beqa guramis Ze...

ivane javaxiSvilis saxelobis Tbilisis saxelmwifo universiteti

beqa guramis Ze boWoriSvili

kvanturi wertilebis eleqtronuli mdgomareobebisa da kvanturi wertilebis eleqtronuli mdgomareobebisa da kvanturi wertilebis eleqtronuli mdgomareobebisa da kvanturi wertilebis eleqtronuli mdgomareobebisa da zonaTaSorisi gadasvlebis Sesabamisi oscilatoruli Zalebis zonaTaSorisi gadasvlebis Sesabamisi oscilatoruli Zalebis zonaTaSorisi gadasvlebis Sesabamisi oscilatoruli Zalebis zonaTaSorisi gadasvlebis Sesabamisi oscilatoruli Zalebis

kvleva sasrul elementTa meTodiT kvleva sasrul elementTa meTodiT kvleva sasrul elementTa meTodiT kvleva sasrul elementTa meTodiT

zust da sabunebismetyvelo mecnierebaTa fakulteti fizikis mimarTuleba

s a d o q t o r o d i s e r t a c i a

programis xelmZRvaneli: fiz.-maT. mecn. doqtori, Tsu sruli profesori aleqsandre Sengelaia

samecniero xelmZRvaneli, fizikis mecnierebaTa doqtori, Tesalonikis aristoteles universitetis asocirebuli profesori xariton polatoglu

Tbilisi 2010

avtoris stili daculia

2

s a r C e v is a r C e v is a r C e v is a r C e v i

Sesavali - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 3

Tavi pirveli Tavi pirveli Tavi pirveli Tavi pirveli

$1.1 naxevargamtaruli kvanturi wertilebi mimoxilva da motivacia- - - - - -7

$1.2 sasrul elementTa meTodi Sredingeris gantolebisaTvis - -- - - - - - -17

$1.3 zogierTi martivi kvanturi sistemis energetikuli speqtri - - - - - - -21

Tavi Tavi Tavi Tavi meoremeoremeoremeore

$2.1 sferuli, cilindruli da konusuri kvanturi wertilebi - - - - - - - -23

$2.2 sferuli kvanturi wertilebis talRuri funqciebi - - - - - - - - - - - 27

$2.3 cilindruli kvanturi wertilebis talRuri funqciebi - - - - - - - - - 29

$2.4 konusuri kvanturi wertilebis talRuri funqciebi - - - - - - - - - - - 32

$2.5 fuZisa da simaRlis cvlilebis gavlena konusuri kvanturi

wertilebis eleqtronul mdgomareobebze - - - - - - - - - - - - - - - - - - - - 35

$2.6 urTierTqmedi / cilindruli kvanturi

wertilebi - xelovnuri molekula - - - - - - - - - - - - - - - - - - - - - - - - - 42

$2.7 kvanturi wertilebis zonaTaSorisi gadasvlebis

Sesabamisi oscilatoruli Zalebi - - - - - - - - - - - - - - - - - - - - - - - - - 50

Tavi Tavi Tavi Tavi mesamemesamemesamemesame

$3.1 kvanturi wertili kvanturi ormo (QDQW) - - - - - - - - - - - - - - - - - - 57

$3.2 QDQW sistemis talRuri funqciebi - - - - - - - - - - - - - - - - - - - - - - 61

$3.3 QDQW-is eleqtron-xvrelis kulonuri urTierTqmedeba - - - - - - - - - 65

$3.4 QDQW-is zonaTaSorisi gadasvlebis oscilatoruli Zalebi - - - - - - 67

miRebuli SedegebimiRebuli SedegebimiRebuli SedegebimiRebuli Sedegebi da daskvnda daskvnda daskvnda daskvnebiebiebiebi - - - - - - - - - - - - - - - - - - - - - - - - - - 69

danarTidanarTidanarTidanarTi - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -72

literaturliteraturliteraturliteraturaaaa - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -75

3

SesavaliSesavaliSesavaliSesavali

nano mecnierebebi es is mecnierebebia, romlebic dakavSirebuli arian nanometruli zomiT gansazRvrul nivTierebis kvlevasTan da manipulaciasTan. sityva nano aq metris erT memiliarded anu 10-9 nawils aRniSnavs. ukanaskneli ori aTwleulis ganmavlobaSi nano mecnierebebi gansakuTrebulad swrafad viTardebian. maTi ganviTarebis tempis kargi maCvenebelebia im statiaTa da maTi citirebebis ricxvi, romelTa sakvanZo sityvac “nano”-s Seicavs. msoflios mraval universitetsa da kvleviT institutSi uamravi adamiani muSaobs sxvadasxva nano mecnierebaSi. rogorc Cveulebriv xdeba xolme ama Tu im mecnierebis ganviTarebas didwilad sazogadoebriv industriuli moTxovna ganapirobebs, nanomecnierebebis SemTxvevaSic swored asea. mecnierul-teknikuri progresis pirobebSi mzardi moTxovnebis dakmakofilebis erT-erTi mniSvnelovani faqtori maRal teqnologiebze modis. am ukanasknelis mniSvnelovani Semadgeneli nawili ki swored nanoteqnologiebia. “nano” termini saocrad multi-disciplinarulia da igi iseT sxvadasxva mecnierebebs aerTianebs, rogorebicaa magaliTad, fizika, qimia, inJineria, biologia, medicina, kompiuteruli mecniereba da a. S. nanoteqnologiaSi nanomecnierebebis da inJineriis saSualebiT Seqmnili axali struqturebi, masalebi da mowyobilobebi moiazreba. miCneulia, rom istoriulad nanoteqnologiis pirvelad winaswarmetyveleba cnobil amerikel mecniers riCard feinmans ekuTvnis, roca man gaaJRera Tavisi cnobili fraza [1]: “There is a plenty of room at the bottom” romelic qarTulad daaxloebiT Semdegnairad SeiZleba iTargmonos: “iq qvemoT didi adgilia”. nanoteqnologiis gamoyenebis pirveli magaliTebi naxevargamtaruli heterostruqturebi da nanostruqturebi iyo, romelTac meorenairad mcire ganzomilebian struqturebsac uwodeben. es sistemebi warmodgenili arian 2, 1 an 0 sivrcul ganzomilebaSi, aseT sistemebs myari sxeulebis fizikaSi kvantur kedlebs (quantum well), kvantur zonrebs (quantum wire) da kvantur wertilebs (quantum dot) uwodeben Sesabamisad [2-9]. aRniSnuli mcire zomis sistemebis gavrecelebis areali mxolod myari sxeulebis fizikiT ar Semoifargleba, aramed igi aseve gvxvdeba iseT dargebSi rogoricaa magaliTad: qimia [10-13], metalurgia [14-15], katalizi [16-17] da sxv.

nano zomis nimuSebs gaaCniaT zogierTi iseTi Tviseba, romelic Sesabamisi nivTierebis makroskopul nimuSs ar axasiaTebs. swored am Tvisebebisa da mcire zomebis gamo rac Tavis mxriv xelsawyos an aqtiuri elementis zomis, nivTierebisa da energiiis dazogvis karg SesaZleblobas iZleva, nanofizika uaRresad aqtualuria Tanamedrove maRali teqnologiebis Seqmnis saqmeSi. mcire zomis sistemebis upiratesoba eleqtronikaSi resursebis dazogvis garda maTi moqmedebis maRali siCqarecaa rac Zalian principulia. naxevargamtaruli kvanturi wertilebi (semiconductor quantum dots) erT-erTi jgufia zemoT aRniSnuli nanosistemebisa, isini warmoadgenen SedarebiT viwro zonian naxevargamtarebs (TviT kvanturi wertili) romlebic “Cadgmuli” arian ufro farTo zonian naxevargamtarSi.

4

kvantur wertilebis kvleva aqtualuria maTi sxvadasxva sferoSi gamoyenebis mizniT magaliTad Suq diodebSi [18-19], biologiur sistemebSi [20], lazerebisaTvis [21], kvantur kompiuterebSi [22], ganaxlebadi energiebis mimarTulebiT [23] da sxv. G

iseve rogorc fizikis yvela mimarTulebiT, Teoriuli kvlevebi nano fizikaSic did rols asruleben. Teoriul gaTvlebs gansakuTrebuli mniSvneloba eniWeba vinaidan maTi gamoyenebis Sedegad SesaZlebelia winaswar, fizikuri sistemis eqperimentalurad Seqmnis gareSe nawinaswarmetyvelebi iqnas sistemis esa Tu is Tviseba da miTiTebuli iqnas is parametrebi, romelic unda qondes sistemas ama Tu im Tvisebis misaRebad. xSiria SemTxveva, rodesac Teoriulad nawinaswarmetyvelebia sruliad axali Tviseba an movlena, romelic adre eqsperimentalurad damzerili ar yofila. kompiuteruli resursebis swrafma ganviTarebam didad Seuwyo xeli kvlevebs nano fizikis dargSic. dResdReobiT SesaZlebelia nanosistemebis sxvadasxva Tvisebebis kvleva sruliad Teoriulad - yovelgvari eqsperimentaluri parametris gamoyenebis gareSe [24-25]. Tumca aseTi gamoTvlebi did kompiuterul resursebs moiTxoven da amitom xSirad nanosistemebis kvlevisas mimarTaven sxvadasxva miaxloebebs da modelebs rogorebicaa magaliTad efeqturi masis miaxloeba [26-28], simkvrivis potencialis Teoria (density functional theory, DFT) [29], empiriuli fsevdopotencialis meTodi [30], Zlieri bmis meTodi [31-32] da sxv. zemoT CamoTvili meTodebidan naSromSi gamokvleuli sakiTxebis Sesaswavlad efeqturi masis miaxloeba yvelaze martivi, gamosayeneblad mosaxexebuli da amave dros efeqturi miaxloebaa, radgan misi gamoyenebisaTvis sakmarisia sistemis ramodenime parametris codna rogorebicaa nawilakTa efeqturi masebi, akrZaluli zonis sigane da potencialebi, aseve es miaxloeba saSualebas gvaZlevs martivad gamoviyenoT sasrul elementTa meTodi sxvadasxva geometriuli formis mqone sistemebisaTvis; aseve gamoTvlebisaTvis kompiuteruli resursebic SedarebiT mcire moiTxoveba, vidre sxva miaxloebebis SemTxvevaSi. yovelive amis Sesaxeb dawvrilebiTaa naTqvami 1.2$-Si. rac Seexeba iseT mZlavr meTodebs rogoricaa simkvrivis potentialis Teoria (spT) da empiriuli fsevdopotencialis meTodi (efm) maTi ZiriTadi sisuste naSromSi Seswavlili sakiTxebis SemTxvevaSi aris is, rom spT ar gvaZlevs saSualebas vipovoT sistemis talRuri funqciebi, xolo efm-s gamosayeneblad garda konkretuli empiriuli monacemebis codnisa aseve saWiroa am meTodis gamoyeneba daxrili potencialis SemTxvevaSi (kvanturi wertili kvanturi ormos Sesaswavlad - naSromis me-3 Tavi) rac sakmaod rTuli gasakeTebelia. unda iTqvas rom spT da efm didi sizustiT gamoirCevian da zogadad efeqturi masis miaxloebaze zustad muSaoben, Tumca rogorc qveviT SevniSnavT efeqturi masis miaxloeba naSromSi gamokvleuli sistemebisaTvis sakmaod zust Sedegebs iZleva. winamdebare naSromSi Teoriulad Seswavlilia sxvadasxva Sedgenilobisa da geometriuli formis kvanturi sistemebis (kvanturi wertili - quantum dot, kvantur wertiluri molekula - quantum dot molecule, kvanturi wertili kvanturi ormo - quantum dot quantum well) energetikuli speqtri, talRuri funqciebi da zonaTSorisi gadasvlebis Sesabamisi oscilatoruli simZlavreebi, aseve eqsitonTa kulonuri urTierTqmedeba. gamoTvlebi Sesrulebulia koordinatze damokidebul efeqturi masis

5

miaxloebaSi momvlebi talRuri funqciis gamoyenebiT, Seswavlilia sistemebis sxvadasxva saintereso Tvisebebi da gamokvleulia sistemebis yofaqceva maTi geometriuli parametrebis cvlilebisas. miuxedavad imisa, rom kvanturi wertilebis Tvisebebis maT zomebze damokidebuleba sakmaod kargad aris Seswavlili, kvlevebis simcirea kvanturi wertilebis formaze damokidebulebis Seswavlis mxriv. vinaidan kvanturi wertilebi mcire zomis obieqtebia, amitom maTi zedapiris farTobsa da formas didi zegavlenis moxdena SeuZlia maT Tvisebebze. am mizniT naSromSi Seswavlilia identuri moculobisa da sxvadasxva geometriuli formis (sferuli, cilindruli da konusuri) mqone kvanturi wertilebi da gamokvleulia maTi Tvisebebis damokidebuleba geometriul formebze. Seswavlili kvanturi sistemebis mdgomareobis sakuTari mniSvnelobebis (energiebis) da sakuTari funqciebis (talRuri funqciebis) gamosaTvlelad winamdebare naSromSi gamoyenebulia sasrul elementTa meTodi, romlis saSualebiTac amoxsnilia Sredingeris stacionaluri gantoleba sferul da cilindrul koordinatebSi konkretuli sistemis simetriis Sesabamisad. simetriis Sedegad Sredingeris gantolebis ganzomileba erTi (cilindruli simetriis SemTxveva) an ori (sferuli simetriis SemTxveva) ganzomilebiT mcirdeba da 3 ganzomilebiani amocana or (cilindruli simetriis SemTxveva) da erT (sferuli simetriis SemTxveva) ganzomilebian amocanaze daiyvaneba, rac mniSvnelovanad amartivebs amocanis amoxsnas kompiuteruli resursebis TvalsazrisiT. sasrul elementTa meTodis upiratesoba aseve mdgomareos mis moxerxebulobaSi aRweros sxvadasxva geometriuli formebis kvanturi sistemebi da saSualebas gvaZlevs pirdapir gaviTvaliswinoT, potencialisa da nawilakTa efeqturi masebis koordinatze damokidebuleba. naSromSi Seswavlilia daxrili potencialis mqone kvanturi wertili kvanturi ormo, romelic axali tipis sistemas warmoadgens. gamokvleulia Tu rogor aris damokidebuli sistemis energetikuli speqtri da zonaTaSorisi gadasvelbis Sesabamisi oscilatoruli Zalebi potencialis daxris siganeze. aseve Sefasebulia sistemisaTvis eleqtron-xvrelis kulonuri urTierTqmedeba. miRebuli Sedegebi sainteresoa sistemis sxvadasxva Tvisebebis marTvis TvalsazrisiT.

6

TTTT aaaa vvvv iiii pppp iiii rrrr vvvv eeee llll iiii

7

$1.1 $1.1 $1.1 $1.1 naxevargamtaruli kvanturi wertilebi mimoxilva da naxevargamtaruli kvanturi wertilebi mimoxilva da naxevargamtaruli kvanturi wertilebi mimoxilva da naxevargamtaruli kvanturi wertilebi mimoxilva da

motivaciamotivaciamotivaciamotivacia

nanostruqturebSi SeiZleba arsebobdes SezRudva muxtis Tavisufali matareblebis moZraobisa erTi an meti mimarTulebiT. muxtis matarebelebis moZraoba moculobiT kristalSi xdeba Tavisuflad yvela mimarTlebiT, egred wodebul kvantur ormoebSi (kvantur kedlebSi) moZraoba SezRudulia - 1, kvantur zonrebSi - 2, kvantur wertilebSi ki samive mimarTulebiT. Tu rogor aris muxtis matarebelTa mdgomareobaTa simkvrive damokidebuli maTi moZraobis SezRudvaze Tvisobrivad damokidebulia, imaze Tu moZraobis ra SezRudvebia Sesaswavl nanostruqturaSi [33].

ganvixiloT moculobiTi (3D) kristali, am SemTxvevaSi eleqtronebi (Tu eleqtronebze visaubrebT) warmoadgenen Tavisufal nawilakebs da isini efeqturi masis miaxloebaSi Sredingeris gantolebis mixedviT aRiwerebian Semdegi talRuri funqciiT:

Ψ , , (1)

xolo energia

!"#"$%& !"

$%& '($ ) ($ ) ($* (2)

sadac # talRuri veqtoria, +& -eleqtronis efeqturi masa, ! plankis dayvanili mudmiva, , , , , , – kristalis Sesabamisad , , – ReZebis gaswvriv zomebia. (1) paraboluri dispersiis damokidebulebis SemTxvevaSi izoenergetikuli zedapiri warmoadgens ( radiusian sferos. Tu gamoviyenebT born-karmanis sasazRvro pirobebs maSin # talRuri veqtoris komponentebisaTvis miiReba:

( $- ( $-

$ ( $- / , $, / 1,2,3, ..

vinaidan 4( sisqis sferuli Sris moculoba 45 47($4( tolia, am moculobaze mosuli 48 mdgomareobaTa ricxvi spinis gaTvaliswinebiT (spinis gamo 2 ze vamravlebT) aris:

48 2 · 47($4(27/ ($4(

7$

(2)-s gaTvaliswinebT mdgomareobaTa simkvrivisaTvis vRebulobT:

: /; <=<> $%&?

"$-"!? $@ (3)

rogorc vxedavT mdgomareobaTa simkvrive moculobiTi kristalis

SemTxvevaSi $@ kanonis mixedviT izrdeba (nax. 1.1). ganvixiloT 2A SemTxveva roca eleqtronebi garda RerZis gaswvriv mimarTulebisa moZraoben kristaluri mesris periodul velSi, xolo RerZis gaswvriv maTi moZraoba SezRudulia B potencialiT:

B 0 Tu 0 D D E B ∞ Tu 0 G G E

am SemTxvevaSi eleqtronTa energia Sedgeba Tavisuflad moZravi eleqtronis energiisa da RerZis gaswvriv datyvevebuli eleqtronis energiaTa jamisagan:

8

', (, (* !"$%& '($ ) ($* ) !"

$%& H-IJ$ $ (4)

sadac 1,2,3, … (4) gamosaxulebis mixedviT 2A SemTxvevaSi eleqtronTa energia

ganisazRvreba ', ( , (* ricxvebiT.

