Nghiên cứu cấu trúc vùng năng lượng của vật liệu hai chiều snse2 đơn lớp bằng lí thuyết phiếm hàm mật độ

✣❸■ ❍➴❈ ❍❯➌
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✖✖✖✖✖✖

◆●❯❨➍◆ ◗❯❆◆● ❈×❮◆●
◆●❍■➊◆ ❈Ù❯ ❈❻❯ ❚❘Ó❈ ❱Ò◆● ◆❿◆●
▲×Ñ◆● ❈Õ❆ ❱❾❚ ▲■➏❯ ❍❆■ ❈❍■➋❯ ❙♥❙❡2 ✣❒◆
▲❰P ❇➀◆● ▲Þ ❚❍❯❨➌❚ P❍■➌▼ ❍⑨▼ ▼❾❚ ✣❐
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❱❾❚ ▲Þ
❚❍❊❖ ✣➚◆❍ ❍×❰◆● ◆●❍■➊◆ ❈Ù❯
◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈
❚❙✳ ▲➊ ❚❍➚ ❚❍❯ P❍×❒◆●

❚❤ø❛ ❚❤✐➯♥ ❍✉➳✱ ♥➠♠ ✷✵✶✾


✣❸■ ❍➴❈ ❍❯➌
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✖✖✖✖✖✖

◆●❯❨➍◆ ◗❯❆◆● ❈×❮◆●
◆●❍■➊◆ ❈Ù❯ ❈❻❯ ❚❘Ó❈ ❱Ò◆● ◆❿◆●
▲×Ñ◆● ❈Õ❆ ❱❾❚ ▲■➏❯ ❍❆■ ❈❍■➋❯ ❙♥❙❡2 ✣❒◆
▲❰P ❇➀◆● ▲Þ ❚❍❯❨➌❚ P❍■➌▼ ❍⑨▼ ▼❾❚ ✣❐
❈❤✉②➯♥ ♥❣➔♥❤✿ ❱❾❚ ▲Þ ▲Þ ❚❍❯❨➌❚ ❱⑨ ❱❾❚ ▲Þ ❚❖⑩◆
▼➣ sè

✿ ✽✹ ✹✵ ✶✵✸

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❱❾❚ ▲Þ
❚❍❊❖ ✣➚◆❍ ❍×❰◆● ◆●❍■➊◆ ❈Ù❯

◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈
❚❙✳ ▲➊ ❚❍➚ ❚❍❯ P❍×❒◆●

❚❤ø❛ ❚❤✐➯♥ ❍✉➳✱ ♥➠♠ ✷✵✶✾
✐



ổ ổ tr ự ừ r tổ
số t q ự tr tr tỹ ữủ
ỗ t sỷ ử ữ tứ ữủ ổ ố tr
t ý ởt ổ tr ự

ứ t



ữớ





ổ ỡ sỹ ú ù t t ừ ữớ ữợ
ồ Pữỡ tr sốt q tr tỹ
õ t t tốt tốt ổ tọ
ỏ t ỡ ố ợ ữỡ ồ tt
sỹ ú ù ộ trủ tổ tr q tr tỹ t t
ổ ọ tr
ổ ợ t ỡ sỹ

ở ộ trủ t tr sốt q tr ồ
õ tổ ỷ ớ ỡ qỵ t ổ t ỵ
ỗ t tổ tr sốt q tr ồ t
tỹ

ứ t



ữớ




▼Ö❈ ▲Ö❈
❚r❛♥❣ ♣❤ö ❜➻❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✐

▲í✐ ❝❛♠ ✤♦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✐✐

▲í✐ ❝↔♠ ì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✐✐✐

▼ö❝ ❧ö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶

❉❛♥❤ ♠ö❝ ❦➼ ❤✐➺✉ ✈➔ ❝→❝ ❝❤ú ✈✐➳t t➢t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

❉❛♥❤ ♠ö❝ ❝→❝ ❝æ♥❣ ❜è ❧✐➯♥ q✉❛♥ ✤➳♥ ❧✉➟♥ ✈➠♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

❉❛♥❤ s→❝❤ ❤➻♥❤ ✈➩ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

▼Ð ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

◆❐■ ❉❯◆● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
❈❤÷ì♥❣ ✶✳ ❚✃◆● ◗❯❆◆ ❱➋ ◆●❍■➊◆ ❈Ù❯ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✶✳ ●r❛♣❤❡♥❡ ✈➔ ✈➟t ❧✐➺✉ ❤❛✐ ❝❤✐➲✉ ❦✐♠ ❧♦↕✐ ❝❤✉②➸♥ t✐➳♣ ❞✐❝❤❛❧❝♦✲
❣❡♥✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✷✳ ❚ê♥❣ q✉❛♥ ✈➲ ❝→❝ t➼♥❤ ❝❤➜t ✤✐➺♥ tû ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2

✳ ✳ ✶✹

❈❤÷ì♥❣ ✷✳ ▲Þ ❚❍❯❨➌❚ P❍■➌▼ ❍⑨▼ ▼❾❚ ✣❐ ✳ ✳ ✳ ✶✼
✷✳✶✳ P❤÷ì♥❣ ♣❤→♣ ❧þ t❤✉②➳t ❝➜✉ tró❝ ✤✐➺♥ tû ❝õ❛ ❝❤➜t r➢♥ ✳ ✳ ✳ ✶✼
✷✳✷✳ ▲þ t❤✉②➳t ♣❤✐➳♠ ❤➔♠ ♠➟t ✤ë ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷✳✷✳✶✳ ❚➼♥❤ ❝❤➜t ❧÷ñ♥❣ tû ❝õ❛ ❝❤➜t r➢♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷✳✷✳✷✳ ❉❋❚ ❞ü❛ tr➯♥ ♠æ ❤➻♥❤ ❚❤♦♠❛s✕❋❡r♠✐ ✈➔ ♣❤÷ì♥❣

♣❤→♣ ❣➛♥ ✤ó♥❣ ✤à❛ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✷✳✸✳ ❈→❝ ✤à♥❤ ❧þ ❍♦❤❡♥❜❡r❣✲❑♦❤♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✷✳✹✳ P❤÷ì♥❣ ♣❤→♣ ❑♦❤♥✕❙❤❛♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✷✳✷✳✺✳ P❤✐➳♠ ❤➔♠ ❣➛♥ ✤ó♥❣ ♠➟t ✤ë ✤à❛ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✷✳✷✳✻✳ P❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ❣r❛❞✐❡♥t s✉② rë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✶


✷✳✸✳ ▲þ t❤✉②➳t ♣❤✐➳♠ ❤➔♠ ♠➟t ✤ë tr♦♥❣ ◗✉❛♥t✉♠ ❊s♣r❡ss♦ ✳ ✳ ✷✽

❈❤÷ì♥❣ ✸✳ ❑➌❚ ◗❯❷ ◆●❍■➊◆ ❈Ù❯ ❱⑨ ❚❍❷❖ ▲❯❾◆ ✸✶
✸✳✶✳ ❈➜✉ tró❝ ❤➻♥❤ ❤å❝ ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✷✳ ❚➼♥❤ ❝❤➜t ✤✐➺♥ tû ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✸✳✸✳ ❷♥❤ ❤÷ð♥❣ ❝õ❛ ❜✐➳♥ ❞↕♥❣ ❧➯♥ t➼♥❤ ❝❤➜t ✤✐➺♥ tû ❝õ❛ ✤ì♥
❧î♣ ❙♥❙❡2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸✳✹✳ ✣ì♥ ❧î♣ ❙♥❙❡2 ✤➦t tr♦♥❣ ✤✐➺♥ tr÷í♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
P❍Ö ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P✳✶
P❤ö ❧ö❝ ✶✿ ❈❤÷ì♥❣ tr➻♥❤ ♠æ ♣❤ä♥❣ ❝➜✉ tró❝ ✈ò♥❣ ♥➠♥❣ ❧÷ñ♥❣
❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P✳✶

