦s ♦r❜✐t❛✐spx ❡ py
s❡❣✉♥❞♦ ❛ ❡str✉t✉r❛ ❡❧❡trô♥✐❝❛ ❞♦ át♦♠♦ ❞❡ ❝❛r❜♦♥♦✿ ✶s2 ✷s2 ✷p2✳ ❊ss❡ t✐♣♦ ❞❡ ❧✐❣❛çã♦ ❡♥tr❡ ♦r❜✐t❛✐ssp2 ♥♦ ♣❧❛♥♦ ❞❛ ❝❛♠❛❞❛ sã♦ ❞❡♥♦♠✐♥❛❞❛s ❝♦♠✉♠❡♥t❡ ❝♦♠♦ ❧✐❣❛çõ❡s ❞♦ t✐♣♦ σ ❡ sã♦ r❡s♣♦♥sá✈❡✐s ♣❡❧❛ ❢♦rt❡ ❡str✉t✉r❛ ❞♦ ❣r❛❢❡♥♦✳ ❖✉tr♦ t✐♣♦ ❞❡ ❧✐❣❛çã♦ ♠❛✐s ❢r❛❝❛ é ❛ ❞❡♥♦♠✐♥❛❞❛ ❧✐❣❛çã♦ t✐♣♦ π q✉❡ ❛❝♦♥t❡❝❡ ❡♥tr❡ ♦s ♦r❜✐t❛✐s pz✱ ♦s q✉❛✐s ✜❝❛♠ ❡♠ ❞✐r❡çã♦
♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ ❞❛ ❝❛♠❛❞❛✳
❋✐❣✉r❛ ✶✳✻✿ ❋♦r♠❛çã♦ ❞♦s ♦r❜✐t❛✐s sp2✱ ❛tr❛✈és ❞❛ ❝♦♠❜✐♥❛çã♦ ❞♦ ♦r❜✐t❛❧ s ❝♦♠ ♦r❜✐t❛✐s px ❡ py✱ ❡♥q✉❛♥t♦ ♦ ♦r❜✐t❛❧ pz ♣❡r♠❛♥❡❝❡ ✐♥✈❛r✐❛♥t❡ ❬✷✵❪✳
❆s ❧✐❣❛çõ❡s σ ❡πr❡s✉❧t❛♠ ❡♠ ❜❛♥❞❛s ❞❡ ❡♥❡r❣✐❛s ♠♦str❛❞❛s ♥❛ ❋✐❣✉r❛ ✶✳✼ ❝❛❧❝✉✲ ❧❛❞❛s ♠❡❞✐❛♥t❡ ♦ ♠♦❞❡❧♦ t✐❣❤t✲❜✐♥❞✐♥❣ ❬✷✷❪✳ ❈♦♠♦ ♣♦❞❡♠♦s ♥♦t❛r✱ ♦ ❡s♣❡❝tr♦ ❞❛ ❋✐❣✉r❛ ✶✳✼ ❡stá ❝♦♠♣♦st♦ ♣♦r ✉♠❛ sér✐❡ ❞❡ ❜❛♥❞❛s ❞❡♥♦♠✐♥❛❞❛s π✱ π∗✱ σ ♦✉ σ∗ ❞❡♣❡♥❞❡♥❞♦ ❞❛
❧✐❣❛çã♦ ❛ q✉❛❧ ❛❝❤❛✲s❡ r❡❧❛❝✐♦♥❛❞❛✳ ■st♦ é✱ ❛s ❜❛♥❞❛sσ ❡σ∗ sã♦ ♣r♦✈❡♥✐❡♥t❡s ❞❛ ❧✐❣❛çã♦ ♥♦
♣❧❛♥♦ ❞♦s ♦r❜✐t❛✐ssp2✱ ❡♥q✉❛♥t♦ ❛s ❜❛♥❞❛s π ❡π∗ r❡s✉❧t❛♠ ❞❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦s ♦r❜✐t❛✐s
pz✳
✻❆s ❝♦♦r❞❡♥❛❞❛s q✉❡ ❧♦❝❛❧✐③❛♠ ♦s ♣♦♥t♦s ❞❡ ❉✐r❛❝ ❑ ❡ ❑✬ ♣♦❞❡♠ s❡r ♦❜s❡r✈❛❞❛s ♥♦ ❆♣ê♥❞✐❝❡ ❞♦
✶✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❡str✉t✉r❛✐s ❞❛ r❡❞❡ 10
❋✐❣✉r❛ ✶✳✼✿ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞❡ ❡♥❡r❣✐❛s ♣❛r❛ ♦ ❣r❛❢❡♥♦ ❬✷✷❪✳
❆s ❝❛r❛❝t❡ríst✐❝❛s ♠❛✐s ✐♥t❡r❡ss❛♥t❡s ❞♦ ❣r❛❢❡♥♦ sã♦ ♦❜t✐❞❛s ♣❛r❛ ❜❛✐①❛s ❡①❝✐t❛çõ❡s ♣❡rt♦ ❞❛ r❡❣✐ã♦ ❞♦s ♣♦♥t♦s ❑ ✭♦✉ ❑✬✮ ♥❛ ♣r✐♠❡✐r❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥✱ ❝♦♥s✐❞❡r❛♥❞♦ só ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦s ♦r❜✐t❛✐s pz✳
