KARDAR-PARISI-ZHANG UNIVERSALITY, ANOMALOUS SCALING … completo.pdf · RENAN AUGUSTO LISBÔA ALMEIDA KARDAR-PARISI-ZHANG UNIVERSALITY, ANOMALOUS SCALING AND CROSSOVER EFFECTS IN

❯❯ Ô

P❩ ❯❱❨ ❯ ❱ ❲

ssrtt♦♥ ♣rs♥t t♦ t ❯♥rs r ❱ç♦s s ♣rt ♦ trt Pr♦r♠ rqr♠♥ts ♥ ♣♣ P②ss ♥ ♦rr t♦ ♦t♥ t tt♦ str ♥t

❱ ❩

Ficha catalográfica preparada pela Biblioteca Central daUniversidade Federal de Viçosa - Câmpus Viçosa

T

Almeida, Renan Augusto Lisbôa, 1990-A447k2015

Kardar-Parisi-Zhang universality, anomalous scalingand crossover effects in the growth of CdTe thin films /Renan Augusto Lisbôa Almeida. - Viçosa, MG, 2015.

xv, 129f. : il. (algumas color.) ; 29 cm.

Inclui apêndices.Orientador : Sukarno Olavo Ferreira.Dissertação (mestrado) - Universidade Federal de

Viçosa.Referências bibliográficas: f. 112-129.

1. Telureto de cádmio. 2. Filmes finos. 3. Kardar-Parisi-Zhang, Equação de. I. Universidade Federal de Viçosa.Departamento de Física. Programa de Pós-graduação emFísica Aplicada. II. Título.

CDD 22. ed. 530

FichaCatalografica :: Fichacatalografica https://www3.dti.ufv.br/bbt/ficha/cadastrarficha/visua...