4( sifarTis rgolis farTobi L 27($;4($; vinaidan erT mdgomareobas

( ( , ( ) sibrtyeSi Seesabameba farTobi: 4L $-" , amitom eleqtronul

mdgomareobaTaA ricxvi rgolSi, mosuli kristalis moculobis erTeulze (spinis gaTvaliswinebiT) iqneba:

48 $MN<M "O<"O

-I <"O"$-I (5)

(4)-is Sesabamisad gvaqvs:

($;$ $%&!" P Q (6)

aq Q !"$%& H-

IJ$ $ - -uri qvezonis Sesabamisi energia. (5) da (6)-is

gaTvaliswinebiT 2A mdgomareobis simkvrivisaTvis vRebulobT:

: $; ∑ <=<> %&

-I!" ∑ Θ P QQ (7)

Θp hevisaidis safexurovani funqciaa, Θp 1 Tu U V 0 da Θp 0 Tu U D 0. (7) ajamva warmoebs qvezonebis mixedviT. zogjer (7)-s Semdegi saxiT warmoadgenen:

: $; %&-I!" WX >

>YZ (8)

sadac WX >>YZ aris mTeli nawili X >

>Y -sa da udris qvezonebis ricxvs. (7) da

(8) dan Cans, rom 2A SemTxvevaSi nebismier qvezonaSi mdgomareobaTa

simkvrive mudmivia da araa damokidebuli energiaze. roca sidide WX >>YZ

icvleba erTi erTeuliT, anu icvleba qvezona, maSin energia icvleba naxtomiseburad da amitom : $; damokidebulebas aqvs safexuriani saxe (nax. 1.1). Tu : $; damokidebulebas gamovsaxavT moculobiTi kristalis - : /; mdgomareobaTa simkvriviT (3) gamosaxulebis Sesabamiad, miviRebT:

: $; : /;WX [

[YZX [

[Y (9)

ukanaskneli gamosaxulebidan Cans, rom roca Q , maSin 2A sistemis kvantur mdgomareobaTa simkvrive moculobiTi kristalis kvantur mdgomareobaTa simkvrivis tolia (es wertilebi aRniSnulia , \, ] nax.1.1-ze).

9

nax. 1.1 moculobiTi kristalisa (nax. 1.1 moculobiTi kristalisa (nax. 1.1 moculobiTi kristalisa (nax. 1.1 moculobiTi kristalisa (^_`a) da kvanturi ormos () da kvanturi ormos () da kvanturi ormos () da kvanturi ormos (^_ba) eleqtronul ) eleqtronul ) eleqtronul ) eleqtronul mdgomareobaTA simkvrivis damikidebuleba energiaze mdgomareobaTA simkvrivis damikidebuleba energiaze mdgomareobaTA simkvrivis damikidebuleba energiaze mdgomareobaTA simkvrivis damikidebuleba energiaze

gamoviTvaloT mdomareobaTa simkvrive 1A SemTxvevisaTvis (kvanturi zonari) anu roca eleqtronTa moZraoba SezRudulia ori mimarTulebiT da maT Tavisufali moZraoba SeuZliaT mxolod erTi RerZis gaswvriv, am RerZad Cven zogadobis SeuzRudavad RerZs avirCevT. zemoT ganxiluli SemTxvevis msgavsad eleqtronTa energiisaTvis gvaqvs:

, +, ( !"$%& ($ ) !"

$%& WH-<J$ +$ ) H-

IJ$ $Z (10)

4 da E Sesabamisad da RerZis gaswvriv potencialuri ormoebis siganeebia. +, 1,2,3, …romlebic axasiaTeben qvezonebs.

mdgomareobaTa ricxvi 4( intervalze, mosuli moculobis erTeulze aris:

48 4(527 ,@ 4(

27E4

(10)-is mixedviT gvaqvs:

( X$%&!" ' P Q,%* sadac, Q,% !"

$%& WH-<J$ +$ ) H-

IJ$ $Z qvezonis energia

mocemuli da +-iT.

amgvarad gvaqvs

4( √$%&<>$!X'>d>e,f*

(11)

10

(11)-is gaTvaliswinebiT mdgomareobaTa simkvrivis (mosuli moculobis erTeulze) energiaze damokidebuleba 1A SemTxvevaSi Semdegnairad moicema:

: ; √$%&-!I< ∑ ∑ g'>d>e,f*

>d>e,fQ% (12)

(12)-is miRebisas gaTvaliswinebulia spini da is, rom 4 erT intervals Seesabameba TiToeuli qvezonis ori intervali: h( romelTaTvisac

' P Q,%* G 0.

(12)-is mixedviT Cans, rom calkeul qvezonaSi energiis zrdiT

mdgomareobaTa simkvrive ' P Q,%*dY" saxiT mcirdeban (nax.1.1) roca Q,%

maSin mdgomareobaTa simkvrive : ; usasruloba xdeba. Tumca, unda aRiniSnos, rom nebismieri energiis intervalis sasruli mniSvnelobisaTvis dasaSveb mdgomareobaTa ricxvi sasrulia.

nax. 1.2 kvanturi zonaris mdgomareobaTnax. 1.2 kvanturi zonaris mdgomareobaTnax. 1.2 kvanturi zonaris mdgomareobaTnax. 1.2 kvanturi zonaris mdgomareobaTa a a a simkvrivis damokidebuleba energiazesimkvrivis damokidebuleba energiazesimkvrivis damokidebuleba energiazesimkvrivis damokidebuleba energiaze

dabolos, ganvixiloT 0A sistema, sadac eleqtronTa moZraoba

SezRudulia samive mimarTulebiT, aseT sistemebs kvanturi wertilebi

ewodebaT. zemoT ganxilulis msgavsad radgan muxtis matareblebi

datyvevebuli arian samive mimarTulebiT, energia yvela mimarTulebiT

dakvantulia:

, +, i !"$%& WH-

<J$ +$ ) H-IJ$ $ ) H-

jJ$ i$Z (13)

sadac 4 , E, k kvanturi wertilis zomebia, , +, i 1,2,3, … qvezonis nomrebia.

(13) aCvenebs rom kvantur wertilSi eleqtronebi Rebuloben energies

mxolod da mxolod diskretul mniSvnelobebs, anu gagvaCnia

11

mdgomareobaTa simkvrivis mkveTri pikebi, romlebic , +, i fiqsirebuli

mniSvnelobebisaTvis SegviZlia aRvweroT dirakis delta funqciiT:

: l; $j<I ∑ ∑ ∑ m P Q,%,nn%Q (14)

nax. 1.3 kvanturi wertilis mdgomareobaTnax. 1.3 kvanturi wertilis mdgomareobaTnax. 1.3 kvanturi wertilis mdgomareobaTnax. 1.3 kvanturi wertilis mdgomareobaTa a a a simkvrivis damokidebuleba energiazesimkvrivis damokidebuleba energiazesimkvrivis damokidebuleba energiazesimkvrivis damokidebuleba energiaze

naxevargamtaruli kvanturi wertilebi, romlebsac xSirad xelovnur atomebsac uwodeben warmoadgenen 0 ganzomilebian naxevargamtarul nanosistemebs, sadac erTi tipis naxevargamtari, romlis akrZaluli zona mcirea, “Cadgmulia” ufro farTo akrZaluli zonis mqone naxevargamtarSi an nawilobriv imyofeba vakuumSi. sxvadasxva naxevargamtarTa savalento da gamtareblobis zonaTa sxvaobis gamo kvanturi wertilebi warmoadgenen kvantur ormoebs maTSi myofi eleqtronebisa da xvrelebisaTvis da swored am potencialuri ormos arsebobis Sedegia kvanturi wertilebis diskretuli energetikuli speqtri da mdgomareobaTa simkvrivis xazovani saxe (nax. 1.1) kvanturi datyvevebis efeqti kvantur wertilebSi kvanturi meqanikis gamovlenis erT-erTi TvalsaCino magaliTia, amitomac Teoriuli TvalsazrisiT kvanturi wertilebis kvleva kvanturi meqanikis fundamentaluri sakiTxebis Seswavlis unkikalur SesaZleblobebs qmnis. kvanturi wertilebis zoma SeiZleba ramdenime nanometridan ramdenime aseul nanometramde icvlebodes, xolo eleqtronebis raodenoba kvantur wertilebSi - erTidan ramodenime aTasamde. TavianTi zomebisa da Tvisebebis gamo SeiZleba iTqvas, rom kvanturi wertilebi warmoadgenen Sualedur sistemebs atomebsa da makroskopul nimuSebs Soris. arseboben rogorc sxvadasxva nivTierebisagan Sedgenili, aseve sxvadasxva geometriuli zomis da formis kvanturi wertilebi [34-37].

nax. 1.4–ze warmodgenilia ramodenime wertilis gamosaxuleba

nax. nax. nax. nax. 1.1.1.1.4444 sxvadasxva formis kvanturi wertilebisxvadasxva formis kvanturi wertilebisxvadasxva formis kvanturi wertilebisxvadasxva formis kvanturi wertilebi

arsebobs kvantur wertilTa zonebis ori tipis ganlageba ix. nax. 1.5. I tipis zonaTa ganlagebisas xvrelebi sivrcis erTsa da imaveganlagebisas – eleqtronebi da xvrelebi sivrculubnebSi arian lokalizebuli. winamdebare naSromSi yvela ganxiluli kvanturi sistema I tipisaa zonaTa ganlagebis mixedviT.

nax. nax. nax. nax. 1.1.1.1.5555 aaaa))))

ze warmodgenilia ramodenime sxvadasxva formis

sxvadasxva formis kvanturi wertilebisxvadasxva formis kvanturi wertilebisxvadasxva formis kvanturi wertilebisxvadasxva formis kvanturi wertilebi [3[3[3[38888]]]]....

arsebobs kvantur wertilTa zonebis ori tipis ganlageba - IzonaTa ganlagebisas SemTxvevaSi eleqtronebi da

xvrelebi sivrcis erTsa da imave ubanSi imyofebian. II tipis zonaTa eleqtronebi da xvrelebi sivrculad gancalkevebul

ubnebSi arian lokalizebuli. winamdebare naSromSi yvela ganxiluli kvanturi sistema I tipisaa

zonaTa ganlagebis mixedviT.

nax. nax. nax. nax. 1.1.1.1.5555 bbbb))))

12

sxvadasxva formis kvanturi

....

I da II tipi [39] eleqtronebi da

ubanSi imyofebian. II tipis zonaTa ad gancalkevebul

winamdebare naSromSi yvela ganxiluli kvanturi sistema I tipisaa

13

nax.nax.nax.nax. 1.1.1.1.5555 aaaa)))) –––– I tipis bI tipis bI tipis bI tipis b) ) ) ) –––– IIIIIIII tipis zonaTatipis zonaTatipis zonaTatipis zonaTa ganlageba kvantur wertilebSi. ganlageba kvantur wertilebSi. ganlageba kvantur wertilebSi. ganlageba kvantur wertilebSi. _^– akrZaluli zona,akrZaluli zona,akrZaluli zona,akrZaluli zona, _o– gamtareblobis zona,gamtareblobis zona,gamtareblobis zona,gamtareblobis zona, _p – savalento zona,savalento zona,savalento zona,savalento zona, qr– gamtareblobis gamtareblobis gamtareblobis gamtareblobis

zonaTa Soris sxvaobazonaTa Soris sxvaobazonaTa Soris sxvaobazonaTa Soris sxvaoba, qs– savalento zonaTa Soris sxvaobasavalento zonaTa Soris sxvaobasavalento zonaTa Soris sxvaobasavalento zonaTa Soris sxvaoba

imisaTvis, rom Teoriulad Seswavlili iqnas kvanturi wertilebi mimarTaven sxvadasxva miaxloebiT meTodebs radgan kvanturi wertilebi warmoadgenen mravali nawilakisagan Sedgenil rTul sistemebs, amas emateba kvanturi wertilebis geometriuli formis da sasazRvro pirobebis gamo gamowveuli sirTuleebi. zogadad, kvanturi wertilebis modelirebas axdenen Semdegi etapebis mixedviT: a) struqturis modelireba zomis, formisa da Semadgenlobis mixedviT b) eleqtronebisa da xvrelebis erTnawilakovani Tvisebebis povna xorcieldeba Sredingeris gantolebis amoxsniT – efeqturi masis, k.p, Zlieri bmis an fsevdopotencialis gamoyenebiT. g) xdeba erTnawilakovan modelze nawilakTa Soris kulonuri urTierTqmedebis damateba SeSfoTebis Teoriis, TviTSeTanxmebuli velis an egredwodebuli urTierTqmedebis sruli amoxsnis mixedviT. sruli amoxsna xorcieldeba kvanturi montekarlos gziT an konfiguraciuli urTierTqmedebis gamoyenebiT [40]. Tumca k.p, Zlieri bmis da fsevdopotencialis miaxloebebi ufro zustad aRweren fizikur rialobas isini moiTxoven gacilebiT met kompiuterul resursebs. rac Seexeba winamdebare naSroms aq kvanturi wertilebis Teoriuli kvlevisas gamoiyeneba erTnawilakobrivi Sredingeris gantoleba sasruli potencialisa da koordinatze damokidebuli efeqturi masiT [26-28,41]. Sredingeris gantolebis amoxsna xorcieldeba sasrul elementTa meTodiT, romelic Sesaswavli sistemis simetriis Sesabamisad warmodgenilia erT da or ganzomilebaSi. gantolebis ganzomilebis Semcirebis Sedegad mkveTrad mcirdeba misi amoxsnisaTvis saWiro kompiuteruli resursebi. cnobilia, rom kvanturi wertilebisaTvis akrZaluli zonis sigane damokidebulia kvanturi wertilebis zomaze es ukanaskneli izrdeba kvanturi wertilis zomis SemcirebiT. Tumca Teoriuli kvlevis Sedegebi, romlebic miiReba efeqturi masisi miaxloebisa da sasruli simaRlis mqone potencialuri barieris gamoyenebiT winamdebare naSromSi ganxiluli zomis kvanturi wertilebisaTvis karg TanxvedraSia Sesabamis eqsperimentalur monacemebTan [42-44]. aRsaniSnavia, rom efeqturi masis miaxloeba ver iZleva zust Sedegebs Zalian mcire kvanturi wertilebisaTvis, Tumca naSromSi gamokvleuli zomis kvanturi wertilebisaTvis aRniSnuli miaxloeba kargi sizustiT gamoirCeva. ase rom naSromSi gamokvleuli sistemebisaTvis, iseTi martivi meTodis gamoyeneba, rogorsac efeqturi masis miaxleoba warmoadgens, sakmaod zusti Sedegebis miRebis saSualebas gvaZlevs, rac zogadad ufro kompleqsuri miaxloebebis gamoyenebiT miiRweva. literaturaSi vxvdebiT kvanturi wertilebis Teoriuli kvlvebis sxva modelebs, roca damatyvevebeli potenciali warmoadgens sxvadasxva potencialebis jams. magaliTad, cilindruli simetriis mqone 3 ganzomilebiani eleqtronTa sistemis - kvanturi wertilis potenciali

14

SeiZleba modelirebuli iqnas rogorc ori: gaswvrivi da ganivi potencialis jami: Btu Bv , ) Bw . Sedegad eleqtronis talRuri funqcia iqneba ori: ganivi da grZivi talRuri funqciis namravli

Ψtu Ψv , · Ψw Bv , da Bw potencialebs garkveuli konkretuli saxe gaaCniaT [45]. rogorc ukve avRniSneT nanostruqturebis mcire zomebis gamo maTi uSualo Seswavla xdeba kvanturi meqanikis gamoyenebiT. marTalia nano sistemebi mcire zomisani arian, magram maTi simcire fardobiTia. Tu ramdenad mcired SegviZlia ganvixiloT esa Tu is nanosistema damokidebulia gansaxlivel konteqstze. nanostruqturebis Seswavlisas mniSvnelovan rols TamaSobs makroskopuli sistemis eqsitonis boris radiusi. Tu kvanturi sistemis zoma naklebia eqsitonis boris radiusze, maSin sistemaSi datyvevebuli nawilakebis eleqtronul mdgomareobebs mniSvnelovnad ganapirobebs damatyvevebeli potenciali, xolo nawilakTa kulonuri urTierTqmedeba nakleb mniSvnelovan rols TamaSobs. winamdebare naSromSi Seswavlili kvanturi wertilebisaTvis kvanturi efeqti arsebiTad ganapirobebs sistemis eleqtronul mdgomareobebs xolo nawilakTa kulonuri urTierTqmedeba Tavisi simciris gamo ugulvebelyofilia. naSromis bolo TaviSi axali tipis sistemisaTvis rogorsac daxrili kvanturi wertili kvanturi ormo (QDQW) warmoadgens Sefasebulia nawilakTa kulonuri urTierTqmedeba, romlis efeqtic erTi rigiT mcire aRmoCnda damatyvevebeli potencialTan SedarebiT. winamdebare naSromSi Teoriulad gamokvleulia Semdegi saxis naxevargamtaruli kvanturi wertilebi: CdSe/ZnSe, InAs/GaAs da HHgS/CdS romelTa miRebis eqsperimentaluri xerxebi dReisaTvis kargadaa damuSavebuli [36, 46-54]. dReisaTvis kargadaa Seswavlili aRniSnuli naxevargamtaruli kvanturi wertilebis sxvadasxva struqturuli, eleqtronuli, optikuri da sxva Tvisebebi.

vinaidan kvantur wertilebSi kvantur efeqtebTan gvaqvs saqme, cxadia kvanturi wertilebis Tvisebebi Zlierad aris damokidebuli maT zomebze. aqamde Seswavlil sxvadasxva Semadgenlobis, zomis da formis kvanturi wertilebis Soris erT-erTi yvelaze kargad gamokvleul sistemas Seadgenen sferuli kvanturi wertilebi usasrulo potencialis SemTxvevaSi [55]. naCvenebia, rom aseTi sistemis amonaxsnebi anu nawilakTa energetikuli doneebi beselis sferuli funqciebis nulebis [56] proporciulebi arian, anu donis nomris zrdasTan erTad kvadratulad izrdebian. igive energiebi sistemis radiusis kvadratis ukuproporciulad mcirdebian. aseve kargadaa Seswavlili sferuli kvanturi wertilebi sasruli potencialis da koordinatze damokidebuli efeqturi masis gaTvaliswinebiT [57]. garda umartivesi sferuli simetriis mqone kvanturi wertilebisa aseve Seswavlilia diskis, linzis, konusis da piramidis formis kvanturi wertilebi [58-61].

kvanturi wertilebSi datyvevebul nawilakTa energiis damokidebuleba maT zomebze zogadi fizikuri mosazrebiTac gasagebia, vinaidan rac mcirea nawilakTa lokalizaciis are miT metia maTi energia da piriqiT, es uSualod gamomdinareobs haizenbergis ganusazRvrelobis principidan impulsa da koordinats Soris.

