Chemically Tunable Properties of Graphene Covered Simultaneously

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Chemically Tunable Properties of Graphene Covered Simultaneously with Hydroxyl and Epoxy Groups Ivan Guilhon Mitoso Rocha, Friedhelm Bechstedt, Silvana Botti, Marcelo Marques, and Lara Kühl Teles J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b09513 • Publication Date (Web): 17 Nov 2017 Downloaded from http://pubs.acs.org on November 21, 2017

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The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

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The Journal of Physical Chemistry

❈❤❡♠✐❝❛❧❧② ❚✉♥❛❜❧❡ Pr♦♣❡rt✐❡s ♦❢ ●r❛♣❤❡♥❡ ❈♦✈❡r❡❞ ❙✐♠✉❧t❛♥❡♦✉s❧② ✇✐t❤ ❍②❞r♦①②❧ ❛♥❞ ❊♣♦①② ●r♦✉♣s †✱‡

■✳ ●✉✐❧❤♦♥✱

‡

❋✳ ❇❡❝❤st❡❞t✱

‡

❙✐❧✈❛♥❛ ❇♦tt✐✱

†

▼✳ ▼❛rq✉❡s✱

∗✱†

❛♥❞ ▲✳ ❑✳ ❚❡❧❡s

†●r✉♣♦ ❞❡ ▼❛t❡r✐❛✐s ❙❡♠✐❝♦♥❞✉t♦r❡s ❡ ◆❛♥♦t❡❝♥♦❧♦❣✐❛✱ ■♥st✐t✉t♦ ❚❡❝♥♦❧ó❣✐❝♦ ❞❡

❆❡r♦♥á✉t✐❝❛✱ ❉❈❚❆✱ ✶✷✷✷✽✲✾✵✵ ❙ã♦ ❏♦sé ❞♦s ❈❛♠♣♦s✱ ❇r❛③✐❧ ‡■♥st✐t✉t ❢ür ❋❡st❦ör♣❡rt❤❡♦r✐❡ ✉♥❞ ✲♦♣t✐❦✱ ❋r✐❡❞r✐❝❤✲❙❝❤✐❧❧❡r✲❯♥✐✈❡rs✐tät ❛♥❞ ❊❚❙❋✱

▼❛①✲❲✐❡♥✲P❧❛t③ ✶✱ ❉✲✵✼✼✹✸ ❏❡♥❛✱ ●❡r♠❛♥② ❊✲♠❛✐❧✿ ❧❦t❡❧❡s❅✐t❛✳❜r

✶

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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❆❜str❛❝t ❲❡ ✐♥✈❡st✐❣❛t❡ t❤❡ ❝❤❡♠✐❝❛❧❧② t✉♥❛❜❧❡ ♣r♦♣❡rt✐❡s ♦❢ ✉♥❞❡rst♦✐❝❤✐♦♠❡tr✐❝ ❣r❛♣❤❡♥❡ ♦①✐❞❡ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡ ❛♥❞ t❤❡ ❛♠♦✉♥t ♦❢ ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s ❛❞s♦r❜❡❞ ♦♥ ❣r❛♣❤❡♥❡✳ ❚❤❡ str✉❝t✉r❛❧ ❞✐s♦r❞❡r ✐s ♠♦❞❡❧❡❞ ✇✐t❤✐♥ ❛ st❛t✐st✐❝❛❧ ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ ❛ ❝❧✉st❡r ❡①♣❛♥s✐♦♥✳ ❆ s❡t ♦❢ ✸✵✽ ❝❧✉st❡r s②♠♠❡tr② ❝❧❛ss❡s ✐s ✐♥✲ ✈❡st✐❣❛t❡❞ ✇✐t❤ ❛❜ ✐♥✐t✐♦ ❝❛❧❝✉❧❛t✐♦♥s ❜❛s❡❞ ♦♥ ❞❡♥s✐t② ❢✉♥❝t✐♦♥❛❧ t❤❡♦r②✳ ❆ ❝♦♠♣❧❡t❡ s❝❡♥❛r✐♦ ♦❢ ❡♥❡r❣❡t✐❝s ❛♥❞ ♣❤❛s❡ st❛❜✐❧✐t② ✐s ❞❡✈❡❧♦♣❡❞✱ ②✐❡❧❞✐♥❣ ✐♥s✐❣❤t ✐♥t♦ t❤❡ str✉❝✲ t✉r❡ ❛♥❞ ❡❧❡❝tr♦♥✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❣r❛♣❤❡♥❡ ♦①✐❞❡✳ ❖✉r r❡s✉❧ts s❤♦✇ t❤❛t t❤❡ t❡♥❞❡♥❝② t♦ ❛❣❣❧♦♠❡r❛t✐♦♥ ♦❢ ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡✐r r❡❧❛t✐✈❡ ♣r♦♣♦rt✐♦♥✱ ❛♥❞ ✐♥❞✐❝❛t❡ t❤❡ ❢❛✈♦r❛❜❧❡ ❢♦r♠❛t✐♦♥ ♦❢ ❛ ✉♥✐q✉❡ ♦①②❣❡♥✲r✐❝❤ ♣❤❛s❡ ✇✐t❤ ❜♦t❤ ❣r♦✉♣s✳ ❙tr✉❝t✉r❛❧ ❛♥❞ ❡❧❡❝tr♦♥✐❝ ♣r♦♣❡rt✐❡s ❛r❡ ♣r❡❞✐❝t❡❞ ❢♦r t❤❡ ✇❤♦❧❡ r❛♥❣❡ ♦❢ ❝❤❡♠✐❝❛❧ ❝♦♠✲ ♣♦s✐t✐♦♥s✳ ❚❤❡ ♦♣t✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ♦①②❣❡♥✲r✐❝❤ ♣❤❛s❡s ❛r❡ ❛❧s♦ ❞✐s❝✉ss❡❞ ✐♥ ❞❡t❛✐❧ ❢♦r ❞✐✛❡r❡♥t ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥s✳

■♥tr♦❞✉❝t✐♦♥ ●r❛♣❤❡♥❡ ✐s ❛ ♠❛t❡r✐❛❧ ✇✐t❤ ❡①❝❡♣t✐♦♥❛❧ ♠❡❝❤❛♥✐❝❛❧✱ ❡❧❡❝tr♦♥✐❝ ❛♥❞ t❤❡r♠❛❧ ♣r♦♣❡rt✐❡s✳

✶✱✷

❚❤❡ ❞✐s❝♦✈❡r② ♦❢ ❣r❛♣❤❡♥❡ ❛♥❞ t❤❡ ❡♠❡r❣❡♥❝❡ ♦❢ ✐ts ✉♥✐q✉❡ ♣❤②s✐❝❛❧ ♣r♦♣❡rt✐❡s ❤❛s ❛ttr❛❝t❡❞ ❣r❡❛t ✐♥t❡r❡st ❜♦t❤ ❢r♦♠ ❛❝❛❞❡♠✐❛ ❛♥❞ ✐♥❞✉str②✱ ❛❧s♦ ✐♥ ✈✐❡✇ ♦❢ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ♠✐♥✐❛t✉r✲ ✐③❛t✐♦♥ ♦❢ ❞❡✈✐❝❡s✳ ❍♦✇❡✈❡r✱ t❤❡ ❢❛❝t t❤❛t ✐ts ❜❛♥❞ str✉❝t✉r❡ ❡①❤✐❜✐ts ♥♦ ❡♥❡r❣② ❣❛♣ ❧✐♠✐ts ✐ts ❡❧❡❝tr♦♥✐❝ ❛♥❞ ♦♣t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s✳ ❈❤❡♠✐❝❛❧ ❢✉♥❝t✐♦♥❛❧✐③❛t✐♦♥ ❜② ♦①②❣❡♥ ❛♥❞ ❤②❞r♦❣❡♥ ❤❛s ❜❡❡♥ ♣r♦♣♦s❡❞ t♦ ♦✈❡r❝♦♠❡ t❤❡s❡ ❧✐♠✐t❛t✐♦♥s✳ ❋✉❧❧② ♦①✐❞✐③❡❞ ❣r❛♣❤❡♥❡✱ ❝❛❧❧❡❞ ❣r❛♣❤❡♥❡ ♦①✐❞❡ ✭♦r ●❖✮✱ ♦r ❤②❞r♦❣❡♥❛t❡❞ ❣r❛♣❤❡♥❡✱ ❦♥♦✇♥ ❛s ❣r❛♣❤❛♥❡✱ ❞✐s♣❧❛② ♦♣❡♥ ❡♥❡r❣② ❣❛♣s ❛♥❞ ❡①❤✐❜✐t s✐❣♥✐✜❝❛♥t❧② ❞✐✛❡r❡♥t ❡❧❡❝tr♦♥✐❝ ❛♥❞ ♦♣t✐❝❛❧ ♣r♦♣❡rt✐❡s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡✐r ♣❛r❡♥t ❣r❛♣❤❡♥❡✳

✸✕✼

❚❤✐s ♥❡✇ ❝❧❛ss ♦❢

t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ✭✷❉✮ ❣r❛♣❤❡♥❡✲❜❛s❡❞ ♠❛t❡r✐❛❧s ✐s t❤❡r❡❢♦r❡ ❛❝t✐✈❡❧② st✉❞✐❡❞ ❢♦r ❛♣♣❧✐❝❛t✐♦♥s✳ ■♥❞❡❡❞✱ ●❖ ✐s ❛ str♦♥❣ ❝❛♥❞✐❞❛t❡ ❢♦r ❛ ✷❉ ♣❧❛t❢♦r♠ ✇✐t❤ t✉♥❛❜❧❡ ❡❧❡❝tr♦♥✐❝ ♣r♦♣❡rt✐❡s ❝♦♠♣❛t✐❜❧❡ t♦ ❣r❛♣❤❡♥❡ t❡❝❤♥♦❧♦❣②✳ ●❖ ❤❛s ❛❧r❡❛❞② ❜❡❡♥ ❡♠♣❧♦②❡❞ ✐♥ ✷❉ s❡♥s♦r ❞❡✈✐❝❡s✱

✷


✽✱✾

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❡❧❡❝tr♦♥✐❝s✱

✶✵✕✶✷

♦♣t♦❡❧❡❝tr♦♥✐❝s

✹✱✶✷

❛♥❞ s✉♣❡r❝❛♣❛❝✐t♦rs✳

✐❝❛❧❧② ❡①❢♦❧✐❛t❡❞ ❛❜♦✉t ✶✺✵ ②❡❛rs ❛❣♦ ❜② ❇r♦❞✐❡

✶✹

✶✸

❚❤❡ ✜rst ●❖ s❛♠♣❧❡s ✇❡r❡ ❝❤❡♠✲

❢r♦♠ ❣r❛♣❤✐t❡ ♦①✐❞❡✳ ❍♦✇❡✈❡r✱ ❞✉❡ t♦ ✐ts

❛♠♦r♣❤♦✉s ♥❛t✉r❡✱ t❤❡ ❞❡t❛✐❧s ♦❢ ✐ts ❛t♦♠✐❝ str✉❝t✉r❡ ❛r❡ st✐❧❧ ✉♥❞❡r ❞❡❜❛t❡✳

✸✱✺

❉✐✛❡r❡♥t ♠♦❞✲

❡❧s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ t♦ st✉❞② t❤❡ ❛t♦♠✐❝ str✉❝t✉r❡ ♦❢ ●❖ ❛♥❞ ❡①♣❧❛✐♥ ✐ts ♣r♦♣❡rt✐❡s✳

✻✱✶✺✕✶✽

❘❡❝❡♥t ◆▼❘ ♠❡❛s✉r❡♠❡♥ts ❤❛✈❡ s❤❡❞ ❧✐❣❤t ♦♥ t❤❡ ❝❤❡♠✐❝❛❧ ❜♦♥❞✐♥❣ ♦❢ ❤②❞r♦①②❧ ✭−❖❍✮ ❛♥❞ ❡♣♦①② ✭−❖−✮ ❣r♦✉♣s ♦♥ ❣r❛♣❤❡♥❡ s❤❡❡ts✳ ❚❤❡ ❡①♣❡r✐♠❡♥t❛❧ ✜♥❞✐♥❣s ✐♥❞✐❝❛t❡ t❤❛t t❤❡ t✇♦ ❝❤❡♠✐❝❛❧ ❣r♦✉♣s t❡♥❞ t♦ ❜❡ ❛❞s♦r❜❡❞ ❝❧♦s❡ t♦ ❡❛❝❤ ♦t❤❡r✳

✶✾

●❖ ✐s ✉s✉❛❧❧② s②♥t❤❡s✐③❡❞

❢r♦♠ ❣r❛♣❤✐t❡ ♣♦✇❞❡r ✉s✐♥❣ ♦①✐❞❛♥t s♦❧✉t✐♦♥s ❛♥❞ s✉❜s❡q✉❡♥t ❡①❢♦❧✐❛t✐♦♥✳

✺

❚❤❡ ✇❡t ❝❤❡♠✐❝❛❧

tr❡❛t♠❡♥t ♦❢ ❣r❛♣❤❡♥❡ ❛♥❞ t❤❡ ❡①♣♦s✉r❡ t♦ ♦①②❣❡♥ ♣❧❛s♠❛ ❛r❡ ❛❧s♦ s✉❝❝❡ss❢✉❧ ✇❛②s t♦ ♦❜t❛✐♥ ●❖ s❛♠♣❧❡s✳

✷✵

●r❛♣❤❡♥❡ ♦①✐❞❡ s❤❡❡ts s❛♠♣❧❡s ♠❛② st✐❧❧ ❡①❤✐❜✐t t✐♦♥ ♣r♦❝❡ss ✭y+z

< 1✮✳

sp2

❝❛r❜♦♥s ❞✉❡ t♦ ❛♥ ✐♥❝♦♠♣❧❡t❡ ♦①✐❞❛✲

❲❡ r❡❢❡r t♦ t❤❡s❡ s②st❡♠s ❛s ✉♥❞❡rst♦✐❝❤✐♦♠❡tr✐❝ ●❖✱ ✇❤✐❝❤ ❝❛♥ ❛❧s♦

❜❡ ♦❜t❛✐♥❡❞ ❜② ❝❤❡♠✐❝❛❧ r❡❞✉❝t✐♦♥ ♦❢ ❢✉❧❧②✲♦①✐❞✐③❡❞ ●❖✳

✷✶

❚❤❡ ✉♥❞❡rst♦✐❝❤✐♦♠❡tr✐❝ ♦①✐❞❡s

❛❧❧♦✇ t✉♥✐♥❣ t❤❡✐r ♣r♦♣❡rt✐❡s ❜❡t✇❡❡♥ t❤♦s❡ ♦❢ ●❖ ❛♥❞ ❣r❛♣❤❡♥❡✳ ❚❤✐s ✐♥t❡r❡st✐♥❣ ❝♦♥❝❡♣t ♦❢ ❝❤❡♠✐❝❛❧ t✉♥✐♥❣ ♣r♦♣❡rt✐❡s ❤❛s ❜❡❡♥ ❛♣♣❧✐❡❞ t♦ ♠❛♥② ✷❉ ❛❧❧♦②s ✐♥ ❡①♣❡r✐♠❡♥t❛❧ t❤❡♦r❡t✐❝❛❧ ✇♦r❦s✳

✷✽✕✸✶

❛♥❞

❆♥❛❧♦❣♦✉s❧②✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ♣❛rt✐❛❧❧② ❢✉♥❝t✐♦♥❛❧✐③❡ s❤❡❡ts✱ ❛s ❞❡♠♦♥✲

str❛t❡❞ ❢♦r t❤❡ ❛❞s♦r♣t✐♦♥ ♦❢ ❤②❞r♦❣❡♥✱ ❣r❛♣❤❡♥❡ s❤❡❡ts✳

✷✷✕✷✼

✸✷✱✸✸

❤❛❧♦❣❡♥

✸✹✱✸✺

❛♥❞ ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s ♦♥

✹✱✸✻✕✸✽

■♥ ❛ ♣r❡✈✐♦✉s ✇♦r❦✱

✸✾

✇❡ ❤❛✈❡ ✐♥✈❡st✐❣❛t❡❞ t❤❡ t✉♥❛❜❧❡ ♣r♦♣❡rt✐❡s ♦❢ ✉♥❞❡rst♦✐❝❤✐♦♠❡tr✐❝

●❖✱ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❡✛❡❝t ♦❢ ❛ s❡♣❛r❛t❡ ❛❞s♦r♣t✐♦♥ ♦❢

−❖❍

♦r

−❖−

♦♥ ❣r❛♣❤❡♥❡✳ ❚❤❡s❡

t✇♦ ❝❛s❡s ❝♦rr❡s♣♦♥❞✱ r❡s♣❡❝t✐✈❡❧②✱ t♦ ✇❡t ❛♥❞ ❞r② ❡♥✈✐r♦♥♠❡♥t❛❧ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥ ❧✐♠✐ts✳ ❍♦✇❡✈❡r✱ ✐t ✐s ❦♥♦✇♥ t❤❛t ✐♥ ♣r❛❝t✐❝❡ ♠♦r❡ t❤❛♥ ♦♥❡ ❛❞s♦r❜❡❞ ❣r♦✉♣✱ ❡✳❣✳

−❖❍

❛♥❞

−❖−✱

♠❛② ❝♦❡①✐st ❜❡❧♦✇ ♦r ❛❜♦✈❡ t❤❡ ❣r❛♣❤❡♥❡ ❜❛s❛❧ ♣❧❛♥❡ ♦❢ ❛ ✉♥❞❡rst♦✐❝❤✐♦♠❡tr✐❝ ●❖✳

✸✻✕✸✽

❚❤❡ ❡✛❡❝t ♦❢ t❤❡ ❝♦❡①✐st❡♥❝❡ ♦❢ ❞✐✛❡r❡♥t ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s ♦♥ t❤❡ t❤❡r♠♦❞②♥❛♠✐❝ st❛❜✐❧✐t②✱ ❛t♦♠✐❝ ♣♦s✐t✐♦♥s ❛♥❞ r❡s✉❧t✐♥❣ ❡❧❡❝tr♦♥✐❝ ❛♥❞ ♦♣t✐❝❛❧ ♣r♦♣❡rt✐❡s ❛r❡ ❧❡ss st✉❞✐❡❞ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ♥♦t ②❡t ❝♦♠♣❧❡t❡❧② ✉♥❞❡rst♦♦❞✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ st✉❞② t❤❡ s✐♠✉❧t❛♥❡♦✉s ❛❞s♦r♣t✐♦♥ ♦❢ ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s ♦♥

✸



❣r❛♣❤❡♥❡ ✐♥ ❛ ♠♦r❡ r❡❛❧✐st✐❝ s❝❡♥❛r✐♦✳

❚❤❡② ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❛t❡❞ ❛s ❝❤❡♠✐❝❛❧ ✈❛r✐❛t✐♦♥s

❜❡t✇❡❡♥ ♣✉r❡ ❣r❛♣❤❡♥❡✱ ❢✉❧❧② ♦①✐❞✐③❡❞ ❣r❛♣❤❡♥❡ ✇✐t❤ ♦♥❧② ❣r❛♣❤❡♥❡ ✇✐t❤ ♦♥❧②

−❖−

Page 4 of 30

−❖❍

❣r♦✉♣s✱ ❛♥❞ ❢✉❧❧② ♦①✐❞✐③❡❞

❣r♦✉♣s✳ ❚❤❡ t❤❡♦r❡t✐❝❛❧ tr❡❛t♠❡♥t ♦❢ t❤❡ s✐♠✉❧t❛♥❡♦✉s ❛❞s♦r♣t✐♦♥

❜② ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s r❡q✉✐r❡s t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ st❛t✐st✐❝❛❧ ❛♣♣r♦❛❝❤ ❢♦r ❢✉♥❝✲ t✐♦♥❛❧✐③❡❞ ❣r❛♣❤❡♥❡ s❤❡❡ts ✐♥ ❣r❡❛t s✐♠✐❧❛r✐t② t♦ ❛ t❡r♥❛r② ❛❧❧♦②✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❣❡♥❡r❛❧✐③❡ t❤❡ ♠❡t❤♦❞ ❜❛s❡❞ ♦♥ ❝❧✉st❡r ❡①♣❛♥s✐♦♥✱ t❤❛t ✇❡ ❤❛❞ ♣r♦♣♦s❡❞ ❢♦r ✐♥ ❘❡❢✳

✸✾

❢♦r s✐♥❣❧❡ ❛❞s♦r❜❡❞

❣r♦✉♣ s♣❡❝✐❡s✱ t♦ t❤❡ ❝❛s❡ ♦❢ s✐♠✉❧t❛♥❡♦✉s ❞❡❝♦r❛t✐♦♥ ♦❢ ❛ ❣r❛♣❤❡♥❡ s❤❡❡t ✇✐t❤ t✇♦ ❞✐✛❡r❡♥t ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s✳ ❲❡ ❢♦❝✉s ♦♥ ✉♥❛♠❜✐❣✉♦✉s ♥❡✇ ❢❡❛t✉r❡s ♦❢ ●❖ ❡♥❡r❣❡t✐❝s✱ ❢♦r ✐♥✲ st❛♥❝❡ t❤❡ ✐♥❝r❡❛s❡ ♦❢ t❤❡r♠♦❞②♥❛♠✐❝ st❛❜✐❧✐t② ❢♦r ❧♦✇✲❡♥❡r❣② ❢✉❧❧② ♦①✐❞✐③❡❞ ❣r❛♣❤❡♥❡ ✇❤❡♥ ❜♦t❤ ❜♦t❤ ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s ❛r❡ s✐♠✉❧t❛♥❡♦✉s❧② ❛❞s♦r❜❡❞✳ ❚❤❡ ✐♥❝r❡❛s❡❞ ❝❤❡♠✐✲ ❝❛❧ ✢❡①✐❜✐❧✐t② ❣✐✈❡s r✐s❡ t♦ ❛ ❧❛r❣❡r r❛♥❣❡ ♦❢ ❡♥❡r❣② ❣❛♣✱ ❛♥❞ ❤❡♥❝❡ ❛ ❣r❡❛t❡r ♣♦t❡♥t✐❛❧ ❢♦r ❜❛♥❞ ❣❛♣ ❡♥❣✐♥❡❡r✐♥❣✳ ❖r❞❡r❡❞ ❣❡♦♠❡tr✐❡s ♠❛② ✐♥❞✐❝❛t❡ t❤❛t t❤❡ ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s ❛❣❣❧♦♠❡r❛t❡ ✐♥ ❛ ✉♥✐q✉❡ ♦①②❣❡♥✲r✐❝❤ ♣❤❛s❡✳ ❚❤❡ ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✳

