ÂiÑÍÈÊ ÍÀÖiÎÍÀËÜÍÎÃÎ ÓÍiÂÅÐÑÈÒÅÒÓ journal of vivl...

Òåîðåòè÷íà i ïðèêëàäíà ôiçèêà

ÂIÑÍÈÊ ÍÀÖIÎÍÀËÜÍÎÃÎ ÓÍIÂÅÐÑÈÒÅÒÓ�ËÜÂIÂÑÜÊÀ ÏÎËIÒÅÕÍIÊÀ�

�Ôiçèêî-ìàòåìàòè÷íi íàóêè��804, 2014, c. 170�174

JOURNAL OF LVIV POLYTECHNICNATIONAL UNIVERSITY

�Physical and mathematical sciences��804, 2014, p. 170�174

ÅËÅÊÒÐÎÍÍÈÉ ÅÍÅÐÃÅÒÈ×ÍÈÉ ÑÏÅÊÒÐ ÊÐÈÑÒÀËÀ LiF

Cèðîòþê Ñ. Â., Êëèñêî Þ. Â.Íàöiîíàëüíèé óíiâåðñèòåò �Ëüâiâñüêà ïîëiòåõíiêà�

âóë. Ñ. Áàíäåðè 12, 79013, Ëüâiâ, Óêðà¨íà

(Îòðèìàíî 15 æîâòíÿ 2014 ð.)Ðîçðàõîâàíi åëåêòðîííi åíåðãåòè÷íi çîíè, çàãàëüíà i ïàðöiàëüíi ãóñòèíè åëåêòðîííèõ

ñòàíiâ êðèñòàëà LiF ç âèêîðèñòàííÿì ãðàäi¹íòíèõ ïîïðàâîê ó ôóíêöiîíàëi îáìiííî-êîðåëÿöiéíî¨ åíåðãi¨. Íà îñíîâi îòðèìàíèõ ðåçóëüòàòiâ áóëè çíàéäåíi êâàçi÷àñòèíêîâi ïî-ïðàâêè äëÿ îäíî÷àñòêîâèõ çîííèõ åíåðãié. Ðîçðàõóíêè âèêîíàíi çà ìåòîäîì ïðîåêöiéíèõïðè¹äíàíèõ õâèëü. Âèïðàâëåíå çíà÷åííÿ øèðèíè çàáîðîíåíî¨ çîíè äóæå äîáðå çiñòàâëÿ¹-òüñÿ ç âèìiðÿíèì åêñïåðèìåíòàëüíî.

Êëþ÷îâi ñëîâà: åíåðãåòè÷íèé ñïåêòð åëåêòðîíiâ, ïàðöiàëüíà ãóñòèíà ñòàíiâ, çàãàëüíà ãóñòèíàñòàíiâ, ôóíêöiÿ Ãðiíà, åíåðãiÿ êâàçi÷àñòèíîê.