✷


❉❆◆❍ ▼Ö❈ ❑➑ ❍■➏❯ ❱⑨ ❈⑩❈ ❈❍Ú
❱■➌❚ ❚➁❚

✷❉ ✭❚✇♦✲❞✐♠❡♥s✐♦♥❛❧✮

❍❛✐ ❝❤✐➲✉

❉❋❚ ✭❉❡♥s✐t② ❢✉♥❝t✐♦♥❛❧ t❤❡♦r②✮

▲þ t❤✉②➳t ♣❤✐➳♠ ❤➔♠ ♠➟t ✤ë

P❇❊ ✭P❡r❞❡✇✲❇✉r❦❡✲❊r♥③❡r❤♦❢✮

❍➔♠ P❡r❞❡✇✲❇✉r❦❡✲❊r♥③❡r❤♦❢

●●❆ ✭●❡♥❡r❛❧✐③❡❞ ❣r❛❞✐❡♥t

●➛♥ ✤ó♥❣ ❣r❛❞✐❡♥t tê♥❣ q✉→t

❛♣♣r♦①✐♠❛t✐♦♥✮
▲❉❆ ✭▲♦❝❛❧ ❞❡♥s✐t② ❛♣♣r♦①✐♠❛t✐♦♥✮ ●➛♥ ✤ó♥❣ ♠➟t ✤ë ✤à❛ ♣❤÷ì♥❣
❍❋❙ ✭❍❛rtr❡❡✕❋♦❝❦✖✲❙❧❛t❡r✮

P❤÷ì♥❣ ♣❤→♣
❍❛rtr❡❡✲❋♦❝❦✕❙❧❛t❡r

❙❈❋ ✭❙❡❧❢ ❝♦♥s✐st❡♥t ❢✐❡❧❞✮

❚r÷í♥❣ tü ❤ñ♣

P❲✾✶

❍➔♠ P❡r❞❡✇ ✈➔ ❲❛♥❣ ✭✶✾✾✶✮

✸


❉❆◆❍ ▼Ö❈ ❈⑩❈ ❈➷◆● ❇➮ ▲■➊◆
◗❯❆◆ ✣➌◆ ▲❯❾◆ ❱❿◆
✶✳ ◆❣✉②❡♥ ❉✳ ❍✐❡♥✱ ◆❣✉②❡♥ ◗✳ ❈✉♦♥❣✱ ▲❡ ▼✳ ❇✉✐✱ P❤❛♠ ❈✳ ❉✐♥❤✱
❈❤✉♦♥❣ ❱✳ ◆❣✉②❡♥✱ ❍✉②♥❤ ❱✳ P❤✉❝✱ ◆❣✉②❡♥ ❱✳ ❍✐❡✉✱ ❍✳❘✳ ❏❛♣✲
♣♦r✱ ▲❡ ❚✳❚✳ P❤✉♦♥❣✱ ❇✉✐ ❉✳ ❍♦✐✱ ▲❡ ❈✳ ◆❤❛♥✱ ❛♥❞ ◆❣✉②❡♥ ◆✳ ❍✐❡✉
✭✷✵✶✾✮✱ ✏❋✐rst ♣r✐♥❝✐♣❧❡s st✉❞② ♦❢ s✐♥❣❧❡✲❧❛②❡r ❙♥❙❡2 ✉♥❞❡r ❜✐❛①✲
✐❛❧ str❛✐♥ ❛♥❞ ❡❧❡❝tr✐❝ ❢✐❡❧❞✿ ▼♦❞✉❧❛t✐♦♥ ♦❢ ❡❧❡❝tr♦♥✐❝ ♣r♦♣❡rt✐❡s✑✱

P❤②s✐❝❛ ❊✿ ▲♦✇✲❞✐♠❡♥s✐♦♥❛❧ ❙②st❡♠s ❛♥❞ ◆❛♥♦str✉❝t✉r❡s ✭✶✶✶✮✱
♣♣✳ ✷✵✶✕✷✵✺✳

✹


❉❆◆❍ ❙⑩❈❍ ❍➐◆❍ ❱➇
❍➻♥❤ ✶✳✶

❈➜✉ tró❝ ♥❣✉②➯♥ tû ❝õ❛ ✈➟t ❧✐➺✉ ▼❳2 ✿ ✭❛✮ ✤ì♥ ❧î♣
✈➔ ✭❜✮ ❤❛✐ ❧î♣ ❬✻❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

❍➻♥❤ ✶✳✷

❈➜✉ tró❝ ✈ò♥❣ ♥➠♥❣ ❧÷ñ♥❣ ✤✐➺♥ tû ❝õ❛ ❝→❝ ✤ì♥ ❧î♣
▼♦❙2 ✈➔ ▼♦❙❡2 ❬✼❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

❍➻♥❤ ✶✳✸

❈➜✉ tró❝ ❤➻♥❤ ❤å❝ ❝õ❛ ❞à ❝➜✉ tró❝ ✈❛♥ ❞❡r ❲❛❛❧s
●❛❙❡✴▼♦❙❡2 t❤❡♦ ❝→❝ ❣â❝ ♥❤➻♥ ❦❤→❝ ♥❤❛✉✿ ✭❛✮ ♥❤➻♥
tø tr➯♥ ①✉è♥❣✱ ✭❜✮ ♥❤➻♥ tø ♠➦t ❜➯♥✱ ✈➔ ✭❝✮ ♥❤➻♥ tø
♣❤➼❛ tr÷î❝ ❬✷✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

❍➻♥❤ ✷✳✶

❙ì ✤ç t❤✉➟t t♦→♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❑♦❤♥✲❙❤❛♠
❜➡♥❣ ✈á♥❣ ❧➦♣ tü ❤ñ♣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

❍➻♥❤ ✸✳✶

❈➜✉ tró❝ ♥❣✉②➯♥ tû ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ð tr↕♥❣ t❤→✐
❝➙♥ ❜➡♥❣ t❤❡♦ ❝→❝ ❣â❝ ♥❤➻♥ ❦❤→❝ ♥❤❛✉✿ ✭❛✮ ◆❤➻♥ tø
tr➯♥ ①✉è♥❣✱ s✐❞❡ ✈✐❡✇ ✭❜✮ ♥❤➻♥ tø ♠➦t ❜➯♥ ❝↕♥❤✱
✈➔ ✭❝✮ ♥❤➻♥ tø ♠➦t tr÷î❝✳ ❈→❝ q✉↔ ❜â♥❣ ♠➔✉ ①❛♥❤
♥❤↕t ✈➔ ♠➔✉ ✈➔♥❣ ❧➛♥ ❧÷ñt ❞ò♥❣ ✤➸ ❝❤➾ ❝→❝ ♥❣✉②➯♥
tû ❙♥ ✈➔ ❙❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

❍➻♥❤ ✸✳✷

P❤ê ♣❤♦♥♦♥ ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ð tr↕♥❣ t❤→✐ ❝➙♥
❜➡♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✺


❍➻♥❤ ✸✳✸

✭❛✮ ❈➜✉ tró❝ ✈ò♥❣ ♥➠♥❣ ❧÷ñ♥❣ ✤✐➺♥ tû ✈➔ ✭❜✮ ♠➟t ✤ë
tr↕♥❣ t❤→✐ r✐➯♥❣ ♣❤➛♥ ✭P❉❖❙✮ ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2
ð tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

❍➻♥❤ ✸✳✹

❙ü ♣❤ö t❤✉ë❝ ❝õ❛ ♥➠♥❣ ❧÷ñ♥❣ t♦➔♥ ♣❤➛♥ ❝õ❛ ✤ì♥
❧î♣ ❙♥❙❡2 ✈➔♦ ❜✐➳♥ ❞↕♥❣✳

❍➻♥❤ ✸✳✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

❈➜✉ tró❝ ✈ò♥❣ ♥➠♥❣ ❧÷ñ♥❣ ✤✐➺♥ tû ❝õ❛ ✤ì♥ ❧î♣
❙♥❙❡2 ❦❤✐ ❜à ❜✐➳♥ ❞↕♥❣ ✈î✐✿ ✭❛✮ εb = −5%✱ ✭❜✮ εb =

−3%✱ ✭❝✮ εb = 0% ✭ð tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣✮✱ ✭❞✮ εb =
3%✱ ✈➔ ✭❡✮ εb = 5%✳ ◆➠♥❣ ❧÷ñ♥❣ ❋❡r♠✐ ✤÷ñ❝ ❝❤å♥
t↕✐ ❣✐→ trà ♥➠♥❣ ❧÷ñ♥❣ ❜➡♥❣ ❦❤æ♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
❍➻♥❤ ✸✳✻

❙ü ♣❤ö t❤✉ë❝ ❝õ❛ ♥➠♥❣ ❧÷ñ♥❣ ✈ò♥❣ ❝➜♠ ❝õ❛ ✤ì♥
❧î♣ ❙♥❙❡2 ✈➔♦ ❜✐➳♥ ❞↕♥❣ εb ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