2 de 3 21-10-2015 12:06

s ♦r s t t♦ t♦s ♣♦♣ tt ♥ t ♠♥t② ♥ tt r

t♥ ♦r rs♥ t ♣♦rt② s♦ r♥s ♥ ♦r ♠♣r♦♥ t t♦♥

s r t tr r ♥ ♥ssr② ♦rs ♥ ♦ s s♦ s♦ ♦s ♦♥

❲

♦♥s♦♥ ♦ ts ssrtt♦♥ ♦♥s t t ♥ ♦ ♦♥ s♣ ♣r♦

♥ ♠② ♦s ②♦♥ ♦ ♣②ss s t♥ t s ♣♣r♦♣rt t♦ strt

t♥♥ t♦s ♣♦♣ ♦ ♣ ♠ t ♦t rt♦♥s s♥ t ♥♥♥ ♦

ts ♦r♥② ② ♣r♥ts r ♠② ss ♥ ♥tr ♣rt ♦ t ♠ t♦② s ②

♥ r♥♥ tr♦ ①♠♣s tt tr ♣ss♦♥ ♥ t ♠st ♦r

ttts ♥ ♥ t ♦r s♠s t♦ ♦♥rs② ♥s ♦r s♣♣♦rt

♥ ♠♦tt♦♥ ♠♥② ♥ t t♠s s♦ ♠st t♥s r♠ ♥t♦s ♦r s♦

♠♥② ②rs ♦ ♦♠♣t② t ♠♣♦ss t♦ ♦rt t♠

♠ ♥♦r♠♦s② rt t♦ ♠② s♣rs♦rs r♥♦ rrr ♥ ♦

r r♥♦ s rt ①♣r♠♥tst ♥♦s ♠♦st r②t♥ ♦t ♥②

r♦t ♥ rtr③t♦♥ t♥q s ♥♦s ♦t ♦t tr♦♥ ♦r

t ♠♦st r♠r ♥ r♥♦ s s ♣rs♦♥t② ②s ♦♣♥ t♦ t s ♥

♣ ②♦♥ ♥ ①♣rt ♣②ssts ♦rt♥t② s ♦♣t ② ♠ s♥ t

♥♥♥ ♦ ♠② ♥rrt ♥ t♦② ♠ ♣r♦ ♦ ♥s♥ ts ♥r s

s♣rs♦♥ ♥ t s♠ ♥ ♦ s st t ♠♦st ♦♠♣t♥t ♣rs♦♥ tt

♥♦♥ s rts♠ s ♣② r r♦ ♥ ♠② ♣r♦rss s ♣②sst ♥ t s

r② r tt t♦t t ts ♦r ♦ st ♠♦r ♦♥ ♦♥t♥♥ sr ♠♥

♥ss ①♣♦♥♥ts t♦♥② ♦ s ♥ s♦♥ ♠ s ♠ttr ♦ ①♠♣

rt s♥t ♥trt② ♥ ♠♥② ♥♣r♣r ♣♦♣ tt sr t t♦rs♣

♥ ♣♣rs ② ♦♥♥♥ ♥st ♠rt ♠ rt t♦ ♦t ♦r s♣♣♦rt♥ ♠②

♥ t♦rs t ♦t♦rt s s ♠② ♥tr ♥ ♣♥

② t ② ♦ t♦ t♥ Pr♦ rrr ♦r sr♥ tt ♥tr

t ♠ ♥ ♦r ♥ ♠ ♦st ♥ t ♠ ♦ ②♦t♦ ♦♥ ② ♦r ♠② t

t♦t ♠♣ t t ♥ ♦ t ② ♦r t t ♥♥♥ s t♦

♦♦ t ♥② t♥s s♦ ♦r r♦♠♠♥♥ ♠ t♦ ♥ r♦ P P❩

♦rs♦♣ s rt ①♣r♥ ♠ t♦ ♠t r ♣♥② ♠

②②s P ❨♥r P rrr ♥ s♦ ♠♥② ♥s♣rt♦♥ ♣②ssts

♥ ♠t♠t♥s ♦r♠r t♦ ♥ ♠♣♦rt♥t ♥sts ♦t ♠② ♦r

s s s s ♦t ♠② ♠ ♣♥s t s ♣r t♦ r t♠

♦t P ❨♥r ♦♣ tt ♥ ♦r t♦tr s♦♦♥ s rs t♦ ♠② r♥s

♦rí③♥ ♥t ♥ ❱♦ ♠ tt t♠ ♥ ♣♥

♥tst ♦♥ ❲ ♥t ♠t ♣♥s t♠♣ t ♥♦ ♦r♦ ♥ ♥♦♥

♦ s ♣♥s ♣rt②

♦t ♠② r♥s r♦♠ ❱ç♦s ①♣rss ♠② s♣ t♥s t♦ s♥r rr③ ♥