■♥ ▼❡t❤♦❞s s❡❝t✐♦♥ ✇❡ ♣r❡s❡♥t t❤❡ st❛t✐st✐❝❛❧ ❛♣✲

♣r♦❛❝❤ t❤❛t ✇❡ ✉s❡ t♦ ❞❡s❝r✐❜❡ t❤❡ s✐♠✉❧t❛♥❡♦✉s ✐♥❝♦♠♣❧❡t❡ ♦①✐❞❛t✐♦♥ ✇✐t❤ t✇♦ ♦①✐❞❛♥t ❣r♦✉♣s✳

❈♦♠♣✉t❛t✐♦♥❛❧ ❞❡t❛✐❧s ❛r❡ ❛❧s♦ ✐♥❝❧✉❞❡❞✳

❆ ❢✉❧❧ ♣✐❝t✉r❡ ♦❢ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ●❖

❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✐ts ❝❤❡♠✐❝❛❧ ❝♦♠♣♦s✐t✐♦♥ ✐s ❣✐✈❡♥ ✐♥ ❘❡s✉❧ts ❛♥❞ ❞✐s❝✉ss✐♦♥ s❡❝t✐♦♥✳

❚❤❡

t❤❡r♠♦❞②♥❛♠✐❝ ❛♥❛❧②s✐s ♦❢ ●❖ ❞❡♠♦♥str❛t❡s t❤❛t t❤❡ ♣r❡✈✐♦✉s❧② r❡♣♦rt❡❞ str♦♥❣ t❡♥❞❡♥❝② t♦ ♣❤❛s❡ s❡♣❛r❛t✐♦♥ ✐♥t♦ ♦①②❣❡♥✲♣♦♦r ❛♥❞ ✲r✐❝❤ ❞♦♠❛✐♥s

✸✾

r❡❧❛t✐✈❡ ❛❜✉♥❞❛♥❝❡ ♦❢ t❤❡ ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s✳

✐s r♦❜✉st ❛❣❛✐♥st ✈❛r✐❛t✐♦♥s ♦❢ t❤❡ ❙tr✉❝t✉r❛❧✱ ❡❧❡❝tr♦♥✐❝ ❛♥❞ ♦♣t✐❝❛❧

♣r♦♣❡rt✐❡s ❢♦r t②♣✐❝❛❧ ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡s ❛r❡ ❝❛❧❝✉❧❛t❡❞ ❛♥❞ ❡①♣❧❛✐♥❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ♠♦st st❛❜❧❡ ❧♦❝❛❧ ❛t♦♠✐❝ ❛rr❛♥❣❡♠❡♥ts✳ ❚❤❡ t❤❡♦r❡t✐❝❛❧ r❡s✉❧ts ❛r❡ ❝♦♠♣❛r❡❞ ✇✐t❤ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛ ✇❤❡♥❡✈❡r ✐t ✐s ♣♦ss✐❜❧❡✳ ❋✐♥❛❧❧②✱ t❤❡ ♠❛✐♥ r❡s✉❧ts ❛♥❞ ❝♦♥❝❧✉s✐♦♥s ❛r❡ ♦r❣❛♥✐③❡❞ ✐♥ t❤❡ ❙✉♠♠❛r② s❡❝t✐♦♥✳

✹


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▼❡t❤♦❞s ❙t❛t✐st✐❝❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❡✛❡❝ts ♦❢ ❜♦♥❞✐♥❣ ❛♥❞ ❞✐s♦r❞❡r ♦♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s ❛❞s♦r❜❡❞ ♦♥ ❛ ❣r❛♣❤❡♥❡ ❧❛②❡r ❛r❡ ❤❡r❡ st✉❞✐❡❞ ✉s✐♥❣ ❛ ❝❧✉st❡r ❡①♣❛♥s✐♦♥ ❛♣♣r♦❛❝❤✱ ❦♥♦✇♥ ❛s ●❡♥❡r❛❧✲ ✐③❡❞ ◗✉❛s✐ ❈❤❡♠✐❝❛❧ ❆♣♣r♦①✐♠❛t✐♦♥ ✭●◗❈❆✮✳ t♦ s❡✈❡r❛❧ ✷❉ ❛♥❞ ✸❉ ❛❧❧♦②s✱ s❤❡❡ts✳

✸✾

✷✽✱✷✾✱✹✶✱✹✷

✹✵

❚❤✐s ❢♦r♠❛❧✐s♠ ❤❛s ❜❡❡♥ s✉❝❝❡ss❢✉❧❧② ❛♣♣❧✐❡❞

❛♥❞✱ ♠♦r❡ r❡❝❡♥t❧②✱ ❣❡♥❡r❛❧✐③❡❞ t♦ ❢✉♥❝t✐♦♥❛❧✐③❡❞ ✷❉

❚❤❡ ❞✐s♦r❞❡r❡❞ s②st❡♠ ✐s ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ ❝❧✉st❡rs ✇✐t❤

n ❝❛r❜♦♥ ❛t♦♠s✱ ✇❤✐❝❤ ❛r❡

❝♦♥s✐❞❡r❡❞ st❛t✐st✐❝❛❧❧② ❛♥❞ ❡♥❡r❣❡t✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ ❡❛❝❤ ♦t❤❡r✳

✹✵

❚❤✐s ❛♣♣r♦①✐♠❛✲

t✐♦♥ ✐s ♠♦r❡ r❡❛❧✐st✐❝ ❛♥❞ ❛❝❝✉r❛t❡ ✇❤❡♥ t❤❡ ❝❧✉st❡rs ❜❡❝♦♠❡ ❧❛r❣❡r✳ ❍♦✇❡✈❡r✱ t❤❡ ♥✉♠❜❡r ♦❢ ❛t♦♠✐❝ ❝♦♥✜❣✉r❛t✐♦♥s t❤❛t ♠✉st ❜❡ ❝♦♥s✐❞❡r❡❞ ❞r❛♠❛t✐❝❛❧❧② ✐♥❝r❡❛s❡s ✇✐t❤ t❤❡ s✉♣❡r❝❡❧❧ s✐③❡✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ t❤❡ ❝❧✉st❡r s✐③❡ ♠✉st ❜❡ ❝❤♦s❡♥ ❛s ❛ ❝♦♠♣r♦♠✐s❡ ❜❡t✇❡❡♥ ♣r❡❝✐s✐♦♥ ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥✜❣✉r❛t✐♦♥s t♦ ❜❡ ❝❛❧❝✉❧❛t❡❞✳ ❍❡r❡✱ ✇❡ ♥❡❡❞ t♦ ❣❡♥❡r❛❧✐③❡ t❤❡ ♠❡t❤♦❞ t♦ t❤❡ ❝❛s❡ ♦❢ s✐♠✉❧t❛♥❡♦✉s ❛❞s♦r♣t✐♦♥ ♦❢ ❛♥❞

−❖−

❣r♦✉♣s✳ ❲❡ ❜✉✐❧❞ ✷❉ s✉♣❡r❝❡❧❧s ✇✐t❤

nS = 8

−❖❍

❝❛r❜♦♥ s✐t❡s✿ ❛s ✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣✳

✶ ❤②❞r♦①②❧s ❝❛♥ ❜❡ ❛❞s♦r❜❡❞ ❛t ❡❛❝❤ ❝❛r❜♦♥ s✐t❡✱ ✇❤✐❧❡ ❛♥ ❡♣♦①② ❣r♦✉♣s ❝❛♥ ❜r✐❞❣❡ ❡❛❝❤ ♦❢ t❤❡

nB = 12

❝❛r❜♦♥✲❝❛r❜♦♥ ❜♦♥❞s✳

❲❡ r❡str✐❝t t❤❡ ♣♦ss✐❜❧❡ ❛rr❛♥❣❡♠❡♥ts ♦❢ ♦①②❣❡♥✲

❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s t♦ t❤♦s❡ t❤❛t ❢✉❧✜❧❧ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❝❛r❜♦♥ ❛t♦♠s s❤♦✉❧❞ ❜❡ t❤r❡❡✲ ♦r ❢♦✉r❢♦❧❞✲❝♦♦r❞✐♥❛t❡❞ ❛❢t❡r ❛❞s♦r♣t✐♦♥✱ ❛♥❞ ✭✐✐✮ ❤②❞r♦①②❧ ❣r♦✉♣s ❝❛♥♥♦t ❜❡ ❛❞s♦r❜❡❞ ♦♥ ✜rst✲♥❡❛r❡st✲♥❡✐❣❤❜♦r ❛t♦♠✐❝ s✐t❡s ♦♥ t❤❡ s❛♠❡ s✐❞❡ ♦❢ t❤❡ ❣r❛♣❤❡♥❡ ❜❛s❛❧ ♣❧❛♥❡✳ ❚❤❡s❡ ❝♦♥❞✐t✐♦♥s ❛❧❧♦✇ t♦ ❡①❝❧✉❞❡ ✐rr❡❧❡✈❛♥t ❛t♦♠✐❝ ❛rr❛♥❣❡♠❡♥ts ✇✐t❤ ❡①tr❡♠❡❧② ❤✐❣❤ ❢♦r♠❛t✐♦♥ ❡♥❡r❣✐❡s✱ ❛s ♣r❡✈✐♦✉s❧② ❞❡♠♦♥str❛t❡❞ ✐♥ ❘❡❢✳

✸✾

❚❤❡ ❞❡s❝r✐❜❡❞ ❝♦♥str❛✐♥ts ❧❡❛❞ t♦ ❛ ✜♥❛❧ s❡t

♦❢ ✶✻✸✹✸ ♣♦ss✐❜❧❡ s✉♣❡r❝❡❧❧s✱ ✐♥❝❧✉❞✐♥❣ ✽ ❝❛r❜♦♥ ❛t♦♠s✱ ❣r♦✉♣s✳ ❚❤❡s❡ ❝♦♥✜❣✉r❛t✐♦♥s ❝❛♥ ❜❡ r❡♦r❣❛♥✐③❡❞ ✐♥ ❞❡❣❡♥❡r❛❝✐❡s

gj ✱

nOH,j −❖❍

❣r♦✉♣s ❛♥❞

nO,j −❖−

J =✸✵✽ s②♠♠❡tr② ❡q✉✐✈❛❧❡♥t ❝❧❛ss❡s ✇✐t❤

❛❝❝♦✉♥t✐♥❣ ❢♦r ❛❧❧ ♣♦ss✐❜❧❡ s②♠♠❡tr② ♦♣❡r❛t✐♦♥s✳ ❋♦r ❢✉rt❤❡r ❞✐s❝✉ss✐♦♥ ♦❢

s♣❡❝✐✜❝ ❛t♦♠✐❝ ❛rr❛♥❣❡♠❡♥ts✱ ❛♥ ❡①❛♠♣❧❡ ♦❢ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ❧❛❜❡❧✐♥❣ ♣r♦❝❡❞✉r❡ ✐s ❣✐✈❡ ✐♥ t❤❡ ❝❛♣t✐♦♥s ♦❢ ❋✐❣✳ ✶✳

✺



(a)

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(b)

❋✐❣✉r❡ ✶✿ ✭❈♦❧♦r ♦♥❧✐♥❡✮ ✭❛✮ ❚❤❡ ❛t♦♠✐❝ s✐t❡s ❛♥❞ t❤❡ ❝❛r❜♦♥✲❝❛r❜♦♥ ❜♦♥❞s ❛r❡ ❧❛❜❡❧❡❞ ✇✐t❤ ✐♥t❡❣❡r ♥✉♠❜❡rs ❢r♦♠ ✶ t♦ ✽ ✭✐♥ ❜❧❛❝❦✮ ❛♥❞ ✶ t♦ ✶✷ ✭✐♥ ✇❤✐t❡✮✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ❧❛❜❡❧ ✐s ♦❜t❛✐♥❡❞ ❜② ❝♦♠❜✐♥✐♥❣ t❤❡ ✽ ❛t♦♠✐❝ s✐t❡ ❧❛❜❡❧s ❛♥❞ t❤❡ ✶✷ ❝❛r❜♦♥✲❝❛r❜♦♥ ❜♦♥❞ ❧❛❜❡❧s✳ ❙✐t❡s ✇✐t❤♦✉t ♦①②❣❡♥ ❣r♦✉♣s ❛r❡ r❡♣r❡s❡♥t❡❞ ❛s ✵✱ ♦①②❣❡♥ ❣r♦✉♣s ❛❜♦✈❡ t❤❡ ❣r❛♣❤❡♥❡ ❧❛②❡r ❛s ✏✉✑ ❛♥❞ ♦①②❣❡♥ ❣r♦✉♣s ❜❡❧♦✇ t❤❡ ❣r❛♣❤❡♥❡ ❧❛②❡r ❛s ✏❞✑✳

✭❜✮ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❝❧✉st❡r

❝♦♥✜❣✉r❛t✐♦♥✳ ❈❛r❜♦♥✱ ♦①②❣❡♥ ❛♥❞ ❤②❞r♦❣❡♥ ❛r❡ r❡s♣❡❝t✐✈❡❧② ✐❧❧✉str❛t❡❞ ❜② ❜r♦✇♥✱ r❡❞ ❛♥❞ ✇❤✐t❡ s♣❤❡r❡s✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛❜❡❧✐♥❣ s②st❡♠ ❞❡s❝r✐❜❡❞ ✐♥ ✭❛✮✱ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ t❤❡ r❡♣r❡s❡♥t❡❞ ❛t♦♠✐❝ ❛rr❛♥❣❡♠❡♥t ✐s ✵✵✉❞✵✵✉❞✲❞✵✵✵✵✉✵✵✵✵✵✵✳

❊❛❝❤ ❝❧✉st❡r ❛rr❛♥❣❡♠❡♥t ✐s t❤❡♥ tr❡❛t❡❞ ❛s ❛ ✉♥✐t ❝❡❧❧ ♦❢ ❛♥ ❛rt✐✜❝✐❛❧ ✷❉ ❝r②st❛❧ ♦❜t❛✐♥❡❞ ❢r♦♠ ❛ r❡♣❡t✐t✐♦♥ ♦❢ t❤❡ ❝♦♥s✐❞❡r❡❞ ❝❧✉st❡r ♦✈❡r t❤❡ ✇❤♦❧❡ ❣r❛♣❤❡♥❡ ❜❛s❛❧ ♣❧❛♥❡✳ ❚❤❡♥✱ ❛ s❡t ♦❢ ♣❤②s✐❝❛❧ ♣r♦♣❡rt✐❡s

Pj

✐s ❝❛❧❝✉❧❛t❡❞ ❢♦r ❡❛❝❤ ❝❧✉st❡r s②st❡♠

♦❢ t❤❡ ❣r❛♣❤❡♥❡ ♦①✐❞❡ s❛♠♣❧❡s ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❢r❛❝t✐♦♥s

j✳

x✱ y ✱

❚❤❡ ❝❤❡♠✐❝❛❧ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞

z

✭✇✐t❤

x + y + z = 1✮✱

✇❤✐❝❤ ❛r❡ ❞❡✜♥❡❞ ❛s t❤❡ ❢r❛❝t✐♦♥s ♦❢ ❝❛r❜♦♥ ❛t♦♠s t❤❛t ❛r❡ ❜♦♥❞❡❞ ♦♥❧② t♦ ❝❛r❜♦♥ ❛t♦♠s✱ t♦ ❤②❞r♦①②❧ ❣r♦✉♣s✱ ❛♥❞ t♦ ❡♣♦①② ❣r♦✉♣s✱ r❡s♣❡❝t✐✈❡❧②✳

❚❤❡ ❛✈❡r❛❣❡ st♦✐❝❤✐♦♠❡tr② ♦❢ t❤❡

❝♦♠♣♦✉♥❞ ❜❡ ❡①♣r❡ss❡❞ ❛s ❈❖y+z/2 ❍y ✳ ▲❡t

xj (x, y, T )✱

❝❧✉st❡r ❝❧❛ss

j

✇❤❡r❡

❜❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♦❝❝✉rr❡♥❝❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡

✐♥ ❛ r❡❛❧✐③❡❞ ●❖ s❛♠♣❧❡ ✇✐t❤ ❛✈❡r❛❣❡ ❝❤❡♠✐❝❛❧ ❝♦♠♣♦s✐t✐♦♥

❣r♦✇t❤ t❡♠♣❡r❛t✉r❡

Mj

j = 1, 2, . . . , J ✱

T✳

■❢ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝❧✉st❡rs ✐s

❡q✉✐✈❛❧❡♥t ❝❧✉st❡rs✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♦❝❝✉rr❡♥❝❡

✻

xj

M✱

❛♥❞ t❤❡ ❝❧✉st❡r ❝❧❛ss

✐s ❡q✉❛❧ t♦


x✱ y ✱

xj = Mj /M ✳

❛♥❞ ❛t t❤❡

j

❝♦♥t❛✐♥s

■♥ ❛ s♦❧✐❞

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✉♥❞❡r ♥♦r♠❛❧ ♣r❡ss✉r❡ ✭ ✶ ❛t♠✮ t❤❡ ●✐❜❜s ❢r❡❡ ❡♥❡r❣② ♦❢ ♠✐①✐♥❣ ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ♠✐①✐♥❣ ❍❡❧♠❤♦❧t③ ❢r❡❡ ❡♥❡r❣②

∆F = ∆U − T ∆S ✳ ✹✵ ∆U

✐s t❤❡ ✐♥t❡r♥❛❧ ❡♥❡r❣② ❛♥❞ ✐t ❝❛♥ ❜❡

✇r✐tt❡♥ ❛s

∆U =

J X

Mj Ej − M (xEC + yECOH + zECO1/2 ),

✭✶✮

i=1

✇❤❡r❡

Ej ✱ EC ✱ ECOH ✱

❛♥❞

ECO1/2

❛r❡ t❤❡ t♦t❛❧ ❡♥❡r❣✐❡s ♦❢ t❤❡ ❝❧✉st❡r ❝♦♥✜❣✉r❛t✐♦♥

j✱

t❤❡

❝❧✉st❡r ♦❢ ♣r✐st✐♥❡ ❣r❛♣❤❡♥❡ ✭❈✮✱ t❤❡ ❝❧✉st❡r ♦❢ ❣r❛♣❤❡♥❡ ✇✐t❤ ❝♦♠♣❧❡t❡ ❤②❞r♦①②❧ ❝♦✈❡r❛❣❡ ✭❈❖❍✮✱ ❛♥❞ t❤❡ ❝❧✉st❡r ♦❢ ❣r❛♣❤❡♥❡ ❢✉❧❧② ❞❡❝♦r❛t❡❞ ✇✐t❤ ❡♣♦①② ❣r♦✉♣s ✭❈❖1/2 ✮✳ ❚❤❡ ❝♦♥✜❣✲ ✉r❛t✐♦♥❛❧ ❡♥tr♦♣②

∆S

✐s ❝❛❧❝✉❧❛t❡❞ ❢r♦♠ t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❙❤❛♥♥♦♥ ❡♥tr♦♣② ❛s ❛ ❢✉♥❝t✐♦♥

♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥

xj

❛♥❞ ❞❡❣❡♥❡r❛❝✐❡s

∆S = kb M

J X

gj

❛s

✸✾

xj (ln gj − ln xj ),

✭✷✮

j=1

✇❤❡r❡

kb

✐s t❤❡ ❇♦❧t③♠❛♥♥ ❝♦♥st❛♥t✳

❋✐♥❛❧❧②✱ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♦❝❝✉rr❡♥❝❡ ♣r♦❜❛❜✐❧✐t②

xj

❢♦r t❤❡ ❞✐✛❡r❡♥t ❝❧✉st❡r ❝❧❛ss❡s ✐s

❞❡t❡r♠✐♥❡❞ ❜② ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤❡ ❍❡❧♠❤♦❧t③ ❢r❡❡ ❡♥❡r❣②✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ J X

xj = 1✱

♠✉st ❢✉❧✜❧❧ s♦♠❡ ❝♦♥str❛✐♥ts✿ ✭✐✮ t❤❡ ♥♦r♠❛❧✐③❛t✐♦♥ ♦❢ t❤❡ t♦t❛❧ ♣r♦❜❛❜✐❧✐t②✱

✭✐✐✮ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛✈❡r❛❣❡ ❝❤❡♠✐❝❛❧ ❝♦♠♣♦s✐t✐♦♥✱

J X

xj nOH,j = nOH

❛♥❞

nOH = 8y

❛♥❞

nO = 4z

❛♥❞

j=1 J X

xj nO,j = nO ✱

j=1

j=1

✇❤❡r❡

xj

❛r❡ t❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s ♣❡r

❝❧✉st❡r ✐♥ ❛ ❈❖y+z/2 ❍y s②st❡♠✳

❚❤✐s ❝♦♥str❛✐♥❡❞ ♠✐♥✐♠✐③❛t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❛t ♦❢ ❛

t❡r♥❛r② ❛❧❧♦② ♦❜t❛✐♥❡❞ ❜② ♠✐①✐♥❣ ❈✱ ❈❖❍✱ ❛♥❞ ❈❖1/2 ✱ ❛s ❞❡s❝r✐❜❡❞ ✐♥ ❘❡❢✳

✹✸

❚❤❡ ❝♦♥str❛✐♥❡❞

♠✐♥✐♠✐③❛t✐♦♥ ❧❡❛❞s t♦ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♦❝❝✉rr❡♥❝❡ ♣r♦❜❛❜✐❧✐t②

n n E gj λ1 OH,j λ2 O,j exp − kBjT xj = J , X Ej nOH,j nO,j g j λ1 λ2 exp − kB T j=1 ✇❤❡r❡

λ1

❛♥❞

λ2

✭✸✮

❛r❡ t✇♦ ♥✉♠❡r✐❝❛❧ ❝♦♥st❛♥ts ♦❜t❛✐♥❡❞ ❛s ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡rs ✐♥ t❤❡ ❝♦♥✲

str❛✐♥❡❞ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✳ ❚❤❡② ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ❜② s♦❧✈✐♥❣ t❤❡ ♣❛✐r ♦❢ ♣♦❧②♥♦♠✐❛❧

✼



Page 8 of 30

❡q✉❛t✐♦♥s

J X

gj (nOH,j −

n n nOH )λ1 OH,j λ2 O,j exp

Ej − kB T

= 0,

✭✹✮

n n nO )λ1 OH,j λ2 O,j exp

Ej − kB T

= 0.