❍➻♥❤ ✸✳✼

❷♥❤ ❤÷ð♥❣ ❝õ❛ ✤✐➺♥ tr÷í♥❣ ♥❣♦➔✐ ❧➯♥ ♥➠♥❣ ❧÷ñ♥❣
t♦➔♥ ♣❤➛♥ ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

❍➻♥❤ ✸✳✽

❈➜✉ tró❝ ✈ò♥❣ ♥➠♥❣ ❧÷ñ♥❣ ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ❦❤✐
❝â ♠➦t ❝õ❛ ✤✐➺♥ tr÷í♥❣ E ✿ E = −5 ❱✴♥♠ ✭❛✮✱ E =

−3 ❱✴♥♠ ✭❜✮✱ E = 3 ❱✴♥♠ ✭❝✮✱ ✈➔ E = 5 ❱✴♥♠
✭❞✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
❍➻♥❤ ✸✳✾

❙ü ♣❤ö t❤✉ë❝ ❝õ❛ ♥➠♥❣ ❧÷ñ♥❣ ✈ò♥❣ ❝➜♠ ❝õ❛ ✤ì♥
❧î♣ ❙♥❙❡2 ✈➔♦ ✤✐➺♥ tr÷í♥❣ E ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✻


é
ỵ ồ t
tứ ữủ õ t t ổ tỹ
r t õ tữợ
tỷ tr t ởt tr ỳ t ữủ ở ỗ
ồ q t ự t tr sốt q r
õ t t t ỵ tt ớ ữủ ý ồ s õ ự ử
tr ổ tỷ ữ õ ự ử tr
tỹ t ỳ õ t ừ
r õ ũ ổ sỹ ổ tữỡ t ợ
tỷ s t ỵ õ t tử
ự ữ r ự ử ữớ t q
t t õ trú tữỡ tỹ ợ ồ ú
tữỡ t ỡ ợ tt t õ ự ử
ỡ t t ợ ỳ t t tỷ
q ồ tr ũ ủ ự ử t út

sỹ q t rt ợ ừ ở ỗ ự t
õ trú tỹ r s ừ
t 2 tr õ tố
tố t út ữủ sỹ q t rt ợ ợ ỳ t
t t ỵ õ ồ ự ừ ú tr ự ử
tỷ q tỷ tữỡ ợ r ợ t
2 ợ ở rở ũ t



ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ❧✐♥❤ ❦✐➺♥ ♥❛♥æ q✉❛♥❣ ✤✐➺♥ tû✳ ●➛♥ ✤➙②✱ ✤ì♥ ❧î♣
❙♥❙❡2 ✕ ♠ët ❞↕♥❣ ✈➟t ❧✐➺✉ t❤✉ë❝ ♥❤â♠ ▼❳2 ✤➣ ✤÷ñ❝ tê♥❣ ❤ñ♣ t❤➔♥❤
❝æ♥❣ ✈➔ t❤✉ ❤ót sü q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ r➜t ❧î♥✳ ❈→❝ ♥❤➔ ❦❤♦❛ ❤å❝
✤➣ ❝❤➾ r❛ r➡♥❣ ✤ì♥ ❧î♣ ❙♥❙❡2 ❧➔ ✈➟t ❧✐➺✉ t❤➙♥ t❤✐➺♥ ♠æ✐ tr÷í♥❣ ✈➔ ❜➲♥
✈ú♥❣ ð ♥❤✐➺t ✤ë ♣❤á♥❣✳ ❚✉② ♥❤✐➯♥✱ ❝❤♦ ✤➳♥ ♥❛② ♥❤ú♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲
✤ì♥ ❧î♣ ❙♥❙❡2 ♥➔② ❝❤÷❛ ✤÷ñ❝ s➙✉ ✈➔ rë♥❣ ♥❤÷ ♥❤ú♥❣ ❣➻ ♥❣÷í✐ t❛ ✤➣ ❧➔♠
✈î✐ ❣r❛♣❤❡♥❡✳ ▼✉è♥ ✤÷❛ ✈➟t ❧✐➺✉ ♥➔② ✈➔♦ tr♦♥❣ ù♥❣ ❞ö♥❣ t❤ü❝ t✐➵♥ ❝õ❛
❝æ♥❣ ♥❣❤➺ ♠î✐✱ ❝➛♥ ♣❤↔✐ ❝â t❤➯♠ ♥❤✐➲✉ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❤✐➸✉ ♥❤✐➲✉ ❤ì♥
✈➲ ❝➜✉ tró❝ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✈➟t ❧þ ❝õ❛ ♥â✳ ❱➻ ✈➟②✱ tæ✐ ✤➣ ❧ü❛ ❝❤å♥ ✤➲
t➔✐ ✧◆❣❤✐➯♥ ❝ù✉ ❝➜✉ tró❝ ✈ò♥❣ ♥➠♥❣ ❧÷ñ♥❣ ❝õ❛ ✈➟t ❧✐➺✉ ❤❛✐ ❝❤✐➲✉ ❙♥❙❡2
✤ì♥ ❧î♣ ❜➡♥❣ ❧þ t❤✉②➳t ♣❤✐➳♠ ❤➔♠ ♠➟t ✤ë✧ ❝❤♦ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛
♠➻♥❤✳

■■✳ ▼ö❝ t✐➯✉ ❝õ❛ ✤➲ t➔✐
▼ö❝ t✐➯✉ ❝õ❛ ✤➲ t➔✐ ❧➔ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t➼♥❤ t♦→♥ ❝➜✉ tró❝ ✈ò♥❣ ♥➠♥❣
❧÷ñ♥❣ ✤✐➺♥ tû ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ❜➡♥❣ ❧þ t❤✉②➳t ♣❤✐➳♠ ❤➔♠ ♠➟t ✤ë✳
◆❣❤✐➯♥ ❝ù✉ ↔♥❤ ❤÷ð♥❣ ❝õ❛ ❜✐➳♥ ❞↕♥❣ ✈➔ ✤✐➺♥ tr÷í♥❣ ❧➯♥ ❝→❝ t➼♥❤ ❝❤➜t
✤✐➺♥ tû ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ❝ô♥❣ ❧➔ ♠ët ♠ö❝ t✐➯✉ ❝õ❛ ✤➲ t➔✐ ♥➔②✳

■■■✳ ◆ë✐ ❞✉♥❣ ♥❣❤✐➯♥ ❝ù✉

✣➸ ✤↕t ✤÷ñ❝ ❝→❝ ♠ö❝ t✐➯✉ tr➯♥ ❝❤ó♥❣ tæ✐ ①→❝ ✤à♥❤ rã ❜❛ ❜➔✐ t♦→♥
❝➛♥ ❣✐↔✐ q✉②➳t ✤â ❧➔✿

• ❳➙② ❞ü♥❣ ❝➜✉ tró❝ ♥❣✉②➯♥ tû ❝õ❛ ✈➟t ❧✐➺✉ ❤❛✐ ❝❤✐➲✉ ✤ì♥ ❧î♣ ❙♥❙❡2 ✳
❳→❝ ✤à♥❤ ❝→❝ t❤❛♠ sè ❝➜✉ tró❝ ❝õ❛ ✈➟t ❧✐➺✉ ð tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣✳
❑✐➸♠ tr❛ t➼♥❤ ❜➲♥ ✈ú♥❣ ❝õ❛ ✈➟t ❧✐➺✉ ✈➲ ♠➦t ✤ë♥❣ ❤å❝ t❤æ♥❣ q✉❛
✽


t➼♥❤ t♦→♥ ♣❤ê ♣❤♦♥♦♥ ❝õ❛ ♥â✳

• ❚➼♥❤ t♦→♥ ✈➔ ♠æ ♣❤ä♥❣ ❝→❝ t➼♥❤ ❝❤➜t ✤✐➺♥ tû ♥❤÷ ❝➜✉ tró❝ ✈ò♥❣
♥➠♥❣ ❧÷ñ♥❣✱ ♠➟t ✤ë tr↕♥❣ t❤→✐ ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ð tr↕♥❣ t❤→✐ ❝➙♥
❜➡♥❣✳

• ❑❤↔♦ s→t ↔♥❤ ❤÷ð♥❣ ❝õ❛ ❜✐➳♥ ❞↕♥❣ ✈➔ ✤✐➺♥ tr÷í♥❣ ❧➯♥ ❝→❝ t➼♥❤ ❝❤➜t
✤✐➺♥ tû ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ✳