15

garda geometriuli zomebisa kvanturi wertilebis Tvisebebi aseve damokidebulia maT geometriul formebzec. [62-63]-Si Seswavlilia sxvadasxva formis Sesabamisad: cilindruli, elifsuri da konusuri erTis mxriv da cilindruli, elifsuri, elifsuri linzis, wakveTili sferuli da konusuri formis kvanturi wertilebi – meores mxriv. miRebulia da Sedarebulia energetikuli speqtris: ZiriTadi i 0 da aRgznebuli i 1 (sadac i - orbitaluri kvanturi ricxvia) donis damokidebuleba kvanturi wertilebis moculobaze samive ganxiluli geometriuli formis SemTxvevaSi. kvlevebi Catarebulia iseTnairad, rom TiToeuli kvanturi wertilis fuZis radiusebi erTmaneTis tolia iseve rogorc maTi simaRleebi. aRniSnuli konfiguraciisTvis (cilindruli, elifsuri, elifsuri linzis, wakveTili sferuli da konusuri formis kvanturi wertilebi) kvanturi wertilis yoveli fiqsirebuli moculobisaTvis energiebi zrdis mixedviT (mciredan didisken) kvanturi wertilebis formebis mixedviT Semdegnairad arian ganlagebuli: konusuri, wakveTili sferuli, elifsuri linzis da diskis formis. [64]-s avtorTa mier gamokvleulia cilindruli, konusuri da linzis formis kanturi wertilebis mdgomareobebis sakuTari mniSvnelobebi da funqciebi. Tumca gamokvleul sxvadasxva zomis kvantur wertilebs gansxvavebuli moculobebi gaaCniaT amitom ver visaubrebT kvanturi wertilebis geometriuli formis gavlenaze, radganac rogorc zeviT ukve aRvniSneT kvanturi wertilebis Tvisebebi Zlierad aris damokidebuli maT zomaze. interess imsaxurebs [65], radganac aq naCvenebia gza, romlis mixedviTac SesaZlebelia energetikuli speqtris analitikuri gziT miReba, naCvenebia rom nawilakTa energias cilindris parametrebze fesvuri xasiaTis damokidebuleba gaaCnia Tumca unda aRvniSnoT, rom am naSromSi gamoyenebulia usasrulo potenciali rac arafizikuria Sedegad aseTi miaxloeba naklebad zusti Sedegebis momcemia gansakuTrebiT es iTqmis mcire zomis kvanturi wertilebize. sainteresoa Sromaa Sesrulebuli [59]-Si, sadac Seswavlilia cilindruli da konusuri kvanturi wertilebis eleqtronuli mdgomareobebi da aseve Seswavlilia damasvelebeli fenis (wetting layer) gavlena am mdgomareobebze. naCvenebia rom damasvelebeli fena Zlierad moqmedebs kvanturi wertilebis eleqtronul mdgomareobebze. aq Seswavlil cilindrul da konusur da aseve wakveTili konusur da piramidul kvanturi wertilebs sxvadasxva moculobebi gaaCniaT amitom eleqtronul mdgomareobaTa gansxvaveba gamowveulia kvantur wertilTa ara mxolod sxvadasxva geometriuli formis aramed sxvadasxva moculobebis qoniTac. [66]-Si Seswavlilia CdSe cilindruli kvanturi wertilis energetikuli doneebis damokidebuleba sxvadasxva fuZe simaRlis fardobisas. 5 nm is radiusis kvanturi wertilisaTvis avtorebi cvlian simaRles 4 dan 14 nm-mde. warmodgenili umciresi energetikuli doneebis Sesabamisi energia izrdeba simaRlis zrdis mixedviT, es zrda Tavidan swrafad xdeba xolo Semdeg neldeba. garda ramodenime umartivesi kvantur-meqanikuri sistemisa romelTa nawili $1.3 Si aris warmodgenili analitikurad aseve ixsneba cilindruli kvanturi ormos SemTxvevac sasruli potencialis pirobebSi efeqturi masis koordinatze damokidebulebisas. am SemTxvevaSi radialuri

16

talRuri funqciebi beselis pirveli da meore rigis funqciebs warmoadgenen, rac Seexeba energiebs isini miiRebian sasazRvro pirobebis Sesabamisi gantolebebis ricxviTi amoxsniT [67], Tumca zemoT Tqmuli Seexeba mxolod ZiriTad mdgomareobas. xolo zogad SemTxvevaSi amocana analitikurad saerTod (arc talRuri funqciebisaTvis) ar ixsneba. rac Seexeba konusuri kvanturi ormos SemTxvevas (konusuri kvanturi wertili, romelic aseve Seswavlilia winamdebare naSromSi) misTvis arc talRuri funqciebi da arc energetikuli doneebi analizuri gziT ar ixsneba. rogorc viciT, sxeulis xazovani zomis SemcirebasTan erTad izrdeba fardoba am sxeulis zedapiris farTobsa da moculobas Soris. es ki niSnavs, rom rac ufro mcirdeba sxeulis zoma, sxeulis zedapirs miT meti gavlena aqvs am sxeulis Tvisebebze. nanonawilakebisaTvis TavianTi mcire zomidan gamomdinare zedapirs didi mniSveneloba eniWeba, vinaidan nanonawilakebis Semadgeneli nawilakebis didi raodenobaa moTavsebuli arian zedapirTan uSualo siaxloveSi. sainteresoa gamovikvlioT Tu rogor aris damokidebuli kvanturi wertilebis sxvadasxva Tvisebebi identuri moculobisa da gansxvavebuli zedapiris mqone kvanturi wertilebisaTvis. am mizniT naSromSi Seswavlilia identuri moculobisa da sxvadasxva zedapiris farTobis mqone kvanturi wertilebi kerZod ki: sferuli cilindruli da konusuri formisa. zedapiris farTobis mixedviT yvelaze mcire farTobi sferul kvantur wertils gaaCnia xolo udidesi – konusur kvantur wertils. rogorc zemoT iyo arniSnuli kvanturi wertilebis Tvisebebis zomaze damokidebuleba sakmaod kargadaa Seswavlili. sainteresoa SeviswavloT Tu rogor aris damokidebuli kvanturi wertilebis geometriul formaze energetikuli speqtri. kerZod fiqsirebuli i 0 da i 1 orbitaluri ricxvebisaTvis (orbitaluri ricxvis Sesaxeb cilindruli simetriis Sesaxeb ixile $1.2-Si) Tu rogor icvleba nawilakTa energia mTavari kvanturi ricxvis mixedviT. radganac kvanturi wertilebSi nawilakTa energiebi maT zomebze Zlier aris damokidebuli, formis gavlenis gasarkvevad vafiqsirebT maT (sxvadasxva formis kvanturi wertilebis) moculobebs. winamdebare naSromSi Seswavlilia identuri moculobis mqone sferuli, cilindruli da konusuri formis kvanturi wertilebi da Sedarebulia maTi energetikuli speqtri. aseve sainteresoa fiqsirebuli moculobis mqone konusuri kvanturi wertilis energetikuli speqtris Seswavla kvanturi wertilis sxvadasxva fuZe-simaRlis Sefardebis SemTxvevaSi. winamdebare naSromSi Seswavlilia aRniSnuli SemTxveva da warmodgenilia aramarto energetikuli speqtri, aramed Sesabamisi mdgomareobebis talRuri funqciebic. winamdebare naSromSi agreTve gamokvleulia kvanturi wertilebis zonaTaSorisi gadasvlebis oscilatoruli Zalebi. dRemde Seswavlilia oscilatoruli Zalebi sxvadasxvagvari kvanturi wertilebisaTvis mag. [68-70]. aRsaniSnavia, rom zonaTaSorisi gadasvlebis Sesabamisi oscilatoruli Zalebi kvanturi wertilebisaTvis maT energetikul speqtrTan SedarebiT naklebad aris Seswavlili.

17

$1.2 sasrul elementTa meTodi$1.2 sasrul elementTa meTodi$1.2 sasrul elementTa meTodi$1.2 sasrul elementTa meTodi Sredingeris gantolebisTvisSredingeris gantolebisTvisSredingeris gantolebisTvisSredingeris gantolebisTvis

kompiuteruli resursebis mkveTri zrdis wyalobiT kondensirebuli garemos fizikis problemebis kompiuterulma simulaciam axali fundamenturi SesaZleblobebi gamoaCina aRniSnuli dargis Teoriuli kvelevis TvalsazrisiT. simulacia iZleva miRebuli eqsperimentaluri Sedegebis detalur interpretacias da SeuZlia iwinaswarmetyvelos garkveuli Tvisebebi da procesebi, romlebic ama Tu im eqsperimentis Catarebisas warmoCindeba. dResdReobiT SesaZlebelia sxvadasxva sistemebis eleqtronuli da struqturuli Tvisebebis gamokvleva mxolod Teoriuli midgomiT – Sredingeris gantolebis amoxsniT ise, rom ar iqnas gamoyenebuli arcerTi empiriuli parametri [24-25]. vinaidan kompiuteruli resursebic SemosazRvrulia, xolo nanosistemebis kompiuteruli kvleva did kompiuterul resursebs moiTxovs maTi Seswavlis sirTulis gamo, rac dakavSirebulia imasTan, rom aseTi sistemebi Sedgebain didi raodenobiT nawilakebisagan da maTi geometriuli formebic rTulia, amitom xSirad mimarTaven sxvadasxva miaxloebebs da meTodebs. imisaTvis, rom kristalSi myofi kvazinawilakebi – eleqtronebi da xvrelebi ganvixiloT rogorc “kvanturad datyvevebulebi”, romelTaTvisac gamoviyenebT Sredingeris erTnawilakobriv gantolebas da ugulvebelvyofT kulonur urTierTqmedebas nawilakebs Soris, sistemis zoma mcire unda iyos eqsitonis boris radiusze an rac igivea eleqtron-xvrelis debroilis talRis sigrZeze es ori formulireba eqvivalenturia [71]. winamdebare naSromSi ganxiluli sistemebi maTi zomebis mixedviT akmayofileben kvanturi datyvevebis pirobebs, vinaidan maTi zomebi mcirea boris eqsitonis radiusze. magaliTisTvis -Tvis eqsitonis boris radiusi 35 nm is tolia [72]. winamdebare naSromSi gamoyenebulia koordinatze damokidebuli efeqturi masis miaxloeba [41] da amoxsnilia Sredingeris stacionaluri gantoleba kvanturi wertilebisaTvis momvlebi talRuri funqciis gamoyenebiT. naSromSi ganxilulia kvanturi wertilebi, romlebsac gaaCniaT sferuli an cilindruli simetria, amitom maTi SeswavlisaTvis mosaxerxebelia gamoviyenoT Sredingeris gantoleba Cawerili sistemis simetriis Sesabamisad sferul an cilindrul koordinatebSi. kvanturi meqanikidan cnobilia, rom nebismieri kvanturi sistema SesaZlebelia srulad aRiweros Sredingeris gantolebiT. es ukanaskneli droze damoukidebel (stacionalur) SemTxvevaSi zogadi saxiT Semdegnairad Caiwereba:

xP !"$% y$ ) Btuz Ψtu Ψtu (1.1)

sadac ! - plankis mudmivaa m - nawilakis masa, Btu- potencialuri funqcia, E - energia da Ψtu - talRuri funqcia. laplasis operatoris sferul da cilindrul koordinatebSi warmodgenis Semdeg [73] (1.1) gantoleba sferul koordinatebSi kuTxuri nawilis gancalkevebis Semdeg ase moicema:

18

P !"$% H<"

<|" ) $|

<<|J ) xBt ) nn!"

$%|" z (1.2)

xolo igive (1.1) gantolebas cilindrul koordinatebSi kuTxuri nawilis gareSe Semdegi saxe aqvs:

P !"$% x

|~

~| Ht ~~| J P n"

|" Φ ) ~"~" z ) Bt, Φ EΦ (1.3)

(1.2) da (1.3) gantolebebis Sesabamisi sruli talRuri funqciis misaRebad

saWiroa (1.2) da (1.3) gantolebis amonaxsnebi: , da Φ gamravlebuli iqnan

TavTavianT kuTxur nawilebze: n%, (sferuli harmonikebi) da n –ze Sesabamisad, sadac i orbitaluri kvanturi ricxvia. aqve SevniSnavT, rom i kvantur ricxvs cilindruli simetriis SemTxvevaSic sferuli SemTxvevis analogiurad orbitalur kvantur ricxvs vuwodebT [74]. (1.2) da (1.3) gantolebis amoxsnisaTvis gamoviyenebT sasrul elementTa meTods (The Finite Element Method). iTvleba, rom pirveli publikacia sasrul elementTa meTodis Sesaxeb gamoqveynebuli iyo 1956 wels [75] am meTodis ganviTarebaSi didi roli iTamaSa berZenma mecnierma argirisma. dasawyisSi sasrul elementTa meTodi struqturul meqanikaSi gamoiyeneboda. dRes sasrul elementTa meTods ZiriTadad inJinrebi mimarTaven, Tumca aRniSnuli meTodi kvanturi meqanikis amocanebis gadawyvetisaTvisac gamoiyeneba [76-77]. sasrul elementTa meTodis arsis wamoCinebisaTvis ganvixiloT Sredingeris gantoleba Cawerili zogadi operatoruli saxiT hamiltonis operatoris dakonkretebis gareSe:

Ψ Ψ (1.4) (1.4) gantolebis amoxsnis mizniT misi amonaxseni warmovadginoT Semdegi saxiT:

Ψ ∑ Cφ (1.5)

sadac Cj jer ucnobi koeficientebia xolo φ - egred wodebuli bazisuri

funqciebia. φ funqciebs qveviT davakonkretebT. (1.5) amonaxsnis (1.4) gantolebaSi Casmis Semdeg vRebulobT:

∑ Cφ E ∑ Cφ (1.6)

ukanasknel gamosaxulebas φ - ze vamravlebT da vaintegrebT amocanis gansazRvris areze. ajamvis da integrebis gadasmis Semdeg vRebulobT:

φ ∑ Cφ dΩ φE ∑ Cφ dΩ (1.7) Tu SemoviRebT Semdeg aRniSvnebs:

φφdΩ (1.8)

φφdΩ L (1.9) (1.7) gantoleba SeiZleba CavweroT matriculi gantolebis saxiT:

] L] (1.10) (1.10) gantolebis amoxsna eqvivalenturia (1.4) gantolebis amoxsnisa. (1.10) gantolebis amoxsniT vpoulobT sistemis energetikul speqtrsa da Cj koeficientebis gamoyenebiT (1.5) is mixedviT vagebT sistemis mdgomareobis funqciebs. vinaidan Cven vxsniT Sredingeris gantolebas ara analizurad, aramed ricxviTi meTodebis gamoyenebiT kompiuteris meSveobiT, amitom sistemis sakuTar mdgomareobebs Sesabamiss mTavar kvanturi ricxvis Semdegnairad vusabamebT: Sredingeris gantolebis amoxsnisas fiqsirebuli i kvanturi

ricxvisaTvis (1.10) gantolebis amoxsnis Sedegad miRebulsistemis sakuTar mniSvnelobebs)udidesisken. Sedegad miRebul energiaTa mimdevrobis mixedviT valagebT am energiebis mqone mdgomareobebs. anu umciresi energies Sesabamisi iqneba ZiriTadi mdgomareoba –amgvarad xdeba nebismieri mdgomarebis danomvra kvanturi ricxvebiT. Caweris simartivisTvis ganvixilavT erT ganzomilebian Sredingeris gantolebas da Semdgom gavakeTebT ganzogadebas or an met SemTxvevebisaTvis. (1.8)-(1.9) matricebis gamosaTvlelad saWiroa avbazisuri funqciebi. gonivruliaSevarCioT, romlebic lokalizebuli da SemosazRvruli iqnebianadvilad vuzrunvelvyofT (funqciaa egred wodebuli qudmaTematikurad Semdegnairad moicema:

(1.11) gansazRvravs erT ganzomilebian qud warmodgenilia nax. 1.4

nax. nax. nax. nax.

x0, x1, x2, x3 . . . kvanZebiTiToeuli funqcia nulisagan gansxvavebulia mxolod ori elementis farglebSi. qudi funqciebis maqsimaluri mniSvneloba, romlkvanZebSi aRweven 1-is tolia. MmagaliTad nomer mesame qudnulisagan gansxvavebulia mniSvnelobas - 1 aRwevs x (1.8) matricis matriculi elementebis kinetikuri wevrs aqvs saxe:

(1.10) gantolebis amoxsnis Sedegad miRebul mniSvnelobebs) valagebT zrdis mixedviT

udidesisken. Sedegad miRebul energiaTa mimdevrobis mixedviT valagebT am energiebis mqone mdgomareobebs. anu umciresi energies Sesabamisi iqneba

– mTavari kvanturi ricxviT 0 nebismieri mdgomarebis danomvra mTavari da orbitalur

ivisTvis ganvixilavT erT ganzomilebian Sredingeris gavakeTebT ganzogadebas or an met ganzomilebiani 9) matricebis gamosaTvlelad saWiroa av

funqciebi. gonivrulia, rom bazisuri funqciebad iseTi funqciebi romlebic lokalizebuli da SemosazRvruli iqnebian

advilad vuzrunvelvyofT (1.5) amonaxsnis krebadobas. erT-erTi aseTi tipis funqciaa egred wodebuli qudi funqcia (hat functionmaTematikurad Semdegnairad moicema:

dYdY

, Tu d

YdYd, Tu

0, Tu D d an G

) gansazRvravs erT ganzomilebian qud funqciebs rom

nax. nax. nax. nax. 1.1.1.1.4444 qudi funqciebis grafikiqudi funqciebis grafikiqudi funqciebis grafikiqudi funqciebis grafiki

. . . kvanZebi, xolo (x0, x1), (x2, x3) . . . elementebi ewodebaT. TiToeuli funqcia nulisagan gansxvavebulia mxolod ori elementis

di funqciebis maqsimaluri mniSvneloba, romlis tolia. MmagaliTad nomer mesame qud

nulisagan gansxvavebulia (x2, x3) da (x3, x4) elementze da maqsimalur x3 kvanZze.