✭✺✮

j=1

J X

gj (nO,j −

j=1

❆❢t❡r t❤❛t t❤❡ ♦❝❝✉rr❡♥❝❡ ♣r♦❜❛❜✐❧✐t✐❡s ❛r❡ ❝❛❧❝✉❧❛t❡❞ ✇✐t❤ ❊q✳ ✭✸✮✱ ❛♥② ❛✈❡r❛❣❡ ♣❤②s✐❝❛❧ ♦r ❝❤❡♠✐❝❛❧ ♣r♦♣❡rt②

P

♦❢ t❤❡ ❞✐s♦r❞❡r❡❞ s②st❡♠ ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❛s ❛♥ ❛✈❡r❛❣❡ ♦❢ t❤❡ ♣r♦♣✲

❡rt✐❡s ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ❝❧✉st❡rs

Pj ✱

✇❡✐❣❤t❡❞ ❜② t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦❝❝✉rr❡♥❝❡ ♣r♦❜❛❜✐❧✐t✐❡s

xj ✿ ✹✵ P =

J X

x j Pj .

✭✻✮

j=1

❆❜ ✐♥✐t✐♦ ❝❛❧❝✉❧❛t✐♦♥s ❚❤❡ ✐♥♣✉t ❞❛t❛ ❢♦r ♦✉r st❛t✐st✐❝❛❧ ♠♦❞❡❧ ❛r❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❞❡♥s✐t② ❢✉♥❝t✐♦♥❛❧ t❤❡♦r② ✭❉❋❚✮✱ ❛s ✐♠♣❧❡♠❡♥t❡❞ ✐♥ t❤❡

❛❜ ✐♥✐t✐♦

❝❛❧❝✉❧❛t✐♦♥s ❜❛s❡❞ ♦♥

❱❆❙P✳ ✹✹✱✹✺ ❲❡ ✉s❡ t❤❡ ❣❡♥❡r❛❧✐③❡❞

❣r❛❞✐❡♥t ❛♣♣r♦①✐♠❛t✐♦♥ ✭●●❆✮✱ ❛s ♣r♦♣♦s❡❞ ❜② P❡r❞❡✇✲❇✉r❦❡✲❊r♥③❡r❤♦❢ ✭●●❆✲P❇❊✮✱ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❡①❝❤❛♥❣❡ ❛♥❞ ❝♦rr❡❧❛t✐♦♥ ✭❳❈✮ ❢✉♥❝t✐♦♥❛❧✳

✹✻✱✹✼

❚❤❡ ♣s❡✉❞♦♣♦t❡♥t✐❛❧s ♦❢ ❈✱ ❖

❛♥❞ ❍ ❛t♦♠s ❛r❡ ❣❡♥❡r❛t❡❞ ✇✐t❤✐♥ t❤❡ Pr♦❥❡❝t♦r✲❆✉❣♠❡♥t❡❞ ❲❛✈❡ ✭P❆❲✮ ♠❡t❤♦❞✳

✹✽✱✹✾

❚❤❡

❦✐♥❡t✐❝✲❡♥❡r❣② ❝✉t♦✛ ♦❢ t❤❡ ♣❧❛♥❡✲✇❛✈❡ ❜❛s✐s ✐s s❡t ❛s ✹✺✵ ❡❱✳ ❚❤❡ t♦t❛❧ ❡♥❡r❣② ❝❛❧❝✉❧❛t✐♦♥s ❛r❡ ♣❡r❢♦r♠❡❞ ✉s✐♥❣ ❛

9①9①1 Γ✲❝❡♥t❡r❡❞

❦✲♣♦✐♥t ❣r✐❞ t♦ ❝♦♠♣✉t❡ t❤❡ ✐♥t❡❣r❛❧s ♦✈❡r t❤❡ ❇r✐❧❧♦✉✐♥ ③♦♥❡ ✭❇❩✮✳

✺✵

▼♦♥❦❤♦rst✲P❛❝❦

■♥ ♦r❞❡r t♦ ✜♥❞ t❤❡

❡q✉✐❧✐❜r✐✉♠ ❝♦♥✜❣✉r❛t✐♦♥✱ ❛❧❧ ❛t♦♠✐❝ ❝♦♦r❞✐♥❛t❡s ❛r❡ r❡❧❛①❡❞ ✉♥t✐❧ t❤❡ ❍❡❧❧♠❛♥♥✲❋❡②♥♠❛♥

−1 ❢♦r❝❡s ❛r❡ s♠❛❧❧❡r t❤❛♥ ✵✳✵✶ ❡❱ ➴ ✇✐t❤✐♥ ❛ s♣✐♥✲✉♥♣♦❧❛r✐③❡❞ ❢♦r♠❛❧✐s♠✳ ❚❤❡ ✷❉ s②st❡♠s ❛r❡ s✐♠✉❧❛t❡❞ ❜② t❤❡ r❡♣❡t✐t✐♦♥ ♦❢ t❤❡ ❝♦♥s✐❞❡r❡❞ ❝❧✉st❡r ✐♥ t❤❡ t❤r❡❡ ❞✐♠❡♥s✐♦♥s✱ ✇✐t❤ ❛ s♣❛❝✐♥❣ ♦❢

L = 20

❜❡t✇❡❡♥ t✇♦ ♥❡✐❣❤❜♦r✐♥❣ ❣r❛♣❤❡♥❡ ♣❧❛♥❡s t♦ ❛✈♦✐❞ s♣✉r✐♦✉s ✐♥t❡r❛❝t✐♦♥s✳

❚❤❡ ❑♦❤♥✲❙❤❛♠ ❜❛♥❞ ❣❛♣ ❝❛❧❝✉❧❛t❡❞ ❢r♦♠ ❉❋❚ ✉s✐♥❣ st❛♥❞❛r❞ ❡①❝❤❛♥❣❡✲❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥❛❧s ✐s str♦♥❣❧② ✉♥❞❡r❡st✐♠❛t❡❞✳

✺✶

❍♦✇❡✈❡r✱ ♦♥❡ ❝❛♥ ❡①tr❛❝t ❛ ❛ q✉❛❧✐t❛t✐✈❡❧② ❝♦rr❡❝t

✽


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❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❡♥❡r❣② ❣❛♣ ❛♥❞ ❜♦✇✐♥❣ ♣❛r❛♠❡t❡rs ♦♥ t❤❡ ❝❤❡♠✐❝❛❧ ❝♦♠♣♦s✐t✐♦♥✳

✷✽✱✷✾✱✺✷

❚❤✐s ❢❛❝t ❤❛s ❛❧s♦ ❜❡❡♥ ❝♦♥✜r♠❡❞ ✐♥ ♦✉r r❡❝❡♥t st✉❞② ♦❢ ❛❞s♦r♣t✐♦♥ ♦❢ ❞✐✛❡r❡♥t ♦①②❣❡♥✲ ❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s ♦♥ ❣r❛♣❤❡♥❡✳

✸✾

❉✉❡ t♦ t❤❡ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❝♦♥✜❣✉r❛t✐♦♥s ❝♦♥s✐❞❡r❡❞ ❛♥❞

t❤❡ s✐❣♥✐✜❝❛♥t ❡♥❤❛♥❝❡♠❡♥t ♦❢ ❝♦♠♣✉t❡r ❝♦st r❡q✉✐r❡❞ ❜② ♠♦r❡ s♦♣❤✐st✐❝❛t❡❞ ♠❡t❤♦❞s ❢♦r ❛❝❝✉r❛t❡ q✉❛s✐♣❛rt✐❝❧❡ st❛t❡s✱ ✇❡ ❢♦❝✉s ♦♥ t❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❛✈❡r❛❣❡ ❡♥❡r❣② ❣❛♣ t♦ ❛♥❛❧②③❡ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ ❡❧❡❝tr♦♥✐❝ ♣r♦♣❡rt✐❡s ♦♥ t❤❡ ❝❤❡♠✐❝❛❧ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ●❖✳ ❚❤❡ ❛t♦♠✐❝ ❛rr❛♥❣❡♠❡♥ts t❤❛t ❣✐✈❡ r✐s❡ t♦ ♠❡t❛❧❧✐❝ ♣r♦♣❡rt✐❡s ❛r❡ ❛ttr✐❜✉t❡❞ ❛ ③❡r♦ ❜❛♥❞ ❣❛♣ ✈❛❧✉❡ ✐♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ st❛t✐st✐❝❛❧ ❛✈❡r❛❣❡s✳ ❖♣t✐❝❛❧ ♣r♦♣❡rt✐❡s ❛r❡ ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❢r❡q✉❡♥❝②✲❞❡♣❡♥❞❡♥t ❞✐❡❧❡❝tr✐❝ ♠❛tr✐①✱ ✇❤✐❝❤ ✐s ❝❛❧❝✉❧❛t❡❞ ✇✐t❤✐♥ t❤❡ ✐♥❞❡♣❡♥❞❡♥t✲q✉❛s✐♣❛rt✐❝❧❡ ❛♣♣r♦①✐♠❛t✐♦♥✳ ♠❛tr✐① ❡❧❡♠❡♥ts

Mcv (~k, qˆ)

✺✸

❚❤❡ ♦♣t✐❝❛❧ tr❛♥s✐t✐♦♥

❛r❡ ❞❡s❝r✐❜❡❞ ❛❞♦♣t✐♥❣ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ❣❛✉❣❡✳

❞✐❡❧❡❝tr✐❝ ❢✉♥❝t✐♦♥ ❛❧❧♦✇s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✷❉ ♦♣t✐❝❛❧ ❛❜s♦r❜❛♥❝❡ ✐♥❝✐❞❡♥❝❡ ❛s ❞❡✜♥❡❞ ✐♥ ❘❡❢✳✿

Aj (ω)

ω L Im[ǫkk (ω)] (k = x, y), c

c ✐s t❤❡ s♣❡❡❞ ♦❢ ❧✐❣❤t ✐♥ ✈❛❝✉✉♠✳

✭✼✮

✺✻

❍♦✇❡✈❡r✱ ❛ t❡♥❞❡♥❝②

❢♦r ❝♦♠♣❡♥s❛t✐♦♥ ♦❢ t❤❡s❡ t✇♦ ❡✛❡❝ts ❤❛s ❜❡❡♥ ❞❡♠♦♥str❛t❡❞ ❢♦r ❣r❛♣❤❡♥❡✱

h❇◆✳ ✺✽

❢♦r ♥♦r♠❛❧

❆♥ ❛❝❝✉r❛t❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ❛❜s♦r♣t✐♦♥ s♣❡❝tr❛ ✇♦✉❧❞

r❡q✉✐r❡ t❤❡ ✐♥❝❧✉s✐♦♥ ♦❢ q✉❛s✐♣❛rt✐❝❧❡ ❝♦rr❡❝t✐♦♥s ❛♥❞ ❡①❝✐t♦♥✐❝ ❡✛❡❝ts✳

✷❉ s②st❡♠s s✉❝❤ ❛s

❚❤❡ ✐♥✲♣❧❛♥❡

✺✺

Aj (ω) = ✇❤❡r❡

✺✹

✺✼

❛♥❞ ✐♥s✉❧❛t✐♥❣

❚❤❡r❡❢♦r❡✱ ❛ ❣♦♦❞ q✉❛❧✐t❛t✐✈❡ ♣✐❝t✉r❡ ♦❢ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡

♦♣t✐❝❛❧ ♣r♦♣❡rt✐❡s ♦♥ t❤❡ ❝❤❡♠✐❝❛❧ ❝♦♠♣♦s✐t✐♦♥ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ✇✐t❤✐♥ t❤❡ ✐♥❞❡♣❡♥❞❡♥t q✉❛s✐♣❛rt✐❝❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r ♦✉r ❞✐s♦r❞❡r❡❞ ✷❉ s②st❡♠✳

✾


✷✾


Page 10 of 30

❘❡s✉❧ts ❛♥❞ ❞✐s❝✉ss✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ st❛❜✐❧✐t② ❚❤❡ ❡♥❡r❣❡t✐❝s ♦❢ t❤❡ ❝❧✉st❡rs ❝❧❛ss❡s ✐s ✐♥✈❡st✐❣❛t❡❞ ❜② ❞❡✜♥✐♥❣ t❤❡ ❡①❝❡ss t♦t❛❧ ❡♥❡r❣② ♦❢ ❡❛❝❤ ❛t♦♠✐❝ ❛rr❛♥❣❡♠❡♥t ❛s

∆Ej = Ej − ✇❤❡r❡

nC,j

nC,j nOH,j nO,j EC − ECOH − ECO1/2 , 8 8 4

✭✽✮

r❡♣r❡s❡♥ts t❤❡ ♥✉♠❜❡r ♦❢ t❤r❡❡✲❢♦❧❞ ❝♦♦r❞✐♥❛t❡❞ ❝❛r❜♦♥s ✐♥ t❤❡ ❝❧✉st❡r ❝❧❛ss

❡①♣r❡ss✐♦♥ ✭✽✮ r❡♣r❡s❡♥ts t❤❡ ❢♦r♠❛t✐♦♥ ❡♥❡r❣② ♦❢ ❛ ❝❧✉st❡r ❝❧❛ss ❡♥❞ ❝♦♠♣♦♥❡♥ts✿ ♣✉r❡ ❣r❛♣❤❡♥❡ ✇✐t❤ t♦t❛❧ ❡♥❡r❣② ❣r♦✉♣s ❛♥❞ ❡♥❡r❣②

EOH ✱

EC ✱

j✱

j✳

✇✐t❤ r❡s♣❡❝t t♦ t❤❡ t❤r❡❡

❢✉❧❧② ♦①✐❞✐③❡❞ ●❖ ✇✐t❤ ♦♥❧②

❛♥❞ ❢✉❧❧② ♦①✐❞✐③❡❞ ●❖ ✇✐t❤ ♦♥❧②

−❖−

❚❤❡

❣r♦✉♣s ❛♥❞ ❡♥❡r❣②

−❖❍

ECO1/2 ✳

❚❤❡ ❛t♦♠✐❝ ❣❡♦♠❡tr② ♦❢ t❤❡ ❡♥❞ ❝♦♠♣♦♥❡♥ts ❛r❡ ❞✐s♣❧❛②❡❞ ✐♥ ❋✐❣s✳ ✷ ❛✱ ❜✱ ❛♥❞ ❝✱ ❢♦r ❣r❛♣❤❡♥❡ ✭x❂✶✮✱ ❛ ❝❧✉st❡r ❢✉❧❧② ❞❡❝♦r❛t❡❞ ✇✐t❤ ❤②❞r♦①②❧ ✭y ❂✶✮✱ ❛♥❞ ✵✵✵✵✵✵✵✵✲✵✵✉❞✵✵✵✵✵✉❞✵✱ t❤❡ ❝❧✉s✲ t❡r ❝❧❛ss ✇✐t❤ ❢♦✉r ❡♣♦①② ❣r♦✉♣s ♣❡r ❝❧✉st❡r ✭z ❂✶✮ ❛♥❞ t❤❡ ❧♦✇❡st ✐♥t❡r♥❛❧ ❡♥❡r❣②✱ r❡s♣❡❝t✐✈❡❧②✳ ❈♦♥✜❣✉r❛t✐♦♥s ✇✐t❤ ❧❛r❣❡ ❡①❝❡ss t♦t❛❧ ❡♥❡r❣✐❡s

∆Ej

❛r❡ ✐♥❤✐❜✐t❡❞ ❜② t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤❡

❍❡❧♠❤♦❧t③ ❢r❡❡ ❡♥❡r❣② ♦❢ ♠✐①✐♥❣✳ ❚❤❡② ❛r❡ r❡❛❧✐③❡❞ ♦♥❧② ❛t ❤✐❣❤ ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡s✱ ✇❤✐❧❡ ❝♦♥✜❣✉r❛t✐♦♥s ✇✐t❤ ♥❡❣❛t✐✈❡ ❢♦r♠❛t✐♦♥ ❡♥❡r❣② ✐♥❞✐❝❛t❡ t❤❛t s✉❝❤ ❛t♦♠✐❝ ❛rr❛♥❣❡♠❡♥ts ❛r❡ t❤❡r♠♦❞②♥❛♠✐❝❛❧❧② ❢❛✈♦r❡❞ ❛♥❞ r❡❛❧✐③❡❞ ❡✈❡♥ ❛t ❧♦✇ t❡♠♣❡r❛t✉r❡s✳ ■♥ ♦✉r ♣r❡✈✐♦✉s ✇♦r❦✱

✸✾

✇❡ r❡♣♦rt❡❞ ♦♥❧② ♣♦s✐t✐✈❡ ❡①❝❡ss ❡♥❡r❣✐❡s ❢♦r ♦①✐❞❛t✐♦♥ ♦❢ ❣r❛♣❤❡♥❡

s❤❡❡ts ✇✐t❤ ♦♥❧② ❤②❞r♦①②❧ ♦r ❡♣♦①② ❣r♦✉♣s✱ ✇❤✐❝❤ r❡s♣❡❝t✐✈❡❧② ❝♦rr❡s♣♦♥❞ t♦

y = 0✳

z = 0

❛♥❞

■♥ t❤❡s❡ t✇♦ ❧✐♠✐t ❝❛s❡s✱ t❤❡ ●❖ s②st❡♠ t❡♥❞s t♦ ❞❡❝♦♠♣♦s❡ ✐♥t♦ ♦①②❣❡♥✲♣♦♦r ❛♥❞

✲r✐❝❤ ❞♦♠❛✐♥s✳ ❍❡r❡✱ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ s✐♠✉❧t❛♥❡♦✉s ♦①✐❞❛t✐♦♥ ✇✐t❤

−❖❍ ❛♥❞ −❖−

❣r♦✉♣s

❛♥❞ ❡①♣❧♦r❡ ♥❡✇ ❢❡❛t✉r❡s t❤❛t ❡♠❡r❣❡ ❢r♦♠ t❤❡ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ ❜♦t❤ ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s✳ ❚❤❡ ✈❛st ♠❛❥♦r✐t② ♦❢ ❝❧✉st❡r ❝♦♥✜❣✉r❛t✐♦♥s st✐❧❧ ❡①❤✐❜✐ts ♣♦s✐t✐✈❡ ❡①❝❡ss❡s ❡♥❡r❣✐❡s✱ ❡s♣❡✲ ❝✐❛❧❧② ❢♦r ✉♥❞❡rst♦✐❝❤✐♦♠❡tr✐❝ ❝♦♥✜❣✉r❛t✐♦♥s✳ ❍♦✇❡✈❡r✱ ❡✐❣❤t ❞✐✛❡r❡♥t ❝❧❛ss❡s ♦❢ ❢✉❧❧② ♦①✐❞✐③❡❞ ❝♦♥✜❣✉r❛t✐♦♥s ✇✐t❤ ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s ❡①❤✐❜✐t ♥❡❣❛t✐✈❡ ❡①❝❡ss ❡♥❡r❣✐❡s✱ ✐♥❞✐❝❛t✐♥❣

✶✵


Page 11 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


❣r❡❛t ❡♥❡r❣❡t✐❝ st❛❜✐❧✐t② ❡✈❡♥ ❛t ❧♦✇ t❡♠♣❡r❛t✉r❡s✳ ❚❤❡s❡ ❝♦♥✜❣✉r❛t✐♦♥s ❛r❡ t❛❜❡❧❡❞ ✐♥ ❚❛✲ ❜❧❡ ✶✳ ❋♦✉r ❝♦♥✜❣✉r❛t✐♦♥s ✇✐t❤ ♥❡❣❛t✐✈❡ ❡①❝❡ss ❡♥❡r❣✐❡s ♦❝❝✉r ❛t

(nC,j /8, nOH,j /8, nO,j /4) =

(0, 0.5, 0.5)✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❢♦✉r −❖❍ ❛♥❞ t✇♦ −❖− ❣r♦✉♣s ♣❡r ✽✲❝❛r❜♦♥ ❝❧✉st❡r✳ ❢♦✉r ❡♥❡r❣❡t✐❝❛❧❧② ❢❛✈♦r❛❜❧❡ ❛rr❛♥❣❡♠❡♥ts ♦❝❝✉r ❛t ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✇♦

❚❤❡ ♦t❤❡r

(nC,j /8, nOH,j /8, nO,j /4) = (0, 0.25, 0.75)✱

−❖❍ ❛♥❞ t❤r❡❡ −❖− ♣❡r ❝❧✉st❡r✳

❚❤❡s❡ ✜♥❞✐♥❣s ❛❣r❡❡ ✇✐t❤ t❤❡ ✐♥❞✐❝❛✲

t✐♦♥ ♦❢ ❤✐❣❤❧② ♦①✐❞✐③❡❞ ❞♦♠❛✐♥s ✇✐t❤ s✐♠✉❧t❛♥❡♦✉s ❛❞s♦r♣t✐♦♥ ♦❢ ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s ❜② ♥✉❝❧❡❛r ♠❛❣♥❡t✐❝ r❡s♦♥❛♥❝❡ ♠❡❛s✉r❡♠❡♥ts✱