■❱✳ ✣è✐ t÷ñ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉
✣è✐ t÷ñ♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ✤➲ t➔✐ ❧➔ ❝→❝ t➼♥❤ ❝❤➜t ✤✐➺♥ tû ❝õ❛ ✤ì♥
❧î♣ ❙♥❙❡2 ✳ ❚r♦♥❣ ✤➲ t➔✐ ♥➔②✱ ❝❤ó♥❣ tæ✐ ❝❤➾ t➟♣ tr✉♥❣ ✈➔♦ ♥❣❤✐➯♥ ❝ù✉ ❝➜✉
tró❝ ✈ò♥❣ ♥➠♥❣ ❧÷ñ♥❣ ✤✐➺♥ tû✱ ♠➟t ✤ë tr↕♥❣ t❤→✐ ✈➔ ❝→❝ ✤➦❝ tr÷♥❣ ❝ì
❜↔♥ ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ✳ ❈❤ó♥❣ tæ✐ ❝❤➾ ♥❣❤✐➯♥ ❝ù✉ ↔♥❤ ❤÷ð♥❣ ❝õ❛ ❜✐➳♥
❞↕♥❣ ♥❤ä ✈➔ ✤✐➺♥ tr÷í♥❣ ✈✉æ♥❣ ❣â❝ ❧➯♥ ❝→❝ t➼♥❤ ❝❤➜t ✤✐➺♥ tû ❝õ❛ ✤ì♥
❧î♣ ❙♥❙❡2 ✳

❱✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉
P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❝❤➼♥❤ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❧➔ ♣❤÷ì♥❣ ♣❤→♣
❧þ t❤✉②➳t ♣❤✐➳♠ ❤➔♠ ♠➟t ✤ë✳ ❈→❝ t➼♥❤ t♦→♥ ❜➡♥❣ ❧þ t❤✉②➳t ♣❤✐➳♠ ❤➔♠
♠➟t ✤ë tr♦♥❣ ✤➲ t➔✐ ✤÷ñ❝ t❤ü❝ ❤✐➺♥ t❤æ♥❣ q✉❛ ❣â✐ ♠æ ♣❤ä♥❣ ◗✉❛♥t✉♠

❊s♣r❡ss♦✳ P❤✐➳♠ ❤➔♠ tr❛♦ ✤ê✐ ✕ t÷ì♥❣ q✉❛♥ P❇❊ ❞♦ P❡r❞❡✇✱ ❇✉r❦❡
✈➔ ❊r♥③❡r❤♦❢ ✤➲ ①✉➜t ✤➣ ✤÷ñ❝ sû ❞ö♥❣ tr♦♥❣ ❝→❝ t➼♥❤ t♦→♥ ❝õ❛ ✤➲ t➔✐✳
❇➯♥ ❝↕♥❤ ✤â✱ ♣❤÷ì♥❣ ♣❤→♣ ❧þ t❤✉②➳t ❤➔♠ ♠➟t ✤ë ❝↔✐ t✐➳♥ ❉❋❚✕❉✷ ❝õ❛
●r✐♠♠❡ ✤➣ ✤÷ñ❝ sû ❞ö♥❣ ✤➸ ①û ❧þ ♠ët sè ✈➜♥ ✤➲ tr♦♥❣ q✉→ tr➻♥❤ t➼♥❤
t♦→♥ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ t÷ì♥❣ t→❝ ✈❛♥ ❞❡r ❲❛❛❧s tr♦♥❣ ✈➟t ❧✐➺✉✳

✾


ố ử
ữủ tr ữợ ởt ổ tr
ử ử t t ữủ


P r ỵ ồ t ử t ừ t
ở ự ố tữủ ự ữỡ
ự ố ử

P ở ỗ ữỡ ữỡ tr tờ q
ự ữỡ tr ữỡ ỵ tt
t ở tự ỡ s q ữỡ ữỡ
ự ở ừ t

P t r t rút r ỹ tr t q
t ữủ ữợ t tr ừ t
r ữỡ tr ổ ọ trú ũ ữủ
tỷ ừ ỡ ợ 2 ữủ ữ r tr ử ử ố

ữỡ

r ữỡ ú tổ tr tờ q
q ự ừ t ú tổ t
tr tr t q ự trữợ q
t ụ ữ ỳ ỏ ọ ọ

r t
t
r ỳ t õ trú ợ
t út ữủ sỹ ú ỵ tứ ở ỗ ồ ú õ
t t t ỵ õ ồ ờ trở sỹ ỳ ữủ tỷ
tứ ữủ õ t t ổ tỹ
r tr t ởt tr ỳ t ữủ ự
t tr sốt q õ t t t ỵ õ
ồ t r ữủ ự ử tr tt
õ ù t õ ữủ ũ
ổ ữớ t rt õ tr ự ử
r tr tt q tỷ t
ử ữủ r ồ t
t õ trú tữỡ tỹ r ữ õ ũ



(a)
a
c

b

(b)

X
M

Hai lớp

X

c
a

Đơn lớp

b

❍➻♥❤ ✶✳✶✿ ❈➜✉ tró❝ ♥❣✉②➯♥ tû ❝õ❛ ✈➟t ❧✐➺✉ ▼❳2 ✿ ✭❛✮ ✤ì♥ ❧î♣ ✈➔ ✭❜✮ ❤❛✐ ❧î♣ ❬✻❪✳

❝➜♠ ❦❤→❝ ❦❤æ♥❣✳ ◆❤✐➲✉ ❧♦↕✐ ✈➟t ❧✐➺✉ ❧î♣ ❤❛✐ ❝❤✐➲✉ ✤ì♥ ❧î♣ ✤➣ ✤÷ñ❝ ♣❤→t
❤✐➺♥ ♥❤÷ s✐❧✐❝❡♥❡ ❬✶❪✱ ❝→❝ ✈➟t ❧✐➺✉ ❦✐♠ ❧♦↕✐ ❝❤✉②➸♥ t✐➳♣ ❞✐❝❤❛❧❝♦❣❡♥✐❞❡
✭❝â ❝æ♥❣ t❤ù❝ ❤â❛ ❤å❝ ❧➔ ▼❳2 ✈î✐ ▼ ❧➔ ♥❣✉②➯♥ tè ❦✐♠ ❧♦↕✐ ❝❤✉②➸♥ t✐➳♣
✈➔ ❳ ❧➔ ♥❣✉②➯♥ tè ❝❤❛❧❝♦❣❡♥✐❞❡✮ ❬✻✱✼✱✷✶✱✸✷❪ ❤❛② ♣❤♦s♣❤♦r❡♥❡ ❬✸✹❪✳ ❚➼♥❤
❝❤➜t ✤✐➺♥ tû ✈➔ tr✉②➲♥ ❞➝♥ ❝õ❛ ❝→❝ ✈➟t ❧✐➺✉ ♥➔② t❤÷í♥❣ r➜t ♥❤↕② ✈î✐
❝→❝ ↔♥❤ ❤÷ð♥❣ tø ❜➯♥ ♥❣♦➔✐ ♥❤÷ ❜✐➳♥ ❞↕♥❣ ❝ì ❤å❝ ❤❛② ✤✐➺♥ tr÷í♥❣✳ ✣➦❝
❜✐➺t✱ ❝→❝ ❞à ❝➜✉ tró❝ ✈❛♥ ❞❡r ❲❛❛❧s ✤÷ñ❝ ❤➻♥❤ t❤➔♥❤ tø ❝→❝ ❧♦↕✐ ✈➟t ❧✐➺✉
❤❛✐ ❝❤✐➲✉ ✤➣ ✤÷ñ❝ ❝❤ó trå♥❣ tr♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙② ❬✸✵✱ ✸✸✱ ✸✹❪ ✈➔ ❝â
♥❤✐➲✉ t✐➲♠ ♥➠♥❣ ù♥❣ ❞ö♥❣ ✈➔♦ tr♦♥❣ ❝→❝ t❤✐➳t ❜à ✤✐➺♥ tû ✈➔ q✉❛♥❣ ✤✐➺♥