ÓÄÊ: 537.311.322

Âñòóï

Ôòîðèä ëiòiþ LiF ¹ äóæå ïåðñïåêòèâíèì ÷óò-ëèâèì äî âèïðîìiíþâàííÿ ìàòåðiàëîì, ÿêèé øèðî-êî âèêîðèñòîâó¹òüñÿ â ðàäiàöiéíié äîçèìåòði¨, îïòî-åëåêòðîíiöi òà iíòåãðàëüíié îïòèöi [1]. Ìîíîêðèñòà-ëè LiF çàñòîñîâóþòü äëÿ îòðèìàííÿ ïàðàìåòðè÷íî-ãî ðåíòãåíiâñüêîãî âèïðîìiíåííÿ (PXR) [2], ÿêå ìà¹ïðàêòè÷íå çíà÷åííÿ â ìåäèöèíi [3]. Óñïiøíå çàñòî-ñóâàííÿ êðèñòàëà ïîòðåáó¹ ÿêîìîãà òî÷íiøèõ çíàíüïðî éîãî åëåêòðîííèé åíåðãåòè÷íèé ñïåêòð. Óÿâëåí-íÿ ïðî öi çíàííÿ äàþòü ðåçóëüòàòè òåîðåòè÷íèõ ðîç-ðàõóíêiâ øèðèíè çàáîðîíåíî¨ çîíè. �¨ âåëè÷èíà, îòðè-ìàíà çà äîïîìîãîþ ðiçíèõ ïiäõîäiâ, ìà¹ òàêi çíà÷åí-íÿ [4]: 8,9 åÂ � ó íàáëèæåííi ôóíêöiîíàëà ëîêàëü-íî¨ åëåêòðîííî¨ ãóñòèíè (LDA); 9,3 åÂ � ó íàáëèæåí-íi õàîòè÷íî¨ ôàçè (RPA), ïî¹äíàíî¨ ç òî÷íèì îáìií-íèì ïîòåíöiàëîì (EXX); 11,2 åÂ � ç òî÷íèì îáìií-íèì ïîòåíöiàëîì (EXX); 11,5 åÂ � ó íàáëèæåííi LDA,ïî¹äíàíîìó ç òî÷íèì îáìiííèì ïîòåíöiàëîì (EXX);13,5 åÂ � ó íàáëèæåííi GW [5], ÿêå ðóíòó¹òüñÿ íà çà-ñòîñóâàííi ìàòðèöåâîãî ïåðøîãî ïîðÿäêó òåîði¨ çáó-ðåííÿ, ïîáóäîâàíî¨ íà îñíîâi ôóíêöi¨ Ãðiíà G òà åêðà-íîâàíîãî ïîòåíöiàëà êðèñòàëà W. Îñòàííié ðåçóëüòàò¹ íàéáëèæ÷èì äî âèìiðÿíîãî çíà÷åííÿ [4], ÿêå äîðiâ-íþ¹ 14,2 åÂ.

Ìåòîþ äîñëiäæåííÿ ¹ ðîçðàõóíîê åëåêòðîííî¨ñòðóêòóðè êðèñòàëà LiF çà ìåòîäîì óçàãàëüíåíîãîãðàäi¹íòíîãî íàáëèæåííÿ (GGA) [6, 7] ôóíêöiîíà-ëà îáìiííî-êîðåëÿöiéíî¨ åíåðãi¨, ùî ¹ ïåðøèì åòàïîìðîáîòè. Äðóãèé åòàï ðóíòó¹òüñÿ íà ðåçóëüòàòàõ ïåð-øîãî i ïîëÿãà¹ ó çàñòîñóâàííi êâàçi÷àñòèíêîâîãî íà-áëèæåííÿ GW. Äëÿ äîñÿãíåííÿ öi¹¨ ìåòè íåîáõiäíîâèêîíàòè òàêi çàâäàííÿ: 1) âèêîíàòè çà äîïîìîãîþïðîãðàìè atompaw [8] ãåíåðóâàííÿ ïñåâäîïîòåíöià-

ëiâ ëiòiþ i ôòîðó ó ôîðìàëiçìi PAW; 2) ðîçðàõóâàòèñàìîóçãîäæåíi ãóñòèíó òà ïîòåíöiàë êðèñòàëà; 3) îá-÷èñëèòè âëàñíi çíà÷åííÿ åíåðãi¨ i õâèëüîâi ôóíêöi¨åëåêòðîíiâ ó êðèñòàëi; 4) îòðèìàòè ïàðöiàëüíi i ïîâ-íi ãóñòèíè åëåêòðîííèõ ñòàíiâ; 5) çíàéòè âèïðàâëåíiçîííi åíåðãi¨ åëåêòðîíiâ ó íàáëèæåííi GW çà äîïî-ìîãîþ ïðîãðàìè ABINIT [9].

Íèæ÷å êîðîòêî îïèñàíèé ôîðìàëiçì ìåòîäó ïðî-åêöiéíèõ ïðè¹äíàíèõ õâèëü PAW òà ðîçâèíåíî¨ íà¨õíié îñíîâi òåîði¨ GW. Ó ðîçäiëi II ïðîàíàëiçîâà-íî ðåçóëüòàòè ðîçðàõóíêó åëåêòðîííèõ åíåðãåòè÷íèõçîí òà ãóñòèí åëåêòðîííèõ ñòàíiâ. Ïåðåõîäèìî äî âè-êîíàííÿ îêðåñëåíèõ çàâäàíü.