rr♦ ❱s♦♥♦s ❨♦ tr♥s♦r♠ t ♥ t ♠♦♠♥ts ♦t ♦ t ♠

♥r s♦ t♥ t♦s ♦ s♥r② ♣ ♠ ♦♥ t ② t ♥s

P♦ s♦ ♣ ♥♦ ♥♦ st♦s r♠♥ ❱ rt♦ ♦♠♥ s♠

rrs♦ ♥ ♠♥② ♦tr ♦♦ r♥s

♠ t♥ t♦ Pr♦ s ♦r ♥tr♦♥ ♠ ♥ ts t♦ ②rs

♦ ♥ t♦ Pr♦ r♥♦ ♦r s t② t♦ ♣t t ts ssrtt♦♥

❯♥♦rt♥t② t s♦rt t♠ tt ♦r rt♥ t ♥ ♦r♥ ♦rrt ♣r♦♠s

♣r♥t ♦t ♦r ♣♥s ts t♠ ♣ t♥s r sr t♦ Pr♦ ①♠♥♦

s ♥♦r ♥ Pr♦ ♦ qrq sss ♦r ♣t♥ t ts

ssrtt♦♥ ♥ ♦r ♥ ♦♥trt♦♥s ♦r t ♥ rs♦♥ ♦ ts t①t

♥s t ♦♦r♥çã♦ ♣rç♦♠♥t♦ Pss♦ í ♣r♦r

P ② ♦♥ ♥ ②r ♦ str ♦rs♣

♦♥t♥ts

❯ ①

①

❯ ①

①

♥tr♦t♦♥

rts ♥r♥ ♥ ❯♥rst② ♥ ♥tr r♦t

r♦♠ rtt② t♦ t ♠②❱s ♥sät③

♦rrt♦♥ ♥t♦♥s

♦♥t♥♠ qt♦♥s ♥ ❯♥rst② sss

rs❲♥s♦♥ ♥ t ♥r qt♦♥

♥r ♣♦st♦♥s♦r♣t♦♥s♦♥ qt♦♥ ♥ r♥

♦s r♦♠ r ♥ s

rrPrs❩♥ ❯♥rst② ss r st♦r ♥

tt♦trt

♥ ♣♣♥s t strt♦♥s

❯♥rs qr ♦♥ss strt♦♥s

❯♥rs ①♠ t t strt♦♥s

tr ♥ ①♣r♠♥t t♦s

♦t ❲ ♥q

t♦♠ ♦r r♦s♦♣②

sr ♥♥

t♥ ♠s ♥♥ r♦t ♥ rtr③t♦♥

❯ ❯♥♦r♥ t P❩ ❯♥rst② ♥ ♥ ♠s

♠♥ttt ♦r♣♦♦ ♥②ss

♦ tt♦♥s ♦♥❯♥rs ♥ ❯♥rs ♥ ①♣♦♥♥ts

♦ ♥

♦r♥ ♦ t P❩ ♠♥s♠ ♥ ♦♥s♦♥s

❯ t ♦ ♠♣rtr ♦♥ r♦t ②♥♠

♥trr♥ ♠♦r♣♦♦② ♥ ♦ tt♦♥s

♥t ♦♥t r♦ ♦

❯♥rs ①♣♦♥♥ts

Prt ♦♥srt♦♥s

❯♥rs strt♦♥s

sr tt♦♥s t T = 150 C P♦ss♦♥♥ r♦t

♥♦♠t♦P❩ r♦ss♦r ♥ r tt♦♥s t T =

200 C

P❩ r♦t t ♣♦st♦♥ s r t

t♦♥s t T = 300 C

♥ ♠rs

♦♥s♦♥s ♥ Prs♣ts

♦r ts ♦t ♦♥t♥♠ r♦t qt♦♥s ♥ ❯♥rst②

sss

♥♦♠ r♦t qt♦♥

rt♦♥ ♦r t ♥r qt♦♥

♦♥♥r qt♦♥ ♥ t ❱ ss

♥♦♠♦s ♥

♦♥♣ts ♦r rs ♥ qr♠ ♥ ♦r t ♥ ♠ r♦t

r ♥s♦♥ ♥ qr♠ ♣

t♦♥

t♦♥ ♣♥♥ ♦♥ sstrt t♠♣rtr

t♦♥ ♣♥♥ ♦♥ ♠♦r ①

r♦t ♥ strtr ♦ ♠s

r♦t ♦s

♦♠♠♥srt② ♥ P♦②r②st♥t②

♥t P♥♦♠♥ ♥ ♣r trtrs

❨ ❨

st ♦ rs

♥tr♦t♦♥

rs s♥♦ ♥tr

♠ ♦ ♠ r♦♥ ♦♥

①♠♣ ♦ r♦ss♦rs ♦rr♥ ♥ ♥tr ♦s

rts ♥r♥ ♥ ❯♥rst② ♥ ♥tr r♦t