tû ♥❛♥♦ ❬✹✱ ✶✷✱ ✹✷❪✳
❑❤→❝ ✈î✐ ❣r❛♣❤❡♥❡✱ ✈➟t ❧✐➺✉ ❝â ❝➜✉ tró❝ ❤➻♥❤ ❧ö❝ ❣✐→❝ ♣❤➥♥❣ ✤÷ñ❝
❤➻♥❤ t❤➔♥❤ tø ❝→❝ ♥❣✉②➯♥ tû ❝❛r❜♦♥✱ ❝→❝ ✤ì♥ ❧î♣ ▼❳2 ❝â ❝➜✉ tró❝ ❞↕♥❣
①➳♣ ❧î♣ ✤÷ñ❝ ❤➻♥❤ t❤➔♥❤ tø ❤❛✐ ❧♦↕✐ ♥❣✉②➯♥ tû ❦❤→❝ ♥❤❛✉✳ ❱➲ ♠➦t ❤➻♥❤
❤å❝✱ ✈➟t ❧✐➺✉ ▼❳2 ❝â t❤➸ tç♥ t↕✐ ð ❤❛✐ ❞↕♥❣ ❝➜✉ tró❝ ❦❤→❝ ♥❤❛✉✱ ✤â ❧➔
❝➜✉ tró❝ ❤➻♥❤ ❧ö❝ ❣✐→❝ ✭❤❛② ❝á♥ ❣å✐ ❧➔ ❝➜✉ tró❝ ✷✕❍✮ ✈➔ ❝➜✉ tró❝ ❤➻♥❤
✶✷


t ỏ ồ trú r ú tổ
q t trú ừ t 2 r ộ ỡ ợ
2 ợ tỷ ỗ t trt tỹ
trú tỷ ừ t t
2 õ tr ữủ ợ r t
õ ũ ổ t ỡ ợ 2 ợ
ũ tữỡ ố ợ ù ợ ở rở ũ tữỡ
ố ợ ữ t t 2 ữủ ý ồ s õ ự
ử tr ổ t tr ổ s t r tứ
ữợ trú ũ ữủ tỷ ừ ỡ ợ 2
2 ữủ tr t t ỵ tt
t ở ở sỹ r r ữ
ừ t s q t t tỷ ừ t 2
tữỡ ố ợ õ ữ ừ t ở
ữ trữớ st t t
tỷ ừ ỡ ợ 2 tữỡ ố ợ
ồ t tr ự trú r s ữủ
t tứ t
trú r s ữủ t tứ ỡ ợ 2 t
õ ỳ t t t ỵ tữớ ổ õ tr
tứ t r t trú tỷ ừ

trú r s 2 ữủ
ự trữợ ụ r r r ợ ởt t
t t ởt trú r s ụ
ởt tr t ũ tr r




Nng lng, eV

Nng lng, eV

ked in Fig. 2 by h, l, and d. After fully optimized, our calculated interlayer distances
nd GaS MoSe2 heterojunctions are 3.45 and 3.54 , respectively. These interlayer

n energy band structures of perfect (a) GaSe, (b) GaS, (c) MoS2 , and (d) MoSe2 monolayers at the
trú ũ ữủ tỷ ừ ỡ ợ 2 2
spectively.

ờ q t t tỷ ừ ỡ
ợ 2
ỡ ợ t s 2 ởt tr số t
trữ ừ ồ t õ t
ữủ sỹ ú ỵ ừ ợ ồ tr tớ õ
t ỵõ tú 2 ố ữủ ự
tứ rt sợ t ọ 2 ù
ữủ tờ ủ t ổ ữỡ tỹ
sỷ ử ỵ tt t ở
ồ r r trữ trú ừ ợ 2
ữ ổ ử tở số ợ ừ t tự ừ

t ỹ tr t t ờ ở
sỹ r r ỡ ợ 2 t ở ồ t
2 t ợ r 2 ợ ữủ t ợ
ỹ t r s ỹ t ỳ
ợ õ ú t ý ồ r õ t t ỡ ợ 2 tứ t
ố ởt ữỡ tỹ tổ



Chemical Physics 521 (2019) 92–99

nd theoretically the physical properties, that may
he GaSe and MoSe2 monolayer to form the GaSe/
present work, we investigate systematically the
f the GaSe/MoSe2 vdWH using DFT method. We
ese properties of the GaSe/MoSe2 vdWH can be
ng the interlayer distance or by applying the
dings could provide the fundamental insights of
H for designing high-performance optoelectronic

ethods

quantum mechanical package, namely Quantum
ch is based on the density functional theory (DFT)
all the structural and electronic properties of the
n addition, to describe the ion-electron and the
energy, we used the projected augmented wave
the generalized gradient approximation (GGA)
urke, and Ernzerhof (PBE) [44]. Also, for deW force, existing mostly in the layered materials,
DFT-D2 method by adding the London dispersion

energy cutoff of 500 eV is applied for the plane
a grid of 9 × 9 × 1 k-point mesh is used for the
n zone (BZ). In addition, for layered GaSe and
in-orbit coupling (SOC) plays an important role
ting in the valence band maximum (VBM) and
mum (CBM), resulting in a decrease in their band
ffects hardly change the shape of the bands of the
efore, the SOC effects are not considered in our
Moreover, it should be noted that the traditional
estimate the band gap values of semiconductors
a-Ernzerhof (HSE06) hybrid function is also used
band-gap of the heterostructure. The geometric
ronic properties calculations are performed until
ergy less than 10 3 eV/Å and 10 6 eV/atom, remetric optimization to find the most energetically
2
all the atoms and lattice parameters of the heto relax. A vacuum layer thickness of 20 Å along
ed to avoid the interaction between two adjacent

❍➻♥❤ ✶✳✸✿ ❈➜✉ tró❝ ❤➻♥❤ ❤å❝ ❝õ❛ ❞à ❝➜✉ tró❝ ✈❛♥ ❞❡r ❲❛❛❧s ●❛❙❡✴▼♦❙❡2 t❤❡♦ ❝→❝ ❣â❝
♥❤➻♥ ❦❤→❝ ♥❤❛✉✿ ✭❛✮
♥❤➻♥ tø tr➯♥ ①✉è♥❣✱ ✭❜✮ ♥❤➻♥ tø ♠➦t ❜➯♥✱ ✈➔ ✭❝✮ ♥❤➻♥ tø ♣❤➼❛
[34,36–38], SiC/TMDs [24,27], phosphorene/TMDs [29,30], SnSe2/
tr÷î❝ ❬✷✶❪✳

MX2 (M = Mo, W, Zr) [46], which are typical vdW interactions. It
implies a vdW type interaction between the GaSe and MoSe2 layers in
the GaSe/MoSe2 vdWH. This finding is in good agreement with the
previous experimental report of the GaSe/MoSe2 vdWH [41].
We further calculate the binding
2 energy (BE) of the GaSe/MoSe2

vdWH to check its stability. The BE of the vdWH is calculated as:
Eb = EGaSe / MoSe2 EGaSe E MoSe2 , where EGaSe / MoSe2 is the total energy of
2 2 vdWH, EGaSe and E MoSe2 represent the total energies of
the GaSe/MoSe
the isolated GaSe and MoSe2 monolayers, respectively. Our calculated
−1 BE−1of the GaSe/MoSe2 vdWH is −40.75 meV. The negative binding
energy indicates that the GaSe/MoSe2 vdWH is stable. In addition, we
find that this BE has the same magnitude as other 2D vdWHs, for instance, graphene/phosphorene [47], InSe/black phosphorus [48], black
phosphorus/MoS2 [49], SiC/graphene [27] and so forth, which are
typical vdW interactions. Thus, in the GaSe/MoSe2 vdWH the interaction between different 2D materials is mainly contributed by vdW interactions.
Fig. 2(a,b) shows
the electronic band structures of the isolated GaSe
2
and MoSe2 monolayers. One can observe that both GaSe and MoSe2
monolayers are semiconductors. The isolated GaSe monolayer has an
indirect band gap of 1.90 eV. Its conduction band minimum (CBM)
locates at the point, whereas its valence band maximum (VBM) locates at the -M path. In contrary, the isolated MoSe22 monolayer has a
direct band gap of 1.53 eV, which is coming from both the CBM and
VBM at the K point. The composed electronic band structure of the
GaSe/MoSe2 vdWH is displayed in Fig. 2(c) using HSE06 and PBE
methods. We find that the GaSe/MoSe2 vdWH has an indirect band gap,
forming between
2 the CBM at the point and the VBM at the K point. In
addition, when the GaSe monolayer couples with the MoSe2 monolayer
to form the vdWH, the band structure of the GaSe/MoSe2 vdWH seems
2 structures of the isolated GaSe and
to be a combination of the band
MoSe2 monolayers. It results on the preservation of electronic properties of both the GaSe and MoSe2 monolayers upon contact owing to the
weak vdW interactions, existing between the GaSe and MoSe2 layers in
the vdWH. Moreover, it is obvious that the GaSe/MoSe2 vdWH is an

indirect semiconductor. Our
2 calculated band gap of the vdWH is
1.42 eV (2.05 eV) obtained from PBE (HSE06) method at the equilibrium state with the interlayer distance d = 3.50 Å. This band gap is
smaller than that of both freestanding GaSe and MoSe2 monolayers.