I. Òåîðiÿ

A Âëàñíi ôóíêöi¨ é âëàñíi åíåðãi¨, îòðèìàíiçà äîïîìîãîþ ìåòîäó ïðîåêöiéíèõ ïðè¹ä-íàíèõ õâèëü

Ìåòîä ïðîåêöiéíèõ ïðè¹äíàíèõ õâèëü (PAW,projector augmented waves) [10] ïî¹äíó¹ îçíàêè ïñåâ-äîïîòåíöiàëà i âñååëåêòðîííîãî ìåòîäó ïðè¹äíàíèõïëîñêèõ õâèëü. Õâèëüîâà |ψn(r)⟩ òà ïñåâäîõâèëüîâà|ψ̃n(r)⟩ ôóíêöi¨ çâ'ÿçàíi ìiæ ñîáîþ òàê:

|ψn(r)⟩=|ψ̃n(r)⟩+∑a

∑i

(|ϕai (r)⟩−|ϕ̃ai (r)⟩)⟨p̃ai |ψ̃n⟩, (1)

äå |ϕai (r)⟩ � àòîìíà õâèëüîâà ôóíêöiÿ, |ϕ̃ai (r)⟩ � ïñåâ-äîõâèëüîâà ôóíêöiÿ, à ⟨p̃ai | � ôóíêöiÿ ïðîåêòîð. Ïiä-ñóìîâóþòü â (1) çà ñôåðàìè ïðè¹äíàííÿ, ÿêi íóìå-ðóþòüñÿ iíäåêñîì a, à iíäåêñ i = {n, l,m} âiäïîâiäà¹êâàíòîâèì ÷èñëàì � ãîëîâíîìó, îðáiòàëüíîìó é ìà-ãíiòíîìó, âiäïîâiäíî.

c⃝ Cèðîòþê Ñ. Â., Êëèñêî Þ. Â., 2014

Lviv Polytechnic National University Institutional Repository http://ena.lp.edu.ua

Åëåêòðîííèé åíåðãåòè÷íèé ñïåêòð êðèñòàëà LiF

Ç ðiâíÿííÿ (1) áà÷èìî, ùî

|ψn(r)⟩ = τ |ψ̃n(r)⟩, (2)

äå îïåðàòîð τ ïåðåòâîðþ¹ ïñåâäîõâèëüîâó |ψ̃n(r)⟩ óâñååëåêòðîííó õâèëüîâó ôóíêöiþ |ψn(r)⟩.

ßâíà ôîðìà îïåðàòîðà âèïëèâà¹ ç ðiâíÿííÿ (1):

τ = 1 +∑a

∑i

(|ϕai ⟩ − |ϕ̃ai ⟩)⟨p̃ai |. (3)

Ñòàöiîíàðíå ðiâíÿííÿ Øðåäiíãåðà

H|ψnk⟩ = |ψnk⟩εnk, (4)

äå H � îïåðàòîð Ãàìiëüòîíà, à |ψnk⟩ � âñååëåêòðîííàõâèëüîâà ôóíêöiÿ, ç óðàõóâàííÿì (2) íàáóâà¹ òàêîãîâèãëÿäó:

τ+Hτ |ψ̃nk⟩ = τ+τ |ψ̃nk⟩εnk, (5)

ó ÿêîìó øóêàíèé ñïåêòð åëåêòðîíiâ εnk òàêèé ñàìèé,ÿê i â ðiâíÿííi (4). Òóò n îçíà÷à¹ íîìåð åíåðãåòè÷íî¨çîíè, à k � âåêòîð ç ïåðøî¨ çîíè Áðèëëþåíà.

B Êâàçi÷àñòèíêîâi åíåðãi¨, îòðèìàíi çà äîïî-ìîãîþ ôóíêöi¨ Ãðiíà

Íà îñíîâi çíàéäåíèõ ó ïiäõîäi PAW âëàñíèõ åíåð-ãié εnk òà ôóíêöié ñòàíó ψnk(r) (4) áóäó¹òüñÿ ôóí-êöiÿ Ãðiíà â ðåàëüíîìó ïðîñòîði òà â óÿâíîìó ÷àñi [5],

GLDA(r, r′; iτ)=

iocc∑nk

Ψnk(r)Ψ∗nk(r

′) exp(εnkτ), τ>0,

−iunocc∑nk

Ψnk(r)Ψ∗nk(r

′) exp(εnkτ), τ<0.