✶✾

❛♥❞ ❛❧s♦ ✇✐t❤ ♦t❤❡r t❤❡♦r❡t✐❝❛❧ ✇♦r❦s t❤❛t

✐♥❞✐❝❛t❡ t❤❡ t♦t❛❧ ❡♥❡r❣✐❡s ❛r❡ ❝♦♥s✐❞❡r❛❜❧② ❧♦✇❡r❡❞ ✇❤❡♥ ❡♣♦①② ❛♥❞ ❤②❞r♦①②❧ ❣r♦✉♣s ❛r❡ ❛❞✲ s♦r❜❡❞ t♦❣❡t❤❡r✳

✸✼✱✸✽

❖①✐❞✐③❡❞ ❞♦♠❛✐♥s ✇✐t❤ ❜♦t❤ ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s ❛r❡✱ t❤❡r❡❢♦r❡✱

t❤❡r♠♦❞②♥❛♠✐❝❛❧❧② ❢❛✈♦r❡❞ ♦✈❡r ❛ s♣❛t✐❛❧ s❡❣r❡❣❛t✐♦♥ ♦❢ t❤❡ ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s✳ ❚❤❡ ❧♦✇❡st ❡♥❡r❣② ❝♦♥✜❣✉r❛t✐♦♥s ❛t

(nC,j /8, nOH,j /8, nO,j /4) = (0, 0.5, 0.5)

❛♥❞

(0, 0.25, 0.75)

❛r❡

✉✵❞✵✵❞✵✉✲✵✵✵✵✵✉❞✵✵✵✵✵ ❛♥❞ ❞✵✵✵✵✵✵❞✲✵✵✵✵✉✉✵✵✵✵✉✵✱ r❡s♣❡❝t✐✈❡❧②✱ ✇❤✐❝❤ ❛r❡ ✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣s✳ ✷ ❞ ❛♥❞ ❡✳

❚❛❜❧❡ ✶✿ ❈❧✉st❡r ❧❛❜❡❧s✱ t❤❡✐r ♥✉♠❜❡rs ♦❢ ❤②❞r♦①②❧ ✭❡♣♦①②✮ ❣r♦✉♣s ♣❡r ❝❧✉st❡r nOH,j ✭nepo,j ✮✱ ❞❡❣❡♥❡r❛❝✐❡s gj ✱ ❛♥❞ ❡①❝❡ss ❡♥❡r❣✐❡s ∆Ej ✳ ❆❧❧ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥s ✇✐t❤ ♥❡❣❛t✐✈❡ ❡①❝❡ss ❡♥❡r❣② ❛r❡ ❧✐st❡❞✳ ▲❛❜❡❧

nOH,j

nepo,j

gj

∆Ej

❞✉✵✵✵✵✵✵✲✵✵✵✵✵✵✉✉✵✵✵❞

✷

✸

✾✻

✲✵✳✹✵

✉✵✵✵✵❞✵✵✲✵✵✵✵✵✉✉✵✵✵✵❞

✷

✸

✹✽

✲✵✳✵✸

❞✵✵✵✵✵✵❞✲✵✵✵✵✉✉✵✵✵✵✉✵

✷

✸

✶✻

✲✵✳✻✼

❞✵✵✵✵✵✵❞✲✵✵✵✵✉✉✵✵✵✵❞✵

✷

✸

✹✽

✲✵✳✶✷

❞✉✵✉❞✵✵✵✲✵✵✵✵✵✵✵✵✵✵✉❞

✹

✷

✹✽

✲✵✳✹✼

✉❞✉❞✵✵✵✵✲✵✵✵✵✵✵✵✵✵✉❞✵

✹

✷

✹✽

✲✵✳✺✽

✉✵✉✵✵❞✵❞✲✵✵✵✵✵✉❞✵✵✵✵✵

✹

✷

✷✹

✲✵✳✷✾

✉✵❞✵✵❞✵✉✲✵✵✵✵✵✉❞✵✵✵✵✵

✹

✷

✷✹

✲✵✳✼✷

✭❡❱✴❝❧✉st❡r✮

❚❤❡ t❤❡r♠♦❞②♥❛♠✐❝ st❛❜✐❧✐t② ♦❢ t❤❡ t❡r♥❛r② s②st❡♠ ✇✐t❤ r❡s♣❡❝t t♦ ✐ts ❝❤❡♠✐❝❛❧ ❝♦♠♣♦s✐✲ t✐♦♥ ✐s ✐♥✈❡st✐❣❛t❡❞ ❜② ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ♠✐①✐♥❣ ❢r❡❡ ❡♥❡r❣② ♣❡r❛t✉r❡s ♦❢

y✱

❛♥❞

z✳

200o C

❛♥❞

1100o C ✱ ✻✱✺✾✕✻✶

∆F (x, y, T ) ❛t t②♣✐❝❛❧ ❣r♦✇t❤ t❡♠✲

❝♦♥s✐❞❡r✐♥❣ t❤❡ ✇❤♦❧❡ ❝♦♠♣♦s✐t✐♦♥ r❛♥❣❡

x = 1−y −z ✱

❚❤❡ r❡s✉❧ts ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ t✇♦ ❝♦❧♦r ♠❛♣s ❞❡♣✐❝t❡❞ ✐♥ ❋✐❣✳ ✸✳ ❚❤❡ ♠♦st

❢❛✈♦r❛❜❧❡ ❝♦♥✜❣✉r❛t✐♦♥s ♦❢ t❤❡ ❡♥❞ ❝♦♠♣♦♥❡♥ts ❈ ✭❣r❛♣❤❡♥❡✮✱ ❈❖❍ ✭●❖ ✇✐t❤ ❤②❞r♦①②❧✮✱

✶✶



(a)

(b)

Page 12 of 30

(c)

(d)

(e)

❋✐❣✉r❡ ✷✿ ✭❈♦❧♦r ♦♥❧✐♥❡✮ ■♠♣♦rt❛♥t ❝❧✉st❡r ❝♦♥✜❣✉r❛t✐♦♥s r❡♣r❡s❡♥t✐♥❣ ✭❛✮ ♣r✐st✐♥❡ ❣r❛♣❤❡♥❡✱ ✭❜✮ ❈❖❍✱ ✭❝✮ ✵✵✵✵✵✵✵✵✲✵✵✉❞✵✵✵✵✵✉❞✵✱ t❤❡ ❧♦✇❡st✲❡♥❡r❣② ❝❧✉st❡r ❢✉❧❧② ♦①✐❞✐③❡❞ ❜② ❡♣♦①② ❣r♦✉♣s✱ ✭❞✮ ✉✵❞✵✵❞✵✉✲✵✵✵✵✵✉❞✵✵✵✵✵✱ ✇✐t❤ ✵✵✵✵✉✉✵✵✵✵✉✵✱ ✇✐t❤

∆Ej =✲✵✳✻✼

∆Ej =✲✵✳✼✷

❡❱✴❝❧✉st❡r✱ ❛♥❞ ✭❡✮ ❞✵✵✵✵✵✵❞✲

❡❱✴❝❧✉st❡r✳

❛♥❞ ❈❖1/2 ✭●❖ ✇✐t❤ ❡♣♦①②✮ ❛r❡ ❝❤♦s❡♥ ❛s t❤❡ ❝♦r♥❡r ♣♦✐♥ts ♦❢ t❤❡ ❞✐s♣❧❛②❡❞ tr✐❛♥❣❧❡s✳ ❚❤❡✐r ❡❞❣❡s ♦❢ t❤❡ tr✐❛♥❣❧❡ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ ♦♥❡ s✐♥❣❧❡ ♣❛r❛♠❡t❡r ✭x✱

y✱

♦r

z ✮✱

✇❤✐❧❡

t❤❡ ♦t❤❡r t✇♦ ❛r❡ s❡t t♦ ③❡r♦✳ ❚❤❡ ❛❞s♦r♣t✐♦♥ ♣r♦❝❡ss❡s ✇✐t❤ s✐♥❣❧❡ ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣ s♣❡❝✐❡s✱ ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❈−❈❖❍ ❛♥❞ ❈−❈❖1/2 ❡❞❣❡s ♦❢ t❤❡ ❝♦❧♦r ♠❛♣s✱ ❡①❤✐❜✐t ♣♦s✲ ✐t✐✈❡ ♠✐①✐♥❣ ❡♥❡r❣✐❡s ❢♦r t❤❡ ❝♦♥s✐❞❡r❡❞ ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡s✱ ✐♥❞✐❝❛t✐♥❣ ❛ str♦♥❣ t❡♥❞❡♥❝② t♦ ♣❤❛s❡ ❞❡❝♦♠♣♦s✐t✐♦♥✱ ✉♥❞❡rst♦♦❞ ❤❡r❡ ❛s t❤❡ ❛❣❣❧♦♠❡r❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛❧✐③✐♥❣ ❣r♦✉♣s ✭s❡❡ ♦✉r ♣r❡✈✐♦✉s r❡s✉❧ts r❡♣♦rt❡❞ ✐♥ ❘❡❢✳

✸✾

✮✳ ❚❤❡ r❡❞ ❛♥❞ ♦r❛♥❣❡ ❛r❡❛s ✐♥❞✐❝❛t❡ ❛♥ ❡♥❤❛♥❝❡✲

♠❡♥t ♦❢ t❤❡ ♠✐①✐♥❣ ❢r❡❡ ❡♥❡r❣②✱ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❛ str♦♥❣ t❡♥❞❡♥❝② t♦ ♣❤❛s❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥t♦ ♣r✐st✐♥❡ ❣r❛♣❤❡♥❡ ✭❈ ✈❡rt❡①✮ ❛♥❞ ❢✉❧❧② ♦①✐❞✐③❡❞ ❞♦♠❛✐♥s ✇✐t❤ ❜♦t❤ ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s ✭❈❖1/2 ✲❈❖❍ ❡❞❣❡✮ ✐s ✐♥❞✐❝❛t❡❞✳ ❲❤❡♥ ❤✐❣❤❡r t❡♠♣❡r❛t✉r❡s ❛r❡ ❝♦♥s✐❞❡r❡❞✱ t❤❡ ❡♥✲ tr♦♣② t❡r♠ ❜❡❝♦♠❡s ♠♦r❡ ✐♠♣♦rt❛♥t ❛♥❞ ❝♦♥tr✐❜✉t❡s t♦ t❤❡ s②st❡♠ st❛❜✐❧✐t② ❛t ✐♥t❡r♠❡❞✐❛t❡ ♦①✐❞❛t✐♦♥ ❧❡✈❡❧✳ ❚❤❡s❡ r❡s✉❧ts ❛r❡ ✐♥ ❛❣r❡❡♠❡♥t ✇✐t❤ ♦t❤❡r t❤❡♦r❡t✐❝❛❧ ✐♥✈❡st✐❣❛t✐♦♥s

✸✼✱✸✽✱✻✷

❛♥❞ ❡①♣❡r✐♠❡♥t❛❧ ✜♥❞✐♥❣s✱ t❤❛t ♣♦✐♥t t♦ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ♥♦♥✲♦①✐❞✐③❡❞ ❞♦♠❛✐♥s ✐♥ ●❖ s❛♠✲ ♣❧❡s✳

✶✻✱✻✸

✶✷


Page 13 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(a) 200 oC

CO1/2

0.34 0.18 0.02 -0.14 -0.30 -0.46

C

ΔF(x,y,T) (eV/cluster) COH

(b) 1100 oC

CO1/2

-0.62

0.08 0.05 0.02 -0.01 -0.02 -0.05

C

ΔF(x,y,T) (eV/cluster) COH

-0.08

❋✐❣✉r❡ ✸✿ ✭❈♦❧♦r ♦♥❧✐♥❡✮ ❚❤❡ ♠✐①✐♥❣ ❢r❡❡ ❡♥❡r❣② ♦❢ t❤❡ ✉♥❞❡rst♦✐❝❤✐♦♠❡tr✐❝ ●❖ s②st❡♠s ❛s ❛ o o ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❛✈❡r❛❣❡ ❝♦♠♣♦s✐t✐♦♥s ❛t ❛ s②♥t❤❡s✐s t❡♠♣❡r❛t✉r❡ ♦❢ ✭❛✮ 200 C ❛♥❞ ✭❜✮ 1100 C ✳

❚❤❡ r❡❛s♦♥s ❢♦r t❤❡ ✐♥❝r❡❛s❡ ♦❢ t❤❡ ♠✐①✐♥❣ ❢r❡❡ ❡♥❡r❣② ❛❧♦♥❣ t❤❡ ❈❖❍−❈❖1/2 ❡❞❣❡s ♦❢ t❤❡ ❝♦❧♦r ♠❛♣s ♦❢ ❋✐❣✳ ✸ ❛r❡ ✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣✳ ✹✳ ❚❤❡ ♠✐①✐♥❣ ❢r❡❡ ❡♥❡r❣② t❤❡ ✐♥t❡r♥❛❧ ❡♥❡r❣② ❝♦♥tr✐❜✉t✐♦♥ ✐♥❣ ❢r❡❡ ❡♥❡r❣②

∆F

∆U

∆F

✐s ❞❡❝♦♠♣♦s❡❞ ✐♥t♦

❛♥❞ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥❛❧ ❡♥tr♦♣② t❡r♠

✐s ❞♦♠✐♥❛t❡❞ ❜② t❤❡ ❡♥tr♦♣② ❝♦♥tr✐❜✉t✐♦♥

−T ∆S

−T ∆S ✳

❚❤❡ ♠✐①✲

❛t ❤✐❣❤ t❡♠♣❡r❛t✉r❡s✳

❍♦✇❡✈❡r✱ ❛t ❧♦✇ t❡♠♣❡r❛t✉r❡s✱ t❤❡ ♦❝❝✉rr❡♥❝❡ ♦❢ ❛t♦♠✐❝ ❛rr❛♥❣❡♠❡♥ts ✇✐t❤ ♥❡❣❛t✐✈❡ ❡①❝❡ss ❡♥❡r❣✐❡s str♦♥❣❧② ❛✛❡❝ts

∆U ✳

❆❝❝♦✉♥t✐♥❣ ❢♦r t❤❡ ❝❧✉st❡r st❛t✐st✐❝s✱ ❢✉❧❧② ♦①✐❞✐③❡❞ ❞♦♠❛✐♥s

❞❡❝♦r❛t❡❞ ✇✐t❤ ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s t✉r♥ ♦✉t t♦ ❜❡ st❛❜❧❡ ❛t t❤❡ ❝♦♥s✐❞❡r❡❞ ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡s✳ ■♥ t❤❡ ❤②❞r♦①②❧✲r✐❝❤ ❧✐♠✐t ✭y ♣♦♥❡♥t✳ ❚❤✐s s✐t✉❛t✐♦♥ ❧❡❛❞s t♦

→ 1✮✱

♥❡❛r❧② ❛❧❧ ❝❧✉st❡rs ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❈❖❍ ❡♥❞ ❝♦♠✲

∆U = 0✱

❝❛❧❝✉❧❛t❡❞ ❜② t❤❡ ❝❧✉st❡r ❡①♣❛♥s✐♦♥✱ s✐♥❝❡

✇❤✐❧❡ ❛♥ ❡♥tr♦♣② ♣❡r ❝❧✉st❡r ♦❢

gCOH = 2✳

✶✸

∆S = kb ln(2)

✐s

■♥ t❤❡ ●◗❈❆ ♣✐❝t✉r❡✱ t❤❡ ❈❖❍ ❝❧✉st❡rs



❛r❡ ❢r❡❡ t♦ ❵✢✐♣✬ ✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠ t❤❡ ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s ✐♥ t❤❡ ♥❡✐❣❤❜♦r❤♦♦❞✱ ❞✉❡ t♦ t❤❡ ❛ss✉♠❡❞ ❡♥❡r❣❡t✐❝ ❛♥❞ st❛t✐st✐❝❛❧ ✐♥❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝❧✉st❡rs✳ ■♥ t❤❡ ❡♣♦①②✲ r✐❝❤ ❧✐♠✐t ✭z

→ 1✮

t❤❡r❡ ✐s ❛ ❝♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ ❞✐✛❡r❡♥t ♣♦ss✐❜❧❡ ❡♣♦①② ❛rr❛♥❣❡♠❡♥ts ✐♥

t❤❡ s❤❡❡ts✳ ❆t ❧♦✇ t❡♠♣❡r❛t✉r❡ ✭s❡❡ t❤❡ s♦❧✐❞ ❧✐♥❡s ✐♥ ❋✐❣✳ ✹✮✱ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥s ✇✐t❤ ❧♦✇❡r ✐♥t❡r♥❛❧ ❡♥❡r❣② ♦❢ ♠✐①✐♥❣ ❛r❡ str♦♥❣❧② ❢❛✈♦r❡❞✱ ②✐❡❧❞✐♥❣ ❧♦✇

∆U

❛♥❞

∆S ✳

❆s t❤❡ t❡♠♣❡r❛✲

t✉r❡ ✐♥❝r❡❛s❡s✱ t❤❡ ♦❝❝✉rr❡♥❝❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❝❧✉st❡r ❝❧❛ss❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ♠♦r❡ ❡♥❡r❣❡t✐❝ ❝♦♥✜❣✉r❛t✐♦♥s ✐s ❡♥❤❛♥❝❡❞✱ ❧❡❛❞✐♥❣ t♦ ❧❛r❣❡r

∆U

❛♥❞

T ∆S

❝♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ t♦t❛❧ ❡①❝❡ss

❢r❡❡ ❡♥❡r❣②✳

Energy (eV/cluster)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 30

0.25 0.00 -0.25 -0.50 ΔU -TΔS ΔF

-0.75 -1.00 0.00

0.25

0.50

0.75

1.00

Composition y ❋✐❣✉r❡ ✹✿ ❡♥❡r❣②

✭❈♦❧♦r ♦♥❧✐♥❡✮ ❚❤❡ ✐♥t❡r♥❛❧ ❡♥❡r❣②

−T ∆S ✱

∆U ✱

t❤❡ ❡♥tr♦♣✐❝ ❝♦♥tr✐❜✉t✐♦♥ t♦ t❤❡ ❢r❡❡

❛♥❞ t❤❡ ❍❡❧♠❤♦❧t③ ❢r❡❡ ❡♥❡r❣② ♦❢ ♠✐①✐♥❣

∆F

❢♦r ❛ ❢✉❧❧② ♦①✐❞✐③❡❞ s②st❡♠ o ✭❈❖❍✮y ✭❈❖1/2 ✮1−y ✳ ❙♦❧✐❞ ❧✐♥❡s ❛r❡ ♦❜t❛✐♥❡❞ ❛t ❛ ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡ ♦❢ 200 C ✱ ✇❤✐❧❡ ❞❛s❤❡❞ o ❧✐♥❡s ✐♥❞✐❝❛t❡ ❛ ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡ ♦❢ 1100 C ✳

❙tr✉❝t✉r❛❧ ♣r♦♣❡rt✐❡s ❲❡ ♦❜t❛✐♥❡❞ t❤❡ ❧❛tt✐❝❡ ❝♦♥st❛♥ts ♦❢ ➴

aC = 2.47

➴✱

aCOH = 2.63

❢♦r t❤❡ ♣✉r❡ ❡♥❞ ❝♦♠♣♦✉♥❞s✳ ❆ ♠❛①✐♠✉♠ ❧❛tt✐❝❡ ♠✐s♠❛t❝❤ ♦❢

❣r❛♣❤❡♥❡✱ ❛♥❞ ●❖ ✇✐t❤ ♦♥❧②

−❖❍ ❣r♦✉♣s✳

➴

7%

❛♥❞

aCO1/2 = 2.57

✐s ❝❛❧❝✉❧❛t❡❞ ❜❡t✇❡❡♥

❚❤❡ ❛✈❡r❛❣❡ ❧❛tt✐❝❡ ❝♦♥st❛♥ts

a(y, z, T ) ♦❢ t❤❡ ●❖

s②st❡♠s ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❝❤❡♠✐❝❛❧ ♣❛r❛♠❡t❡rs ❛r❡ ❝❛❧❝✉❧❛t❡❞ ✇✐t❤✐♥ t❤❡ ●◗❈❆ st❛t✐st✐❝s ❛♥❞ ❞❡♣✐❝t❡❞ ✐♥ ❋✐❣✳ ✺✳ ❆ ❣r❛❞✉❛❧ ❝❤❛♥❣❡ ❜❡t✇❡❡♥ t❤❡ ❡♥❞ ❝♦♠♣♦♥❡♥t ❧❛tt✐❝❡ ❝♦♥st❛♥t ✐s ♦❜s❡r✈❡❞ ❛♥❞ t❤❡ ❧✐♥❡❛r ✜t

a = xaC + yaCOH + zaCO1/2

✶✹

❣✐✈❡s t❤❡ ❝❛❧❝✉❧❛t❡❞ ❧❛tt✐❝❡ ❝♦♥st❛♥t


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✇✐t❤ ❛ ♠❛①✐♠❛❧ ❡rr♦r ❜❡❧♦✇ ✶✳✺✪ ❛t ❜♦t❤ ❝♦♥s✐❞❡r❡❞ t❡♠♣❡r❛t✉r❡s✳ ❚❤❛t ♠❡❛♥s t❤❛t t❤❡ ❱❡❣❛r❞✬s ❧❛✇ ✐s ✇✐❞❡❧② ❢✉❧✜❧❧❡❞✳

CO1/2

2.625 2.599 2.573 2.548 2.522 2.500

C

Lattice constant a (Å) COH

❋✐❣✉r❡ ✺✿ ✭❈♦❧♦r ♦♥❧✐♥❡✮ ❚❤❡ ❛✈❡r❛❣❡ ❧❛tt✐❝❡ ❝♦♥st❛♥t

2.470

a ♦❢ t❤❡ ●❖ s②st❡♠s ❝❛❧❝✉❧❛t❡❞ ✇✐t❤✐♥

t❤❡ ●◗❈❆ ❛♣♣r♦❛❝❤ ❛s t❤❡ s❡❝♦♥❞✲♥❡❛r❡st ♥❡✐❣❤❜♦r ❈ ❞✐st❛♥❝❡✳ ❆ s②♥t❤❡s✐s t❡♠♣❡r❛t✉r❡ ♦❢ 200o C ✐s ❝♦♥s✐❞❡r❡❞✳