◆♦ ❣r❛❢❡♥♦ ❡①✐st❡♠ ♣♦rt❛❞♦r❡s ❞❡ ❝❛r❣❛s ✭❡❧étr♦♥s ♦✉ ❜✉r❛❝♦s✮ q✉❡ s❡ ♣r♦♣❛❣❛♠ ❝♦♠♦ s❡ ❢♦ss❡♠ ❢ér♠✐♦s s❡♠ ♠❛ss❛ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡s ❞❛ ♦r❞❡♠ ❞❡ 106 ♠✴s ❛♦ ✐♥✈és ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③3×108 ♠✴s✳ ❆ss✐♠✱ ♥❡ss❡ s✐st❡♠❛ ❡❧❡s ❞❡✈❡♠ s❡r ❡s❝r✐t♦s ♣❡❧❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ ❛♦ ✐♥✈és ❞❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ✉s✉❛❧♠❡♥t❡ ✉s❛❞❛✳ ❆ ❞❡s❝r✐çã♦ ❞♦s ♣♦rt❛✲ ❞♦r❡s ❞❡ ❝❛r❣❛ ♣♦r ❢✉♥çõ❡s ❞❡ ♦♥❞❛ ❝♦♠ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s é s❡♠❡❧❤❛♥t❡ ❛ ✉s❛❞❛ ♣❛r❛ ❡s❝r❡✈❡r ❢✉♥çõ❡s ❞❡ ♦♥❞❛ ❞❡ s♣✐♥✳ ◆♦ ❡♥t❛♥t♦✱ ❛s ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❛ ❝❛❞❛ ✉♠❛ ❞❛s s✉❜r❡❞❡s ❞❛ ❡str✉t✉r❛ ❞♦ ❣r❛❢❡♥♦ ❡ ♥ã♦ ❛♦ s♣✐♥ ❞❛s ♣❛rtí❝✉❧❛s✳ ❉❡ss❛ ❢♦r♠❛✱ ❞✐③✲s❡ q✉❡ ♦s ♣♦rt❛❞♦r❡s ❞❡ ❝❛r❣❛ ♥♦ ❣r❛❢❡♥♦ t❡♠ ❛ss♦❝✐❛❞♦ ❛ ❡❧❡s ✉♠ ♣s❡✉❞♦s♣✐♥✱ σ✳ ❖ ♣s❡✉❞♦s♣✐♥ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s s✉❜r❡❞❡s ♣❛r❛ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❞♦s ♣♦rt❛❞♦r❡s ❞❡ ❝❛r❣❛✱ ❛ss✐♠✱ ❡❧étr♦♥s ❡ ❜✉r❛❝♦s ❝✉❥♦s ♣s❡✉❞♦s♣✐♥s ❛♣♦♥t❛♠ ❡♠ ✉♠❛ ♠❡s♠❛ ❞✐r❡çã♦ ♣♦❞❡♠ s❡r ✐♠❛❣✐♥❛❞♦s ❝♦♠♦ t❡♥❞♦ ♦r✐❣❡♠ ❡♠ ✉♠❛ ♠❡s♠❛ s✉❜r❡❞❡✳ ❊♠ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❞❡s❝r✐çã♦ ❛tr❛✈és ❞❡ ✉♠❛ ❡q✉❛çã♦ ❝♦♠♦ ❛ ❞❡ ❉✐r❛❝ ❡ ❞♦ ❡s♣❡❝tr♦ ❞❡ ❡♥❡r❣✐❛ s❡♠❡❧❤❛♥t❡ ❛♦ ❞❡ ♣❛rtí❝✉❧❛s r❡❧❛t✐✈íst✐❝❛s✱ ♦s ♣♦rt❛❞♦r❡s ❞❡ ❝❛r❣❛ ♥♦ ❣r❛❢❡♥♦ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❢ér♠✐♦♥s ❞❡ ❉✐r❛❝ s❡♠ ♠❛ss❛ ♦✉ ❞❡ ♣❛rtí❝✉❧❛s q✉✐r❛✐s s❡♠ ♠❛ss❛✳
❖ ❡s♣❡❝tr♦ ❧✐♥❡❛r ❞❡ ❡♥❡r❣✐❛ ❞♦ ❣r❛❢❡♥♦ ❡ ❛ ❞❡s❝r✐çã♦ ❞♦s ♣♦rt❛❞♦r❡s ❞❡ ❝❛r❣❛ ♣♦r ✉♠❛ ❡q✉❛çã♦ ❞♦ t✐♣♦ ❞❡ ❉✐r❛❝ ❧❡✈❛♠ ❛ ✈ár✐❛s ❛♥❛❧♦❣✐❛s ❡♥tr❡ ♦s ❢❡♥ô♠❡♥♦s ❞❡ tr❛♥s♣♦rt❡ ♥♦ ❣r❛❢❡♥♦ ❡ ❢❡♥ô♠❡♥♦s ❡st✉❞❛❞♦s ♣❡❧❛ ❡❧❡tr♦❞✐♥â♠✐❝❛ q✉â♥t✐❝❛✳ ❉❛❞❛ ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ s❡♠ ♠❛ss❛ ❡♠ ✷✰✶ ❞✐♠❡♥sõ❡s✿