(6)Òåïåð ìîæåìî çàïèñàòè ó íàáëèæåííi õàîòè÷íî¨

ôàçè (RPA) íåçâiäíèé îïåðàòîð ïîëÿðèçàöi¨ â ðåàëü-íîìó ïðîñòîði òà óÿâíîìó ÷àñi

P 0(r, r′; iτ) = −iGLDA(r, r′; iτ)GLDA(r′, r;−iτ). (7)

Çíàéøîâøè ïåðåòâîðåííÿ Ôóð'¹ äëÿ P 0 ó îáåðíå-íîìó ïðîñòîði òà äëÿ óÿâíî¨ åíåðãi¨, áóäó¹ìî ñèìåòðè-÷íó äiåëåêòðè÷íó ìàòðèöþ â îáåðíåíîìó ïðîñòîði

ε̃(k,G,G′; iω)=δGG′− 4π

|k+G||k+G′|P 0(k,G,G′; iω).

(8)Äàëi âèêîíó¹ìî iíâåðñiþ ñèìåòðè÷íî¨ äiåëåêòðè-

÷íî¨ ìàòðèöi äëÿ êîæíî¨ k òî÷êè i êîæíî¨ óÿâíî¨åíåðãi¨ é îá÷èñëþ¹ìî åêðàíîâàíó Êóëîíîâó âçà¹ìî-äiþ â îáåðíåíîìó ïðîñòîði çà òàêîþ ôîðìóëîþ:

W (k,G,G′; iω)=4π

|k+G||k+G′|ε̃−1(k,G,G′; iω).

(9)Íàñòóïíèì êðîêîì ¹ ïåðåòâîðåííÿ Ôóð'¹ W äî

ðåàëüíîãî ïðîñòîðó i óÿâíîãî ÷àñó. Òåïåð ìîæíà îá-÷èñëèòè îïåðàòîð âëàñíî¨ åíåðãi¨

Σ(r, r′; iτ) = iG(r, r′; iτ)W (r, r′; iτ). (10)

Äàëi âèçíà÷à¹ìî ñåðåäíi çíà÷åííÿ âëàñíî¨ åíåðãi¨

⟨Ψnk|Σ(iτ)|Ψnk⟩. (11)

Íàðåøòi, çíàõîäèìî êâàçi÷àñòèíêîâi ïîïðàâêè äîâëàñíèõ çíà÷åíü LDA ó ïåðøîìó ïîðÿäêó òåîði¨ çàçáóðåííÿì ⟨Σ−V LDAxc ⟩ . Äëÿ öüîãî çàñòîñîâó¹ìî ðîç-âèíåííÿ Òåéëîðà äëÿ Σ(ω) âiäíîñíî ω = εLDAnk

εqpnk = εLDAnk +1

Znk⟨Ψnk|Σ(εLDAnk )− Vxc|Ψnk⟩, (12)

äå ñòàëà íîðìóâàííÿ

Znk = 1− d

dω⟨Ψnk|Σ(ω)|Ψnk⟩ |ω=εLDAnk

(13)

Ñàìîóçãîäæåíi âëàñíi åíåðãi¨ εnk òà ôóíêöi¨ ñòàíóΨnk(r) êðèñòàëà LiF ó íàáëèæåííi LDA áóëè âèêîðè-ñòàíi ÿê íóëüîâå íàáëèæåííÿ äëÿ ïîäàëüøîãî ðîçðà-õóíêó òî÷íiøèõ êâàçi÷àñòèíêîâèõ åíåðãié εqpnk ó ôîð-ìàëiçìi GW.