❞ö♥❣ ♥❤÷ ✤➣ →♣ ❞ö♥❣ ✤è✐ ✈î✐ ❙♥❙ ❬✶✽❪✳ ❙❤❛❢✐q✉❡ ✈➔ ❝→❝ ❝ë♥❣ sü ✤➣ ❝❤➾
r❛ r➡♥❣✱ ✤ì♥ ❧î♣ ❙♥❙❡ ❝â ✤ë ❧✐♥❤ ✤ë♥❣ ❝õ❛ ✤✐➺♥ tû ❦❤→ ❝❛♦✱ ❧➯♥ ✤➳♥
✹✻✷✳✻✶ ❝♠ ❱ s

✈➔ ❝â ✤ë ❞➝♥ ♥❤✐➺t ❦❤→ t❤➜♣✱ t➛♠ ✸✳✽✷ ❲♠−1 ❑−1 ❬✸✼❪✳

✣➙② ❧➔ ♥❤ú♥❣ ✤➦❝ t➼♥❤ q✉❛♥ trå♥❣ ❝❤♦ ✈✐➺❝ ❝â t❤➸ sû ❞ö♥❣ ✤ì♥ ❧î♣ ❙♥❙❡2

tion spectra is calculated by using the GW0 apgenvalues and wave functions of the PBE funcmpute the quasiparticle energies and solve the
n. These calculations are performed by using a kENCUT of 350 eV, an NBAND of 300 and 10
e(conduction) bands for excitonic eigenstates. In
mputational cost of GW0 calculations, different
are adopted.

ssion

Fig. 1. (a) Top view (b) side view and (c) front view of the relaxed geometric
structure of the GaSe/MoSe2 vdWHs.

tr♦♥❣ ❝→❝ t❤✐➳t ❜à ♥❤✐➺t ✤✐➺♥✳ ❉♦ ♥❤ú♥❣ t➼♥❤ ❝❤➜t ✤✐➺♥ tû ✈➔ ♥❤✐➺t ✤✐➺♥
✈÷ñt trë✐✱ ✤ì♥ ❧î♣ ❙♥❙❡ ✤÷ñ❝ ❦ý ✈å♥❣ s➩ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝

t❤✐➳t ❜à ♥❤✐➺t ✤✐➺♥ tû ❬✶✾✱ ✷✻✱ ✹✵✱ ✹✶❪✳

❚➼♥❤ ❝❤➜t ✤✐➺♥ tû ❝õ❛ ❝→❝ ✤ì♥ ❧î♣ ❙♥❙❡ ✤➣ ✤÷ñ❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤→

c structure of the GaSe/MoSe2 vdWH is displayed
on, the optimized lattice parameters of GaSe and
spectively, are 3.742 Å and 3.278 Å. These values
e previous theoretical reports and experimental
Se2 vdWH is built by matching a ( 3 × 3 ) suonolayer with a (2 × 2) supercell of the MoSe2
e mismatch between the GaSe and MoSe2 monois less than 2.5%. After relaxation, the obtained
ckness h, l and the interlayer distance (d), as ilspectively, are 4.77 Å, 3.37 Å and 3.5 Å. The
d = 3.5 Å is greater than the sum of the covalent
he GaSe and MoSe2 layers, indicating that the Se
d the Se atoms of MoSe2 layer in the GaSe/MoSe2
bonding range. Such interlayer distance is comer 2D vdWHs, such as graphene/GaX (X = Se, S)

♥❤✐➲✉ ❣➛♥ ✤➙② ❬✶✺✱ ✹✸❪✳ ▲✐ ✈➔ ❝→❝ ❝ë♥❣ sü ✤➣ ❝❤➾ r❛ r➡♥❣✱ ð tr↕♥❣ t❤→✐
❝➙♥ ❜➡♥❣✱ ✤ì♥ ❧î♣ ❙♥❙❡ ❧➔ ✈➟t ❧✐➺✉ ❜→♥ ❞➝♥ ❝â ✈ò♥❣ ❝➜♠ ①✐➯♥ ❬✶✾❪✳ ✣ë
rë♥❣ ✈ò♥❣ ❝➜♠ ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡ ✱ tò② ✈➔♦ ♣❤÷ì♥❣ ♣❤→♣ t➼♥❤ t♦→♥✱ ♥➡♠
tr♦♥❣ ❦❤♦↔♥❣ tø ✵✳✼✾ ❡❱ ✤➳♥ ✶✳✵✼ ❡❱ ❬✶✺✱ ✶✼✱ ✶✾❪✳ ❇➯♥ ❝↕♥❤ ✤â✱ ❝→❝ t➼♥❤
❝❤➜t ✤✐➺♥ tû ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡ ♣❤ö t❤✉ë❝ r➜t ❧î♥ ✈➔♦ ❝→❝ ②➳✉ tè ❜➯♥

♥❣♦➔✐ ♥❤÷ ✤✐➺♥ tr÷í♥❣ ❤❛② ❜✐➳♥ ❞↕♥❣✳ ❈❤➥♥❣ ❤↕♥ ♥❤÷✱ ♥❣÷í✐ t❛ ✤➣ q✉❛♥
93

✶✺


s→t ✤÷ñ❝ sü ❝❤✉②➸♥ ♣❤❛ tø ❜→♥ ❞➝♥ s❛♥❣ ❦✐♠ ❧♦↕✐ tr♦♥❣ ✤ì♥ ❧î♣ ❙♥❙❡2
❦❤✐ ♥â ❝❤à✉ ↔♥❤ ❤÷ð♥❣ ❝õ❛ ❜✐➳♥ ❞↕♥❣ ♥➨♥ ❬✶✼❪ ❤❛② sü ❤②❞r♦ ❤â❛ ❜➲ ♠➦t
✤ì♥ ❧î♣ ❙♥❙❡2 ✤➣ ❞➝♥ ✤➳♥ sü t❤❛② ✤ê✐ ✤→♥❣ ❦➸ ❝→❝ t➼♥❤ ❝❤➜t ✤✐➺♥ tû
❝õ❛ ♥â✱ ✤➦❝ ❜✐➺t ❧➔ ✤ë rë♥❣ ✈ò♥❣ ❝➜♠ ❬✹✸❪✳ ✣➦❝ ❜✐➺t✱ ♥➠♥❣ ❧÷ñ♥❣ ✈ò♥❣
❝➜♠ ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ❜à ❣✐↔♠ ♠ët ❝→❝❤ ♥❤❛♥❤ ❝❤â♥❣ ❦❤✐ ♥â ❝❤à✉ ↔♥❤