υF(σxpx+σypy)Ψ(r, t) = i~
∂Ψ(r, t)
✶✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❡str✉t✉r❛✐s ❞❛ r❡❞❡ 11
♦♥❞❡ σi✱ sã♦ ❛s ♠❛tr✐③❡s ❞❡ P❛✉❧✐ ❝♦♠ i = x, y, z✱ ❡ pj é ❛ ❝♦♠♣♦♥❡♥t❡ ❞♦ ♦♣❡r❛❞♦r
♠♦♠❡♥t♦p =−i~∇ ❝♦♠ j =x, y✳ ❚❡♥❞♦ ❡♠ ✈✐st❛ ❡st❛❞♦s ❡st❛❝✐♦♥ár✐♦s é ♣♦ssí✈❡❧ ❢❛③❡r
❛ s✉❜st✐t✉✐çã♦ Ψ(r, t) = ψ(r)e−iǫt/~
✳ ■ss♦ tr❛♥s❢♦r♠❛ ❛ ❡q✳ ❞❡ ❉✐r❛❝ ♥✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
υF(σxpx+σypy)ψ(r) =ǫψ(r). ✭✶✳✷✮
❆q✉✐ ❝♦♥s✐❞❡r❛♠♦s ❛ ✉♥✐❞❛❞❡ r❡❧❛t✐✈íst✐❝❛ e = 1✳ ❉❡ss❛ ❢♦r♠❛ ♦ ❍❛♠✐❧t♦♥✐❛♥♦
q✉❡ ❞❡s❝r❡✈❡ ♦s ♣♦rt❛❞♦r❡s ❞❡ ❝❛r❣❛ é ❞♦ t✐♣♦ ❞❡ ❉✐r❛❝✱ ❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
H =υF
0 px−ipy
px+ipy 0
!
=υFσ·♣. ✭✶✳✸✮
❉❡✈✐❞♦ ❛♦ ❢❛t♦ ❞❡ σ s❡r ♦ ♣s❡✉❞♦s♣✐♥ ❞❛ s✉❜r❡❞❡✱ ♦ ❛✉t♦❡st❛❞♦ |ΨKi = (ψA, ψB)
é ❝❤❛♠❛❞♦ ❞❡ ♣s❡✉❞♦s♣✐♥♦r✱ ♦♥❞❡ ψA ❢♦r♥❡❝❡ ❛ ❛♠♣❧✐t✉❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡♥❝♦♥tr❛r
♦ ❡❧étr♦♥ ♥❛ s✉❜r❡❞❡ ❆✱ ❛ss✐♠ ❝♦♠♦ψB ♦ ❢❛③✱ ♥❛ s✉❜r❡❞❡ ❇✳ ❖ ❍❛♠✐❧t♦♥✐❛♥♦ q✉❡ ❞❡s❝r❡✈❡
♦s ❡st❛❞♦s ❡♠ t♦r♥♦ ❞♦s ♣♦♥t♦s ❞❡K ❞❛ ③♦♥❛ ❞❡ ❇r✐❧❧♦✐✉♥ é✿
HK =~υF
0 kx−iky
kx+iky 0
!
. ✭✶✳✹✮
❉✐❛❣♦♥❛❧✐③❛♥❞♦ ♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❛❝✐♠❛✱ ❡♥❝♦♥tr❛♠♦s ♦s ❛✉t♦✈❡t♦r❡s✿
E =±~υFk, ✭✶✳✺✮
❡ ♦s ❛✉t♦❡st❛❞♦s✿
Ψ±K
= √1
2 1
±eiθ
!
✭✶✳✻✮
❝♦♠ θ = arctg(ky
kx)✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦s ❡st❛❞♦s ♣♦r ✈♦❧t❛ ❞♦ ♣♦♥t♦ ❑✬✱ ❡♥❝♦♥tr❛♠♦s✱ ❞❡
❢♦r♠❛ ❛♥á❧♦❣❛✿
HK′ =υFσ∗·♣
Ψ±K′
= √1 2
1
±e−iθ
!
. ✭✶✳✼✮
P♦r ✜♠✱ ✈❡♠♦s t❛♠❜é♠ q✉❡ ❛ r❡❧❛çã♦ ❡♥tr❡ ♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❛♦ r❡❞♦r ❞♦s ♣♦♥t♦s ❞❡ ❉✐r❛❝ s❡ ❞á ❛tr❛✈és ❞❛ r❡❧❛çã♦ ❍K = ❍∗−K✱ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❡①✐❜❡♠ ♦ ♠❡s♠♦
❡s♣❡❝tr♦ ❞❡ ❡♥❡r❣✐❛✳