❚❤❡ ❜✉❝❦❧✐♥❣ ❛♠♣❧✐t✉❞❡

∆(y, z, T ) ♦❢ t❤❡ ❝❛r❜♦♥ s❤❡❡t ✐s ❛❧s♦ ✐♥✈❡st✐❣❛t❡❞✳

❧❛②❡r ❡①❤✐❜✐ts ❛ ❤♦♥❡②❝♦♠❜ ♣❧❛♥❛r str✉❝t✉r❡ ✇✐t❤ ❛ ❜✉❝❦❧✐♥❣ ❛♠♣❧✐t✉❞❡ ♦❢

∆j = 0.51

❚❤❡ ❣r❛♣❤❡♥❡

∆j = 0.00 ➴ ✱ ✇❤✐❧❡ t❤❡ ❈❖❍ s②st❡♠ s❤♦✇s

➴ ✱ ✇❤✐❝❤ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ♣✉r❡

sp3

❤②❜r✐❞✐③❛t✐♦♥

♦❢ t❤❡ ❝❛r❜♦♥ ❛t♦♠✐❝ ♦r❜✐t❛❧s✳ ❚❤❡ ❈❖1/2 ❡♥❞ ❝♦♠♣♦♥❡♥t ❢✉❧❧② ❞❡❝♦r❛t❡❞ ❜② ❡♣♦①② ❣r♦✉♣s ✉♥❞❡r❣♦❡s ❛ ❞✐✛❡r❡♥t ❞✐st♦rt✐♦♥ ♦❢ t❤❡ ♣❧❛♥❛r ❤♦♥❡②❝♦♠❜ str✉❝t✉r❡✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ❤♦✇ t❤❡ ❡♣♦①② ❣r♦✉♣s ❛r❡ ❞✐str✐❜✉t❡❞✳ ❇✉❝❦❧✐♥❣ ❛♠♣❧✐t✉❞❡s ❜❡t✇❡❡♥

∆j =

✵✳✵✵ ➴

❛♥❞ ✵✳✺✵ ➴

❛r❡

♦❜s❡r✈❡❞✱ t❤❛t r❡s✉❧ts ✐♥ ❛♥ ❛✈❡r❛❣❡ ❜✉❝❦❧✐♥❣ ❛♠♣❧✐t✉❞❡ ♦❢ ✵✳✸✻ ➴ ❢♦r t❤❡ ❝♦♥s✐❞❡r❡❞ r❛♥❣❡ ♦❢ t❡♠♣❡r❛t✉r❡s✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❛✈❡r❛❣❡ ❜✉❝❦❧✐♥❣ ❛♠♣❧✐t✉❞❡

o ❝♦♠♣♦s✐t✐♦♥s ❛t ✷✵✵ ❈ ✐s ❞✐s♣❧❛②❡❞ ✐♥ ❋✐❣✳

✻✳

∆(y, z, T ) ♦♥ t❤❡ ❝❤❡♠✐❝❛❧

❆ s❧✐❣❤t ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ t❡♠♣❡r❛t✉r❡ ✐s

♦❜s❡r✈❡❞✱ ❤♦✇❡✈❡r t❤❡ q✉❛❧✐t❛t✐✈❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❛♠♣❧✐t✉❞❡ ✇✐t❤ t❤❡ ●❖ ❝❤❡♠✐❝❛❧ ❝♦♥t❡♥t r❡♠❛✐♥s t❤❡ s❛♠❡ ❛t ✶✶✵✵

o

❈✳

❖♥❡ ♦❜s❡r✈❡s ❛ ♣r♦♥♦✉♥❝❡❞ ♠❛①✐♠✉♠ ❜✉❝❦❧✐♥❣ ❛♠♣❧✐t✉❞❡ ♦❢ ✵✳✼✶ ➴ ❢✉❧❧② ♦①✐❞✐③❡❞ ✇✐t❤ ❜♦t❤ ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s ✐♥ ♣r♦♣♦rt✐♦♥s ♥❡❛r

❢♦r ❣r❛♣❤❡♥❡ ♦①✐❞❡

nOH /nO = 1✳

❚❤✐s

✈❛❧✉❡ ♦❢ t❤❡ ❜✉❝❦❧✐♥❣ ✐s ✐s ❧❛r❣❡r t❤❛♥ ❛♥② ❜✉❝❦❧✐♥❣ ❛♠♣❧✐t✉❞❡ ♦❜s❡r✈❡❞ ❢♦r t❤❡ ❡♥❞ ❝♦♠♣♦✲ ♥❡♥ts✳

❚❤✐s ✐s t❤❡ r❡s✉❧t ♦❢ ❧❛r❣❡ st❛t✐st✐❝❛❧ ❝♦♥tr✐❜✉t✐♦♥s ♦❢ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥s ✉✵❞✵✵❞✵✉✲

✵✵✵✵✵✉❞✵✵✵✵✵ ❛♥❞ ✉❞✉❞✵✵✵✵✲✵✵✵✵✵✵✉❞✵✱ ✇❤✐❝❤ ❛r❡ t❤❡r♠♦❞②♥❛♠✐❝❛❧❧② ❢❛✈♦r❡❞ ❞✉❡ t♦ t❤❡✐r

✶✺



♥❡❣❛t✐✈❡ ❡①❝❡ss ❡♥❡r❣✐❡s ✭∆Ej

= −0.72

❛♥❞ ✲✵✳✺✽ ❡✈✴❝❧✉st❡r✱ r❡s♣❡❝t✐✈❡❧②✮✳

❜✉❝❦❧✐♥❣ ❛♠♣❧✐t✉❞❡s s✐❣♥✐✜❝❛♥t❧② ❧❛r❣❡r t❤❛♥ t❤❡ ♣✉r❡ r❡s♣❡❝t✐✈❡❧②✮✱ ❛s s♦♠❡

−❖❍✴−❖−

Page 16 of 30

sp3

❜✉❝❦❧✐♥❣ ✭∆j

❚❤❡② ❡①❤✐❜✐t

= 0.75➴

❛♥❞ ✵✳✻✷✱

❞✐str✐❜✉t✐♦♥s ✐♥❞✉❝❡ ❧♦✇ ❡♥❡r❣② ❧♦♥❣✲r❛♥❣❡ ❞✐st♦rt✐♦♥s ♦❢

t❤❡ ♣❧❛♥❛r ❤♦♥❡②❝♦♠❜ str✉❝t✉r❡✳

CO1/2

0.71 0.60 0.48 0.37 0.26 0.14

C

Buckling amplitude Δ (Å) COH

0.00

∆ ♦❢ t❤❡ 200o C ✳

❋✐❣✉r❡ ✻✿ ✭❈♦❧♦r ♦♥❧✐♥❡✮ ❚❤❡ ❛✈❡r❛❣❡ ❜✉❝❦❧✐♥❣ ❛♠♣❧✐t✉❞❡ ✇✐t❤✐♥ t❤❡ ●◗❈❆ ❛♣♣r♦❛❝❤ ❛t ❛ s②♥t❤❡s✐s t❡♠♣❡r❛t✉r❡ ♦❢

❚❤❡ ♠♦st st❛❜❧❡ ❝♦♥✜❣✉r❛t✐♦♥ ❛t

(x, y, z) = (0, 0.5, 0.5)

●❖ s②st❡♠s ❝❛❧❝✉❧❛t❡❞

r❡✈❡❛❧s ❛❧t❡r♥❛t❡❞ str✐♣❡s ♦❢

❝❛r❜♦♥ ❛t♦♠s ❜♦✉♥❞ t♦ ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s✱ ❛s ✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣✳ ✷✭❞✮✳ ❚❤❡ ♠♦st st❛❜❧❡ ❝♦♥✜❣✉r❛t✐♦♥ ❛t

(x, y, z) = (0, 0.25, 0.75)

✐s ❛ ♥❡✇❧② ❞✐s❝♦✈❡r❡❞ ❢❛✈♦r❛❜❧❡ str✉❝t✉r❡✱

♠❛❞❡ ♦❢ ❤❡①❛❣♦♥s ♦❢ ❝❛r❜♦♥ ❛t♦♠s ❛tt❛❝❤❡❞ t♦ ❡♣♦①② ❣r♦✉♣s✱ s✉rr♦✉♥❞❡❞ ❜② ❝❛r❜♦♥ ❛t♦♠s ❛tt❛❝❤❡❞ t♦

−❖❍

❣r♦✉♣s✳ ❚❤✐s ♣❛tt❡r♥ ❝❛♥ ❜❡ ✈❡r✐✜❡❞ ❜② ❜✉✐❧❞✐♥❣ ❛ ✷❉✲♣❡r✐♦❞✐❝ r❡♣❡t✐t✐♦♥

♦❢ t❤❡ ❝❧✉st❡r ♦❢ ❋✐❣✳ ✷✭❡✮✳

❊❧❡❝tr♦♥✐❝ ❛♥❞ ♦♣t✐❝❛❧ ♣r♦♣❡rt✐❡s ❲❡ ❝❛❧❝✉❧❛t❡❞ t❤❡ ❡❧❡❝tr♦♥✐❝ ❛♥❞ ♦♣t✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ✸✵✽ ❝❧✉st❡r ❝♦♥✜❣✉r❛t✐♦♥s ✉s✐♥❣ t❤❡ ●●❆✲P❇❊ ❡①❝❤❛♥❣❡ ❛♥❞ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥❛❧✳ ❚❤❡ ❉❋❚ ❑♦❤♥✲❙❤❛♠ ❢✉♥❞❛♠❡♥t❛❧ ❜❛♥❞ ❣❛♣s ❤❛✈❡ ✈❛❧✉❡s ♦❢

Eg,C = 0

❡❱✱

t❤❡ ❝♦♥s✐❞❡r❡❞ ❡♥❞ ❝♦♠♣♦♥❡♥ts✳

Eg,COH = 2.25

❡❱✱ ❛♥❞

Eg,CO1/2 = 4.00

❡❱ ❛t ✷✵✵

o

❈ ❢♦r

❲❡ ❞❡♠♦♥str❛t❡❞ t❤❛t ✐♥ ❝❛s❡ ♦❢ ❛❞s♦r♣t✐♦♥ ♦❢ ♦♥❧② ♦♥❡

♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣ ♦♥ ❣r❛♣❤❡♥❡ t❤❡ ❑♦❤♥✲❙❤❛♠ ❡♥❡r❣② ❣❛♣s ♦❜t❛✐♥❡❞ ❢r♦♠ ●●❆✲P❇❊ ❝❛❧❝✉❧❛t✐♦♥s s❤♦✇ t❤❡ s❛♠❡ tr❡♥❞s ❛s t❤♦s❡ ❝♦♠♣✉t❡❞ ✇✐t❤ ♠♦r❡ ❛❝❝✉r❛t❡ s❝r❡❡♥❡❞ ❤②❜r✐❞ ❢✉♥❝t✐♦♥❛❧s✳

✸✾

✶✻


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❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❡♥❡r❣② ❣❛♣ ❞❡♣❡♥❞s ♦♥ t❤❡ st♦✐❝❤✐♦♠❡tr②✱ ✐✳❡✳✱ t❤❡ ❣❡♥❡r❛❧ ♦①✐❞❛t✐♦♥ st❛t❡ ♦❢ ●❖✳ ❲❡ ❝❛♥ t❤❡r❡❢♦r❡ ❝♦♥s✐❞❡r t❤❡ ❛✈❡r❛❣❡ ❣❛♣ ♣❛r❛♠❡t❡rs

y

❛♥❞

z✱

Eg (y, z, T )

❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡

r❡❧❛t❡❞ t♦ ♦①✐❞❛t✐♦♥ ❜② ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s✱ r❡s♣❡❝t✐✈❡❧② ❛♥❞

♦♥ t❤❡ ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡

T✳

❚❤❡ ❛✈❡r❛❣❡ ❣❛♣ ✐s ♥♦t ❛ s✐♠♣❧❡ ✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ♦❢ t❤❡

❜❛♥❞ ❣❛♣s ♦❢ t❤❡ ❡♥❞ ♠❛t❡r✐❛❧s✱ ❛♥❞ t❤❡ ❞❡✈✐❛t✐♦♥ ❢r♦♠ t❤✐s s✐♠♣❧❡ ❧✐♥❡❛r ❜❡❤❛✈✐♦r ❣✐✈❡s t❤❡ s♦✲❝❛❧❧❡❞ ❣❛♣ ❜♦✇✐♥❣

∆Eg (y, z, T )✱

❞❡✜♥❡❞ ❛s

Eg (y, z, T ) = xEg,C + yEg,COH + zEg,CO1/2 + ∆Eg (y, z, T ).

✭✾✮

❆♥ ❡st✐♠❛t❡ ♦❢ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡s ♦❢ t❤❡ ❡♥❡r❣② ❣❛♣ ❝❛♥ ❜❡ ♠❛❞❡ ❜② ❛♣♣❧②✐♥❣ ❛ s❝✐ss♦r ♦♣❡r❛t♦r ❛♣♣r♦①✐♠❛t✐♦♥✱ ❝♦♥s✐❞❡r✐♥❣ q✉❛s✐♣❛rt✐❝❧❡ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❡♥❡r❣② ❣❛♣s ♦❢ t❤❡ ❈❖❍ ❛♥❞ ❈❖1/2 ✱ ❛♥❞ t❤❡ ❣❛♣ ❜♦✇✐♥❣ ❡st✐♠❛t❡❞ ❢r♦♠ t❤❡ ●●❆✲P❇❊ ❝❛❧❝✉❧❛t✐♦♥s✳

o o ❚❤❡ ❛✈❡r❛❣❡ ❣❛♣s ❛r❡ ❝❛❧❝✉❧❛t❡❞ ✇✐t❤✐♥ ●◗❈❆ ❛t ✷✵✵ ❈ ❛♥❞ ✶✶✵✵ ❈✳ ❚❤❡ ❡♥❡r❣② ❣❛♣ ❜♦✇✐♥❣s

∆Eg (y, z, T )

♦❜t❛✐♥❡❞ ✉s✐♥❣ t❤❡ ❣❛♣s

✐♥s❡rt✐♥❣ t❤❡ st❛t✐st✐❝❛❧ ✇❡✐❣❤s

xj

Eg,j

❢♦r ❡❛❝❤ ❝♦♥✜❣✉r❛t✐♦♥ ❝❧❛ss

j

❛♥❞ t❤❡ ❢♦r♠✉❧❛ ✭✻✮✱

❛t t❤❡ ❝♦♥s✐❞❡r❡❞ ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡s ❛r❡ ❞❡♣✐❝t❡❞ ✉s✐♥❣

❝♦❧♦r ♠❛♣s ✐♥ ❋✐❣✳ ✼✳ ■♥ ❛❣r❡❡♠❡♥t ✇✐t❤ ♣r❡✈✐♦✉s r❡s✉❧ts ❢♦r ❣r❛♣❤❡♥❡ ♦①✐❞❡ s②st❡♠s ❞❡❝♦r❛t❡❞ ✇✐t❤ s✐♥❣❧❡ ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s✱

✸✾

t❤❡ ❜❛♥❞ ❣❛♣ ❜♦✇✐♥❣ ❢♦r

−❖−

r✐❝❤ ♦①✐❞❛t✐♦♥ ❞♦❡s ♥♦t s✐❣♥✐✜✲

❝❛♥t❧② ❝❤❛♥❣❡ ✇❤❡♥ t❤❡ t❡♠♣❡r❛t✉r❡ ✐♥❝r❡❛s❡s✱ ✇❤✐❧❡ t❤❡ ❜❛♥❞ ❣❛♣ ❜♦✇✐♥❣ ❢♦r ❞❛t✐♦♥ ❝❤❛♥❣❡s ❢r♦♠ r❡❣✉❧❛r ❜♦✇✐♥❣ ✭∆Eg ❜♦✇✐♥❣ ♦r ❛♥t✐✲❜♦✇✐♥❣ ✭∆Eg

−❖❍ r✐❝❤ ♦①✐✲

< 0✮ ❛t ✷✵✵o ❈ t♦ ❛ ♣r♦✜❧❡ t❤❛t ♠❛② ❡①❤✐❜✐t r❡❣✉❧❛r

> 0✮ ❛t ✶✶✵✵o ❈✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ♦①✐❞❛t✐♦♥ ❧❡✈❡❧ ♦❢ t❤❡ s②st❡♠✳

❚❤❡ ❣❛♣ ❜♦✇✐♥❣s ❛❧♦♥❣ t❤❡ ❈−❈❖1/2 ❡❞❣❡ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s

∆Eg (x, T ) = −b(T )x(1 − x)✱

✇❤✐❧❡ ❛❧♦♥❣ t❤❡ ❈−❈❖❍ ❡❞❣❡ t❤❡ ❣❛♣ ♠❛② ❡①❤✐❜✐t ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ❜♦✇✐♥❣ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝❤❡♠✐❝❛❧ ❝♦♠♣♦s✐t✐♦♥ ❛t ❧❛r❣❡ ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡s ✭s❡❡ ❚❤❡ ❡♥❡r❣② ❣❛♣ ❜♦✇✐♥❣ ❛❧♦♥❣ t❤❡ ❈❖1/2 −❈❖❍ ❡❞❣❡ ✭y ♥❡♦✉s ❞❡♣♦s✐t✐♦♥ ♦❢

−❖❍

❛♥❞

❢✉❧❧② ♦①✐❞✐③❡❞ s②st❡♠s ✇✐t❤

−❖−

✸✾

✮✳

= 1 − z ✮✱

✐✳❡✳✱ ❢♦r t❤❡ s✐♠✉❧t❛✲

❣r♦✉♣s ✐♥ t❤❡ ❧✐♠✐t ♦❢ ❝♦♠♣❧❡t❡ ♦①✐❞❛t✐♦♥✱ ✐s ❧❛r❣❡r ❢♦r

−❖−✴−❖❍

r❛t✐♦s ❜❡t✇❡❡♥ ✶ ❛♥❞ ✸✱ ❛t ❜♦t❤ ❣r♦✇t❤ t❡♠♣❡r❛✲

t✉r❡s✳ ❚❤✐s ❡♥❤❛♥❝❡♠❡♥t ♦❢ t❤❡ ❡♥❡r❣② ❣❛♣s ✐s ❡①♣❧❛✐♥❡❞ ❜② t❤❡ s✐❣♥✐✜❝❛♥t ❝♦♥tr✐❜✉t✐♦♥s ♦❢

✶✼



(a) 200 oC

CO1/2

Page 18 of 30

1.55 1.25 0.96 0.67 0.38 0.09

C

ΔEg (eV)

COH

-0.21

(b) 1100 oC CO1/2

1.07 0.85 0.63 0.40 0.18 -0.04

C

ΔEg (eV)

COH

-0.26

❋✐❣✉r❡ ✼✿ ✭❈♦❧♦r ♦♥❧✐♥❡✮ ❚❤❡ ❛✈❡r❛❣❡ ❡♥❡r❣② ❣❛♣ ❜♦✇✐♥❣ ♦❢ t❤❡ ●❖ s②st❡♠s ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ o o t❤❡ ❛✈❡r❛❣❡ ❝♦♠♣♦s✐t✐♦♥s ❛t ❛ ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡ ♦❢ ✭❛✮ 200 C ❛♥❞ ✭❜✮ 1100 C ✳

❢❛✈♦r❛❜❧❡ ❝♦♥✜❣✉r❛t✐♦♥s ✇✐t❤ ♥❡❣❛t✐✈❡ ❡①❝❡ss ❡♥❡r❣✐❡s ❛♥❞ ❧❛r❣❡ ❡♥❡r❣② ❣❛♣ ✈❛❧✉❡s✱ ❡✳❣✳✱ ✹✳✹✵ ❡❱ ✭✉✵❞✵✵❞✵✉−✵✵✵✵✵✉❞✵✵✵✵✵✮ ❛♥❞ ✹✳✻✽ ❡❱ ✭❞✵✵✵✵✵✵❞−✵✵✵✵✉✉✵✵✵✵✉✵✮✳ ❙✐❣♥✐✜❝❛♥t ❝❤❛♥❣❡s ❝❛♥ ❜❡ ♦❜s❡r✈❡❞ ❛t ❞✐✛❡r❡♥t t❡♠♣❡r❛t✉r❡s✳ ❚❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ st❛t✐st✐❝❛❧ ❛♣♣r♦❛❝❤ ♦♥ ❛ ❧❛r❣❡r s❡t ♦❢ ❝♦♥✜❣✉r❛t✐♦♥s r❡s✉❧ts ❤❡♥❝❡ ✐♥t♦ ❛♥ ❡✈❡♥ ❧❛r❣❡r ❡♥❡r❣② ❣❛♣ r❛♥❣❡ ❢♦r ❜❛♥❞ ❣❛♣ ❡♥❣✐♥❡❡r✐♥❣ t❤❛♥ t❤❡ ♦♥❡ ♦❜t❛✐♥❡❞ ❜② ❨❛♥