II. Ðåçóëüòàòè òà ¨õ îáãîâîðåííÿ

Íà ðèñ. 1 çîáðàæåíi çàêîíè äèñïåðñi¨, îòðèìàíiçà äîïîìîãîþ òåîði¨ PAW ó íàáëèæåííi GGA, îïè-ñàíîìó ôîðìóëàìè (1)�(5). Çàóâàæó¹ìî, ùî êðèñòàëLiF ¹ ïðÿìîçîííèì, i éîãî ìiæçîííà ùiëèíà ðîçòà-øîâàíà â òî÷öi Ã (GM). Çíà÷åííÿ ìiæçîííèõ ùiëèííàïiâïðîâiäíèêiâ é äiåëåêòðèêiâ, îòðèìàíi ó íàáëè-æåííi LDA òà éîãî âäîñêîíàëåíîìó âàðiàíòi GGA,¹ çàíèæåíèìè ïîðiâíÿíî ç âèìiðÿíèìè åêñïåðèìåí-òàëüíî. Çíà÷åííÿ ïðÿìî¨ ìiæçîííî¨ ùiëèíè â òî÷öiÃ (GM) 9,13 åÂ, òîäi ÿê åêñïåðèìåíòàëüíå çíà÷åííÿäîðiâíþ¹ 14,2 åÂ. Îòæå, îòðèìàíå â îäíî÷àñòèíêîâî-ìó ïiäõîäi çíà÷åííÿ øèðèíè çàáîðîíåíî¨ çîíè ìåíøåâiä åêñïåðèìåíòàëüíîãî íà 5,07 åÂ. Âiäíîñíà ïîõèáêàöüîãî âàðiàíòà ðîçðàõóíêó äîðiâíþ¹ 35, 7 %.

Íà ðèñ. 2 çîáðàæåíi çàêîíè äèñïåðñi¨, îòðèìàíi çàäîïîìîãîþ òåîði¨ PAW ó íàáëèæåííi GW, îïèñàíî-ìó ôîðìóëàìè (6)�(13). Çíà÷åííÿ ïðÿìî¨ ìiæçîííî¨ùiëèíè â òî÷öi Ã (GM) 14,00 åÂ, òîäi ÿê åêñïåðè-ìåíòàëüíå çíà÷åííÿ äîðiâíþ¹ 14,2 åÂ. Îòæå, îòðè-ìàíå â áàãàòî÷àñòèíêîâîìó ïiäõîäi çíà÷åííÿ øèðèíèçàáîðîíåíî¨ çîíè ìåíøå âiä åêñïåðèìåíòàëüíîãî íà0,2 åÂ. Âiäíîñíà ïîõèáêà öüîãî âàðiàíòà ðîçðàõóíêó1, 4 %.

Çàêîíè äèñïåðñi¨, çîáðàæåíi íà ðèñ. 1, 2, íå äàþòüçìîãè çðîáèòè âèñíîâêè ïðî ñòðóêòóðó åëåêòðîííèõåíåðãåòè÷íèõ çîí êðèñòàëà. Òîìó áóëè îá÷èñëåíi ãó-ñòèíè åëåêòðîííèõ ñòàíiâ, ÿêi ìè ïðîàíàëiçó¹ìî.

Íà ðèñ. 3, 4 íàâåäåíi ïàðöiàëüíi ãóñòèíè åëåêòðîí-íèõ ñòàíiâ íà àòîìàõ ëiòiþ òà ôòîðó, âiäïîâiäíî. Áà-÷èìî, ùî äíî âàëåíòíî¨ çîíè (åíåðãi¨ ïîáëèçó 20 åÂ)óòâîðþþòü ïåðåâàæíî s-ñòàíè F, à âíåñêè s-, p- òàd-ñòàíiâ Li äîâîëi ìàëi. Ïåðåõîäèìî áëèæ÷å äî ñòåëiâàëåíòíî¨ çîíè, òîáòî â iíòåðâàë åíåðãié åëåêòðîíiâ−3, 0 ≤ E ≤ 0, 0 åÂ. Òóò äîìiíóþòü p-ñòàíè F, à âíå-ñêè s-, p- òà d-ñòàíiâ Li òàêîæ äîâîëi ìàëi.

Theoretical and applied physics 171


Cèðîòþê Ñ. Â., Êëèñêî Þ. Â.