8) matricis matriculi elementebis kinetikuri wevrs aqvs saxe:

19

energiebs (anu - umciresidan

udidesisken. Sedegad miRebul energiaTa mimdevrobis mixedviT valagebT am energiebis mqone mdgomareobebs. anu umciresi energies Sesabamisi iqneba

da ase Semdeg. da orbitaluri

ivisTvis ganvixilavT erT ganzomilebian Sredingeris ganzomilebiani

9) matricebis gamosaTvlelad saWiroa avirCioT φ funqciebad iseTi funqciebi

romlebic lokalizebuli da SemosazRvruli iqnebian. amiT erTi aseTi tipis

hat function), romelic

(1.11)

unqciebs romelTa grafiki

. . . elementebi ewodebaT. TiToeuli funqcia nulisagan gansxvavebulia mxolod ori elementis

di funqciebis maqsimaluri mniSvneloba, romlebsac isini is tolia. MmagaliTad nomer mesame qudi funqcia

da maqsimalur

8) matricis matriculi elementebis kinetikuri wevrs aqvs saxe:

20

4$ 4 $ 4

romelic nawilobrivi integrebis gamoyenebis Sedegad Semdegnairad warmodgeba:

<"<" 4 <

< P <<

<< 4 (1.12)

(1.12) gamosaxulebis pirveli wevri integrebis sazRvrebze nuls gvaZlevs, fizikuri mosazrebidan gamomdinare, vinaidan kvanturi wertilbSi datyvevebuli eleqtronebis da xvrelebis talRuri funqcia kvanturi wertilidan Sors nulis tolia, amitom (1.12)-Si marjvena mxaris pirveli wevri sazRvrebze Casmisas nuls gvaZlevs da vRebulobT:

<"<" 4 P <

<<

< 4 (1.13)

(1.13) gardaqmna saSualebas gvaZlevs SevamciroT meore rigis warmoebuli pirveli rigis warmoebulamde da gamoviyenoT qudi funqciebi, romlebic wrfivi funqciebia. vinaidain (1.3) Sredingeris gantoleba 2 ganzomilebiania, amitom misTvis H da S matricebis gamosaTvlelad saWiroa organzomilebiani qudi funqciebis gamoyeneba. or da sam ganzomilebiani qudi funqciebi miiReba erTganzomilebiani qudi funqciebis pirdapiri gadamravlebiT, magaliTad Tu da erTganzomilebiani qudi funqciebia maSin or

ganzomilebiani qudi funqcia iqneba: , · analogiurad moviqceviT samganzomilebian SemTxvevaSic. zemoT aRniSnuli meTodi erTze met ganzomilebiani amocanis SemTxvevaSi msgavsi iqneba erTganzomilebianisa im gansxvavebiT, rom (1.8) da (1.9) integralebi or da met jeradi integrebiT Seicvleba Sesabamisad. zemoT aRwerili meTodi Zalian mosaxerxebelia sxvadasxva geometriuli formebis mqone kvanturi wertilebis SeswavlisaTvis da zogadad Sredingeris gantolebis amosaxsnelad nebismieri potencialis, sistemis nebismieri geometriisa da nawilakis masis koordinatze damokidebulebis pirobebSi. sasrul elementTa meTodis gamoyeneba Sredingeris gantolebisaTvis erTganzomilebiani marTkuTxa potencialuri ormos SemTxveva vrceli saxiT mocemulia danarTSi. danarTSi aseve warmodgenilia cilindruli da konusuri kvanturi wertilebis amoxsna.

21

$1.3 z$1.3 z$1.3 z$1.3 zogierTi martivi kvanturi sistemis energetikuli speqtriogierTi martivi kvanturi sistemis energetikuli speqtriogierTi martivi kvanturi sistemis energetikuli speqtriogierTi martivi kvanturi sistemis energetikuli speqtri

kvanturi meqanikidan cnobila, rom nebismieri kvanturi sistema sruliad xasiaTdeba mdgomareobis funqciiT, romelic miiReba Sredingeris gantolebis amoxsniT. kvanturi meqanikaSi cotaa iseTi amocanebi, romelTaTvisac SesaZlebelia Sredingeris gantolebis amoxsna analitikurad. mxolod martivi kvanturi amocanebisaTvisaa SesaZlebeli analitikuri amonaxsnis miReba. aseTi sistemis magaliTia + masis nawilakis moZraoba usasrulo siRrmis marTkuTxa potencialur ormoSi. erTganzomilebian SemTxvevaSi nawilakis talRuri funqciebi da sakuTari energiebi Semdegnairad moicema:

ΨQ X$ -Q

(1.14)

Q Q"-"!"$%" (1.15)

sadac potenciuri ormos siganea, 1,2,3 …. meore magaliTi analitikurad amoxsnadi SemTxvevisa aris nawilakis moZraoba sferuli simetriis usasrulo siRrmis kvantur ormoSi. am SemTxvevaSi sistemis talRuri funqciebi da energetikuli speqtria [55]:

Ψ,,t X $?

He| ⁄ JYHeJ n,%, (1.16)

Q,n !"$%" H QnJ$

(1.17)

sadac Qn beselis sferuli funqciis ¡nH QnJ 0 -is i rigis nulebia, -

sferuli ormos radiusii. morigi magaliTia usasrulo cilindruli ormo [78]:

Φ¢, , ∑ £% Hfe ¤J sin H-

J %Qcos+ ) \%Qsin+ª,Q«%«l (1.18)

%Q !"$% WH-

J$ ) fe"" Z (1.19)

sadac %Q cilindruli beselis funqciis nulebia, , -cilindruli ormos simaRle, ki misi radiusi. arsebobs kidev sxva amocanebic, rodesac Sredingeris gantoleba analizurad ixneba, Tumca maT aq ar warmovadgenT. imisaTvis, rom dagvedgina Cvens mier gamoyenebuli meTodis samarTlianoba da sizuste, igi SevamowmeT am paragrafSi ganxiluli sistemebisaTvis. anu SevadareT meTodis gamoyenebiT miRebuli Sedegebi analizurad miRebul Sedegebs. (vinaidan aq warmodgenili kvantur meqanikuri amocanebis amoxsnebi Seesabameba sistemaSi usasrulo potencialis arsebobas, meTodis gamoyenebisas CavsviT fizikurad didi potenciali) meTodiT da analitikurad miRebul Sedegebs Soris gansxvaveba (naSromSi warmodgenili sizustis SemTxvevaSi) energetikuli speqtrisaTvis 1%-s ar aRemateba.

22

TTTT aaaa vvvv iiii mmmm eeee oooo rrrr eeee

23

$2.1 sferuli$2.1 sferuli$2.1 sferuli$2.1 sferuli,,,, cilindruli da konusuri kvanturi wertilebicilindruli da konusuri kvanturi wertilebicilindruli da konusuri kvanturi wertilebicilindruli da konusuri kvanturi wertilebi

winamdebare paragrafSi warmodgenilia ]4L/¬L sferuli, cilindruli da konusuri formis kvanturi wertilebis energetikuli doneebi. ]4L/¬L Seqmnis eqsperimentaluri meTodebi kargadaa ganviTarebuli [77-80]. Sesaswavli kvanturi wertilebis zomebi SerCeulia iseTnairad, rom samive sxvadasxva geometriuli formis kvantur wertils gaCnia identuri moculoba, amasTan cilindruli da konusuri formis kvantur wertilebs aqvT erTi da igive simaRle rac sferuli kvanturi wertilis diametris tolia da 8 nm Seadgens. Cven SeviswavleT eleqtronebis da xvrelebis energetikuli speqtri zemoTaRniSnuli kvanturi wertilebisaTvis, naSromSi aseve mocemulia Sesabamisi kvanturi mdgomareobebis talRuri funqciebi. Teoriuli gamoTvlebisaTvis gamoyenebulia Semdegi parametrebi: ]4L - isaTvis [81] eleqtronis efeqturi masa +& 0.13+ xvrelis efeqturi masa +j& 0.44+ , xolo ¬L -isaTvis [82] +& 0.21+ , +j& 0.6+ , sadac + Tavisufali eleqtronis masaa ( + 9.109 · 10d/(: ). gamtareblobis zonaTa Soris sxvaoba: 5 0.745, savalento zonaTa Soris sxvaoba 5j 0.235. naSromSi yvelgan warmodgenili xvrelebis energiebi aTvlili arian savalento zonidan da dadebiT sidideebad arian mocemuli. sferul kvantur wertilSi gamtareblobis zonaSi gvaqvs eleqtronTa or-ori energetikuli done rogorc i 0 aseve i 1 orbitaluri kvanturi ricxvis mixedviT. rac Seexeba konusur da cilindrul kvantur wertilebs, maTTvis ramodenime bmuli mdgomareoba arsebobs. rogorc naxazebidan Cans fiqsirebuli donis nomris SemTxvevaSi mcire gamonaklisis garda sferuli kvanturi wertilis Sesabamisi energiebi aRemateba konusuri kvanturi wertilis energiebs da es ukanaskneli – cilindruli kvanturi wertilis Sesabamis energiebs. nax. 2.1-2.2 idan Cans saintereso Tavisebureba konusuri da cilindruli kvanturi wertilebisa, rac imaSi mdgomareobs, rom aRniSnuli sistemebis energetikuli speqtri TiTqmis wrfivi damokidebulebiT izrdeba energetikuli donis (mTavari kvanturi ricxvis) zrdis mixedviT.

24

nax. nax. nax. nax. 2.2.2.2.1111 a)a)a)a)

nax. nax. nax. nax. 2.2.2.2.1111 b)b)b)b)

nax. nax. nax. nax. 2.2.2.2.1111 sferuli cilindruli da konusuri kvanturi wertilebis eleqtronTa sferuli cilindruli da konusuri kvanturi wertilebis eleqtronTa sferuli cilindruli da konusuri kvanturi wertilebis eleqtronTa sferuli cilindruli da konusuri kvanturi wertilebis eleqtronTa

energetikuli speqtri orbitaluri kvanturi ricxviT a) energetikuli speqtri orbitaluri kvanturi ricxviT a) energetikuli speqtri orbitaluri kvanturi ricxviT a) energetikuli speqtri orbitaluri kvanturi ricxviT a) ± ² b) b) b) b) ± ³

25



nax. nax. nax. nax. 2.2.2.2.2222 sferuli cilindruli da konusuri kvanturi wertilebis xvrelTa sferuli cilindruli da konusuri kvanturi wertilebis xvrelTa sferuli cilindruli da konusuri kvanturi wertilebis xvrelTa sferuli cilindruli da konusuri kvanturi wertilebis xvrelTa

energetikuli speqtri orbitaluri kvanturi ricxviT a) energetikuli speqtri orbitaluri kvanturi ricxviT a) energetikuli speqtri orbitaluri kvanturi ricxviT a) energetikuli speqtri orbitaluri kvanturi ricxviT a) ± ² b) b) b) b) ± ³

igive iTqmis xvrelebis energetikul doneebTan dakavSirebiTac. aRsaniSnavia, rom kvanturi wertilebis sxvadasxva geometriuli formis miuxedavaT, samive tipis kvantur wertilSi umciresi donis Sesabamisi

26

energia TiTqmis igivea. rac gvaZlevs safuZvels davaskvnaT, rom kvanturi wertilis umdablesi energetikuli doneebi mciredad arian damokidebuli kvanturi wertilis formaze, magram cxadia rom damokidebuli arian kvanturi wertilis moculobaze.

27

$$$$2.2 sferuli kvanturi wertilebis talRuri funqciebi2.2 sferuli kvanturi wertilebis talRuri funqciebi2.2 sferuli kvanturi wertilebis talRuri funqciebi2.2 sferuli kvanturi wertilebis talRuri funqciebi

nax. 2.3-2.4 warmodgenilia sferuli kvanturi wertilebis kvanturi

mdgomareobebis radialuri talRuri funqciebi Sesabamisad i 0 da i 1

orbitaluri kvanturi ricxvebisaTvis (warmodgenil naxazebze mocemulia

daunormirebeli talRuri funqciebi).

a) b)a) b)a) b)a) b)

nax. nax. nax. nax. 2.32.32.32.3 sferuli kvanturi wertilis sferuli kvanturi wertilis sferuli kvanturi wertilis sferuli kvanturi wertilis eleqtronis talRuri eleqtronis talRuri eleqtronis talRuri eleqtronis talRuri

funqciebifunqciebifunqciebifunqciebi orbitaluri kvanturi ricxvisTvisorbitaluri kvanturi ricxvisTvisorbitaluri kvanturi ricxvisTvisorbitaluri kvanturi ricxvisTvis ± ²

a) mTavari kvanturi ricxviT a) mTavari kvanturi ricxviT a) mTavari kvanturi ricxviT a) mTavari kvanturi ricxviT ´ ³, b) , b) , b) , b) ´ b

a) a) a) a) b)b)b)b)

nax. nax. nax. nax. 2.2.2.2.4444 sferuli kvanturi wertilis sferuli kvanturi wertilis sferuli kvanturi wertilis sferuli kvanturi wertilis eleqtronis talRurieleqtronis talRurieleqtronis talRurieleqtronis talRuri

funqciebi orbitaluri kvanturi ricxvisTvis funqciebi orbitaluri kvanturi ricxvisTvis funqciebi orbitaluri kvanturi ricxvisTvis funqciebi orbitaluri kvanturi ricxvisTvis ± ³

a) mTavari kvanturi ricxviT a) mTavari kvanturi ricxviT a) mTavari kvanturi ricxviT a) mTavari kvanturi ricxviT ´ ³, b) , b) , b) , b) ´ b

rogorc zemoT warmodgenili naxazebidan Cans eleqtronebis ganawilebis

albaToba sferul kvantur wertilSi orbitaluri ricxviT i 0

maqsimaluria kvanturi wertilis geometriul centrSi, xolo i 1 Tvis

eleqtronis aRmoCenis albaToba kvanturi wertilis centrSi nulis tolia

da igi Tavis maqsimalur mniSvnelobas centridan daaxloebiT 2 nm is

manZilze aRweven. naxazebidan aseve SegviZlia davaskvnaT, rom dabali

28

energiebis mqone nawilakebi ZiriTadad lokalizebuli arian kvanturi

wertilis ubanSi – centridan 4 nm is manZilze.

2.5 naxazze warmodgenilia xvrelebis Sesabamisi radialuri talRuri

funqciebi i 0 orbitaluri kvanturi ricxvisaTvis


nax. nax. nax. nax. 2.2.2.2.5555 sferuli kvanturi wertilis sferuli kvanturi wertilis sferuli kvanturi wertilis sferuli kvanturi wertilis xvrelxvrelxvrelxvreliiiis talRuri s talRuri s talRuri s talRuri

funqciebi orbitaluri kvanturi ricxvisTvis funqciebi orbitaluri kvanturi ricxvisTvis funqciebi orbitaluri kvanturi ricxvisTvis funqciebi orbitaluri kvanturi ricxvisTvis ± ²

a) a) a) a) mTavari kvanturi mTavari kvanturi mTavari kvanturi mTavari kvanturi ricxviT ricxviT ricxviT ricxviT ´ ³, b) , b) , b) , b) ´ b

nax. 2.3 da nax. 2.5 erTi SexedviT identuri naxazebia, magram Tu kargad davukvirdebiT SevamCnevT, rom xvrelebis Sesabamisi radialuri talRuri funqciebi kvanturi wertilis gareT ufro male miileva vidre eleqtronis Sesabamisi talRuri funqciebi rac fizikurad xvrelis didi efeqturi masiT aris gamowveuli.

29

$2.3$2.3$2.3$2.3 cilindruli kvanturi wertilebis talRuri funqciebicilindruli kvanturi wertilebis talRuri funqciebicilindruli kvanturi wertilebis talRuri funqciebicilindruli kvanturi wertilebis talRuri funqciebi

nax. 2.6 ze warmodgenilia i 0 orbitaluri kvanturi ricxvis Sesabamisi cilindruli kvanturi wertilebis talRuri funqciebi.

a) a) a) a) b) b) b) b)

g) g) g) g) d)d)d)d)

30

e)e)e)e)

nax. nax. nax. nax. 2.2.2.2.6666 cilindriuli kvanturi wertilis eleqtronis talRuri funqciebi cilindriuli kvanturi wertilis eleqtronis talRuri funqciebi cilindriuli kvanturi wertilis eleqtronis talRuri funqciebi cilindriuli kvanturi wertilis eleqtronis talRuri funqciebi ± ²

orbitaluri kvanturi ricxviT da orbitaluri kvanturi ricxviT da orbitaluri kvanturi ricxviT da orbitaluri kvanturi ricxviT da ´ mTavari kvanturi ricxvebiT: a) mTavari kvanturi ricxvebiT: a) mTavari kvanturi ricxvebiT: a) mTavari kvanturi ricxvebiT: a) ´ ³ b) b) b) b) ´ b

g) g) g) g) ´ ` d) d) d) d) ´ µ e) e) e) e) ´ ¶

nax. 2.7 ze warmodgenilia orbitaluri kvanturi ricxvis i 1-is Sesabamisi cilindruli kvanturi wertilebis talRuri funqciebi.


31

g)g)g)g) d)d)d)d)

nax.nax.nax.nax. 2.2.2.2.7777 cilindriuli kvanturi wertilcilindriuli kvanturi wertilcilindriuli kvanturi wertilcilindriuli kvanturi wertilis eleqtronis talRuri funqciebi is eleqtronis talRuri funqciebi is eleqtronis talRuri funqciebi is eleqtronis talRuri funqciebi ± ³

orbitaluri kvanturi ricxviT da orbitaluri kvanturi ricxviT da orbitaluri kvanturi ricxviT da orbitaluri kvanturi ricxviT da ´ mTavari kvanturi ricxvebiT: a) mTavari kvanturi ricxvebiT: a) mTavari kvanturi ricxvebiT: a) mTavari kvanturi ricxvebiT: a) ´ ³ b) b) b) b) ´ b

g) g) g) g) ´ ` d) d) d) d) ´ µ

rogorc nax. 2.6-ze vxedavT, pirveli sami energetikuli donis talRur funqciebs, i 0 orbitaluri kvanturi ricxvisaTvis gaaCniaT erTi piki t RerZis gaswvriv xolo RerZis (kvanturi wertilis RerZi) gaswvriv pikebis raodenoba mTavari kvanturi ricxvis tolia. i 0 is Sesabamis rigiT meoTxe talRuri funqcias ki t RerZis gaswvriv 2 piki aqvs da RerZis gaswvriv 1 piki. vinaidan erTi RerZis gaswvriv talRur funqciaTa fiqsirebuli pikebis raodenobis SemTxvevaSi meore RerZis gaswvriv pikebis met raodenobas meti energia Seesabameba, amitom SegviZlia davaskvnaT, rom dabali energiebis SemTxvevaSi eleqtronis talRur funqcias urCevnia metad daikvantos RerZis gaswvriv vidre radialuri mimarTulebiT. i 1 orbitaluri ricxvis SemTxvevaSi arsebuli kvanturi mdgomareobebis talRur funqciebs gaaCniaT TiTo piki radialuri mimarTulebiT, xolo kvanturi wertilis RerZis gaswvriv pikTa raodenoba mTavari kvanturi ricxvis tolia. naSromis naxazebiT gadatvirTvis Tavidan acilebis mizniT xvrelebis Sesabamis talRur funqciebs aRar warmovadgenT, vinaidan maTi talRuri funqciebic nax. 2.6-2.7 warmodgenili eleqtronTa talRuri funqciebis analogiuri kanonzomierebebiT xasiaTdebian (pikebis ganlageba mTavari da orbitaluri ricxvebis mixedviT). zemoT warmodgenil naxazebidan kargaT Cans, rom nawilakebis talRuri funqciebi simetriulia cilindris geometriuli centris mimarT cilindris RerZis gaswvriv, es faqti cilindris simetriulobidan gamomdinare fizikuri Sedegia.