❡t ❛❧✳ ✸✼✱✸✽

❚❤❡s❡ ❛✉t❤♦rs ❤❛❞ ✐♥ ❢❛❝t r❡♣♦rt❡❞ ❛♥

❡♥❡r❣② ❣❛♣ ❜♦✇✐♥❣ ❜❡t✇❡❡♥ ✵✳✽✲✶✳✶ ❡❱ ❢♦r ❢✉❧❧② ♦①✐❞✐③❡❞ ❝♦♥✜❣✉r❛t✐♦♥s✱ ✇❤❡♥ t❤❡ s②st❡♠ ✐s ♣r❡♣❛r❡❞ ❛t ❧♦✇ ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡✳ ❚❤❡ ✐♥❝r❡❛s❡ ♦❢ t❤❡ ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡ ✐♥❞✉❝❡s ♠♦r❡ ❝♦♥✜❣✉r❛t✐♦♥❛❧ ❞✐s♦r❞❡r ✐♥ t❤❡ ●❖ s②st❡♠ ❛♥❞ t❤❡ ❝♦♥tr✐❜✉t✐♦♥s ♦❢ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥s ✇✐t❤ ❧♦✇❡r ❡♥❡r❣✐❡s ❞❡❝r❡❛s❡ ✇✐t❤ t❤❡ r✐s✐♥❣ t❡♠♣❡r❛t✉r❡✳ ❆ s✐❣♥❛t✉r❡ ♦❢ t❤✐s ❡✛❡❝t ✐s t❤❡ s❤r✐♥❦❛❣❡ ♦❢ t❤❡ r❡❞ ❛r❡❛ ✐♥ ❋✐❣✳ ✼✭❜✮✱ ❝♦♠♣❛r❡❞ ✇✐t❤ ❋✐❣✳ ✼✭❛✮✳ ❚❤✐s r❡s✉❧t s❤♦✇s t❤❡ ❜✐❣ ♣♦t❡♥t✐❛❧ ♦❢ ❣r❛♣❤❡♥❡ ♦①✐❞❡ t♦ ♣r♦✈✐❞❡ ❛ ✷❉ ♣❧❛t❢♦r♠ ❢♦r ♥♦✈❡❧ ✷❉ ❡❧❡❝tr♦♥✐❝s✱ ✇❤♦s❡ ❡❧❡❝tr♦♥✐❝ ♣r♦♣❡rt✐❡s ❝❛♥ ❜❡ t✉♥❡❞ ❝♦♥tr♦❧❧✐♥❣

✶✽


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❜♦t❤ t❤❡ ❝❤❡♠✐❝❛❧ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥s✳ ❚❤❡r❡❜②✱ t❤❡ ❝♦❛❞s♦r♣t✐♦♥ ♦❢ ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s ♣❛✈❡s t❤❡ ✇❛② ❢♦r ✐♠♣r♦✈❡❞ t✉♥✐♥❣ str❛t❡❣✐❡s✱ ♥♦t ♦♥❧② ✐♥ t❤❡ ❧✐♠✐t ♦❢ ❢✉❧❧ ♦①✐❞❛t✐♦♥ ❜✉t ❛❧s♦ ✐♥ t❤❡ ✐♥t❡r♠❡❞✐❛t❡ r❡❣✐♠❡✱ ✇✐t❤ str✉❝t✉r❡s s✐♠✐❧❛r t♦ t❤♦s❡ ✐♥ ❋✐❣s✳ ✷ ❞ ❛♥❞ ❡✳ ❚❤❡ ❣❛♣ ❜♦✇✐♥❣

∆Eg (y, z, T )

✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣✳ ✼ ❝❛♥ ❜❡ ✜tt❡❞ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①✲

♣r❡ss✐♦♥

∆Eg (y, z, T ) =c0 + c1 (1 − x)yz + c2 (1 − y)zx + (1 − z)xy(c3 x + c4 y) + c5 xyz,

✇❤❡r❡ ❡rr♦r✳

c0 ✱ c1 ✱ c2 ✱ c3 ✱ c4 ✱

❛♥❞

c5

✭✶✵✮

❛r❡ ♣❛r❛♠❡t❡rs ❝❛❧❝✉❧❛t❡❞ ❜② ♠✐♥✐♠✐③✐♥❣ t❤❡ t♦t❛❧ sq✉❛r❡❞

❲❡ ❝❛❧❝✉❧❛t❡❞ t❤❡s❡ ✜tt✐♥❣ ♣❛r❛♠❡t❡rs ❢♦r

❑ ✉s✐♥❣ ◆❡❧❞❡r✲▼❡❛❞ ❛❧❣♦r✐t❤♠✳

✻✹

T =✸✵✵

❑✱ ✹✼✸ ❑✱ ✶✵✵✵ ❑✱ ❛♥❞ ✶✹✼✸

❚❤❡ ♦❜t❛✐♥❡❞ ❝♦❡✣❝✐❡♥ts ❛♥❞ t❤❡ ✜tt✐♥❣ ❛✈❡r❛❣❡ ❡rr♦r

❛t ❡❛❝❤ t❡♠♣❡r❛t✉r❡ ❛r❡ ❧✐st❡❞ ✐♥ ❚❛❜❧❡ ✷✳

❚❤❡ ♠❛①✐♠❛❧ ❡rr♦r ❢♦r ❛❧❧ ❝♦♥s✐❞❡r❡❞ ❝❤❡♠✐❝❛❧

❝♦♠♣♦s✐t✐♦♥s ❛♥❞ t❡♠♣❡r❛t✉r❡s ✇❛s ✵✳✷✼ ❡❱✳

❚❛❜❧❡ ✷✿ ❋✐tt✐♥❣ ♣❛r❛♠❡t❡rs ❢♦r ❣❛♣ ❜♦✇✐♥❣ ∆Eg (y, z, T ) ❛s ❞❡✜♥❡❞ ✐♥ ❊q✳✶✵ ❢♦r ❞✐✛❡r❡♥t ❣r♦✇t❤ t❡♠♣❡r❛t✉r❡s✳ ❚❤❡ ❛✈❡r❛❣❡ ✜tt✐♥❣ ❡rr♦r e¯ ✐s ❣✐✈❡♥ ❢♦r ❡❛❝❤ t❡♠♣❡r❛t✉r❡✳ P❛r❛♠❡t❡rs ❢♦r ✐♥t❡r♠❡❞✐❛t❡ t❡♠♣❡r❛t✉r❡s ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❜② ✐♥t❡r♣♦❧❛t✐♦♥✳ ❚❡♠♣❡r❛t✉r❡ ✭❑✮

c0

c1

c2

c3

c4

c5

e¯ ✭❡❱✮

✸✵✵

✵✳✵✶

✺✳✽✼

✵✳✸✷

✵✳✵✸

✲✹✳✺✷

✶✻✳✻✵

✵✳✵✾

✹✼✸

✵✳✵✹

✺✳✼✻

✵✳✶✽

✲✵✳✵✹

✲✷✳✸✶

✾✳✹✹

✵✳✵✹

✶✵✵✵

✵✳✵✽

✹✳✼✽

✵✳✵✼

✷✳✸✵

✲✷✳✸✻

✵✳✶✶

✵✳✵✼

✶✸✼✸

✵✳✵✽

✹✳✷✽

✵✳✶✶

✸✳✷✷

✲✸✳✺✹

✲✷✳✸✾

✵✳✵✼

❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ♦♣t✐❝❛❧ ❛❜s♦r❜❛♥❝❡ ♦♥ t❤❡ s②st❡♠ ✐s r❡♣r❡s❡♥t❡❞ ✐♥ ❋✐❣✳ ✽✳ ❚❤❡ s♣❡❝tr❛

Aj (ω)

−❖−✴−❖❍

r❛t✐♦ ♦❢ ❛ ❢✉❧❧② ♦①✐❞✐③❡❞

❢♦r ❛❧❧ ❝❧✉st❡r ❝❧❛ss❡s ❤❛✈❡ ❜❡❡♥ ❝♦♠✲

♣✉t❡❞ ✐♥ t❤❡ ✐♥❞❡♣❡♥❞❡♥t✲q✉❛s✐♣❛rt✐❝❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♥❞ ❛✈❡r❛❣❡❞ ✉s✐♥❣ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s

xj (y, z, T )

✐♥ ✭✻✮✳ ❖♥❡ ♦❜s❡r✈❡s t❤❛t t❤❡ t❡♠♣❡r❛t✉r❡ ✐♥❝r❡❛s❡ ❞♦❡s ♥♦t str♦♥❣❧② ❛✛❡❝t t❤❡

♦♣t✐❝❛❧ ❛❜s♦r♣t✐♦♥ ♦❢ ●❖ ✐♥ t❤❡ ❢✉❧❧② ♦①✐❞✐③❡❞ ❧✐♠✐t ✇✐t❤ ❝♦✲❛❞s♦r♣t✐♦♥ ♦❢

−❖❍

❛♥❞

−❖−

❣r♦✉♣s✳ ❚❤✐s ✐♥s❡♥s✐t✐✈✐t② s✉❣❣❡sts t❤❛t t❤❡ st❛t✐st✐❝s ✐s ❞♦♠✐♥❛t❡❞ ❜② t❤❡ ❧♦✇ ❡♥❡r❣② ❝♦♥✲

✶✾




Page 20 of 30

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t✐♠❡ ♦♥ ❣r❛♣❤❡♥❡✳ ❚❤✐s ✐s s✐♠✐❧❛r t♦ ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥ ♦♥❧② ♦♥❡ ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣ ✐s ❝♦♥s✐❞❡r❡❞✳ ❚❤❡ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ t❤❡ s②st❡♠ ❡♥❡r❣❡t✐❝s ♣r♦✈❡❞ t❤❛t ❛ r❛♥❞♦♠ ❞✐str✐❜✉t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥❛❧✐③✐♥❣ ❣r♦✉♣s ✐♥ t❤❡ ♦①②❣❡♥✲r✐❝❤ ❞♦♠❛✐♥s ✐s ♦❜s❡r✈❡❞ ❛t t②♣✐❝❛❧ ❣r♦✇t❤ t❡♠♣❡r❛✲ t✉r❡s✳ ❲❡ ❝♦✉❧❞ ✐❞❡♥t✐❢② ❢❛✈♦r❡❞ ❢✉❧❧② ♦①✐❞✐③❡❞ ♦r❞❡r❡❞ ❛rr❛♥❣❡♠❡♥ts ♦❢ ❤②❞r♦①②❧ ❛♥❞ ❡♣♦①② ❣r♦✉♣s ✇✐t❤ ♥❡❣❛t✐✈❡ ❡①❝❡ss ❡♥❡r❣②✳ ▼♦r❡♦✈❡r✱ ✇❡ r❡✈❡❛❧❡❞ ❤♦✇ t❤❡ ❞❡♥s✐t② ❛♥❞ t❤❡ ❛rr❛♥❣❡✲ ♠❡♥t ♦❢ t❤❡ ♦①②❣❡♥✲❝♦♥t❛✐♥✐♥❣ ❣r♦✉♣s ❝♦♥tr♦❧ ❜✉❝❦❧✐♥❣ ❛♠♣❧✐t✉❞❡ ♦❢ t❤❡ ❝❛r❜♦♥ ❜❛s❛❧ s❤❡❡t✳ ❚❤❡ ❝❤❡♠✐❝❛❧ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❛t♦♠✐❝ ❛rr❛♥❣❡♠❡♥ts ❛❧s♦ s✐❣♥✐✜❝❛♥t❧② ✐♥✢✉❡♥❝❡ t❤❡ ❛✈❡r❛❣❡ ❢✉♥❞❛♠❡♥t❛❧ ❡♥❡r❣② ❣❛♣ ❛♥❞ t❤❡ ❜❛♥❞ ❣❛♣ ❜♦✇✐♥❣✳ ■♥ ❝❛s❡ ♦❢ ❛ ❢✉❧❧ ♦①✐❞❛t✐♦♥ ✇✐t❤ ❡q✉❛❧ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢

−❖❍

❛♥❞

−❖−

❣r♦✉♣s ❛ str♦♥❣ ❛♥t✐❜♦✇✐♥❣ ✐s ❢♦✉♥❞✱ ❛♥❞ ✐t ✐s ❡①♣❧❛✐♥❡❞ ❛s

❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❧❛r❣❡ ❜❛♥❞ ❣❛♣s ❝❛❧❝✉❧❛t❡❞ ❢♦r t❤❡ ❢❛✈♦r❡❞ ❛t♦♠✐❝ ❣❡♦♠❡tr✐❡s ✐❞❡♥t✐✜❡❞ ✐♥ t❤✐s ✇♦r❦✳ ❋✐♥❛❧❧②✱ ✇❡ ❞✐s❝✉ss❡❞ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ♦♣t✐❝❛❧ ❛❜s♦r❜❛♥❝❡ ♦♥ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ t❡♠♣❡r❛t✉r❡ ♦❢ ❛ ❢✉❧❧② ♦①✐❞✐③❡❞ ●❖ ♣❤❛s❡ ❛♥❞ ❞❡♠♦♥str❛t❡❞ ❛ str♦♥❣ ✐♥✢✉❡♥❝❡ ♦❢ t❤❡ r❡❧❛t✐✈❡ ❛♠♦✉♥t ♦❢

−❖❍

❛♥❞

−❖−

❣r♦✉♣s ♦♥ t❤❡ ❝❛❧❝✉❧❛t❡❞ ♣r♦♣❡rt✐❡s✳

❚❤❡s❡ r❡s✉❧ts ♦✛❡r ✈❛❧✉❛❜❧❡ ✐♥s✐❣❤t ✐♥t♦ t❤❡ str✉❝t✉r❡ ♦❢ ❞✐s♦r❞❡r❡❞ ❣r❛♣❤❡♥❡ ♦①✐❞❡ s❛♠✲ ♣❧❡s✱ ❛♥❞ s✉♣♣♦rt t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ❛tt❡♠♣ts t♦ ♦❜t❛✐♥ s❛♠♣❧❡s ✇✐t❤ ❞❡✜♥❡❞ ❝❤❡♠✐❝❛❧ ❝♦♠✲ ♣♦s✐t✐♦♥✱ ❛ s✉✣❝✐❡♥t ♣❤❛s❡ st❛❜✐❧✐t② ❛♥❞ t❤❡ ❛✈❡r❛❣❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐s♦r❞❡r❡❞ s②st❡♠✳

❆❝❦♥♦✇❧❡❞❣❡♠❡♥t ❲❡ t❤❛♥❦ t❤❡ ❇r❛③✐❧✐❛♥ ❢✉♥❞✐♥❣ ❛❣❡♥❝✐❡s ❋❆P❊❙P ✭●r❛♥t ♥✳ ✷✵✶✷✴✺✵✼✸✽✲✸✮✱ ❈❆P❊❙ ✭P❱❊ ✲ ●r❛♥ts ♥✳ ✽✽✽✽✶✳✵✻✽✸✺✺✴✷✵✶✹✲✶✱ ❛♥❞ ✽✽✽✽✼✳✶✶✻✺✸✺✴✷✵✶✻✲✵✵✮✱ ❛♥❞ ❈◆P◗ ✭●r❛♥ts ♥✳ ✸✵✺✹✵✺✴✷✵✶✹✲ ✹✱ ❛♥❞ ✸✵✽✼✹✷✴✷✵✶✻✲✽✮ ❢♦r t❤❡ ✜♥❛♥❝✐❛❧ s✉♣♣♦rt✳

❚❤❡ ❛✉t❤♦rs ❛❝❦♥♦✇❧❡❞❣❡ t❤❡ ◆❛t✐♦♥❛❧

▲❛❜♦r❛t♦r② ❢♦r ❙❝✐❡♥t✐✜❝ ❈♦♠♣✉t✐♥❣ ✭▲◆❈❈✴▼❈❚■✱ ❇r❛③✐❧✮ ❢♦r ♣r♦✈✐❞✐♥❣ ❍P❈ r❡s♦✉r❝❡s ♦❢ t❤❡ ❙❉✉♠♦♥t s✉♣❡r❝♦♠♣✉t❡r✳ ❈♦♠♣✉t❛t✐♦♥❛❧ r❡s♦✉r❝❡s ✇❡r❡ ❛❧s♦ ♣r♦✈✐❞❡❞ ❜② t❤❡ ▲❡✐❜♥✐③ ❙✉♣❡r❝♦♠♣✉t✐♥❣ ❈❡♥tr❡ t❤r♦✉❣❤ t❤❡ ♣r♦❥❡❝ts ♣r✻✷❥❛✳ ■✳ ●✉✐❧❤♦♥ ✐s ❣r❛t❡❢✉❧ t♦ t❤❡ ❤♦s♣✐t❛❧✐t② ♦❢ t❤❡ ❋r✐❡❞r✐❝❤✲❙❝❤✐❧❧❡r ❯♥✐✈❡rs✐t②✳

✷✶



Page 22 of 30

❘❡❢❡r❡♥❝❡s ✭✶✮ ●❡✐♠✱ ❆✳ ❑✳❀ ◆♦✈♦s❡❧♦✈✱ ❑✳ ❙✳ ❚❤❡ ❘✐s❡ ♦❢ ●r❛♣❤❡♥❡✳

◆❛t✉r❡ ▼❛t✳ ✷✵✵✼✱ ✻✱ ✶✽✸✳

✭✷✮ P❛r❦✱ ❙✳❀ ❘✉♦✛✱ ❘✳ ❙✳ ❈❤❡♠✐❝❛❧ ▼❡t❤♦❞s ❢♦r t❤❡ Pr♦❞✉❝t✐♦♥ ♦❢ ●r❛♣❤❡♥❡s✳

◆❛t✉r❡ ◆❛♥♦

✷✵✵✾✱ ✹✱ ✷✶✼✳ ✭✸✮ ❊✐❣❧❡r✱ ❙✳❀ ❍✐rs❝❤✱ ❆✳ ❈❤❡♠✐str② ✇✐t❤ ●r❛♣❤❡♥❡ ❛♥❞ ●r❛♣❤❡♥❡ ❖①✐❞❡✲❈❤❛❧❧❡♥❣❡s ❢♦r ❙②♥t❤❡t✐❝ ❈❤❡♠✐sts✳

❆♥❣❡✇❛♥❞t❡ ❈❤❡♠✐❡ ■♥t❡r♥❛t✐♦♥❛❧ ❊❞✐t✐♦♥ ✷✵✶✹✱ ✺✸✱ ✼✼✷✵✕✼✼✸✽✳

✭✹✮ ▲♦❤✱ ❑✳ P✳❀ ❇❛♦✱ ◗✳❀ ❊❞❛✱ ●✳❀ ❈❤❤♦✇❛❧❧❛✱ ▼✳ ●r❛♣❤❡♥❡ ❖①✐❞❡ ❛s ❛ ❈❤❡♠✐❝❛❧❧② ❚✉♥❛❜❧❡ P❧❛t❢♦r♠ ❢♦r ❖♣t✐❝❛❧ ❆♣♣❧✐❝❛t✐♦♥s✳

◆❛t✉r❡ ❈❤❡♠✐str② ✷✵✶✵✱ ✷✱ ✶✵✶✺✕✶✵✷✹✳

✭✺✮ ▼❛♦✱ ❙✳❀ P✉✱ ❍✳❀ ❈❤❡♥✱ ❏✳ ●r❛♣❤❡♥❡ ❖①✐❞❡ ❛♥❞ ✐ts ❘❡❞✉❝t✐♦♥✿ ▼♦❞❡❧✐♥❣ ❛♥❞ ❊①♣❡r✐✲ ♠❡♥t❛❧ Pr♦❣r❡ss✳

❘❙❈ ❆❞✈✳ ✷✵✶✷✱ ✷✱ ✷✻✹✸✕✷✻✻✷✳

✭✻✮ ❉r❡②❡r✱ ❉✳ ❘✳❀ P❛r❦✱ ❙✳❀ ❇✐❡❧❛✇s❦✐✱ ❈✳ ❲✳❀ ❘✉♦✛✱ ❘✳ ❙✳ ❚❤❡ ❈❤❡♠✐str② ♦❢ ●r❛♣❤❡♥❡ ❖①✐❞❡✳

❈❤❡♠✳ ❙♦❝✳ ❘❡✈✳ ✷✵✶✵✱ ✸✾✱ ✷✷✽✕✷✹✵✳

✭✼✮ P✉❧❝✐✱ ❖✳❀ ●♦r✐✱ P✳❀ ▼❛rs✐❧✐✱ ▼✳❀ ●❛r❜✉✐♦✱ ❱✳❀ ❉❡❧ ❙♦❧❡✱ ❘✳❀ ❇❡❝❤st❡❞t✱ ❋✳ ❙tr♦♥❣ ❊①❝✐✲ t♦♥s ✐♥ ◆♦✈❡❧ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❈r②st❛❧s✿ ❙✐❧✐❝❛♥❡ ❛♥❞ ●❡r♠❛♥❛♥❡✳

❊P▲ ✭❊✉r♦♣❤②s✐❝s

▲❡tt❡rs✮ ✷✵✶✷✱ ✾✽✱ ✸✼✵✵✹✳ ✭✽✮ ❘♦❜✐♥s♦♥✱ ❏✳ ❚✳❀ P❡r❦✐♥s✱ ❋✳ ❑✳❀ ❙♥♦✇✱ ❊✳ ❙✳❀ ❲❡✐✱ ❩✳❀ ❙❤❡❡❤❛♥✱ P✳ ❊✳ ❘❡❞✉❝❡❞ ●r❛♣❤❡♥❡ ❖①✐❞❡ ▼♦❧❡❝✉❧❛r ❙❡♥s♦rs✳

◆❛♥♦ ▲❡tt❡rs ✷✵✵✽✱ ✽✱ ✸✶✸✼✕✸✶✹✵✳

✭✾✮ ❉✉❛✱ ❱✳❀ ❙✉r✇❛❞❡✱ ❙✳❀ ❆♠♠✉✱ ❙✳❀ ❆❣♥✐❤♦tr❛✱ ❙✳❀ ❏❛✐♥✱ ❙✳❀ ❘♦❜❡rts✱ ❑✳❀ P❛r❦✱ ❙✳❀ ❘✉♦✛✱ ❘✳❀