-20

0

20

40

k

E(k), eV

GMMXGM

Ðèñ. 1. Åëåêòðîííèé åíåðãåòè÷íèé ñïåêòð êðèñòàëàLiF, îòðèìàíèé â íàáëèæåííi GGA

-20

0

20

40

k

E(k), eV

GMMXGM

Ðèñ. 2. Åëåêòðîííèé åíåðãåòè÷íèé ñïåêòð êðèñòàëàLiF, îòðèìàíèé â íàáëèæåííi GW

-20 -10 0 10 20

0.0

0.2

0.4

st./eV

E-EF

, eV

Li, s

-20 -10 0 10 20

0.0

0.5

1.0

Li, p

-20 -10 0 10 20

0.00

0.01

0.02

Li, d

Ðèñ. 3. Ïàðöiàëüíi ãóñòèíè åëåêòðîííèõ ñòàíiâó êðèñòàëi LiF íà àòîìi ëiòiþ, îòðèìàíi

â íàáëèæåííi GGA

-20 -10 0 10 20

0

5

10

st./eV

E-EF, eV

F, s

-20 -10 0 10 20

0

2

4

6

F, p

-20 -10 0 10 20

0.0

0.1

0.2

F, d

Ðèñ. 4. Ïàðöiàëüíi ãóñòèíè åëåêòðîííèõ ñòàíiâó êðèñòàëi LiF íà àòîìi ôòîðó, îòðèìàíi

â íàáëèæåííi GGA

172 Òåîðåòè÷íà i ïðèêëàäíà ôiçèêà


Åëåêòðîííèé åíåðãåòè÷íèé ñïåêòð êðèñòàëà LiF

Ñòðóêòóðà ñòàíiâ çîíè ïðîâiäíîñòi âàæëèâà äëÿàíàëiçó îïòè÷íèõ âëàñòèâîñòåé êðèñòàëiâ. Ñòàíè çî-íè ïðîâiäíîñòi õàðàêòåðèçóþòüñÿ åíåðãiÿìè, âèùèìè9,13 åÂ. Íà âiäìiíó âiä âàëåíòíî¨ çîíè, çàñåëåíî¨ åëå-êòðîíàìè, ïóñòà çîíà ïðîâiäíîñòi õàðàêòåðèçó¹òüñÿs- òà p-ñòàíàìè Li, ç íåâåëèêîþ äîìiøêîþ p-ñòàíiâ Fòà d-ñòàíiâ Li é F. Òàêå çíà÷íå äîìiíóâàííÿ ó âàëåí-òíié çîíi ñòàíiâ F ìîæíà çðîçóìiòè, âçÿâøè äî óâàãèòå, ùî ëiòié ìà¹ ëèøå îäèí âàëåíòíèé åëåêòðîí, òîäiÿê ôòîð äà¹ ó âàëåíòíó çîíó ñiì åëåêòðîíiâ.

-20 -10 0 10 20

0

2

4

6

E-EF

, eV

st./eV

LiF

Ðèñ. 5. Ïîâíà ãóñòèíà åëåêòðîííèõ ñòàíiâó êðèñòàëi LiF, îòðèìàíà â íàáëèæåííi GGA

Ïiñëÿ àíàëiçó ðèñ. 3, 4 ñòà¹ çðîçóìiëîþ ñòðóêòó-ðà ïîâíî¨ ãóñòèíè ñòàíiâ, çîáðàæåíà íà ðèñ. 5. Âiä-ëiê åíåðãi¨ ïðîâàäèòüñÿ âiä ðiâíÿ Ôåðìi. Áà÷èìî,ùî âàëåíòíà çîíà ëîêàëiçîâàíà â äîâîëi âóçüêèõ ií-òåðâàëàõ åíåðãié åëåêòðîíiâ −20, 0 ≤ E ≤ 0, 0 åÂ.Ïîòiì çàáîðîíåíà çîíà, ïðîìiæîê åíåðãi¨ ÿêî¨0, 0 ≤ E ≤ 9, 13 åÂ. Çîíà ïðîâiäíîñòi õàðàêòåðèçó¹-òüñÿ åíåðãiÿìè E > 9, 13 åÂ.