❤÷ð♥❣ ❝õ❛ →♣ s✉➜t ❜➯♥ ♥❣♦➔✐ ❬✽❪✳ ❈→❝ t➼♥❤ t♦→♥ ❜➡♥❣ ❧þ t❤✉②➳t ♣❤✐➳♠
❤➔♠ ♠➟t ✤ë ❣➛♥ ✤➙② ❝ô♥❣ ❝❤➾ r❛ r➡♥❣✱ ✤ì♥ ❧î♣ ❙♥❙❡2 ❝â ✈ò♥❣ ❝➜♠ ❧î♥
❤ì♥ ❦❤→ ♥❤✐➲✉ s♦ ✈î✐ ✈ò♥❣ ❝➜♠ ❝õ❛ ❙♥❙❡2 ð ❞↕♥❣ ❦❤è✐ ❬✶✼❪✳ ❈ò♥❣ ✈î✐
✤â✱ ❜➡♥❣ ❝→❝ t➼♥❤ t♦→♥ sû ❞ö♥❣ ❧þ t❤✉②➳t ♣❤✐➳♠ ❤➔♠ ♠➟t ✤ë✱ ●♦♥③❛❧❡③
✈➔ ❖❧❡②♥✐❦ ✤➣ ❝❤♦ t❤➜② r➡♥❣ ♥➠♥❣ ❧÷ñ♥❣ ✈ò♥❣ ❝➜♠ ❝õ❛ ✈➟t ❧✐➺✉ ❙♥❙❡2
❣✐↔♠ ❦❤✐ t❛ ❣✐↔♠ ❜➲ ❞➔② ❝õ❛ ✈➟t ❧✐➺✉ ✭tù❝ ❧➔ ❣✐↔♠ sè ❧î♣✮ ✈➔ ❙♥❙❡2 ❝â
♥➠♥❣ ❧÷ñ♥❣ ✈ò♥❣ ❝➜♠ ❧î♥ ♥❤➜t ð tr↕♥❣ t❤→✐ ✤ì♥ ❧î♣ ❬✶✺❪✳ ❚✉② ♥❤✐➯♥✱
♥❤✐➲✉ ✈➜♥ ✤➲ tr♦♥❣ ❝→❝ t➼♥❤ ❝❤➜t ✤✐➺♥ tû ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ❝á♥ ❝❤÷❛
rã r➔♥❣ ✈➔ ❝➛♥ t✐➳♣ tö❝ ♥❣❤✐➯♥ ❝ù✉ ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣✳ ❚r♦♥❣ ❧✉➟♥
✈➠♥ ♥➔②✱ ❝❤ó♥❣ tæ✐ s➩ t✐➳♥ ❤➔♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ✤✐➺♥ tû ✈➔ ↔♥❤
❤÷ð♥❣ ❝õ❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜➯♥ ♥❣♦➔✐ ♥❤÷ ❜✐➳♥ ❞↕♥❣ ✈➔ ✤✐➺♥ tr÷í♥❣ ❧➯♥
❝→❝ t➼♥❤ ❝❤➜t ✤✐➺♥ tû ❝õ❛ ✤ì♥ ❧î♣ ❙♥❙❡2 ❜➡♥❣ ❧þ t❤✉②➳t ♣❤✐➳♠ ❤➔♠ ♠➟t
✤ë✳

✶✻


ữỡ
ị P
r ữỡ ú tổ tr ởt tt ữỡ
ỵ tt t ở ũ ự t
t tỷ ừ t Pữỡ
ú tữớ ữủ ũ tr ỵ tt
t ở ữ t ở ữỡ ữỡ
rt s rở ụ ữủ tr tr ữỡ
t ỵ tt t ở tr t srss
ữủ tr tt ụ ữỡ tỹ
ự ừ t

Pữỡ ỵ tt trú tỷ ừ

t r
r ữỡ ự tỹ
t t ữỡ ỵ tt trú tỷ tr strtr
tr t ữợ tỹ ừ ữỡ õ t ữủ
tr ởt ồ ữ s

t ữủ ừ ởt trú t rỗ s õ õ t
ỹ t t q

tố ữ õ trú tỷ ừ t
tr t trú õ ữủ ọ t ỳ t
tr t tr tố ữ õ trú ử tở



❝❤õ ②➳✉ ✈➔♦ ❣r❛❞✐❡♥t ❝õ❛ ♥➠♥❣ ❧÷ñ♥❣ ✭✤↕♦ ❤➔♠ ❜➟❝ ♥❤➜t ❝õ❛ ♥➠♥❣
❧÷ñ♥❣ t❤❡♦ ✈à tr➼ ❝õ❛ ❝→❝ ♥❣✉②➯♥ tû✮✳

• ❚❤ü❝ ❤✐➺♥ ❝→❝ t➼♥❤ t♦→♥ t➛♥ sè ❞❛♦ ✤ë♥❣ ❝õ❛ ❝→❝ ♣❤➙♥ tû ❞♦ sü
❝❤✉②➸♥ ✤ë♥❣ ❝õ❛ ❝→❝ ♥❣✉②➯♥ tû tr♦♥❣ ♣❤➙♥ tû✳ ❈→❝ t➛♥ sè ♥➔② ♣❤ö
t❤✉ë❝ ✈➔♦ ✤↕♦ ❤➔♠ ❜➟❝ ❤❛✐ ❝õ❛ ♥➠♥❣ ❧÷ñ♥❣ ❝â ❧✐➯♥ q✉❛♥ tî✐ ❝➜✉
tró❝ ♥❣✉②➯♥ tû✳
P❤÷ì♥❣ ♣❤→♣ ♥➔② ❞ü❛ tr➯♥ ❝ð sð ❝→❝ ✤à♥❤ ❧✉➟t ❝ì ❜↔♥ ❝õ❛ ❝ì ❤å❝
❧÷ñ♥❣ tû✳ ❚❤➟t ✈➟②✱ ❝❤ó♥❣ t❛ ❝â t❤➸ t❤✉ ✤÷ñ❝ ❝→❝ tr↕♥❣ t❤→✐ ❝õ❛ ❤➺ ❜➡♥❣
❝→❝❤ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❙❝❤r☎
♦❞✐♥❣❡r ❞ø♥❣✳ ❚✉② ♥❤✐➯♥✱ ✤è✐ ✈î✐ ❝→❝ ❤➺ ❝â
sè ❤↕t ❧î♥ t❤➻ ✈✐➺❝ ❣✐↔✐ ❝❤➼♥❤ ①→❝ ♥❣❤✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ ❙❝❤r☎
♦❞✐♥❣❡r ❧➔
❜➜t ❦❤↔ t❤✐✳ ❱➻ ✈➟②✱ ♣❤÷ì♥❣ ♣❤→♣ ❧þ t❤✉②➳t ❝➜✉ tró❝ ✤✐➺♥ tû ❝❤♦ ♣❤➨♣
❝❤ó♥❣ t❛ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❙❝❤r☎

♦❞✐♥❣❡r✳ ❈â ♥❤✐➲✉ ❝→❝❤ ✤➸
t✐➳♥ ❤➔♥❤ ❝→❝ t➼♥❤ t♦→♥ ❣➛♥ ✤ó♥❣ ♥➔②✳ ❈→❝ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥
✈➠♥ ♥➔② ✤÷ñ❝ t❤ü❝ ❤✐➺♥ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♣❤✐➳♠ ❤➔♠ ♠➟t ✤ë ✭❞❡♥s✐t②
❢✉♥❝t✐♦♥❛❧ t❤❡♦r② ✕ ❉❋❚✮✳ ❈❤ó♥❣ tæ✐ s➩ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝❤✐ t✐➳t
♣❤÷ì♥❣ ♣❤→♣ ♥➔② tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝õ❛ ❧✉➟♥ ✈➠♥✳

✷✳✷✳ ▲þ t❤✉②➳t ♣❤✐➳♠ ❤➔♠ ♠➟t ✤ë
▲þ t❤✉②➳t ♣❤✐➳♠ ❤➔♠ ♠➟t ✤ë ❧➔ ♠ët ❧þ t❤✉②➳t ✤÷ñ❝ ❞ò♥❣ ✤➸ ♠æ t↔
❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➺ ❡❧❡❝tr♦♥ tr♦♥❣ ❝→❝ ✈➟t r➢♥✱ tr♦♥❣ ♥❣✉②➯♥ tû✱ ♣❤➙♥
tû✳✳✳ ❞ü❛ tr➯♥ ❧þ t❤✉②➳t ❧÷ñ♥❣ tû✳ ▼➟t ✤ë ✤✐➺♥ tû ✤÷ñ❝ sû ❞ö♥❣ ✤➸ ♠æ
t↔ tr↕♥❣ t❤→✐ ❝õ❛ ❤➺ t❤❛② ✈➻ sû ❞ö♥❣ ❤➔♠ sâ♥❣✳ ❍➔♠ sâ♥❣ ♠æ t↔ ❤➺