▼❛♥♦❤❛r✱ ❙✳ ❆❧❧✲❖r❣❛♥✐❝ ❱❛♣♦r ❙❡♥s♦r ❯s✐♥❣ ■♥❦❥❡t✲Pr✐♥t❡❞ ❘❡❞✉❝❡❞

●r❛♣❤❡♥❡ ❖①✐❞❡✳

❆♥❣❡✇❛♥❞t❡ ❈❤❡♠✐❡ ✷✵✶✵✱ ✶✷✷✱ ✷✷✵✵✕✷✷✵✸✳

✭✶✵✮ ●ó♠❡③✲◆❛✈❛rr♦✱ ❈✳❀ ❲❡✐t③✱ ❘✳ ❚✳❀ ❇✐tt♥❡r✱ ❆✳ ▼✳❀ ❙❝♦❧❛r✐✱ ▼✳❀ ▼❡✇s✱ ❆✳❀ ❇✉r❣❤❛r❞✱ ▼✳❀ ❑❡r♥✱ ❑✳ ❊❧❡❝tr♦♥✐❝ ❚r❛♥s♣♦rt Pr♦♣❡rt✐❡s ♦❢ ■♥❞✐✈✐❞✉❛❧ ❈❤❡♠✐❝❛❧❧② ❘❡❞✉❝❡❞ ●r❛♣❤❡♥❡ ❖①✐❞❡ ❙❤❡❡ts✳

◆❛♥♦ ▲❡tt❡rs ✷✵✵✼✱ ✼✱ ✸✹✾✾✕✸✺✵✸✳ ✷✷


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✭✶✶✮ ❲❡✐✱ ❩✳❀ ❲❛♥❣✱ ❉✳❀ ❑✐♠✱ ❙✳❀ ❑✐♠✱ ❙✳✲❨✳❀ ❍✉✱ ❨✳❀ ❨❛❦❡s✱ ▼✳ ❑✳❀ ▲❛r❛❝✉❡♥t❡✱ ❆✳ ❘✳❀ ❉❛✐✱ ❩✳❀ ▼❛r❞❡r✱ ❙✳ ❘✳❀ ❇❡r❣❡r✱ ❈✳ ❡t ❛❧✳ ◆❛♥♦s❝❛❧❡ ❚✉♥❛❜❧❡ ❘❡❞✉❝t✐♦♥ ♦❢ ●r❛♣❤❡♥❡ ❖①✐❞❡ ❢♦r ●r❛♣❤❡♥❡ ❊❧❡❝tr♦♥✐❝s✳

✭✶✷✮ ❲✉✱ ❳✳❀

❙♣r✐♥❦❧❡✱ ▼✳❀

▲✐✱ ❳✳❀

❙❝✐❡♥❝❡ ✷✵✶✵✱ ✸✷✽✱ ✶✸✼✸✕✶✸✼✻✳ ▼✐♥❣✱ ❋✳❀

❇❡r❣❡r✱ ❈✳❀

❞❡ ❍❡❡r✱ ❲✳ ❆✳ ❊♣✐t❛①✐❛❧✲

●r❛♣❤❡♥❡✴●r❛♣❤❡♥❡✲❖①✐❞❡ ❏✉♥❝t✐♦♥✿ ❆♥ ❊ss❡♥t✐❛❧ ❙t❡♣ ❚♦✇❛r❞s ❊♣✐t❛①✐❛❧ ●r❛♣❤❡♥❡ ❊❧❡❝tr♦♥✐❝s✳

P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✷✵✵✽✱ ✶✵✶✱ ✵✷✻✽✵✶✳

✭✶✸✮ ❩❤❛♥❣✱ ❏✳❀ ❩❤❛♦✱ ❳✳ ❙✳ ❈♦♥❞✉❝t✐♥❣ P♦❧②♠❡rs ❉✐r❡❝t❧② ❈♦❛t❡❞ ♦♥ ❘❡❞✉❝❡❞ ●r❛♣❤❡♥❡ ❖①✐❞❡ ❙❤❡❡ts ❛s ❍✐❣❤✲P❡r❢♦r♠❛♥❝❡ ❙✉♣❡r❝❛♣❛❝✐t♦r ❊❧❡❝tr♦❞❡s✳

❚❤❡ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝❛❧

❈❤❡♠✐str② ❈ ✷✵✶✷✱ ✶✶✻✱ ✺✹✷✵✕✺✹✷✻✳ ✭✶✹✮ ❇r♦❞✐❡✱ ❇✳ ❈✳ ❖♥ t❤❡ ❆t♦♠✐❝ ❲❡✐❣❤t ♦❢ ●r❛♣❤✐t❡✳

P❤✐❧♦s♦♣❤✐❝❛❧ ❚r❛♥s❛❝t✐♦♥s ♦❢ t❤❡

❘♦②❛❧ ❙♦❝✐❡t② ♦❢ ▲♦♥❞♦♥ ✶✽✺✾✱ ✶✹✾✱ ✷✹✾✕✷✺✾✳ ✭✶✺✮ ❍❡✱ ❍✳❀ ❘✐❡❞❧✱ ❚✳❀ ▲❡r❢✱ ❆✳❀ ❑❧✐♥♦✇s❦✐✱ ❏✳ ❙♦❧✐❞✲❙t❛t❡ ◆▼❘ ❙t✉❞✐❡s ♦❢ t❤❡ ❙tr✉❝t✉r❡ ♦❢ ●r❛♣❤✐t❡ ❖①✐❞❡✳

❚❤❡ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝❛❧ ❈❤❡♠✐str② ✶✾✾✻✱ ✶✵✵✱ ✶✾✾✺✹✕✶✾✾✺✽✳

✭✶✻✮ ❍❡✱ ❍✳❀ ❑❧✐♥♦✇s❦✐✱ ❏✳❀ ❋♦rst❡r✱ ▼✳❀ ▲❡r❢✱ ❆✳ ❆ ◆❡✇ ❙tr✉❝t✉r❛❧ ▼♦❞❡❧ ❢♦r ●r❛♣❤✐t❡ ❖①✐❞❡✳

❈❤❡♠✐❝❛❧ P❤②s✐❝s ▲❡tt❡rs ✶✾✾✽✱ ✷✽✼✱ ✺✸ ✕ ✺✻✳ ✭✶✼✮ ❍♦❢♠❛♥♥✱ ❯✳❀ ❍♦❧st✱ ❘✳ Ü❜❡r ❞✐❡ ❙❛❡✉r❡♥❛t✉r ✉♥❞ ❞✐❡ ▼❡t❤②❧✐❡r✉♥❣ ✈♦♥ ●r❛♣❤✐t♦①②❞✳

❇❡r✐❝❤t❡ ❞❡r ❞❡✉ts❝❤❡♥ ❝❤❡♠✐s❝❤❡♥ ●❡s❡❧❧s❝❤❛❢t ✭❆ ❛♥❞ ❇ ❙❡r✐❡s✮ ✶✾✸✾✱ ✼✷✱ ✼✺✹✕✼✼✶✳ ✭✶✽✮ ❘✉❡ss✱ ●✳ Ü❜❡r ❞❛s ●r❛♣❤✐t♦①②❤②❞r♦①②❞ ✭●r❛♣❤✐t♦①②❞✮✳

▼♦♥❛ts❤❡❢t❡ ❢ür ❈❤❡♠✐❡ ✉♥❞

❱❡r✇❛♥❞t❡ ❚❡✐❧❡ ❆♥❞❡r❡r ❲✐ss❡♥s❝❤❛❢t❡♥ ✶✾✹✼✱ ✼✻✱ ✸✽✶✕✹✶✼✳ ✭✶✾✮ ❈❛✐✱ ❲✳❀ P✐♥❡r✱ ❘✳ ❉✳❀ ❙t❛❞❡r♠❛♥♥✱ ❋✳ ❏✳❀ P❛r❦✱ ❙✳❀ ❙❤❛✐❜❛t✱ ▼✳ ❆✳❀ ■s❤✐✐✱ ❨✳❀ ❨❛♥❣✱ ❉✳❀ ❱❡❧❛♠❛❦❛♥♥✐✱ ❆✳❀ ❆♥✱ ❙✳ ❏✳❀ ❙t♦❧❧❡r✱ ▼✳ ❡t ❛❧✳ ❙②♥t❤❡s✐s ❛♥❞ ❙♦❧✐❞✲❙t❛t❡ ◆▼❘ ❙tr✉❝t✉r❛❧ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ✶✸❈✲▲❛❜❡❧❡❞ ●r❛♣❤✐t❡ ❖①✐❞❡✳

✭✷✵✮ ◆♦✉r❜❛❦❤s❤✱ ❆✳❀

❈❛♥t♦r♦✱ ▼✳❀

❱♦s❝❤✱ ❚✳❀

✷✸

❙❝✐❡♥❝❡ ✷✵✵✽✱ ✸✷✶✱ ✶✽✶✺✕✶✽✶✼✳

P♦✉rt♦✐s✱ ●✳❀


❈❧❡♠❡♥t❡✱ ❋✳❀

✈❛♥ ❞❡r


Page 24 of 30

❱❡❡♥✱ ▼✳ ❍✳❀ ❍♦❢❦❡♥s✱ ❏✳❀ ❍❡②♥s✱ ▼✳ ▼✳❀ ●❡♥❞t✱ ❙✳ ❉✳❀ ❙❡❧s✱ ❇✳ ❋✳ ❇❛♥❞●❛♣ ❖♣❡♥✲ ✐♥❣ ✐♥ ❖①②❣❡♥ P❧❛s♠❛✲tr❡❛t❡❞ ●r❛♣❤❡♥❡✳

✭✷✶✮ ❑✐♠✱ ▼✳ ❈✳❀ ●r❛♣❤❡♥❡✿

❍✇❛♥❣✱ ●✳ ❙✳❀

◆❛♥♦t❡❝❤♥♦❧♦❣② ✷✵✶✵✱ ✷✶✱ ✹✸✺✷✵✸✳

❘✉♦✛✱ ❘✳ ❙✳ ❊♣♦①✐❞❡ ❘❡❞✉❝t✐♦♥ ✇✐t❤ ❍②❞r❛③✐♥❡ ♦♥

❆ ❋✐rst Pr✐♥❝✐♣❧❡s ❙t✉❞②✳

❚❤❡ ❏♦✉r♥❛❧ ♦❢ ❈❤❡♠✐❝❛❧ P❤②s✐❝s ✷✵✵✾✱ ✶✸✶✱

✵✻✹✼✵✹✳

✭✷✷✮ ❈✐✱ ▲✳ ❆t♦♠✐❝ ▲❛②❡rs ♦❢ ❍②❜r✐❞✐③❡❞ ❇♦r♦♥ ◆✐tr✐❞❡ ❛♥❞ ●r❛♣❤❡♥❡ ❉♦♠❛✐♥s✳

◆❛t✉r❡

▼❛t❡r✐❛❧ ✷✵✶✵✱ ✾✱ ✹✸✵✕✹✸✺✳ ✭✷✸✮ ❯❞❞✐♥✱ ▼✳ ❘✳❀ ▼❛❥❡t②✱ ❙✳❀ ▲✐✱ ❏✳❀ ▲✐♥✱ ❏✳ ❨✳❀ ❏✐❛♥❣✱ ❍✳ ❳✳ ▲❛②❡r✲❙tr✉❝t✉r❡❞ ❍❡①❛❣♦♥❛❧ ✭❇◆✮❈ ❙❡♠✐❝♦♥❞✉❝t♦r ❆❧❧♦②s ✇✐t❤ ❚✉♥❛❜❧❡ ❖♣t✐❝❛❧ ❛♥❞ ❊❧❡❝tr✐❝❛❧ Pr♦♣❡rt✐❡s✳

❏♦✉r♥❛❧

♦❢ ❆♣♣❧✐❡❞ P❤②s✐❝s ✷✵✶✹✱ ✶✶✺✱ ✵✾✸✺✵✾✳ ✭✷✹✮ ❯❞❞✐♥✱ ▼✳ ❘✳❀ ▲✐✱ ❏✳❀ ▲✐♥✱ ❏✳ ❨✳❀ ❏✐❛♥❣✱ ❍✳ ❳✳ ❈❛r❜♦♥✲r✐❝❤ ❍❡①❛❣♦♥❛❧ ✭❇◆✮❈ ❆❧❧♦②s✳

❏♦✉r♥❛❧ ♦❢ ❆♣♣❧✐❡❞ P❤②s✐❝s ✷✵✶✺✱ ✶✶✼✱ ✷✶✺✼✵✸✳ ✭✷✺✮ ▲✐✱ ❍✳❀ ❉✉❛♥✱ ❳✳❀ ❲✉✱ ❳✳❀ ❩❤✉❛♥❣✱ ❳✳❀ ❩❤♦✉✱ ❍✳❀ ❩❤❛♥❣✱ ◗✳❀ ❩❤✉✱ ❳✳❀ ❍✉✱ ❲✳❀ ❘❡♥✱ P✳❀ ●✉♦✱ P✳ ❡t ❛❧✳ ●r♦✇t❤ ♦❢ ❆❧❧♦② ▼♦❙✷①❙❡✷✭✶✲①✮ ◆❛♥♦s❤❡❡ts ✇✐t❤ ❋✉❧❧② ❚✉♥❛❜❧❡ ❈❤❡♠✐❝❛❧ ❈♦♠♣♦s✐t✐♦♥s ❛♥❞ ❖♣t✐❝❛❧ Pr♦♣❡rt✐❡s✳

❏♦✉r♥❛❧ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ❈❤❡♠✐❝❛❧ ❙♦❝✐❡t② ✷✵✶✹✱

✶✸✻✱ ✸✼✺✻✕✸✼✺✾✳ ✭✷✻✮ ❈❤❡♥✱ ❨✳❀ ❳✐✱ ❏✳❀ ❉✉♠❝❡♥❝♦✱ ❉✳ ❖✳❀ ▲✐✉✱ ❩✳❀ ❙✉❡♥❛❣❛✱ ❑✳❀ ❲❛♥❣✱ ❉✳❀ ❙❤✉❛✐✱ ❩✳❀ ❍✉❛♥❣✱ ❨✳✲❙✳❀ ❳✐❡✱ ▲✳ ❚✉♥❛❜❧❡ ❇❛♥❞ ●❛♣ P❤♦t♦❧✉♠✐♥❡s❝❡♥❝❡ ❢r♦♠ ❆t♦♠✐❝❛❧❧② ❚❤✐♥ ❚r❛♥s✐t✐♦♥✲▼❡t❛❧ ❉✐❝❤❛❧❝♦❣❡♥✐❞❡ ❆❧❧♦②s✳

❆❈❙ ◆❛♥♦ ✷✵✶✸✱ ✼✱ ✹✻✶✵✕✹✻✶✻✳

✭✷✼✮ ❩❤❡♥❣✱ ❩✳❀ ❨❛♦✱ ❏✳❀ ❨❛♥❣✱ ●✳ ❈❡♥t✐♠❡t❡r✲❙❝❛❧❡ ❉❡♣♦s✐t✐♦♥ ♦❢ ▼♦✵✳✺❲✵✳✺❙❡✷ ❆❧❧♦② ❋✐❧♠ ❢♦r ❍✐❣❤✲P❡r❢♦r♠❛♥❝❡ P❤♦t♦❞❡t❡❝t♦rs ♦♥ ❱❡rs❛t✐❧❡ ❙✉❜str❛t❡s✳

❆❈❙ ❆♣♣❧✐❡❞ ▼❛t❡r✐❛❧s

✫ ■♥t❡r❢❛❝❡s ✷✵✶✼✱ ✾✱ ✶✹✾✷✵✕✶✹✾✷✽✳ ✭✷✽✮ ●✉✐❧❤♦♥✱ ■✳❀ ❚❡❧❡s✱ ▲✳ ❑✳❀ ▼❛rq✉❡s✱ ▼✳❀ P❡❧❛✱ ❘✳ ❘✳❀ ❇❡❝❤st❡❞t✱ ❋✳ ■♥✢✉❡♥❝❡ ♦❢ ❙tr✉❝t✉r❡

✷✹


Page 25 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


❛♥❞ ❚❤❡r♠♦❞②♥❛♠✐❝ ❙t❛❜✐❧✐t② ♦♥ ❊❧❡❝tr♦♥✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❙✐❈✱ ❙✐●❡✱ ❛♥❞ ●❡❈ ❆❧❧♦②s✳

P❤②s✳ ❘❡✈✳ ❇ ✷✵✶✺✱ ✾✷✱ ✵✼✺✹✸✺✳

✭✷✾✮ ●✉✐❧❤♦♥✱ ■✳❀ ▼❛rq✉❡s✱ ▼✳❀ ❚❡❧❡s✱ ▲✳ ❑✳❀ ❇❡❝❤st❡❞t✱ ❋✳ ❖♣t✐❝❛❧ ❆❜s♦r❜❛♥❝❡ ❛♥❞ ❇❛♥❞✲●❛♣ ❊♥❣✐♥❡❡r✐♥❣ ♦❢

(BN)1−x (C2 )x

s✐t✐♦♥ ❋❧✉❝t✉❛t✐♦♥ ❊✛❡❝ts✳

❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❆❧❧♦②s✿ P❤❛s❡ ❙❡♣❛r❛t✐♦♥ ❛♥❞ ❈♦♠♣♦✲

P❤②s✳ ❘❡✈✳ ❇ ✷✵✶✼✱ ✾✺✱ ✵✸✺✹✵✼✳

✭✸✵✮ P❛❞✐❧❤❛✱ ❏✳ ❊✳❀ ❙❡✐①❛s✱ ▲✳❀ P♦♥t❡s✱ ❘✳ ❇✳❀ ❞❛ ❙✐❧✈❛✱ ❆✳ ❏✳ ❘✳❀ ❋❛③③✐♦✱ ❆✳ ◗✉❛♥t✉♠ ❙♣✐♥ ❍❛❧❧ ❊✛❡❝t ✐♥ ❛ ❉✐s♦r❞❡r❡❞ ❍❡①❛❣♦♥❛❧ ❙✐①●❡✶✲① ❆❧❧♦②✳

P❤②s✳ ❘❡✈✳ ❇ ✷✵✶✸✱ ✽✽✱ ✷✵✶✶✵✻✳

✭✸✶✮ ❆❦✐②❛♠❛✱ ❚✳❀ ❨♦s❤✐♠✉r❛✱ ●✳❀ ◆❛❦❛♠✉r❛✱ ❑✳❀ ■t♦✱ ❚✳ ❚❤❡♦r❡t✐❝❛❧ ■♥✈❡st✐❣❛t✐♦♥s ♦♥ t❤❡ ❙t❛❜✐❧✐t② ❛♥❞ ❊❧❡❝tr♦♥✐❝ ❙tr✉❝t✉r❡s ♦❢ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ●r♦✉♣✲■❱ ❚❡r♥❛r② ❆❧❧♦② ▼♦♥♦✲ ❧❛②❡rs✳

❏♦✉r♥❛❧ ♦❢ ❱❛❝✉✉♠ ❙❝✐❡♥❝❡ ✫ ❚❡❝❤♥♦❧♦❣② ❇✱ ◆❛♥♦t❡❝❤♥♦❧♦❣② ❛♥❞ ▼✐❝r♦❡❧❡❝tr♦♥✲

✐❝s✿ ▼❛t❡r✐❛❧s✱ Pr♦❝❡ss✐♥❣✱ ▼❡❛s✉r❡♠❡♥t✱ ❛♥❞ P❤❡♥♦♠❡♥❛ ✷✵✶✼✱ ✸✺✱ ✵✹❋✶✵✸✳ ✭✸✷✮ ●❛♦✱ ❍✳❀ ❲❛♥❣✱ ▲✳❀ ❩❤❛♦✱ ❏✳❀ ❉✐♥❣✱ ❋✳❀ ▲✉✱ ❏✳ ❇❛♥❞ ●❛♣ ❚✉♥✐♥❣ ♦❢ ❍②❞r♦❣❡♥❛t❡❞ ●r❛♣❤❡♥❡✿ ❍ ❈♦✈❡r❛❣❡ ❛♥❞ ❈♦♥✜❣✉r❛t✐♦♥ ❉❡♣❡♥❞❡♥❝❡✳

❚❤❡ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝❛❧ ❈❤❡♠✲

✐str② ❈ ✷✵✶✶✱ ✶✶✺✱ ✸✷✸✻✕✸✷✹✷✳ ✭✸✸✮ ❍❛❜❡r❡r✱ ❉✳❀ ❱②❛❧✐❦❤✱ ❉✳ ❱✳❀ ❚❛✐♦❧✐✱ ❙✳❀ ❉♦r❛✱ ❇✳❀ ❋❛r❥❛♠✱ ▼✳❀ ❋✐♥❦✱ ❏✳❀ ▼❛r❝❤❡♥❦♦✱ ❉✳❀ P✐❝❤❧❡r✱ ❚✳❀ ❩✐❡❣❧❡r✱ ❑✳❀ ❙✐♠♦♥✉❝❝✐✱ ❙✳ ❡t ❛❧✳ ❚✉♥❛❜❧❡ ❇❛♥❞ ●❛♣ ✐♥ ❍②❞r♦❣❡♥❛t❡❞ ◗✉❛s✐✲ ❋r❡❡✲❙t❛♥❞✐♥❣ ●r❛♣❤❡♥❡✳

◆❛♥♦ ▲❡tt❡rs ✷✵✶✵✱ ✶✵✱ ✸✸✻✵✕✸✸✻✻✳

✭✸✹✮ ▼❡❞✉r✐✱ P✳❀ ❈❤❡♥✱ ❍✳❀ ❳✐❛♦✱ ❏✳❀ ▼❛rt✐♥❡③✱ ❏✳ ❏✳❀ ❈❛r❧s♦♥✱ ❚✳❀ ❩❤❛♥❣✱ ❏✳✲●✳❀ ❉❡♥❣✱ ❩✳ ❉✳ ❚✉♥❛❜❧❡ ❊❧❡❝tr♦❝❤❡♠✐❝❛❧ Pr♦♣❡rt✐❡s ♦❢ ❋❧✉♦r✐♥❛t❡❞ ●r❛♣❤❡♥❡✳