Âèñíîâêè

Ðîçðàõîâàíi åëåêòðîííi åíåðãåòè÷íi ñïåêòðè éãóñòèíè åëåêòðîííèõ ñòàíiâ øèðîêîçîííîãî äiåëåêò-ðèêà LiF. Çíàéäåíî, ùî â îäíîåëåêòðîííîìó íàáëè-æåííi GGA çíà÷åííÿ øèðèíè çàáîðîíåíî¨ çîíè ìåí-øå, íiæ âèìiðÿíå íà 5,07 åÂ. Îòðèìàíå ó íàáëèæåííiGW çíà÷åííÿ øèðèíè çàáîðîíåíî¨ çîíè êðèñòàëà LiFäîðiâíþ¹ 14,0 åÂ, ùî ïðåêðàñíî çiñòàâëÿ¹òüñÿ ç åêñ-ïåðèìåíòàëüíèì çíà÷åííÿì 14,2 åÂ. Çíà÷åííÿ ïîâ-íî¨ øèðèíè âàëåíòíî¨ çîíè äîðiâíþ¹ 19,89 (GGA) òà21,88 åÂ (GW). Çíà÷åííÿ øèðèíè âàëåíòíî¨ çîíè ïî-áëèçó ¨¨ äíà äîðiâíþ¹ 0,70 (GGA) òà 0,80 åÂ (GW).Çíà÷åííÿ øèðèíè âàëåíòíî¨ çîíè ïîáëèçó ¨¨ ñòåëiäîðiâíþ¹ 3,00 (GGA) òà 3,95 åÂ (GW). Ïîðiâíÿ¹ìîîòðèìàíå íàìè çíà÷åííÿ øèðèíè çàáîðîíåíî¨ çîíè14,00 åÂ ç îòðèìàíèìè íåùîäàâíî çíà÷åííÿìè [11]13,24 åÂ (LDA+GW) òà 15,81 åÂ (QP+GW), QP �êâàçi÷àñòèíêîâå íàáëèæåííÿ òåîði¨. Ìîæëèâî, êðà-ùèé ðåçóëüòàò îòðèìàíèé íàìè çàâäÿêè âèêîðèñòàí-íþ íà ïåðøîìó åòàïi ðîçðàõóíêó ìåòîäó PAW+GGA(ôîðìóëè (1)�(5)), òîäi ÿê àâòîðè [11] áóäóâàëè ïåð-øèé åòàï íà çáåðiãàþ÷èõ íîðìó ïñåâäîïîòåíöiàëàõ óôîðìàëiçìi LDA.

Ëiòåðàòóðà

[1] Somma F., Aloe P., d'Acapito F., Montereali R. M.,Polosan S., Secu M., Vincenti M. A. Optical and X-rayabsorption spectroscopy in lead doped lithium �uori-de crystals, 11th Europhysical Conference on Defects inInsulating Materials (EURODIM 2010) IOP PublishingIOP Conf. Series: Materials Science and Engineering,2010, 15, 012035(5).

[2] Sones B., Danon Y., Block R. Lithium �uoride (LiF)crystal for parametric X-ray (PXR) production, Nucl.Instr. and Meth. in Phys. Res. B., 2005, 227, 22�31.

[3] Piestrup M. A., Wu X., Kaplan V. V., Uglov S. R.,Cremer J. T., Rule D. W., Fiorito R. B. A design ofmammography units using a quasimonochromatic x-raysource, Rev. Scienti�c Inst., 2001, 72, 2159-2170.

[4] Gruning M., Marini A., Rubio A. Density functionalsfrom many-body perturbation theory: The band gap forsemiconductors and insulators // J. Chem. Phys., 2006,124, 154108(9).

[5] Arnaud B., Alouani M. Implementation of an all-electron GW Approximation using the Projector

Augmented Wave method: I. Formulation and appli-cation to the electronic structure of semiconductors //Phys. Rev. B, 2000, 62, 4464�4476.

[6] Perdew J. P., Burke K., Ernzerhof M. GeneralizedGradient Approximation Made Simple // Phys. Rev.Letters. � 1996. � 77. � No. 18. � P. 3865�3868.

[7] Ernzerhof M., Scuseria G. E. Assessment of thePerdew-Burke-Ernzerhof exchange-correlation functi-onal // J. Chem. Phys. 1999, 110, 5029�5036.

[8] Tackett A. R., Holzwarth N. A. W., Matthews G. E.Computer Phys. Comm., 2001, 135, 348.