♦♠♣rs♦♥ t♥ strt♦♥s ♥ t s♠ ①♣tt♦♥

②♣ r♦♥ss rss t♠ ♣♦t ♦ × ♦ r♥ ♥ ♥tr r♦t

②♣ ♦ r♦♥ss ♣♦t ♦ × ♦ ♦r ♥ ♥tr r♦♥ t t♠ t

①♠♣ ♦ ♦♣♦♣ ♦r♥ ♥t♦♥

r♥ s♣s ♥ ♥r ♦ r♦♥ss r

rrPrs❩♥ ❯♥rst② ss

ss ♣r♦s ♦r tr♠♥st P❩ r♦t

❯♥rs strt♦♥s ♦t♥ ② Prä♦r ♥ ♣♦♥

sts ♦r tr♥t q r②sts

s ♦r r♥t ♥rst② sss


❲ s②st♠ s ♥ ts ♦r

s ♣rts ♦ ♥

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

♠s ♦r t♥ ♠s r♦♥ t 250 C

❳② s♣tr ♦ ♣♦②r②st♥ ②rs

♦ r♦♥ss s♥ ♦r t♥ ♠s r♦♥ t T = 250 C

♦r♠③ s♦♣ ♦r♥ ♦r t♥ ♠s r♦♥ t T = 250 C

♦ r♦♥ss ♦ t♥ ♠s s ♥t♦♥ ♦ t♠

s s ♦r sr r♦♥s ♦ t tst ②r

s s ♦r srs r♦♥ t T = 250 C

s s ♦r s♠♣s r♦♥ t T = 250 C

t ♦ ♠♣rtr ♦♥ r♦t ②♥♠

♠s ♦ t♥ ♠s r♦♥ t T = 150 200 ♥ 300 C

②♣ r♥♠♦♥ ♣r♦s t t sr ♦r T = 200 C ♥ 300 C

①tr ♥ t rt♦♥ ♥ ②rs r♦♥ t r♥t T

♦ r♦♥ss ♦r ♠s r♦♥ t T = 150 C 200 C ♥ 300 C

sts r♦♠ t ♥t ♦♥t r♦ ♦

♦ r♦♥ss r ♥ s♦♣s♦♣ ♦r♥

s s ♦r srs r♦♥ t T = 150 C

S ♥ K s s ♥t♦♥ ♦ t ♦① s③ r♦♠ s ♥ s

s s ♦r srs r♦♥ t T = 200 C

s s ♦r s♠♣s r♦♥ t T = 200 C

s s ♦r s♠♣s r♦♥ t T = 200 C

s s ♦r srs r♦♥ t T = 300 C

s s ♦r srs r♦♥ t T = 300 C

S ♥ K s ♥t♦♥ ♦ t ♦① s③ r♦♠ s ♦r T = 300 C


♦♥tr ♦r ♦ λ s ♥t♦♥ ♦ T

♦r ts ♦t ♦♥t♥♠ r♦t qt♦♥s ♥ ❯s

♠t ♦ t ♠ ♣♦t♥t ♣♥♥ ♦♥ t ♦ rtr

①

♥♦♠♦s ♥

①♠♣ ♦ ♥♦♠♦s s♥

♦♥♣ts ♦r rs ♥ qr♠ ♥ ♦r t ♥ ♠ r♦t

♠ ♦ P r②st t qr♠

♠t ♦ ♥t♦♥ ♣r♦sss

♥ ♦ t s r ♥r② s ♥t♦♥ ♦ ♠♥ ♥s s③

♥t♠♦ts

♠s r♦♥ ♦♥

♠t ♦ ♥t ♣r♦sss

♠t ♦ t tt ♣♦t♥t

①

st ♦ s


❯♥rs P❩ s ♦r ♠♥ts ♦ s ♥

❯♥rs P❩ s ♦r ♠♥ts ♦ s ♥

t ♦ ♠♣rtr ♦♥ r♦t ②♥♠

❱s ♦r t ♥♦♥♥rs ①♣♦♥♥t α1(t, T )

❱s ♦r t ♥♦♥♥rs ①♣♦♥♥ts κ(t, T ) ♥ ncoar(t, T )

❱s ♦r t ♥rs ①♣♦♥♥ts β(T ) ♥ 1/z(T )