32

$2.4$2.4$2.4$2.4 konusuri kvanturi wertilebis talRuri funqciebikonusuri kvanturi wertilebis talRuri funqciebikonusuri kvanturi wertilebis talRuri funqciebikonusuri kvanturi wertilebis talRuri funqciebi

nax. 2.8-ze warmodgenilia konusuri kvanturi wertilebis talRuri funqciebi i 0 orbitaluri kvanturi ricxvisaTvis



33

e)e)e)e)

nax. nax. nax. nax. 2.2.2.2.8888 konusuri kvanturi wertilikonusuri kvanturi wertilikonusuri kvanturi wertilikonusuri kvanturi wertilis eleqtronis talRuri funqciebi s eleqtronis talRuri funqciebi s eleqtronis talRuri funqciebi s eleqtronis talRuri funqciebi ± ²

orbitaluri kvanturi ricxviT da orbitaluri kvanturi ricxviT da orbitaluri kvanturi ricxviT da orbitaluri kvanturi ricxviT da ´ mTavari kvanturi ricxvebiT: mTavari kvanturi ricxvebiT: mTavari kvanturi ricxvebiT: mTavari kvanturi ricxvebiT: a) a) a) a) ´ ³, b) , b) , b) , b) ´ ³

g) g) g) g) ´ ` d) d) d) d) ´ µ e) e) e) e) ´ ¶

nax. 2.9 ze mocemulia orbitaluri kvanturi ricxvis i 1 -is Sesabamisi konusuri kvanturi wertilebis talRuri funqciebi eleqtronebisaTvis.


34


nax. nax. nax. nax. 2.2.2.2.9999 konusuri kvanturi wertilkonusuri kvanturi wertilkonusuri kvanturi wertilkonusuri kvanturi wertilis eleqtronis talRuri funqciebi is eleqtronis talRuri funqciebi is eleqtronis talRuri funqciebi is eleqtronis talRuri funqciebi ± ³

orbitaluri kvanturi ricxviT da orbitaluri kvanturi ricxviT da orbitaluri kvanturi ricxviT da orbitaluri kvanturi ricxviT da ´ mTavari kvanturi ricxvebiT: mTavari kvanturi ricxvebiT: mTavari kvanturi ricxvebiT: mTavari kvanturi ricxvebiT: a) a) a) a) ´ ³, b) , b) , b) , b) ´ b

g) g) g) g) ´ ` d) d) d) d) ´ µ

konusuri kvanturi wertilebis talRuri funqciebi cilindruli kvanturi wertilebisgan gansxvavebiT ar xasiaTdebian raime simetriiT. eleqtronis ganawilebis albaTobis maqsimumi i 0 orbitaluri kvanturi ricxvisaTvis konusis simetriis RerZze mdebareobs, xolo i 1 orbitaluri kvanturi ricxvis SemTxvevaSi wanacvlebulia RerZis mimarT ganivad. xvrelebis Sesabamisi talRuri funqciebi eleqtronis talRuri funqciebis msgavsia garda im gansxvavebisa, rom isini kvanturi wertilis aris miRma ufro swrafi tempiT miiswrafian nulisaken, es faqti rogorc zemoT ukve iyo aRniSnuli fizikurad gamowveulia xvrelebis SedarebiT didi efeqturi masiT.

35

$2.5$2.5$2.5$2.5 fuZisa da simaRlis cvlilebis gavlena konusuri kvanturifuZisa da simaRlis cvlilebis gavlena konusuri kvanturifuZisa da simaRlis cvlilebis gavlena konusuri kvanturifuZisa da simaRlis cvlilebis gavlena konusuri kvanturi

wertilebis eleqtronul mdgomareobebzewertilebis eleqtronul mdgomareobebzewertilebis eleqtronul mdgomareobebzewertilebis eleqtronul mdgomareobebze

winamdebare paragrafSi ganxilulia / konusuri kvanturi wertilebi da warmodgenilia Sedegebi, romlebic uCvenebs Tu rogor aris damokidebuli aRniSnuli sistemis eleqtronuli mgomareobebi konusuri kvanturi wertilis fuZisa da simaRlis cvlilebaze. konusuri kvanturi wertilis fuZe da simaRle imdagvarad icvleba, rom mudmivi rCeba kvanturi wertilis moculoba, rac daaxloebiT 450 nm3 Seadgens. vinaidan kvanturi wertilebis Tvisebebi zogadad damokidebulia maT moculobaze amitom imisaTvis, rom gavarkvioT fuZisa da simaRlis cvlilebis mixedviT konusuri kvanturi wertilebis Tvisebebi, kvlevebi Catarebulia mudmivi moculobis SenarCunebiT. ganxilulia sami sxvadasxva fuZe-simaRlis mqone konusuri kvanturi wertili: 1) H=3nm R=12nm, 2) H=6nm R=8.48nm da 3) H=12nm R=6nm. sadac H da R konusuri kvanturi wertilis simaRle da fuZis radiusia Sesabamisad. gamoTvlebisas gamoyenebulia / kvanturi wertilis qvemoT mocemuli parametrebi. InAs: +& 0.023+ , +j& 0.41+ , GaAs: +& 0.067+ ,

+j& 0.45+ , da gamtareblobis zonaTa Soris sxvaoba 5 0.775 da savalento zonaTa Soris sxvaoba 5j 0.335 [83]. qvemoT naxazebze mocemulia / konusuri kvanturi wertilebis energetikuli doneebi eleqtronebisa da xvrelebisaTvis.


36


nax. nax. nax. nax. 2.2.2.2.10101010 konusuri kvanturi wertilkonusuri kvanturi wertilkonusuri kvanturi wertilkonusuri kvanturi wertilis gamtareblobis eleqtronebis energetikuli is gamtareblobis eleqtronebis energetikuli is gamtareblobis eleqtronebis energetikuli is gamtareblobis eleqtronebis energetikuli

speqtri a) speqtri a) speqtri a) speqtri a) ± ² da b) da b) da b) da b) ± ³ orbitaluri kvanturi ricxvebiTorbitaluri kvanturi ricxvebiTorbitaluri kvanturi ricxvebiTorbitaluri kvanturi ricxvebiT


37


nax. nax. nax. nax. 2.2.2.2.11111111 konusuri kvanturi wertilkonusuri kvanturi wertilkonusuri kvanturi wertilkonusuri kvanturi wertilis xvrelebis is xvrelebis is xvrelebis is xvrelebis energetikuli speqtrienergetikuli speqtrienergetikuli speqtrienergetikuli speqtri

a) a) a) a) ± ² da b) da b) da b) da b) ± ³ orbitaluri kvanturi ricxvebiTorbitaluri kvanturi ricxvebiTorbitaluri kvanturi ricxvebiTorbitaluri kvanturi ricxvebiT

Seswavlil konusur kvantur wertilebSi gamtareblobsi zonaSi i 0 orbitaluri kvanturi ricxviT ori, xolo i 1 SemTxvevaSi erTi bmuli mdgomareobaa. xvrelebis SemTxvevaSi ki ramodenime rogorc i 0 aseve i 1 saTvis. 2.1 paragrafSi rogorc vnaxeT konusuri kvanturi wertilebis energetikuli mdgomareobebi fiqsirebuli orbitaluri ricxvis SemTxvevaSi mTavari kvanturi ricxvis zrdasTan erTad daaxloebiT wrfivad izrdeba. analogiur surTs vxedavT axlac - sxvadasxva fuZe-simaRlis Sefardebis mqone kvanturi wertilebisTvisac. konusuri kvanturi wertilebis simaRlis gazrdiT (fuZis radiusis SemcirebiT) kvazinawilakebis - eleqtronebis da xvrelebis energiebi mcirdeba. radganac xvrelebis SemTxvevaSi ramdenime bmuli mdgomareoba gvaqvs, amitom kargad Cans, rom Seswavlili sami sxvadasxva geometriuli konfiguraciis mqone konusuri kvanturi wertilis xverelebis energiebs Soris sxvaoba mTavari kvanturi ricxvis zrdasTan erTad izrdeba. qvemoT naxazebze warmodgenilia konusuri kvanturi wertilebis talRuri funqciebi xvrelebisaTvis. vinaidan eleqtronebis SemTxvevaSi jamSi i 0 da i 1 orbitaluri kvanturi ricxvebisaTvis mxolod 3 bmuli mdgomareoba gvaqvs samive konfiguraciis kvanturi wertilisaTvis, xvrelebisaTvis ki ramodenime, amitom xvrelebis SemTxvevaSi kargad Cans talRuri funqciebis cvlilebis dinamika mTavari kvanturi ricxvis mixedviT. naxazebze Sedarebis mizniT gverdigverd warmodgenilia H=3nm R=12nm da H=12nm R=6nm zomebis mqone konusur kvantur wertilTa xvrelebis talRuri funqciebi.

38

aaaa****) ) ) ) aaaa********))))

bbbb****) ) ) ) bbbb********))))

gggg****) ) ) ) gggg********))))

39

dddd****) ) ) ) dddd********))))

eeee****) ) ) ) eeee********))))

nax. nax. nax. nax. 2.2.2.2.11112222 xvrelebis talRuri funqciebi xvrelebis talRuri funqciebi xvrelebis talRuri funqciebi xvrelebis talRuri funqciebi ± ² orbitaluri ricxvisaTvis, mTavari orbitaluri ricxvisaTvis, mTavari orbitaluri ricxvisaTvis, mTavari orbitaluri ricxvisaTvis, mTavari

kvanturi ricxviT: a) kvanturi ricxviT: a) kvanturi ricxviT: a) kvanturi ricxviT: a) ´ ³ , b) , b) , b) , b) ´ b g) g) g) g) ´ ` d) d) d) d) ´ µ e) e) e) e) ´ ¶, , , , * ---- konusuri konusuri konusuri konusuri

kvanturi wertilebikvanturi wertilebikvanturi wertilebikvanturi wertilebi zomebiT: zomebiT: zomebiT: zomebiT: H=3nm R=12nm, ** ---- konusuri konusuri konusuri konusuri kvanturi wertilebikvanturi wertilebikvanturi wertilebikvanturi wertilebi

zomebiT: zomebiT: zomebiT: zomebiT: H=12nm R=6nm.

40

aaaa****) ) ) ) aaaa********))))

bbbb****) ) ) ) bbbb********))))

g*) g*) g*) g*) g**)g**)g**)g**)

41

d*) d*) d*) d*) d**)d**)d**)d**)

e*) e*) e*) e*) e**)e**)e**)e**)

nax. nax. nax. nax. 2.2.2.2.11113333 xvrelebis talRuri funqciebi xvrelebis talRuri funqciebi xvrelebis talRuri funqciebi xvrelebis talRuri funqciebi ± ³ orbitaluri ricxvisaTvis, mTavari orbitaluri ricxvisaTvis, mTavari orbitaluri ricxvisaTvis, mTavari orbitaluri ricxvisaTvis, mTavari

kvanturi ricxviT: a) kvanturi ricxviT: a) kvanturi ricxviT: a) kvanturi ricxviT: a) ³ , b) , b) , b) , b) ´ b g) g) g) g) ´ ` d) d) d) d) ´ µ e) e) e) e) ´ ¶, , , , * ---- konusuri kvanturi konusuri kvanturi konusuri kvanturi konusuri kvanturi

wertilebi zomebiT: wertilebi zomebiT: wertilebi zomebiT: wertilebi zomebiT: H=3nm R=12nm, ** ---- konusuri kvanturi wertilebi zomebiT: konusuri kvanturi wertilebi zomebiT: konusuri kvanturi wertilebi zomebiT: konusuri kvanturi wertilebi zomebiT:

H=12nm R=6nm.

zogadi gansxvaveba rac SeiniSneba nax. 2.12-2.13 warmodgenili sxvadasxva

fuZe-simaRlis Sefardebis mqone kvanturi wertilebisaTvis aris is, rom

nawilakebis talRuri funqciebis dakvantva xdeba upiratesad im

mimarTulebiT romlis xazovani zomac metia. Sedegad, umdablesi doneebis

garda talRuri funqciebi aranair msgavs formas ar avlenen.

$2.6 urTierTqmedi $2.6 urTierTqmedi $2.6 urTierTqmedi $2.6 urTierTqmedi

wertilebi wertilebi wertilebi wertilebi

ori an meti urTierTqmedi wertilur molekulas [wertiluri molekulebi (vinaidan maTi Tvisebebi damokidebulia aramarto maT SemadgenlobaSi Semaval TiToeul kvantur wertilze, aramed urTierTqmedebazec. aqedan gamomdinareTvisebebis manipulireba SesaZlebelia kvantur wertilTa Soris manZilis (zogadad orientaciis) wertilebis Tvisebebi Zlierad aris damokidebuli maT geometriaze, risi magaliTebi zeviTac vnaxeT. aseve kvanturi wertiuri Tvisebebic Zlierad aris damokidebuli Soris manZilze da es ukanaskneli SeiZdamatebiTi parametri saWiro winamdebare paragrafSi warmodgenilia ori cilindruli kvanturi wertilis urTierTqmedeba. karian erTi meoris Tavze iseemTxveva. orive kvanturi wertilis warmodgenilia naSromSi ganxiluli sqematuri gamosaxuleba. gamoTvlebi CatarebuliawertilTa SorisA 4sistemisaTvis gamoyenebulia paragraf 2

qveviT warmodgenilia kvanturi sistemis energetikuli speqtris

damokidebuleba kvantur wertilTa Soris

$2.6 urTierTqmedi $2.6 urTierTqmedi $2.6 urTierTqmedi $2.6 urTierTqmedi ¸´¹º/»¼¹º cilindruli kvanturicilindruli kvanturicilindruli kvanturicilindruli kvanturi

wertilebi wertilebi wertilebi wertilebi ---- xelovnuri molekulaxelovnuri molekulaxelovnuri molekulaxelovnuri molekula

urTierTqmedi kvanturi wertilis erTobliobas kvantur [84] an xelovnur molekulas uwodeben. kvanturi

molekulebi (quantum dot molecule) sakmaod saintereso sistemebia vinaidan maTi Tvisebebi damokidebulia aramarto maT SemadgenlobaSi Semaval TiToeul kvantur wertilze, aramed am kvanturi

aqedan gamomdinare, kvanturi wertiluri moleTvisebebis manipulireba SesaZlebelia kvantur wertilTa Soris manZilis

orientaciis) cvlilebiTac. rogorc cnobiliawertilebis Tvisebebi Zlierad aris damokidebuli maT geometriaze, risi

zeviTac vnaxeT. aseve kvanturi wertiuri Tvisebebic Zlierad aris damokidebuli masSi Semaval kvantur wertilTa

manZilze da es ukanaskneli SeiZleba gamoviyenoT rogorc damatebiTi parametri saWiro Tvisebebis misaRebad.

winamdebare paragrafSi warmodgenilia ori cilindruli kvanturi wertilis urTierTqmedeba. kvanturi wertilebi moTavsebuli arian erTi meoris Tavze ise, rom maTi geometriuli simetriis RerZebi

kvanturi wertilis geometriuli zomebi identuria. nax.1.1omSi ganxiluli urTierqmedi kvanturi wertilebis

sqematuri gamosaxuleba. gamoTvlebi Catarebulia / sxvadasxva manZilis SemTxvevaSi

sistemisaTvis gamoyenebulia paragraf 2-Si mecemuli parametrebi.

nax. nax. nax. nax. 2222.1.1.1.14444 kvanturi wertilebis sqematuri kvanturi wertilebis sqematuri kvanturi wertilebis sqematuri kvanturi wertilebis sqematuri

gamosaxuleba. gamosaxuleba. gamosaxuleba. gamosaxuleba. H=3nm, R=7nm


damokidebuleba kvantur wertilTa Soris 4 manZilze.