❏✳ ▼❛t❡r✳ ❈❤❡♠✳ ❆ ✷✵✶✸✱

✶✱ ✼✽✻✻✕✼✽✻✾✳ ✭✸✺✮ ❑❛r❧✐❝❦②✱ ❋✳❀ ❑✉♠❛r❛ ❘❛♠❛♥❛t❤❛ ❉❛tt❛✱ ❑✳❀ ❖t②❡♣❦❛✱ ▼✳❀ ❩❜♦r✐❧✱ ❘✳ ❍❛❧♦❣❡♥❛t❡❞ ●r❛♣❤❡♥❡s✿

❘❛♣✐❞❧② ●r♦✇✐♥❣ ❋❛♠✐❧② ♦❢ ●r❛♣❤❡♥❡ ❉❡r✐✈❛t✐✈❡s✳

✻✹✸✹✕✻✹✻✹✳

✷✺


❆❈❙ ◆❛♥♦ ✷✵✶✸✱ ✼✱


Page 26 of 30

✭✸✻✮ ❏✐❛♥❣✱ ❳✳❀ ◆✐s❛r✱ ❏✳❀ P❛t❤❛❦✱ ❇✳❀ ❩❤❛♦✱ ❏✳❀ ❆❤✉❥❛✱ ❘✳ ●r❛♣❤❡♥❡ ❖①✐❞❡ ❛s ❛ ❈❤❡♠✐❝❛❧❧② ❚✉♥❛❜❧❡ ✷✲❉ ▼❛t❡r✐❛❧ ❢♦r ❱✐s✐❜❧❡✲❧✐❣❤t P❤♦t♦❝❛t❛❧②st ❆♣♣❧✐❝❛t✐♦♥s✳

❏♦✉r♥❛❧ ♦❢ ❈❛t❛❧②s✐s

✷✵✶✸✱ ✷✾✾✱ ✷✵✹ ✕ ✷✵✾✳ ✭✸✼✮ ❨❛♥✱ ❏✳✲❆✳❀ ❳✐❛♥✱ ▲✳❀ ❈❤♦✉✱ ▼✳ ❨✳ ❙tr✉❝t✉r❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝ Pr♦♣❡rt✐❡s ♦❢ ❖①✐❞✐③❡❞ ●r❛♣❤❡♥❡✳

P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✷✵✵✾✱ ✶✵✸✱ ✵✽✻✽✵✷✳

✭✸✽✮ ❨❛♥✱ ❏✳✲❆✳❀ ❈❤♦✉✱ ▼✳ ❨✳ ❖①✐❞❛t✐♦♥ ❋✉♥❝t✐♦♥❛❧ ●r♦✉♣s ♦♥ ●r❛♣❤❡♥❡✿ ❙tr✉❝t✉r❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝ Pr♦♣❡rt✐❡s✳

P❤②s✳ ❘❡✈✳ ❇ ✷✵✶✵✱ ✽✷✱ ✶✷✺✹✵✸✳

✭✸✾✮ ●✉✐❧❤♦♥✱ ■✳❀ ❇❡❝❤st❡❞t✱ ❋✳❀ ❇♦tt✐✱ ❙✳❀ ▼❛rq✉❡s✱ ▼✳❀ ❚❡❧❡s✱ ▲✳ ❑✳ ❚❤❡r♠♦❞②♥❛♠✐❝✱ ❊❧❡❝✲ tr♦♥✐❝✱ ❛♥❞ ❖♣t✐❝❛❧ Pr♦♣❡rt✐❡s ♦❢ ●r❛♣❤❡♥❡ ❖①✐❞❡✿ ❆ st❛t✐st✐❝❛❧ ❛❜ ■♥✐t✐♦ ❆♣♣r♦❛❝❤✳

P❤②s✳ ❘❡✈✳ ❇ ✷✵✶✼✱ ✾✺✱ ✷✹✺✹✷✼✳ ✭✹✵✮ ❙❤❡r✱ ❆✳❀ ✈❛♥ ❙❝❤✐❧❢❣❛❛r❞❡✱ ▼✳❀ ❈❤❡♥✱ ❆✳✲❇✳❀ ❈❤❡♥✱ ❲✳ ◗✉❛s✐❝❤❡♠✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ ❜✐♥❛r② ❛❧❧♦②s✳

P❤②s✳ ❘❡✈✳ ❇ ✶✾✽✼✱ ✸✻✱ ✹✷✼✾✕✹✷✾✺✳

✭✹✶✮ ❚❡❧❡s✱ ▲✳ ❑✳❀ ❋✉rt❤♠ü❧❧❡r✱ ❏✳❀ ❙❝♦❧❢❛r♦✱ ▲✳ ▼✳ ❘✳❀ ▲❡✐t❡✱ ❏✳ ❘✳❀ ❇❡❝❤st❡❞t✱ ❋✳ ❋✐rst✲ ♣r✐♥❝✐♣❧❡s ❈❛❧❝✉❧❛t✐♦♥s ♦❢ t❤❡ ❚❤❡r♠♦❞②♥❛♠✐❝ ❛♥❞ ❙tr✉❝t✉r❛❧ Pr♦♣❡rt✐❡s ♦❢ ❙tr❛✐♥❡❞

Inx Ga1−x N

❛♥❞

Alx Ga1−x N

❆❧❧♦②s✳

P❤②s✳ ❘❡✈✳ ❇ ✷✵✵✵✱ ✻✷✱ ✷✹✼✺✕✷✹✽✺✳

✭✹✷✮ ❋r❡✐t❛s✱ ❋✳ ▲✳❀ ❋✉rt❤♠ü❧❧❡r✱ ❏✳❀ ❇❡❝❤st❡❞t✱ ❋✳❀ ▼❛rq✉❡s✱ ▼✳❀ ❚❡❧❡s✱ ▲✳ ❑✳ ■♥✢✉❡♥❝❡ ♦❢ t❤❡ ❈♦♠♣♦s✐t✐♦♥ ❋❧✉❝t✉❛t✐♦♥s ❛♥❞ ❉❡❝♦♠♣♦s✐t✐♦♥ ♦♥ t❤❡ ❚✉♥❛❜❧❡ ❉✐r❡❝t ●❛♣ ❛♥❞ ❖s❝✐❧❧❛t♦r ❙tr❡♥❣t❤ ♦❢ ●❡✶✲①❙♥① ❆❧❧♦②s✳

❆♣♣❧✐❡❞ P❤②s✐❝s ▲❡tt❡rs ✷✵✶✻✱ ✶✵✽ ✳

✭✹✸✮ ▼❛rq✉❡s✱ ▼✳❀ ❚❡❧❡s✱ ▲✳ ❑✳❀ ❋❡rr❡✐r❛✱ ▲✳ ●✳❀ ❙❝♦❧❢❛r♦✱ ▲✳ ▼✳ ❘✳❀ ❋✉rt❤♠ü❧❧❡r✱ ❏✳❀ ❇❡❝❤st✲ ❡❞t✱ ❋✳ ❙t❛t✐st✐❝❛❧ ▼♦❞❡❧ ❆♣♣❧✐❡❞ t♦ ❛♥❞ ❊♥❡r❣② ●❛♣s ♦❢

Ax By C1−x−y D

Alx Gay In1−x−y X ✭X = As✱

◗✉❛t❡r♥❛r② ❆❧❧♦②s✿ ❇♦♥❞ ▲❡♥❣t❤s

P✱ ♦r ◆✮ ❙②st❡♠s✳

P❤②s✳ ❘❡✈✳ ❇ ✷✵✵✻✱

✼✸✱ ✷✸✺✷✵✺✳ ✭✹✹✮ ❑r❡ss❡✱ ●✳❀ ❋✉rt❤♠ü❧❧❡r✱ ❏✳ ❊✣❝✐❡♥❝② ♦❢ ❛❜✲■♥✐t✐♦ ❚♦t❛❧ ❊♥❡r❣② ❈❛❧❝✉❧❛t✐♦♥s ❢♦r ▼❡t❛❧s ❛♥❞ ❙❡♠✐❝♦♥❞✉❝t♦rs ❯s✐♥❣ ❛ P❧❛♥❡✲✇❛✈❡ ❇❛s✐s ❙❡t✳

✷✻

❈♦♠♣✉t✳ ▼❛t❡r✳ ❙❝✐✳ ✶✾✾✻✱ ✻✱ ✶✺✳


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✭✹✺✮ ❑r❡ss❡✱ ●✳❀ ❍❛❢♥❡r✱ ❏✳ ❆❜ ■♥✐t✐♦ ▼♦❧❡❝✉❧❛r ❉②♥❛♠✐❝s ❢♦r ▲✐q✉✐❞ ▼❡t❛❧s✳

P❤②s✳ ❘❡✈✳ ❇

✶✾✾✸✱ ✹✼✱ ❘✺✺✽✳ ✭✹✻✮ P❡r❞❡✇✱ ❏✳ P✳❀ ❇✉r❦❡✱ ❑✳❀ ❊r♥③❡r❤♦❢✱ ▼✳ ●❡♥❡r❛❧✐③❡❞ ●r❛❞✐❡♥t ❆♣♣r♦①✐♠❛t✐♦♥ ▼❛❞❡ ❙✐♠♣❧❡✳

P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✶✾✾✻✱ ✼✼✱ ✸✽✻✺✕✸✽✻✽✳

✭✹✼✮ P❡r❞❡✇✱ ❏✳ P✳❀ ❇✉r❦❡✱ ❑✳❀ ❊r♥③❡r❤♦❢✱ ▼✳ ●❡♥❡r❛❧✐③❡❞ ●r❛❞✐❡♥t ❆♣♣r♦①✐♠❛t✐♦♥ ▼❛❞❡ ❙✐♠♣❧❡ ❬P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✼✼✱ ✸✽✻✺ ✭✶✾✾✻✮❪✳

P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✶✾✾✼✱ ✼✽✱ ✶✸✾✻✕✶✸✾✻✳

✭✹✽✮ ❇❧ö❝❤❧✱ P✳ ❊✳ Pr♦❥❡❝t♦r ❆✉❣♠❡♥t❡❞✲✇❛✈❡ ▼❡t❤♦❞✳

P❤②s✳ ❘❡✈✳ ❇ ✶✾✾✹✱ ✺✵✱ ✶✼✾✺✸✕✶✼✾✼✾✳

✭✹✾✮ ❑r❡ss❡✱ ●✳❀ ❏♦✉❜❡rt✱ ❉✳ ❋r♦♠ ✉❧tr❛s♦❢t Ps❡✉❞♦♣♦t❡♥t✐❛❧s t♦ t❤❡ Pr♦❥❡❝t♦r ❆✉❣♠❡♥t❡❞✲ ✇❛✈❡ ▼❡t❤♦❞✳

P❤②s✳ ❘❡✈✳ ❇ ✶✾✾✾✱ ✺✾✱ ✶✼✺✽✕✶✼✼✺✳

✭✺✵✮ ▼♦♥❦❤♦rst✱ ❍✳ ❏✳❀ P❛❝❦✱ ❏✳ ❉✳ ❙♣❡❝✐❛❧ P♦✐♥ts ❢♦r ❇r✐❧❧♦✉✐♥✲③♦♥❡ ■♥t❡❣r❛t✐♦♥s✳

P❤②s✳ ❘❡✈✳

❇ ✶✾✼✻✱ ✶✸✱ ✺✶✽✽✕✺✶✾✷✳ ✭✺✶✮ ❋❡rr❡✐r❛✱ ▲✳ ●✳❀ ▼❛rq✉❡s✱ ▼✳❀ ❚❡❧❡s✱ ▲✳ ❑✳ ❆♣♣r♦①✐♠❛t✐♦♥ t♦ ❉❡♥s✐t② ❋✉♥❝t✐♦♥❛❧ ❚❤❡♦r② ❢♦r t❤❡ ❈❛❧❝✉❧❛t✐♦♥ ♦❢ ❇❛♥❞ ●❛♣s ♦❢ ❙❡♠✐❝♦♥❞✉❝t♦rs✳

P❤②s✳ ❘❡✈✳ ❇ ✷✵✵✽✱ ✼✽✱ ✶✷✺✶✶✻✳

✭✺✷✮ P❡❧❛✱ ❘✳ ❘✳❀ ▼❛rq✉❡s✱ ▼✳❀ ❚❡❧❡s✱ ▲✳ ❑✳ ❈♦♠♣❛r✐♥❣ ▲❉❆✲✶✴✷✱ ❍❙❊✵✸✱ ❍❙❊✵✻ ❛♥❞ ●✵❲✵ ❆♣♣r♦❛❝❤❡s ❢♦r ❇❛♥❞ ●❛♣ ❈❛❧❝✉❧❛t✐♦♥s ♦❢ ❆❧❧♦②s✳

❏♦✉r♥❛❧ ♦❢ P❤②s✐❝s✿ ❈♦♥❞❡♥s❡❞ ▼❛tt❡r

✷✵✶✺✱ ✷✼✱ ✺✵✺✺✵✷✳ ✭✺✸✮ ❆❞♦❧♣❤✱ ❇✳❀ ●❛✈r✐❧❡♥❦♦✱ ❱✳ ■✳❀ ❚❡♥❡❧s❡♥✱ ❑✳❀ ❇❡❝❤st❡❞t✱ ❋✳❀ ❉❡❧ ❙♦❧❡✱ ❘✳ ◆♦♥❧♦❝❛❧✐t② ❛♥❞ ▼❛♥②✲❜♦❞② ❊✛❡❝ts ✐♥ t❤❡ ❖♣t✐❝❛❧ Pr♦♣❡rt✐❡s ♦❢ ❙❡♠✐❝♦♥❞✉❝t♦rs✳

P❤②s✳ ❘❡✈✳ ❇ ✶✾✾✻✱

✺✸✱ ✾✼✾✼✕✾✽✵✽✳ ✭✺✹✮ ●❛❥❞♦➨✱ ▼✳❀ ❍✉♠♠❡r✱ ❑✳❀ ❑r❡ss❡✱ ●✳❀ ❋✉rt❤♠ü❧❧❡r✱ ❏✳❀ ❇❡❝❤st❡❞t✱ ❋✳ ▲✐♥❡❛r ❖♣t✐❝❛❧ Pr♦♣❡rt✐❡s ✐♥ t❤❡ Pr♦❥❡❝t♦r✲❆✉❣♠❡♥t❡❞ ❲❛✈❡ ▼❡t❤♦❞♦❧♦❣②✳ ✵✹✺✶✶✷✳

✷✼


P❤②s✳ ❘❡✈✳ ❇ ✷✵✵✻✱ ✼✸✱


Page 28 of 30

✭✺✺✮ ▼❛tt❤❡s✱ ▲✳❀ P✉❧❝✐✱ ❖✳❀ ❇❡❝❤st❡❞t✱ ❋✳ ▼❛ss✐✈❡ ❉✐r❛❝ ◗✉❛s✐♣❛rt✐❝❧❡s ✐♥ t❤❡ ❖♣t✐❝❛❧ ❆❜✲ s♦r❜❛♥❝❡ ♦❢ ●r❛♣❤❡♥❡✱ ❙✐❧✐❝❡♥❡✱ ●❡r♠❛♥❡♥❡✱ ❛♥❞ ❚✐♥❡♥❡✳

❏♦✉r♥❛❧ ♦❢ P❤②s✐❝s✿ ❈♦♥❞❡♥s❡❞

▼❛tt❡r ✷✵✶✸✱ ✷✺✱ ✸✾✺✸✵✺✳ ✭✺✻✮ ❇❡❝❤st❡❞t✱ ❋✳

▼❛♥②✲❜♦❞② ❆♣♣r♦❛❝❤ t♦ ❊❧❡❝tr♦♥✐❝ ❊①❝✐t❛t✐♦♥s ❀

❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥

❍❡✐❞❡❧❜❡r❣✱ ✷✵✶✺✳

✭✺✼✮ ❨❛♥❣✱ ▲✳❀ ❉❡s❧✐♣♣❡✱ ❏✳❀ P❛r❦✱ ❈✳✲❍✳❀ ❈♦❤❡♥✱ ▼✳ ▲✳❀ ▲♦✉✐❡✱ ❙✳ ●✳ ❊①❝✐t♦♥✐❝ ❊✛❡❝ts ♦♥ t❤❡ ❖♣t✐❝❛❧ ❘❡s♣♦♥s❡ ♦❢ ●r❛♣❤❡♥❡ ❛♥❞ ❇✐❧❛②❡r ●r❛♣❤❡♥❡✳

P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✷✵✵✾✱ ✶✵✸✱

✶✽✻✽✵✷✳

✭✺✽✮ ❲✐rt③✱ ▲✳❀ ▼❛r✐♥✐✱ ❆✳❀ ❘✉❜✐♦✱ ❆✳ ❊①❝✐t♦♥s ✐♥ ❇♦r♦♥ ◆✐tr✐❞❡ ◆❛♥♦t✉❜❡s✿ ❉✐♠❡♥s✐♦♥❛❧✐t② ❊✛❡❝ts✳

P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✷✵✵✻✱ ✾✻✱ ✶✷✻✶✵✹✳

✭✺✾✮ ▼❛r❝❛♥♦✱ ❉✳ ❈✳❀ ❑♦s②♥❦✐♥✱ ❉✳ ❱✳❀ ❇❡r❧✐♥✱ ❏✳ ▼✳❀ ❙✐♥✐ts❦✐✐✱ ❆✳❀ ❙✉♥✱ ❩✳❀ ❙❧❡s❛r❡✈✱ ❆✳❀ ❆❧❡♠❛♥②✱ ▲✳ ❇✳❀ ▲✉✱ ❲✳❀ ❚♦✉r✱ ❏✳ ▼✳ ■♠♣r♦✈❡❞ ❙②♥t❤❡s✐s ♦❢ ●r❛♣❤❡♥❡ ❖①✐❞❡✳

❆❈❙

◆❛♥♦ ✷✵✶✵✱ ✹✱ ✹✽✵✻✕✹✽✶✹✳ ✭✻✵✮ ❏✉♥❣✱ ■✳❀ ❋✐❡❧❞✱ ❉✳ ❆✳❀ ❈❧❛r❦✱ ◆✳ ❏✳❀ ❩❤✉✱ ❨✳❀ ❨❛♥❣✱ ❉✳❀ P✐♥❡r✱ ❘✳ ❉✳❀ ❙t❛♥❦♦✈✐❝❤✱ ❙✳❀ ❉✐❦✐♥✱ ❉✳ ❆✳❀ ●❡✐s❧❡r✱ ❍✳❀ ❱❡♥tr✐❝❡✱ ❈✳ ❆✳ ❡t ❛❧✳ ❘❡❞✉❝t✐♦♥ ❑✐♥❡t✐❝s ♦❢ ●r❛♣❤❡♥❡ ❖①✲ ✐❞❡ ❉❡t❡r♠✐♥❡❞ ❜② ❊❧❡❝tr✐❝❛❧ ❚r❛♥s♣♦rt ▼❡❛s✉r❡♠❡♥ts ❛♥❞ ❚❡♠♣❡r❛t✉r❡ Pr♦❣r❛♠♠❡❞ ❉❡s♦r♣t✐♦♥✳

❚❤❡ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝❛❧ ❈❤❡♠✐str② ❈ ✷✵✵✾✱ ✶✶✸✱ ✶✽✹✽✵✕✶✽✹✽✻✳

✭✻✶✮ ❊❞❛✱ ●✳❀ ❈❤❤♦✇❛❧❧❛✱ ▼✳ ❈❤❡♠✐❝❛❧❧② ❉❡r✐✈❡❞ ●r❛♣❤❡♥❡ ❖①✐❞❡✿ ❚❤✐♥✲❋✐❧♠ ❊❧❡❝tr♦♥✐❝s ❛♥❞ ❖♣t♦❡❧❡❝tr♦♥✐❝s✳

❚♦✇❛r❞s ▲❛r❣❡✲❆r❡❛

❆❞✈❛♥❝❡❞ ▼❛t❡r✐❛❧s ✷✵✶✵✱ ✷✷✱ ✷✸✾✷✕✷✹✶✺✳

✭✻✷✮ ❩❤♦✉✱ ❙✳❀ ❇♦♥❣✐♦r♥♦✱ ❆✳ ❖r✐❣✐♥ ♦❢ t❤❡ ❈❤❡♠✐❝❛❧ ❛♥❞ ❑✐♥❡t✐❝ ❙t❛❜✐❧✐t② ♦❢ ●r❛♣❤❡♥❡ ❖①✐❞❡✳

❙❝✐❡♥t✐✜❝ ❘❡♣♦rts ✷✵✶✸✱ ✸✱ ✷✹✽✹✳

✭✻✸✮ ▲❡r❢✱ ❆✳❀ ❍❡✱ ❍✳❀ ❋♦rst❡r✱ ▼✳❀ ❑❧✐♥♦✇s❦✐✱ ❏✳ ❙tr✉❝t✉r❡ ♦❢ ●r❛♣❤✐t❡ ❖①✐❞❡ ❘❡✈✐s✐t❡❞✳

❏♦✉r♥❛❧ ♦❢ P❤②s✐❝❛❧ ❈❤❡♠✐str② ❇ ✶✾✾✽✱ ✶✵✷✱ ✹✹✼✼✕✹✹✽✷✳

✷✽


❚❤❡

Page 29 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


✭✻✹✮ ◆❡❧❞❡r✱ ❏✳ ❆✳❀ ▼❡❛❞✱ ❘✳ ❆ ❙✐♠♣❧❡① ▼❡t❤♦❞ ❢♦r ❋✉♥❝t✐♦♥ ▼✐♥✐♠✐③❛t✐♦♥✳

❏♦✉r♥❛❧ ✶✾✻✺✱ ✼✱ ✸✵✽✕✸✶✸✳

✷✾


❚❤❡ ❈♦♠♣✉t❡r


●r❛♣❤✐❝❛❧ ❚❖❈ ❊♥tr② T=473 K

CO1/2

1.5 1.0 0.5 C

ΔEg (eV)

COH

✸✵


Page 30 of 30

Chemically Tunable Properties of Graphene Covered Simultaneously

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