N ✤✐➺♥ tû s➩ ♣❤↔✐ ❝❤ù❛ 3N ❜✐➳♥ tå❛ ✤ë✳ ❚r♦♥❣ ❦❤✐ ✤â✱ ♠➟t ✤ë ✤✐➺♥ tû
❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ ❜❛ ❜✐➳♥ tå❛ ✤ë ✤ë❝ ❧➟♣ ✈î✐ sè ✤✐➺♥ tû✳ ❱➻ t❤➳ ❦❤✐ ❣✐❛
t➠♥❣ sè ✤✐➺♥ tû ❝õ❛ ❤➺✱ ❤➔♠ sâ♥❣ s➩ trð ♥➯♥ ♣❤ù❝ t↕♣ ♥❤÷♥❣ ♠➟t ✤ë
✶✽


tỷ ổ t ờ số ỵ tt t ở õ
ữ ợ tr t t t t t ỵ ử t
t t tứ ỳ ữỡ tr rt ỡ ừ t ỵ ữủ tỷ
ỵ tt t ở ữủ sỷ ử ởt rở r
tr ở ỗ ồ tứ t ỵ õ ồ
ồ t t t



t ữủ tỷ ừ t r

r t r t t tỷ tỷ t t

ợ s ởt õ trt tỹ t t t ừ
t r ộ tỷ t ố ữủ ừ t ừ
ú t tữớ ọ q ố ữủ ừ tỷ
t ừ t r t ỵ tt ữủ tỷ õ t ữủ t
ữ s

h2

H =
2
1
+
8

i

2Ri h2

Mi
2

i=j

i

2Ri
1

me
4

e2
1
+
|ri rj | 8

i,j

e2 Zi2
|Ri rj |


tr õ Mi ố ữủ ừ t t tr Ri me ố ữủ
ừ tỷ t tr ri số tr ổ tự ở
ừ t tỷ số t t t ữỡ
t ỳ t tỷ ỳ tỷ ỳ t
t r ữủ t t tứ N tỷ tỷ t t
t tợ rt sỹ tữỡ t ỳ tỷ t
tỷ tỷ N ợ õ t t ừ t r
tổ q ữỡ tr rr ợ t ữ tr



ữ ổ t t tr õ t t
ự ở ú ợ t



ỹ tr ổ sr ữỡ
ú ữỡ

ổ t ổ sr tr õ
sõ ự t ừ N tỷ ữủ t t ởt ữủ
ợ ỡ ỡ tr ữỡ tr rr õ t
ở tỷ n(r) r ổ sr ở ừ
tỷ ữủ ữ ởt ừ t ở tỷ
tỷ ữủ ỵ tữ õ ữ tỷ ỗ t ỗ t
ổ tữỡ t õ t ở tỷ ợ t ở ữỡ
t ồ trữợ
ữủ ừ ữỡ õ tỷ
ữủ sỷ ổ tữỡ t ợ t ổ t tợ sỹ tr ờ
tữỡ q ừ tỷ r ữ r ữỡ
ú ữỡ sỹ tr ờ Pữỡ
ử ữủ ỳ ừ ổ s r
ữủ sỷ ử P ữủ tỷ
tr ởt trữớ t Vext (r) õ

d3 rn(r)5/3 +

EF T [n] =C1
+

1
2

d3 rd3 r

d3 rVext n(r) + C2

n(r)n(r )

|r r |

d3 rn(r)4/3 +


tr õ t t ú ữỡ ở
ợ C1 =

3
2 2/3

10 (3 )

t tự tr ờ ữỡ C2 =

43 (3/)1/3 t ố ũ ữủ rtr t



◆➠♥❣ ❧÷ñ♥❣ ✈➔ ♠➟t ✤ë tr↕♥❣ t❤→✐ ❝ì ❜↔♥ ❝â t❤➸ t➻♠ ✤÷ñ❝ ❜➡♥❣ ❝ü❝ t✐➸✉
❤â❛ ♣❤✐➳♠ ❤➔♠ EF T [n] tr♦♥❣ t♦➔♥ ❜ë ❝→❝ ♠➟t ✤ë ❦❤↔ ❞➽ n(r) ✤÷ñ❝ ①→❝
✤à♥❤ t❤æ♥❣ q✉❛ tê♥❣ sè ✤✐➺♥ tû t❤❡♦ ❜✐➸✉ t❤ù❝✿

d3 rn(r) = N

✭✷✳✸✮

▼➟t ✤ë ❡❧❡❝tr♦♥ ð tr↕♥❣ t❤→✐ ❝ì ❜↔♥ ✤÷ñ❝ rót r❛ tø ✤✐➲✉ ❦✐➺♥ ❝ü❝
t✐➸✉ ❝õ❛ ♣❤✐➳♠ ❤➔♠ ♥➠♥❣ ❧÷ñ♥❣✱ ❝❤➥♥❣ ❤↕♥ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♥❤➙♥
tû ▲❛❣r❛♥❣❡✳ ❑➳t q✉↔ ❝õ❛ ♣❤➨♣ ❣➛♥ ✤ó♥❣ ♥➔② ❦❤✐ →♣ ❞ö♥❣ ❝❤♦ ❝→❝ ❤➺

❡❧❡❝tr♦♥ tr♦♥❣ ♥❣✉②➯♥ tû✱ ♣❤➙♥ tû ❧➔ ❦❤→ ❤↕♥ ❝❤➳✳ ▼➦❝ ❞ò ❝❤♦ ❞→♥❣
✤✐➺✉ ❝õ❛ ♠➟t ✤ë ❡❧❡❝tr♦♥ t÷ì♥❣ ✤è✐ ❝❤➼♥❤ ①→❝ ✈➲ ♠➦t ✤à♥❤ t➼♥❤✱ ♥❤÷♥❣
✈➲ ♠➦t ✤à♥❤ ❧÷ñ♥❣ t❤➻ ❤♦➔♥ t♦➔♥ ❦❤æ♥❣ ♣❤ò ❤ñ♣✳ ❚ø ✤â ❞➝♥ ✤➳♥ ♥❤ú♥❣
❦➳t q✉↔ ♣❤✐ ✈➟t ❧þ✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ ❦❤æ♥❣ t❤➸ ♠æ t↔ ✤÷ñ❝ ❝➜✉ tró❝ ❧î♣
❝õ❛ ❡❧❡❝tr♦♥ tr♦♥❣ ♥❣✉②➯♥ tû✱ ❦❤æ♥❣ ❞➝♥ tî✐ ❧✐➯♥ ❦➳t ❤â❛ ❤å❝ tr♦♥❣ ♣❤➙♥
tû✱✳✳✳ ◆❤ú♥❣ ❦❤✉②➳t ✤✐➸♠ ♥➔② ♣❤➛♥ ❧î♥ ✤÷ñ❝ ❦❤➢❝ ♣❤ö❝ tr♦♥❣ ♣❤÷ì♥❣
tr➻♥❤ ❝õ❛ ❑♦❤♥ ✈➔ ❙❤❛♠✳ ✣✐➲✉ ♥➔② ✤➣ ❧➔♠ ♥➯♥ t❤➔♥❤ ❝æ♥❣ ❝õ❛ ❧þ t❤✉②➳t
❉❋❚✳

✷✳✷✳✸✳

❈→❝ ✤à♥❤ ❧þ ❍♦❤❡♥❜❡r❣✲❑♦❤♥

▼➟t ✤ë tr↕♥❣ t❤→✐ ❝õ❛ ✤✐➺♥ tû

▼➟t ✤ë tr↕♥❣ t❤→✐ ❝õ❛ ✤✐➺♥ tû n(r) ❧➔ ♠ët ❤➔♠ ❦❤æ♥❣ ➙♠ ❝õ❛ ❝→❝
❜✐➳♥ ❦❤æ♥❣ ❣✐❛♥ ✈➔ ❜à tr✐➺t t✐➯✉ ❦❤✐ ❞➛♥ r❛ ✈æ ❝ò♥❣✳ ▼➟t ✤ë tr↕♥❣ t❤→✐
❝õ❛ ✤✐➺♥ tû n(r) ❧➔ ♠ët ✤↕✐ ❧÷ñ♥❣ ❝â t❤➸ q✉❛♥ s→t ✈➔ ❝â t❤➸ ✤♦ ✤÷ì❝
❜➡♥❣ t❤ü❝ ♥❣❤✐➺♠ ✭❝❤➥♥❣ ❤↕♥ ♥❤÷ ❜➡♥❣ ♥❤✐➵✉ ①↕ t✐❛ ❳✮✳
▼➟t ✤ë tr↕♥❣ t❤→✐ ❝õ❛ ✤✐➺♥ tû ❣✐↔♠ t❤❡♦ t✐➺♠ ❝➟♥ ❝õ❛ ❤➔♠ ♠ô ❦❤✐
r❛ ①❛ t➜t ❝↔ ❝→❝ ❤↕t ♥❤➙♥ ✈➔ ❝â t❤➸ ✈✐➳t ❞÷î✐ ❞↕♥❣✿

√
n(r) ∼ exp(−2 I|r|)
✷✶

✭✷✳✹✮