42

cilindruli kvanturicilindruli kvanturicilindruli kvanturicilindruli kvanturi

s erTobliobas kvantur s uwodeben. kvanturi

) sakmaod saintereso sistemebia vinaidan maTi Tvisebebi damokidebulia aramarto maT SemadgenlobaSi

uri wertilebis kvanturi wertiluri molekulis

Tvisebebis manipulireba SesaZlebelia kvantur wertilTa Soris manZilis . rogorc cnobilia, kvanturi

wertilebis Tvisebebi Zlierad aris damokidebuli maT geometriaze, risi zeviTac vnaxeT. aseve kvanturi wertiuri molekulebis

masSi Semaval kvantur wertilTa leba gamoviyenoT rogorc

winamdebare paragrafSi warmodgenilia ori cilindruli formis ebi moTavsebuli

rom maTi geometriuli simetriis RerZebi triuli zomebi identuria. nax.1.14.

kvanturi wertilebis / kvantur

sxvadasxva manZilis SemTxvevaSi. / Si mecemuli parametrebi.

kvanturi wertilebis sqematuri kvanturi wertilebis sqematuri kvanturi wertilebis sqematuri kvanturi wertilebis sqematuri

H=3nm, R=7nm


43

nax. nax. nax. nax. 2222....11115555 eleqtronis energiebis damokidebuleba kvantur wertilTa Soris eleqtronis energiebis damokidebuleba kvantur wertilTa Soris eleqtronis energiebis damokidebuleba kvantur wertilTa Soris eleqtronis energiebis damokidebuleba kvantur wertilTa Soris d

manZilzemanZilzemanZilzemanZilze l=0 da da da da l=1 orbitaluri ricxvebiTorbitaluri ricxvebiTorbitaluri ricxvebiTorbitaluri ricxvebiT....

nax. nax. nax. nax. 2222....16161616 xvrelebis energiebis damokidebuleba kvantur wertilTa Soris xvrelebis energiebis damokidebuleba kvantur wertilTa Soris xvrelebis energiebis damokidebuleba kvantur wertilTa Soris xvrelebis energiebis damokidebuleba kvantur wertilTa Soris d

manZilze orbitaluri ricxviT manZilze orbitaluri ricxviT manZilze orbitaluri ricxviT manZilze orbitaluri ricxviT l=0=0=0=0

44

nax. nax. nax. nax. 2222....17171717 xvrelebis energiebis damokidebuleba kvantur wertilTa Soris xvrelebis energiebis damokidebuleba kvantur wertilTa Soris xvrelebis energiebis damokidebuleba kvantur wertilTa Soris xvrelebis energiebis damokidebuleba kvantur wertilTa Soris d

manZilze orbitaluri ricxviT manZilze orbitaluri ricxviT manZilze orbitaluri ricxviT manZilze orbitaluri ricxviT l=1=1=1=1

mimdinare paragrafSi ganxiluli / kvanturi wertilebi, romlebic kvantur wertilur molekulas (xelovnur molekulas) Seadgenen erTmaneTisgan fizikurad usasrulod Sor manZilze ise iqcevian, rogorc ori izolirebuli kvanturi wertili. rogorc zemoT warmodgenili naxazebidan Cans fizikurad usasrulod Sori manZili eleqtronebisaTvis daaxloebiT 6 nm ia xolo xvrelebisaTvis - daaxloebiT 2-4 nm. erTmaneTisagan fizikurad Sor manZilze myofi kvanturi wertilebisagan Semdgari sistemis energetikuli doneebi gadagvarebulia. kvanturi wertilebis miaxloebiT irTveba urTierTqmedeba kvantur mdgomareobebs Soris, romlis wyalobiTac xdeba gadagvarebis moxsna, rac fizikurad doneebis Sesabamisi energiebis gaxleCaSi gamoixateba. radgan / – Si eleqtronebis efeqturi masa xvrelis efeqtur masaze erTi rigiT metia, amitom axlo manZilebze eleqtronebi gacilebiT metad urTierTqmedeben vidre xverelibi. rogorc nax. 2.15 dan Cans eleqtronebis urTierTqmedeba imdenad didia, rom 4 1.3 nm manZilze axlos mitanisas i 0 da i 1 is Sesabamisi energetikuli doneebi ise ixliCebian, rom i 0 is Sesabamisi energia i 1 donis energiaze didi xdeba. xvrelebis SemTxvevaSi rogorc nax. 2.17 uCvenebs i 1 orbitaluri kvanturi ricxvebis Sesabamisi energetikuli doneebi mTavari kvanturi ricxvebiT: 2 da 3 erTmaneTs kveTen 4=0.5 nm manZilze.

45

qvemoT naxazebze warmodgenilia xelovnuri molekulis eleqtronTa talRuri funqciebi.


nax. nax. nax. nax. 2.2.2.2.18181818 xelovnuri molekulisxelovnuri molekulisxelovnuri molekulisxelovnuri molekulis eleqtronTa eleqtronTa eleqtronTa eleqtronTa talRuri funqciebitalRuri funqciebitalRuri funqciebitalRuri funqciebi ½ ². ¶ nm,nm,nm,nm, ± ² a) a) a) a)

´ ³,,,, b) b) b) b) ´ b


nax. nax. nax. nax. 2.2.2.2.19191919 xelovnuri molekulis xelovnuri molekulis xelovnuri molekulis xelovnuri molekulis eleqtronTa talRuri funqciebi eleqtronTa talRuri funqciebi eleqtronTa talRuri funqciebi eleqtronTa talRuri funqciebi ½ ². ¶ nm, nm, nm, nm, ± ³

a) a) a) a) ´ ³, b) , b) , b) , b) ´ b

46


nax. nax. nax. nax. 2.2.2.2.20202020 xelovnuri molekulis xelovnuri molekulis xelovnuri molekulis xelovnuri molekulis eleqtronTa eleqtronTa eleqtronTa eleqtronTa talRuri funqciebi talRuri funqciebi talRuri funqciebi talRuri funqciebi ½ ³ nm, nm, nm, nm, ± ² a) a) a) a)

´ ³, b) , b) , b) , b) ´ b


nax. nax. nax. nax. 2.2.2.2.21212121 xelovnuri molekulisxelovnuri molekulisxelovnuri molekulisxelovnuri molekulis eleqtronTa eleqtronTa eleqtronTa eleqtronTa talRuri funqciebi talRuri funqciebi talRuri funqciebi talRuri funqciebi ½ ³ nm, nm, nm, nm, ± ³

a)a)a)a) ´ ³, b) , b) , b) , b) ´ b


nax. nax. nax. nax. 2.2.2.2.22222222 xelovnuri molekulis xelovnuri molekulis xelovnuri molekulis xelovnuri molekulis eleqtronTa talRuri funqciebi eleqtronTa talRuri funqciebi eleqtronTa talRuri funqciebi eleqtronTa talRuri funqciebi ½ ` nm, nm, nm, nm, ± ² a) a) a) a)

´ ³, b) , b) , b) , b) ´ b

47

qvemoT naxazebze warmodgenilia xelovnuri molekulis xvrelebis talRuri funqciebi 4 0.5 nm SemTxvevisTvis.


gggg) ) ) ) dddd))))

e)e)e)e)

nax. nax. nax. nax. 2.2.2.2.22223333 xelovnuri molekulis xelovnuri molekulis xelovnuri molekulis xelovnuri molekulis xvrelTa talRuri funqciebixvrelTa talRuri funqciebixvrelTa talRuri funqciebixvrelTa talRuri funqciebi ½ ². ¶ nm,nm,nm,nm, ± ² a)a)a)a)

´ ³; b); b); b); b) ´ b; g) ; g) ; g) ; g) ´ `; d); d); d); d) ´ µ; e); e); e); e) ´ ¶

48


nax. nax. nax. nax. 2.2.2.2.22224444 g) nax. g) nax. g) nax. g) nax. 2.2.2.2.22224444 d)d)d)d)

nax. nax. nax. nax. 2.2.2.2.22224444 e)e)e)e)

nax. nax. nax. nax. 2.2.2.2.22224444 xelovnuri molekulxelovnuri molekulxelovnuri molekulxelovnuri molekulis xvrelTa talRuri funqciebi is xvrelTa talRuri funqciebi is xvrelTa talRuri funqciebi is xvrelTa talRuri funqciebi ½ ². ¶ nm, nm, nm, nm, ± ³ a) a) a) a)

´ ³; b) ; b) ; b) ; b) ´ b; g) ; g) ; g) ; g) ´ `; d) ; d) ; d) ; d) ´ µ; e) ; e) ; e) ; e) ´ ¶

49

xelovnuri molekulis talRuri funqciebi gamoirCevian simetriiT kvantur wertilTa geometriuli centrebis SemaerTebeli wrfis gaswvriv rac aRniSnuli sistemis geometriuli simetriiT aris gamowveuli.

50

$2.7 $2.7 $2.7 $2.7 kvanturi wertilebikvanturi wertilebikvanturi wertilebikvanturi wertilebissss zonaTaSoriszonaTaSoriszonaTaSoriszonaTaSorisiiii gadasvlebisgadasvlebisgadasvlebisgadasvlebis

Sesabamisi oscilatoruli ZalebiSesabamisi oscilatoruli ZalebiSesabamisi oscilatoruli ZalebiSesabamisi oscilatoruli Zalebi

kvantur meqanikaSi oscilatoruli Zala warmoadgens kvanturi doneebs Soris eleqtronuli gadasvlis sazoms. oscilatoruli Zalebis codna mniSvnelovania rogorc atomuri fizikisaTvis, aseve kondensirebuli garemos fizikisTvis, radganac oscilatoruli Zalebi aRweren eleqtromagnituri talRebis urTierTqmedebas nivTierebasTan, es ukanaskneli ki friad mniSvnelovania, rogorc Teoriuli aseve eqsperimentaluri TvalsazrisiTac. kvanturi wertilis oscilatoruli Zalebi damokidebulia kvanturi wertilebis zomebze da formebze. oscilatorul Zalebis marTva aseve SesaZlebelia eleqtruli veliT [85]. Tu da ¾ kvanturi sistemis sawyisi da sabolo mdgomareobebia Sesabamisad ¿|ΨÁ da ÂΨÃ|¿ talRuri funqciebiT maSin uganzomilebo oscilatoruli Zala, romelic Ä ¾ gadasvlas Seesabameba Semdagnairad

gamoiTvleba [86] ¾,Å $%>Æd> ÇÈΨÃÇÉu · uÇΨÊÇ$

(2.1)

sadac + Tavisufali eleqtronis masaa, Å P - saboloo da sawyiss

kvantur mdgomareobaTa Soris energiaTa sxvaoba, Éu - impulsis operatori xolo u - erTeulovani veqtori.

umartivesi kvanturi sistemebis - erTganzomilebiani ormosa da harmoniuli oscilatoris SemTxvevaSi, uganzomilebo oscilatoruli Zalebis gamoTvla gakeTebulia hensis mier [87]. kvanturi wertilebis zonaTa Sorisi gadasvlebis Sesabamisi uganzomilebo oscilatoruli Zalebis gamosaTvlelad gamoviyenebT (2.1) formulas, sadac kvanturi wertilebis eleqtronebisa da xvrelebis talRuri funiqciebia Casmuli, xolo energiaTa sxvaobis nacvlad eleqtronis da xvrelis Sesabamisi energiaTa sxvaoba. radganac gamoTvlebs vawarmoebT sferuli da cilindruli koordinatebis gamoyenebiT, amitom saWiroa (2.1) formulis warmodgena aRniSnul koordinatul sistemebSi rac qvemoT mocemulia gaSlili saxiT.

Éu · u ) ) (2.2) koordinatul warmodgenaSi gvaqvs:

P! ~~ P! ~

~ P! ~~

dekartuli warmoebulebi sferul koordinatebSi: Ë

Ë ÌÍ ËËt ) 1

t ÌÍÌÍ ËË P 1

t

ËË

ËË Ë

Ët ) 1t ÌÍ Ë

Ë ) 1t

ÌÍ

ËË

ËË ÌÍ Ë

Ët P 1Ët Ë

Ë

moculobis elementi sferul koordinatebSi:

4Ω t$444t

51

dekartuli warmoebulebi cilindrul koordinatebSi: ËË ÌÍ Ë

Ët P 1t Ë

Ë

ËË Ë

Ët ) 1t ÌÍ Ë

Ë

ËË Ë

Ë

moculobis elementi cilindrul koordinatebSi:

4Ω t4t44 (2.2) formulis sferul (cilindrul) koordinatebSi warmodgenis Semdeg SegviZlia gamoviTvaloT uganzomilebo oscilatoruli Zalebi (2.1) gamosaxulebis gamoyenebiT, sadac talRuri funqciebi mocemulia sferul (cilindrul) koordinatebSi. vinaidan Sredingeris gantoleba, romlis amoxsnasac vawarmoebT mocemulia sferul (cilindrul) koordinatebSi, amitom misi amonaxsnebi pirdapir SeiZleba Casmuli iqnas (2.1) gamosaxulebaSi.

sferuli cilindruli da konusuri kvanturi wertilebisferuli cilindruli da konusuri kvanturi wertilebisferuli cilindruli da konusuri kvanturi wertilebisferuli cilindruli da konusuri kvanturi wertilebi

Cven gamovikvlieT zonaTaSorisi gadasvlebis Sesabamisi uganzomilebo oscilatoruli Zalebi ]4L/¬L sferuli, cilindruli da konusuri kvanturi wertilebisaTvis [88]. samive geometriuli formis kvantur wertili imdagvarad SevarCieT, rom maTi moculoba identuri yofiliyo. kerZoT sferuli kvanturi wertilis radiusi 5 nm-ia, aseve 5-5 nm-ia cilndruli da konusuri formis kvanturi wertilebis fuZeebis radiusebi. sasrul elementTa saSualebiT da am paragrafSi zemoT warmodgenili formalizmiT (2.1) gamosaxulebis Sesabamisad Cven gamovTvaleT kvanturi wertilebis oscilatoruli Zalebi. miRebuli Sedegebi warmodgenilia qvemoT mocemul naxazebze.

52

nax. nax. nax. nax. 2.22.22.22.25555

sferuli kvanturi wertilisferuli kvanturi wertilisferuli kvanturi wertilisferuli kvanturi wertilissss zonaTaSorisi gadasvlebis Sesabamisi zonaTaSorisi gadasvlebis Sesabamisi zonaTaSorisi gadasvlebis Sesabamisi zonaTaSorisi gadasvlebis Sesabamisi uganzomilebo uganzomilebo uganzomilebo uganzomilebo

oscilatoruli Zalebioscilatoruli Zalebioscilatoruli Zalebioscilatoruli Zalebi

nax. 2.nax. 2.nax. 2.nax. 2.26262626 a)a)a)a)

nax. 2.nax. 2.nax. 2.nax. 2.26262626 a) a) a) a) konusuri b) cilindruli konusuri b) cilindruli konusuri b) cilindruli konusuri b) cilindruli

gadasvlebis Sesabamisi gadasvlebis Sesabamisi gadasvlebis Sesabamisi gadasvlebis Sesabamisi

sferuli kvanturi wertilebisaTvis gadasvlebi, romlebic Seesabameba eleqtronisa da xvrelis mdgomareobebs erTi da igive orbitaluri ricxviT akrZalulia. rac Seexeba gadasvlebs oscilatorul Zalebs konusuri da cilindruli kvanturi wertilebisaTvis Tu gadasvlis Sesabamis xvrelis da eleqtronis mdgomareobas erTmaneTisagan erTiT gansxvavebuli orbitaluri momentda cilindruli kvanturi wertilebis Sesabamisi oscilatorulisferuli kvanturi wertilis oscilatorul Zalebze daaxloebiT 2 rigididia.

urTierTqmedi kvanturi wertilebiurTierTqmedi kvanturi wertilebiurTierTqmedi kvanturi wertilebiurTierTqmedi kvanturi wertilebi

Cven aseve SeviswavleT ori urTierTqmediformis kvanturi wertilis oscilatoruli Zalebi Soris sxvadasxva manZiliswertilis radiusi da simaRle erTmaneTis tolia da 5nmnaxazebze mocemulia kvlevis Sedegebi:

nax.nax.nax.nax. 2.262.262.262.26 b)b)b)b)

konusuri b) cilindruli konusuri b) cilindruli konusuri b) cilindruli konusuri b) cilindruli kvanturi wertilikvanturi wertilikvanturi wertilikvanturi wertilissss zonaTaSorisi zonaTaSorisi zonaTaSorisi zonaTaSorisi

gadasvlebis Sesabamisi gadasvlebis Sesabamisi gadasvlebis Sesabamisi gadasvlebis Sesabamisi uganzomilebo uganzomilebo uganzomilebo uganzomilebo oscilatoruli Zalebioscilatoruli Zalebioscilatoruli Zalebioscilatoruli Zalebi

sferuli kvanturi wertilebisaTvis gadasvlebi, romlebic Seesabameba eleqtronisa da xvrelis mdgomareobebs erTi da igive orbitaluri ricxviT akrZalulia. rac Seexeba gadasvlebs oscilatorul Zalebs konusuri da cilindruli kvanturi wertilebisaTvis isini maqsimaluriaTu gadasvlis Sesabamis xvrelis da eleqtronis mdgomareobas erTmaneTisagan erTiT gansxvavebuli orbitaluri momenti aqvTda cilindruli kvanturi wertilebis Sesabamisi oscilatoruli

eruli kvanturi wertilis oscilatorul Zalebze daaxloebiT 2 rigi

urTierTqmedi kvanturi wertilebiurTierTqmedi kvanturi wertilebiurTierTqmedi kvanturi wertilebiurTierTqmedi kvanturi wertilebi

Cven aseve SeviswavleT ori urTierTqmedi /kvanturi wertilis oscilatoruli Zalebi kvantur wertilTa

Soris sxvadasxva manZilis SemTxvevaSi [89]. cilindruli kvanturi radiusi da simaRle erTmaneTis tolia da 5nm-s udris. qvemoT

naxazebze mocemulia kvlevis Sedegebi:

nax. 2.nax. 2.nax. 2.nax. 2.27272727 a)a)a)a)

53

zonaTaSorisi zonaTaSorisi zonaTaSorisi zonaTaSorisi

oscilatoruli Zalebioscilatoruli Zalebioscilatoruli Zalebioscilatoruli Zalebi

sferuli kvanturi wertilebisaTvis gadasvlebi, romlebic Seesabameba eleqtronisa da xvrelis mdgomareobebs erTi da igive orbitaluri ricxviT akrZalulia. rac Seexeba gadasvlebs oscilatorul Zalebs

isini maqsimaluria Tu gadasvlis Sesabamis xvrelis da eleqtronis mdgomareobas

i aqvT. konusuri da cilindruli kvanturi wertilebis Sesabamisi oscilatoruli Zalebi

eruli kvanturi wertilis oscilatorul Zalebze daaxloebiT 2 rigiT

cilindruli kvantur wertilTa

cilindruli kvanturi s udris. qvemoT

nax. 2.nax. 2.nax. 2.nax. 2.27272727 bbbb))))

54

55

nax. 2.nax. 2.nax. 2.nax. 2.27272727 g)g)g)g)

nax. 2.nax. 2.nax. 2.nax. 2.27272727 urTierTqmedi cilindruli urTierTqmedi cilindruli urTierTqmedi cilindruli urTierTqmedi cilindruli kvanturi kvanturi kvanturi kvanturi wertilebiwertilebiwertilebiwertilebis zonaTaSorisi s zonaTaSorisi s zonaTaSorisi s zonaTaSorisi

gadasvlebis Sesabamisi gadasvlebis Sesabamisi gadasvlebis Sesabamisi gadasvlebis Sesabamisi uganzomilebo uganzomilebo uganzomilebo uganzomilebo oscilatoruli Zalebi a) oscilatoruli Zalebi a) oscilatoruli Zalebi a) oscilatoruli Zalebi a) ½ Înm b) nm b) nm b) nm b) ½ ¶ g) g) g) g)

½ b

nax. 2.27-ze vxedavT, rom sistemis oscilatoruli Zalebi ganxiluli manZilebisTvis ar arian damokidebuli kvantur wertilTa Soris manZilze, Tumca gadasvlis energiebi kvantur wertilTa urTierTqmedebis gamo wanacvlebuli arian mcire da didi energiebisaken araurTierTqmed kvantur wertilebTan SedarebiT. Cven vnaxeT, rom oscilatoruli Zalebi RerZis perpendikularulad TiTqmis ar arian damokidebuli kvantur wertilTa Soris manZilze, meores mxriv oscilatoruli Zalebi -is gaswvriv 4 2-saTvis 3 rigiT didia vidre 4 7 SemTxvevaSi.