①

❯

♥♥ st♦ sô ❯♥rs r ❱ç♦s rr♦ 2015P❩ ❯❱❨ ❯ ❱ ❲ r♥t♦r r♥♦ ♦ rrr ♦r♥t♦r ♦ ♦sé r

st tr♦ sts ♥â♠ rs♠♥t♦ ♠s ♥♦s rt♦ á

♠♦ ♣r t♠♣rtrs ♣♦sçã♦ ♥tr 150 C 300 C ❯♠ rçã♦

♥tr ♦çã♦ ♦s ♠♦rr♦s tçõs ♦♥♦s ♦♠♣r♠♥t♦s ♦♥ ♥ sr♣rí

♦ ♠ é st ♥♦♥trs q ss rt♦s ♦♠♣r♠♥t♦s

♦♥ sã♦ ts ♣♦r ♠ ♦♠♣tçã♦ ♥tr ♦ rst♦ ♦r♠çã♦ t♦s ♥

♦r rã♦s ③♥♦s ♦♦s ♥tr ♠ ♣r♦ss♦ r①çã♦ ♦r♥♦ sã♦

♣♦sçã♦ ♣rtís ♠♦és s♦r sss rõs ❯♠ ♠♦♦

♦♥t r♦ ♥ét♦ ♦rr♦♦r s ①♣çõs ♠ q T é ss ♦♠

♣tçã♦ á ♦r♠ r♥ts ♥ár♦s ♥ s r♦s ts ♦♠♦ rs♠♥t♦

s♦rr♦♥♦ r♦ss♦r s♦rr♦♥♦ ♣r rs♠♥t♦ ♦rr♦♥♦

s ♥ô♠ tr♥s♥t ♠ ♣rtr ♣r T = 250 C ♠♦strs q tçõs

♥ s♣rí sã♦ srts ♣ ér qçã♦ rrPrs❩♥ P❩

♦ ♠s♠♦ t♠♣♦ q ♥rs s strçõs tr r♦s ♦

tr ♠á①♠ ♣r ss P❩ é ♥♠♥t ①♣r♠♥t♠♥t ♠♦♥str

♥â♠ s tçõs ♥ s♣rí ♠s rs♦s ♦trs t♠♣rtrs ♥

é srt ♣ qçã♦ P❩ ♠s ♦♠ r♥ts ♦rs ♣r t♥sã♦ s♣r ν

①

♣r ♦ ①ss♦ ♦ λ sr ♣r T = 150 C ♥♦♥trs ♠ rs

♠♥t♦ P♦ss♦♥♥♦ q ♥ ν = λ = 0 Pr T = 200 C ♥trt♥t♦ ♠ r♦ss♦r

tór♦♣rP❩ é ♥♦♥tr♦ ♦♠ λ > 0 ♥st s♥♦ r♠ ♦r♠ s

P❩ ♣r ♠s rs♦s T ∈ [200, 250] C ♦rr ♦♠♣① ♥â♠

♠♣♦t♠♥t♦ ♦s rã♦s r♥t q s♣ç♦s ♥s ③♥♥çs ♦s ♠s♠♦s ♥ã♦ sã♦

t♦t♠♥t ♣r♥♦s ss ♠♥s♠♦ rçã♦ t♠ ♦ ♠s♠♦ t♦ r

çã♦ tr ♦ ♠♦♦ ♣♦sã♦ íst ♦ q ♠ ①ss♦ ♦

λ > 0 ♥♠♥t ♣r ♠s rs♦s T = 300 C ♠♦♥strs q λ < 0 ♠

♣rtr ♦ ♠♥s♠♦ P❩ ♣r ♠s rs♦s st t♠♣rtr ♦rr t

t① rs ♣♦sçã♦ ♣rtís q é ♣♥♥t s ♥♥çõs ♦s

st ê♥♦♠♥♦ ♣♦ sr ①♣♦ ♠ tr♠♦s ♦ ♦♥t st♥ ♦ q é tã♦ ♣

q♥♦ qã♦ ♠s ♦♠♥t ♥♥ ♦r s♣rí ♦ t♦s t♠♣♦♥t♦

r♦ss♦r t♠♣♦rs s ♥ô♠ ♦♦rr♥♦ ♠ T = 200 C 300 C ①♣♦♥ts

s ♠ ♠ rr ss ❯♥rs ♦ rs♠♥t♦ ♦♥t♦ ♠

♥♦♦ ♠ét♦♦ s♥♦♦ q ♥ç s♦r s♠♣s ♦♠♣rçã♦ ♥tr ①♣♦♥ts