[9] Gonze X., Amadon B., Anglade P.-M., Beuken J.-M., Bottin F., Boulanger P., Bruneval F., Caliste D.,Caracas R., Cote M., Deutsch T., Genovese L., GhosezPh., Giantomassi M., Goedecker S., Hamann D.R.,Hermet P., Jollet F., Jomard G., Leroux S., ManciniM., Mazevet S., Oliveira M. J. T., Onida G., Poui-llon Y., Rangel T., Rignanese G.-M., Sangalli D.,Shaltaf R., Torrent M., Verstraete M. J., Zerah G.,Zwanziger J. W. ABINIT: First-principles approach

Theoretical and applied physics 173


Cèðîòþê Ñ. Â., Êëèñêî Þ. Â.

of materials and nanosystem properties. ComputerPhysics Communications, 2009, 180, 2582�2615.

[10] Blochl P. E. Projector augmented-wave method, Phys.Rev., 1994, B 50 , No. 23, 17953-17979.

[11] Botti S., Marques M. A. L. Strong renormalizationof the electronic band gap due to lattice polarizationin the GW formalism // Phys. Rev. Lett., 2013, 110,226404(5).

ÝËÅÊÒÐÎÍÍÛÉ ÝÍÅÐÃÅÒÈ×ÅÑÊÈÉ ÑÏÅÊÒÐÊÐÈÑÒÀËËÀ LiF

Ñèðîòþê Ñ. Â., Êëûñêî Þ. Â.Íàöèîíàëüíûé óíèâåpñèòåò �Ëüâèâñêà ïîëèòýõíèêà�

óë. Ñ. Áàíäåpû, 12, 79013, Ëüâîâ, Óêðàèíà

Ðàññ÷èòàíû ýëåêòðîííûå ýíåðãåòè÷åñêèå çîíû, îáùàÿ è ïàðöèàëüíûå ïëîòíîñòè ýëå-êòðîííûõ ñîñòîÿíèé êðèñòàëëà LiF ñ èñïîëüçîâàíèåì ãðàäèåíòíûõ ïîïðàâîê â ôóíêöèî-íàëå îáìåííî-êîððåëÿöèîííîé ýíåðãèè. Íà îñíîâå ïîëó÷åííûõ ðåçóëüòàòîâ áûëè íàéäåíûêâàçè÷àñòè÷íûå ïîïðàâêè äëÿ îäíî÷àñòè÷íûõ çîííûõ ýíåðãèé. Ðàñ÷åòû âûïîëíåíû ìåòî-äîì ïðîåêöèîííûõ ïðèñîåäèíåííûõ âîëí. Èñïðàâëåííàÿ øèðèíà çàïðåùåííîé çîíû î÷åíüõîðîøî ñîïîñòàâëÿåòñÿ ñ ýêñïåðèìåíòàëüíî èçìåðåííûì çíà÷åíèåì.

Êëþ÷åâûå ñëîâà: ýíåðãåòè÷åñêèé ñïåêòð ýëåêòðîíîâ, ïàðöèàëüíàÿ ïëîòíîñòü ñîñòîÿíèé,îáùàÿ ïëîòíîñòü ñîñòîÿíèé, ôóíêöèÿ Ãðèíà, ýíåðãèÿ êâàçè÷àñòèö.

ÓÄÊ: 537.311.322

ELECTRONIC ENERGY BAND SPECTRUMOF THE CRYSTAL LiF

Syrotyuk S. V., Klysko Yu. V.Lviv Polytechnic National University

12, S. Bandery Str., Lviv, 79013, Ukraine

Electronic energy bands, total and partial density of electronic states of the crystal LiF withgradient corrections in the functional of the exchange-correlation energy have been evaluated.On the basis of obtained results the quasiparticle corrections for the single-particle band energi-es have been calculated. The calculations are performed by projector augmented waves. Thecorrected band gap is very good if compared with the experimentally measured value.

Key words: electron energy spectrum, partial density of states, total density of states, Green'sfunction, quasiparticle energy.

UDK: 537.311.322

174 Òåîðåòè÷íà i ïðèêëàäíà ôiçèêà


ÂiÑÍÈÊ ÍÀÖiÎÍÀËÜÍÎÃÎ ÓÍiÂÅÐÑÈÒÅÒÓ journal of vivl...

Documents