56

T a v i m e s a m eT a v i m e s a m eT a v i m e s a m eT a v i m e s a m e

$3.1 $3.1 $3.1 $3.1 kvanturi wertili kvanturi wertili kvanturi wertili kvanturi wertili

teqnologiebis gnviTarebam SesaZlebeli gaxada eqsperimentalurad damzadebuliyo kvanturi wertili, romlis garSemo kvanturi ormo iqneboda. aRniSnuli struqtura pirvelad Seqmnes [90]. maT gaakeTes romelic Tavis mxriv dafares isev wertilis garda, romelsac SuaSi myofi guli warmoadgens gvqondes erTi an ramdenimestruqturebs kvanturi wertili kwell).

winamdebare naSromSi Cven Teoriulad gamovikvlieT kvanturi wertili kvanturi ormoSre, :L -s Sre da sabolood Sedgeba erTi kvanturi wertilisa da mis irgvliv mdebare kvanturi ormosagan. naSromSi Seswavlilia daxrili potencialuri barieris mqone struqtura.sqematuri gamosaxuleba mocemulia nax. struqturis formula Semdegnairad SegviZlia warmovadginoT:sadac icvleba kvanturi wertili kvanturi ormos centridan manZilis mixedviT ise, rom miiReba sxvadasxva daxris potencialis mqone wertili kvanturi ormoeqsperimentalurad xorcielorTqlis qimiuri depoziciis gziT

sistemis parametrebi aR+j& 0.044+ , Ï 0.55gamtareblobis zonaTa Soris sxvaobaa sxvaoba 5j 0.655 [96]. barieris da kvanturi ormos sigane

kvanturi wertili kvanturi wertili kvanturi wertili kvanturi wertili kvanturi ormo (kvanturi ormo (kvanturi ormo (kvanturi ormo (ÐÑÐÒ

teqnologiebis gnviTarebam SesaZlebeli gaxada eqsperimentalurad damzadebuliyo kvanturi wertili, romlis garSemo kvanturi ormo iqneboda. aRniSnuli struqtura pirvelad eiCmiulerma, miusma da velerma

maT gaakeTes ]4L guli, romelic dafares romelic Tavis mxriv dafares isev ]4L -iT. aseT sistemebSi kvanturi wertilis garda, romelsac SuaSi myofi guli warmoadgens

erTi an ramdenime (multi) [91] sferuli kvanturi ormostruqturebs kvanturi wertili kvanturi ormo ewodebaT (quantum dot quantum

Cven Teoriulad gamovikvlieT :L /kvanturi wertili kvanturi ormo [92]. struqturas aqvs :L-is guli,

da sabolood struqtura imyofeba ]4L -SiSedgeba erTi kvanturi wertilisa da mis irgvliv mdebare kvanturi ormosagan. naSromSi Seswavlilia marTkuTxa da sxvadasxva wrfivad daxrili potencialuri barieris mqone struqtura. aRniSnuli struqturissqematuri gamosaxuleba mocemulia nax. 3.1. sistemis wrfivad daxrili struqturis formula Semdegnairad SegviZlia warmovadginoT:

kvanturi wertili kvanturi ormos centridan manZilis ise, rom miiReba sxvadasxva daxris potencialis mqone

wertili kvanturi ormo. daxrili potencialis mqone struqturebi xorcieldeba molekulur-sxivuri epitaqsiis

depoziciis gziT [93-94].

nax. nax. nax. nax. 3.13.13.13.1 kvanturi wertili kvanturi wertili kvanturi wertili kvanturi wertili

kvanturi ormos sqematuti kvanturi ormos sqematuti kvanturi ormos sqematuti kvanturi ormos sqematuti

gamosaxuleba. gamosaxuleba. gamosaxuleba. gamosaxuleba.

daxris siganeadaxris siganeadaxris siganeadaxris siganea

Rebulia Semdegnairad [95] :L - 5 ]4L - +& 0.2+ , +j& 0.7+

gamtareblobis zonaTa Soris sxvaobaa 5 1.355 savalento zonaTa Soris kvanturi wertilis radiusi 3 nm

barieris da kvanturi ormos sigane - 2-2 nm.

57

ÐÑÐÒ))))

teqnologiebis gnviTarebam SesaZlebeli gaxada eqsperimentalurad damzadebuliyo kvanturi wertili, romlis garSemo kvanturi ormo

Cmiulerma, miusma da velerma romelic dafares :L -is feniT,

aseT sistemebSi kvanturi wertilis garda, romelsac SuaSi myofi guli warmoadgens SesaZlebelia

sferuli kvanturi ormo. am tipis quantum dot quantum

/ ]4L / :L / ]4L is guli, ]4L-is

Si. anu sistema Sedgeba erTi kvanturi wertilisa da mis irgvliv mdebare kvanturi

sxvadasxva wrfivad aRniSnuli struqturis

sistemis wrfivad daxrili struqturis formula Semdegnairad SegviZlia warmovadginoT: :d]4L ,

kvanturi wertili kvanturi ormos centridan manZilis ise, rom miiReba sxvadasxva daxris potencialis mqone kvanturi

daxrili potencialis mqone struqturebi sxivuri epitaqsiisa da

kvanturi wertili kvanturi wertili kvanturi wertili kvanturi wertili

kvanturi ormos sqematuti kvanturi ormos sqematuti kvanturi ormos sqematuti kvanturi ormos sqematuti

gamosaxuleba. gamosaxuleba. gamosaxuleba. gamosaxuleba. » barieris barieris barieris barieris

daxris siganeadaxris siganeadaxris siganeadaxris siganea

+& 0.036+ , , Ï 2.55 .

savalento zonaTa Soris kvanturi wertilis radiusi 3 nm-ia, kvanturi

naSromSi gamokvleuliakulonuri urTierTqmedebauganzomilebo oscilatoruli Zalebi barieris daxris sxvadasxva mniSvnelobebisTvis: =0.5 nm, 1.0 nm. 0 nmTvisebebis cvlileba barieris daxris TiToeuli orbitaluri ricxvisTvis QDQW struqturas aqvs mdgomareoba. QDQW sistemis energiebis damokidebuleba barieris daxris siganeze mocemulia nax. 3.2

gamokvleulia sistemis energetikuli doneebikulonuri urTierTqmedeba da zonaTaSorisi gadasvlebis Sesabamisi

oscilatoruli Zalebi barieris daxris sxvadasxva =0.5 nm, 1.0 nm. 0 nm. Seswavlilia aRniSnuli sistemis

barieris daxris siganesTan mimarTebaSiTiToeuli orbitaluri ricxvisTvis i 0 da i 1 saTvis

struqturas aqvs eleqtronebisa da xvrelebisaTvis orsistemis energiebis damokidebuleba barieris daxris

3.2-3.3


58

sistemis energetikuli doneebi, eqsitonTa orisi gadasvlebis Sesabamisi

oscilatoruli Zalebi barieris daxris sxvadasxva aRniSnuli sistemis

siganesTan mimarTebaSi. saTvis Seswavlil

eleqtronebisa da xvrelebisaTvis or-ori bmuli sistemis energiebis damokidebuleba barieris daxris

nax. nax. nax. nax. 3.23.23.23.2 ÐÑÐÒ sistemaSi sistemaSi sistemaSi sistemaSi

daxris siganeze a) daxris siganeze a) daxris siganeze a) daxris siganeze a)

b) b) b) b) ---- orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT

rogorc nax. 3.2-ze maqsimaluria marTkuTxa barieris SemTxvevaSi da barieris daxris ubnis zrdasTan erTad mcirdeba. aSemTxvevaSi mTavari kvanturi ricxvebiT erTmanaTTan axlos imyofebian da potencialerTmaneTs Sordebian. xolo energiebi Tanabari tempiT mcirdebian anu maT Soris sxvaoba TiTqmis ar aris damokidebuli potencialis daxris ubnis siganeze. nax. mocemulia nax. 3.2-is analogiuri m

nax. 3.2nax. 3.2nax. 3.2nax. 3.2 b)b)b)b)

sistemaSi sistemaSi sistemaSi sistemaSi eleqtronis energieleqtronis energieleqtronis energieleqtronis energiebebebebis damokidebuleba is damokidebuleba is damokidebuleba is damokidebuleba

daxris siganeze a) daxris siganeze a) daxris siganeze a) daxris siganeze a) ---- orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT ±orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT ± ³

vxedavT QDQW sistemis eleqtronTmaqsimaluria marTkuTxa barieris SemTxvevaSi da barieris daxris ubnis zrdasTan erTad mcirdeba. aRsaniSnavia, rom eleqtronTa doneebi SemTxvevaSi mTavari kvanturi ricxvebiT 1 , 2 energerTmanaTTan axlos imyofebian da potencialis daxris ubnis zrdiT erTmaneTs Sordebian. xolo i 1 mdgomareobis Sesabamisienergiebi Tanabari tempiT mcirdebian anu maT Soris sxvaoba TiTqmis ar aris damokidebuli potencialis daxris ubnis siganeze. nax.

is analogiuri monacemebi xvrelTaTvis.

59

is damokidebuleba is damokidebuleba is damokidebuleba is damokidebuleba » ---- barieris barieris barieris barieris

²

sistemis eleqtronTa energiebi maqsimaluria marTkuTxa barieris SemTxvevaSi da barieris daxris ubnis

saniSnavia, rom eleqtronTa doneebi i 0 energiis mixedviT

is daxris ubnis zrdiT Sesabamisi eleqtronTa

energiebi Tanabari tempiT mcirdebian anu maT Soris sxvaoba TiTqmis ar aris damokidebuli potencialis daxris ubnis siganeze. nax. 3.3-ze



60

61

nax. nax. nax. nax. 3.33.33.33.3 ÐÑÐÒ sistemaSi xvrelTa energiebis damokidebuleba sistemaSi xvrelTa energiebis damokidebuleba sistemaSi xvrelTa energiebis damokidebuleba sistemaSi xvrelTa energiebis damokidebuleba » ---- barieris daxris barieris daxris barieris daxris barieris daxris

siganeze a) siganeze a) siganeze a) siganeze a) ---- orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT ± ²

b) b) b) b) ---- orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT orbitaluri kvanturi ricxviT ± ³

xvrelebis SemTxvevaSic vxedavT energiebis analogiuri xasiaTis damokidebulebas potencialis daxris siganesTan mimarTebaSi, rac zemoT eleqtronebisTvis aRvniSneT. axsna imisa Tu ratom izrdeba sxvaoba 1 da 2 energetikul mdgomareobebs Soris warmodgenilia momdevno paragrafSi QDQW -is talRuri funqciebis ganxilvisas.

$3.2 $3.2 $3.2 $3.2 ÐÑÐÒ sistemis talRuri funqciebisistemis talRuri funqciebisistemis talRuri funqciebisistemis talRuri funqciebi

qvemoT warmodgenilia QDQW sistemis eleqtronTa talRuri funqciebi.


nax. 3nax. 3nax. 3nax. 3.4.4.4.4 ÐÑÐÒ----s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi » ² nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT

± ² a) a) a) a) ´ ³ b) b) b) b) ´ b

62


nax. 3nax. 3nax. 3nax. 3.5.5.5.5 ÐÑÐÒ----s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi » ² nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT

± ³ a) a) a) a) ´ ³ b) b) b) b) ´ b


nax. 3nax. 3nax. 3nax. 3.6.6.6.6 ÐÑÐÒ----s eleqtronTas eleqtronTas eleqtronTas eleqtronTa talRuri funqciebi talRuri funqciebi talRuri funqciebi talRuri funqciebi » ². ¶ nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT

± ² a) a) a) a) ´ ³ b) b) b) b) ´ b


nax. 3nax. 3nax. 3nax. 3.7.7.7.7 ÐÑÐÒ----s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi » ². ¶ nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT

± ³ a) a) a) a) ´ ³ b) b) b) b) ´ b

63


nax. 3nax. 3nax. 3nax. 3.8.8.8.8 ÐÑÐÒ----s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi » ³ nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT

± ² a) a) a) a) ´ ³ b) b) b) b) ´ b


nax. 3nax. 3nax. 3nax. 3.9.9.9.9 ÐÑÐÒ----s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi s eleqtronTa talRuri funqciebi » ³ nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT nm, kvanturi ricxvebiT

± ³ a) a) a) a) ´ ³ b) b) b) b) ´ b

davubrundeT wina paragrafSi SeniSnul faqts, rom 1 da 2 mTavari kvanturi ricxvebis Sesabamis energiebs Soris sxvaoba i 0 isaTvis barieris daxris ubnis zrdasTan erTad izrdeba, xolo i 1 -is pirobebSi TiTqmis ucvleli rCeba. SevecadoT fizikurad avxsnaT es Sedegi. rogorc kvanturi meqanikidan aris cnobili kvanturi sistemis mdgomareobis energiaze zogad warmodgensas gvaZelvs am mdgomareobis talRuri funqcia. rac ufro swrafad icvleba talRuri funqcia fizikuri sivrcis fiqsirebul areSi miT metia Sesabamisi kvanturi donis energia, vinaidan hamiltonianis operatoris kinetikur wevri, romelic warmoebuls Seicavs swrafad cvalebadi talRuri funqciaze moqmedebiT met energiebs iZleva.

64

Tu davakvirdebiT 0 nm-is Sesabamis talRur funqciebs mTavari kvanturi ricxvebiT - 1 da 2 da orbitaluri kvanturi ricxviT - i 0 davinaxavT, rom am ori kvanturi mdgomareobis talRuri funqciebi daaxloebiT erTnairi siCqariT icvlebian amitom doneTa energiebic erTmaneTisgan mcired gansxvavdebian. xolo rac Seexeba 0.5 nm da 1 nm-is Sesabamis talRur funqciebs, maT SemTxvevaSi Tvali advilad SeniSnavs, rom isini gansxvavebuli siswrafiT icvlebian Sesabamisad maT ufro metad gansxvavebuli energiebi unda SeesabamebodeT. analogiuri msjelobiT aixsneba is daaxloebiT Tanabari sxvaoba 1 da 2 doneebs Soris i 1 SemTxvevaSi. mimdinare paragrafSi warmodgenili naxazebidan kargad Cans, rom i 1 is Sesabamisi talRur funqciebi QDQW areze daaxloebiT erTnairi siCqariT icvlebian. QDQW -is talRur funqciebze dakvirvebiT vadgenT, rom nawilakebis lokalizacia gulSi da ormoSi SesamCnevad gansxvavdeba sxvadasxva kvanturi mdgomareobebisaTvis. kvanturi meqanikidan rogorc cnobilia, rac ufro metia nawilakis energia miT metia misi unari kvantur barierSi gasvlisa da Sesabamisad - misi lokalizaciis are farTe, Tumca rogorc naxazebidan vxedavT i 1 is Sesabamisi mdgomareobebi ufro lokalizebulia gulSi an ormoSi vidre mdgomareobebi i 0 -iT. Tu davakvirdebiT talRuri funqciebis yofaqcevas zemoT warmodgenil naxazebze SevamCnevT, rom igi garkveul ubnebze swrafad icvleba. talRuri funqciebis es Tavisebureba, romelic gansakuTrebiT TvalsaCinoa 0nm-is Sesabamis naxazebze gamowveuli nawilakebis efeqturi masis cvlilebiT kvanturi wertilisa da kvanturi ormos sazRvrebze.

65

$3.3 $3.3 $3.3 $3.3 ÐÑÐÒ----isisisis eleqtroneleqtroneleqtroneleqtron----xvrelis kulonuri urTierTqmedebaxvrelis kulonuri urTierTqmedebaxvrelis kulonuri urTierTqmedebaxvrelis kulonuri urTierTqmedeba

aqamde zeviT yvelgan Cven ar viTvaliswinebdiT kulonur urTierTqmedebas eleqtronsa da xvrels Soris, vinaidan Cvens mier ganxiluli sistemebi eqsitonTa boris radiusze mcire zomisani iyvnen. am SemTxvevaSi ki damatyvevebeli potencialia arsebiTi, kulonuri urTierTqmedeba ki masTan SedarebiT mcire. radgan eleqtronebi da xvrelebi eleqtrulad damuxtuli nawilakebia maT Soris arsebobs eleqtruli anu kulonuri urTierTqmedeba. urTierTqmedi eleqtron-xvrelis kulonuri energia nivTierebaSi Semdegnairad gamoiTvleba [97]:

ÓÔÕn "Ö |×Ø|uØ|"|×Ù|uÙ|"

||uØd|uÙ| 4/tj4/t (3.1)

sadac Ψjtu da Ψjtu xvrelisa da eleqtronis talRuri funqciebia Sesabamisad. Ú - dieleqtrikuli konstantaa.

QDQW amocanas vxsniT sferul koordinatebSi sasrul elementTa meTodis gamoyenebiT, rogorc es pirvel TavSi aris warmodgenili. sferul

koordinatebSi unda gamoisaxos aseve

||uØd|uÙ| da 4/tj4/t . pirdapiri CasmiT

SeiZleba naCvenebi iqnas, rom sferul koordinatebSi t, , da t$, $, $ wertilebs Soris , manZili Semdegnairad gamoisaxeba:

, Ût$ ) t$$ P 2tt$Ü$ÌÍ P $ ) ÌÍÌÍ$ÝÞ $@

anu gvaqvs:

||uØd|uÙ| Ûtj$ ) t$ P 2tjtÜjÌÍj P ) ÌÍjÌÍÝÞdY

" (3.2)

moculobis elementisaTvis gvaqvs:

4/tj4/t tj$t$j4j44j44tj4t (3.3)

Cven gamoviTvaleT eleqtron-xvrelis araekranirebuli kulonuri urTierTqmedebis energia :L / ]4L / :L / ]4L kvanturi wertili kvanturi ormosaTvis potencialis daxris sxvadasxva siganis SemTxvevaSi (3.1) formulis gamoyenebiT masSi (3.2) da (3.3) gamosaxulebebis gaTvaliswinebiT. nawilakTa kulonuri urTierTqmedebis mniSvnelobebi mocemulia 1-2 cxrilebSi:

66

Eeleqtronis donis nomeri

xvrelis donis nomeri

Aaraekranirebuli kulonuri

energia ( 0 nm)

Aaraekranirebuli kulonuri energia

( 0.5 nm)


energia ( 1 nm) 1 1 0.3402 0.3386 0.3957 1 2 0.5908 0.4314 0.4559 2 1 0.3255 0.3074 0.3484 2 2 0.5068 0.5832 0.5561

cxrili. 1 cxrili. 1 cxrili. 1 cxrili. 1 ÐÑÐÒ ----is eleqtronis eleqtronis eleqtronis eleqtron----xvrelxvrelxvrelxvrelis is is is ± ²----is Sesabamisi araekranirebuli is Sesabamisi araekranirebuli is Sesabamisi araekranirebuli is Sesabamisi araekranirebuli

kulonuri kulonuri kulonuri kulonuri urTierTqmedebis urTierTqmedebis urTierTqmedebis urTierTqmedebis energia (energia (energia (energia (rq))))

Eeleqtronis donis nomeri

xvrelis donis nomeri


energia ( 0 nm)


( 0.5 nm)


( 1 nm) 1 1 0.3851 0.3588 0.3669 1 2 0.3035 0.3057 0.3140 2 1 0.3025 0.3084 0.3159 2 2 0.5229 0.5554 0.5193

cxrili. 2 cxrili. 2 cxrili. 2 cxrili. 2 ÐÑÐÒ ----is eleqtronis eleqtronis eleqtronis eleqtron----xvrelxvrelxvrelxvrelis is is is ± ³----is Sesabamisi araekranirebuli is Sesabamisi araekranirebuli is Sesabamisi araekranirebuli is Sesabamisi araekranirebuli

kulonurikulonurikulonurikulonuri urTierTqmedebisurTierTqmedebisurTierTqmedebisurTierTqmedebis energia (energia (energia (energia (rq))))

radgan kulonuri urTierTqmedeba mizidulobis xasiaTisaa igi eqsitonis energias amcirebs. cxrilebSi warmodgenilia araekranirebuli kulonuri urTierTqmedebis energia. Tu gaviTvaliswinebT, rom ÚßÏM 11.4 , ÚÓ<M 5.5 [95] miviRebT, rom kulonuri urTierTqmedebis Sesabamisi energia kvanturi datyvevebis Sesabamis energiaaze erTi rigiT mcirea.

naSromSi mocemuli meTodiT, Cvens mier gamoTvlili kulonuri urTierTqmedebis sisworis Sesamowmeblad gamovTvaleT fereiras da proetos mier analizuri gamoTvlebiT napovni kulonuri urTierTqmedebis energiebi [57] da davrwmundiT, rom Cvens mier gamoTvlili Sedegebi karg TanxvedraSia aRniSnul statiaSi miRebul SedegebTan, rac Cveni gamoTvlebis sizustes adasturebs.

$3.4 $3.4 $3.4 $3.4 ÐÑÐÒ----is zonaTaSorisiis zonaTaSorisiis zonaTaSorisiis zonaTaSorisi

am paragrafSi ganvixilavT Cvens mier Seswavlilkvanturi wertili kvanturi ormos uganzomilebo oscilatorul Zalebsmocemulia Seswavlili sistemis oscilatoruli Zalebis damokidebusistemis potencialis daxri

ÐÑÐÒ----is oscilatoruli Zalebis damokidebuleba potencialisis oscilatoruli Zalebis damokidebuleba potencialisis oscilatoruli Zalebis damokidebuleba potencialisis oscilatoruli Zalebis damokidebuleba potencialis

nax. 3.10 - ze k aRniSnavs xvrels k0 1 aRniSnavs zonaTaSoris gadasvlas, romelic Seesabamebaorbitaluri ricxvis mqone xvrelsa da eleqtronis mdgomareobebs Soris gadasvlas.QDQW struqturis gadasvlebs SorisgaaCia k0 0 gadasvlas, amasTan misi sidide TiTqmis damoukidebelia potencialis daxraze. k0

is zonaTaSorisiis zonaTaSorisiis zonaTaSorisiis zonaTaSorisi gadasvlebis oscilatoruli Zalebigadasvlebis oscilatoruli Zalebigadasvlebis oscilatoruli Zalebigadasvlebis oscilatoruli Zalebi

am paragrafSi ganvixilavT Cvens mier Seswavlil :L /kvanturi wertili kvanturi ormos zonaTaSorisi gadasvlebis Sesabamis uganzomilebo oscilatorul Zalebs. qvemoT warmodgenil naxazze

wavlili sistemis oscilatoruli Zalebis damokidebusistemis potencialis daxris siganeze.

nax. 3nax. 3nax. 3nax. 3.10.10.10.10

is oscilatoruli Zalebis damokidebuleba potencialisis oscilatoruli Zalebis damokidebuleba potencialisis oscilatoruli Zalebis damokidebuleba potencialisis oscilatoruli Zalebis damokidebuleba potencialis

» daxris siganezedaxris siganezedaxris siganezedaxris siganeze

aRniSnavs xvrels da - eleqtrons. magaliTzonaTaSoris gadasvlas, romelic Seesabameba

orbitaluri ricxvis mqone xvrelsa da i 1 orbitaluri ricxvis mqone eleqtronis mdgomareobebs Soris gadasvlas. rogorc nax.

struqturis gadasvlebs Soris yvelaze mcire oscilatoruli Zala , amasTan misi sidide TiTqmis damoukidebelia

0 1 da k1 1 gadasvlebi marTkuTxa potencialis

67

gadasvlebis oscilatoruli Zalebigadasvlebis oscilatoruli Zalebigadasvlebis oscilatoruli Zalebigadasvlebis oscilatoruli Zalebi

/ ]4L / :L / ]4L aTaSorisi gadasvlebis Sesabamis

. qvemoT warmodgenil naxazze wavlili sistemis oscilatoruli Zalebis damokidebuleba

is oscilatoruli Zalebis damokidebuleba potencialisis oscilatoruli Zalebis damokidebuleba potencialisis oscilatoruli Zalebis damokidebuleba potencialisis oscilatoruli Zalebis damokidebuleba potencialis

eleqtrons. magaliTisaTvis zonaTaSoris gadasvlas, romelic Seesabameba i 0

orbitaluri ricxvis mqone rogorc nax. 3.10 gviCvenebs

yvelaze mcire oscilatoruli Zala , amasTan misi sidide TiTqmis damoukidebelia

gadasvlebi marTkuTxa potencialis

68

SemTxvevaSi aRweven maqsimalur mniSvnelobas, xolo potencialis daxris zrdasTan erTad mcirdebian. k1 0 gadasvla potencialis daxrasTan erTad mcirdeba magram SedarebiT nela vidre k0 1 da k1 1.

optikurad aqtiuri QDQW struqturis misaRebad ufro efeqturia marTkuTxa potencialis gamoyeneba, Tumca vinaidan potencialze damokidebulia aseve sistemis energiebi, amitom saWiro energiebis misaRebad konkretuli miznebis misaRwevad SeiZleba gamoyenebuli iqnas daxrili potenciali.

69

mmmmiRebuliRebuliRebuliRebuli i i i SedegebSedegebSedegebSedegebi da daskvni da daskvni da daskvni da daskvnebiebiebiebi

1. naSromSi Teoriulad Seswavlila kvanturi wertilebis da axali - daxrili kvanturi wertili kvanturi ormos (QDQW) eleqtronuli mdgomareobebi da zonaTaSorisi gadasvlebis oscilatoruli Zalebi aseve eqsitonis kulonuri urTierTqmedeba. sistemebisaTvis amoxsnilia Sredingeris stacionaluri gantoleba sasrul elementTa meTodis gamoyenebiT koordinatze damokidebuli efeqturi masis miaxloebaSi potencialis sasrulobis gaTvaliswinebiT. efeqturi masis miaxloeba, romelic gamoyenebulia naSromSi, gamokvleuli zomis kvanturi wertilebis SemTxvevaSi sakmaod karg Sedegebs iZleva da eqsperimentalur SedegebTan karg TanxmobaSia, aman safuZveli mogvca gvemuSava efeqturi masis modeliT da sakmao sizustiT migveRo Sedegebi, romlebic zogadad ufro rTuli modelebis gamoyenebas moiTxoven. Semdeg safexurze anu Sredingeris gantolebis amoxsnisas mivmarTeT sasrul elementTa meTods, romelic gamoviyeneT sferul da cilindrul koordinatebSi sakvlevi sistemis simetriis Sesabamisad. simetriis arsebobis Sedegad mniSvnelovnad Semcirda kompiuterul resursebi. aRsaniSnavia sasrul elementTa meTodis simartive, rac koordinatze damokidebuli efeqturi masis da potencialis moxerxebul manipulaciis saSualebas gvaZlevs, aseve advilad viTvaliswinebT sakvlevi sistemis geometiul formebsac.

2. kvanturi wertilebis Tvisebebis maT geometriul formaze damokidebulebis Seswavlis mizniT gamokvleulia identuri moculobis mqone sferuli, konusuri da cilindruli kvanturi wertilebi, ise rom cilindrul da konusur kvantur wertilebs aqvT identuri simaRle rac sferuli kvanturi wertilis diametris tolia. napovnia, rom aRniSnuli kvanturi wertilebisaTvis mciredi gamonaklisis garda fiqsirebuli orbitaluri da mTavari kvanturi ricxvis SemTxvevaSi sferul kvantur wertilebSi eleqtronTa da xvrelTa energiebi maqsimaluria xolo cilindrul kvantur wertilSi minimaluri. rogorc kvlevam aCvena konusuri da cilindruli kvanturi wertilebis energiebi fiqsirebuli orbitaluri ricxviT daaxloebiT wrfivad izrdeba mTavari kvanturi ricxvis zrdasTan erTad. aseve AaRmoCnda, rom umdablesi energetikuli doneebi eleqtronTa da xvrelTaTvis naklebad aris damokidebuli kvanturi wertilis formaze. cilindruli kvanturi wertilebis talRuri funqciebi xasiaTdebian simetriiT kvanturi wertilis geometriuli centris mimarT. sferuli kvanturi wertilisaTvisaTvis Tu nawilakis orbitaluri kvanturi ricxvi i 0 , maSin nawilakis aRmoCenis albaToba Tavis maqsimalur mniSvnelobas kvanturi wertilis geometriul centrSi aRwevs, xolo Tu i 1 maSin igi centridan wanacvlebulia. cilindruli da konusuri kvanturi wertilebisaTvis ki nawilakebis ganawilebis

70

albaTobis maqsimumi i 0 SemTxvevaSi kvanturi wertilis simetriis RerZze mdebareobs.

3. Tu konusur kvantur wertilebs, erTi da igive moculoba da fuZe-simaRlis sxvadasxva fardoba gaaCniaT, maSin simaRlis SemcirebiT (radiusis gazrdiT) nawilakTa energiebi izrdeba da es cvlileba dabali doneebisaTvis ufro mcirea vidre SedarebiT maRali doneebisaTvis.

4. naSromSi Seswavlili ori urTierTqmedi cilindruli formis identuri / kvanturi wertilebi Tu usasrulod arian daSorebuli maSin maTi energetikuli doneebi gadagvarebulia. kvanturi wertilebis daaxloebiT xdeba gadagvarebis moxsna da energetikuli doneebi ixliCeba. rogorc kvlevam aCvena eleqtronebisTvis urTierTqmedeba daaxloebiT 6 nm manZilze xdeba arsebiTi xolo xvrelebisTvis daaxloebiT 2-4 nm-ze imis mixedviT Tu ra orbitaluri kvanturi ricxvis mqone donezea saubari. xvrelebis urTierTqmedebis manZili mcirea eleqtronTa urTierTqmedebis manZilze imitom, rom xvrelebisTvis gvirabuli efeqti ufro sustia, rac xvrelebis didi efeqturi masiT aixsneba. kvanturi wertilebis daaxloebisas energiis gaxleCa imdenad didi SeiZleba gaxdes, rom i 0 donis energia i 1 donis Sesabamis energiaze didi aRmoCndes.

5. oscilatoruli Zalebi sferuli kvanturi wertilisTvis zonaTaSorisi gadasvlebisaTvis erTi da igive orbitaluri ricxvis mqone doneebs Soris akrZalulia. konusuri da cilindruli kvanturi wertilebis oscilatoruli Zalebi sferuli kvanturi wertilis Sesabamis osciulatorul Zalebze 2 rigiT didia.

6. urTierTqmedi kvanturi wertilebis Sesabamisi oscilatoruli Zalebi naSromSi ganxiluli manZilebisTvis ar arian damokidebuli kvantur wertilTa Soris manZilze, Tumca gadasvlis energiebi kvantur wertilTa urTierTqmedebis gamo wanacvlebuli arian mcire da didi energiebisaken araurTierTqmed kvantur wertilebTan SedarebiT.

7. QDQW sistemis energiebi barieris daxris siganis gazrdiT mcirdeba. marTkuTxa potencialis SemTxvevaSi roca i 0 eleqtronTa doneebi iseve rogorc xvrelTa doneebi energetikulad axlos arian da barieris daxris siganis zrdasTan erTad erTmaneTs Sordebian. anu QDQW sistemis umciresi energetikuli doneebi Zlierad arian damokidebuli potencialis cvlilebaze, gansxvavebiT imisa rac kvantur wertilebSi vnaxeT, sadac umciresi energiebi naklebad iyvnen damokidebuli kvanturi wertilis geometriul formebze. QDQW sistemis maRali energetikuli doneebi ( i 1 ) ufro lokalizebulebia vidre dabali doneebi (i 0).

71

8. Seswavlil QDQW sistemaSi eleqtron-xvrelis urTierTqmedebis kulonuri energia erTi rigiT mcirea maT kvantur datyvevebis Sesabamis energiasTan SedarebiT.

9. QDQW sistemas gaaCnia didi zonaTaSorisi gadasvlebis Sesabamisi oscilatoruli Zalebi da zogierTi maTgani potencialis daxris siganis SemcirebiT mkveTrad izrdebian, maSin roca zogierTi gadasvlis Sesabamisi oscilatoruli Zala TiTqmis ar aris damokidebuli potencialis daxris siganeze.

72

dddd aaaa nnnn aaaa rrrr TTTT iiii

erTganzomilebiani erTganzomilebiani erTganzomilebiani erTganzomilebiani marTkuTxa kvanturi ormomarTkuTxa kvanturi ormomarTkuTxa kvanturi ormomarTkuTxa kvanturi ormossss amoxsnaamoxsnaamoxsnaamoxsna sasrul elementTa meTodiTsasrul elementTa meTodiTsasrul elementTa meTodiTsasrul elementTa meTodiT

amosaxsnelia Semdegi gantoleba:

] L] (1) sadac

φφdx (2)

L φφdx (3)

φ funqcias ganvsazRvravT rogorc nawil-nawil mocemuli funqcias (Piecewise Function):

dYdY

, Tu d

YdYd

, Tu

0, Tu D d an G

φ funqciis gansazRvris are moicavs kvantur wertilis da kvanturi wertilidan fizikurad usasrulod daSorebuli midamos, romlis boloebzec sistemis talRuri funqciebi nulis tolia.

(2) integralSi Semaval hamiltonis funqcias aqvs saxe:

P !$2+ y$ ) B

sadac B potencialuri funqciaa da igi SeiZleba mocemuli iqnas rogorc nawil-nawil mocemuli funqcia:

B 0 kvanturi ormos gareT

B PBl kvanturi ormos SigniT

potencialis analogiurad SeiZleba Caiweros nawilakis masa - + rogorc koordinatze damokidebuli.

(2) da (3) gamosaxulebebis integrebiT vRebulobT da L matricul

elementebs, romelTaganac vadgenT L matricebs. (1) gantolebis amoxsna warmoadgens sakuTari mniSvnelobebisa da sakuTari funqciebis ganzogadoebul amocanas. am amocanis amonaxsnebi - sakuTari mniSvnelobebi warmoadgenen sistemis energiebs, xolo ] sakuTri funqciebi φ funqciebze gadamravlebis da ajamvis Semdeg sistemis talRur funqciebs gvaZleven.

73

cilindruli da konusuri cilindruli da konusuri cilindruli da konusuri cilindruli da konusuri kvanturi wertilebis amoxsnakvanturi wertilebis amoxsnakvanturi wertilebis amoxsnakvanturi wertilebis amoxsna

vinaidan cilindruli koordinatebis SemTxvevaSi sistemebi aRiwereba

ori (r,z) koordinatiT da maT gaaCnia aqsialuri simetria z RerZis mimarT,

amitom sakmarisia realuri (cilindruli, konusuri) kvanturi wertilis

amoxsna Sesabamisi fizikuri aris naxevarze. qvemoT mocemul naxazze es

are muqad aris Seferili.

a) a) a) a) cilindrulicilindrulicilindrulicilindruli da da da da b) b) b) b) konusurikonusurikonusurikonusuri kvanturi wertilebis sqematuri gamosaxulebakvanturi wertilebis sqematuri gamosaxulebakvanturi wertilebis sqematuri gamosaxulebakvanturi wertilebis sqematuri gamosaxuleba. . . .

gamuqebuligamuqebuligamuqebuligamuqebuli zona warmoadgens ares zona warmoadgens ares zona warmoadgens ares zona warmoadgens ares romelzedacromelzedacromelzedacromelzedac ixsneba Sredingeris gantolebaixsneba Sredingeris gantolebaixsneba Sredingeris gantolebaixsneba Sredingeris gantoleba

cilindrisa da konusis SigniT mudmivi potenciali aRvniSnoT 5-iT, xolo

gareT - 5$ iT. cilindrul kvanturi wertilis SemTxvevaSi potencialuri

funqcia SegviZlia ganvsazRvroT rogorc Semdegi saxis nawil-nawil

mocemuli funqcia:

5t, 5 Tu t D da ¬ ¬$

5t, 5$ Tu t G an D ¬ an G ¬$

xolo konusuri kvanturi wertilisaTvis potencialuri funqcia SegviZlia

warmovadginoT rogorc Semdegi saxis nawil-nawil mocemuli funqcia:

5t, 5 Tu 0 D t Yá"dá"dáY da ¬ ¬$

5t, 5$ Tu t G Yá"dá"dáY an D ¬ an G ¬$

74

analogiurad ganisazRvreba koordinatze damokidebuli efeqturi

masac +t, (m1 kvanturi wertilis SigniT da m2 kvanturi wertilis

gareT).

75

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