ss ♦rs t♦r♠♥t s♣r♦s ♣r♠t♥♦s ♦♥r q ♦ rs♠♥t♦

♠ ♠ ♠♣ ① t♠♣rtr ♣rt♥ à ss P❩

①

♥♥ st♦ sô ❯♥rs r ❱ç♦s rr② 2015P❩ ❯❱❨ ❯ ❱ ❲ sr r♥♦ ♦ rrr ♦sr ♦ ♦sé r

♥ ts ♦r ♦♥ r♣♦rts ♦♥ t r♦t ②♥♠ ♦ t♥ ♠s ♦r ♣♦st♦♥

t♠♣rtrs T ♥ t r♥ ♦ 150 C t♦ 300 C rt♦♥ t♥ t ♠♦♥

♦t♦♥ ♥ r♥t tt♦♥s t sr s ♥ sts ♥

♥s tt s♦rt♥t ss r tt ② ♥ ♥tr♣② t♥ t ts ♦ t ♦r

♠t♦♥ ♦ ts t ♦ ♦♥rs ♦ ♥♦r♥ r♥s ♥ r①t♦♥ ♣r♦ss

st♠s r♦♠ t s♦♥ ♥ ♣♦st♦♥ ♦ ♣rts ♠♦s t♦rr

ts r♦♥s ♥t ♦♥t r♦ ♠♦ ♦rr♦♦rts ts rs♦♥♥s s T s

♥rs tt ♦♠♣tt♦♥ s rs t♦ r♥t s♥r♦s ♥ t r♦♥♥ s♥

s s ♥♦rrt r♦t r♦ss♦r r♦♠ r♥♦♠ t♦ ♦rrt r♦t ♥ tr♥

s♥t ♥♦♠♦s s♥ ♥ ♣rtr ♦r T = 250 C ♦♥ s♦s tt tt♦♥s

♦ sr r sr ② t rt rrPrs❩♥ P❩ qt♦♥

♥ t ♠♥t♠ tt t ♥rst② ♦ t ♦ r♦♥ss ♥ ♠①♠ t

strt♦♥s ♦r t P❩ ss s ♥② ①♣r♠♥t② ♠♦♥strt ②♥♠

♦ tt♦♥s t t sr ♦r ♦tr t♠♣rtrs st s sr ② t P❩

qt♦♥ t t r♥t s ♦r t s♣r t♥s♦♥ ν ♥ ①ss ♦ ♦t②

λ ♠② ♦r T = 150 C ♦♥ ♥s P♦ss♦♥♥ r♦t tt ♥ts ν = λ = 0

①

♦r T = 200 C ♦r ♥♦♠t♦P❩ r♦ss♦r s ♦♥ t λ > 0 ♥ t

s♦♥ r♠ ♦r♥ ♦ t P❩ s♥ ♦r ♠s r♦♥ t T ∈ [200, 250] C

st♠s r♦♠ ♦♠♣① ②♥♠ ♦ r♥ ♣♥ r♥ s♣ t t

♥♦r♦♦ ♦ r♥s r ♥♦t s rt♦♥ ♠♥s♠ s t s♠ t

♦ t tr rt♦♥ ♦ t st ♣♦st♦♥ ♠♦ s t♦ ♥ ①ss ♦

♦t② λ > 0 ♥② ♦r ♠s r♦♥ t T = 300 C ♦♥ ♠♦♥strts tt P❩

r♦t t λ < 0 ts ♣ ♥ ♣rtr t P❩ ♠♥s♠ t ts T ♦♠s

r♦♠ t rs rt ♦ t ♣♦st♦♥ ♦ ♣rts ♣♥s ♦♥ t ♦

s♦♣s s ♣♥♦♠♥♦♥ ♥ ①♣♥ ♥ tr♠s ♦ t st♥ ♦♥t s

s♦ s♠r s ♠♦r ♦② ♥♥t s t sr t♦ ♥tt♠ ts t♠♣♦r

r♦ss♦r ♥ ♥♦♠♦s s♥ t♥ ♣ ♥ T = 200 C ♥ T = 300 C s

①♣♦♥♥ts ♥ r t ❯♥rst② ss ♦ t r♦t ♦ttst♥♥ ♥

s♠ ♦♣ ♥s ♦r t s♠♣ ♦♠♣rs♦♥ t♥ ①♣♦♥♥ts ♥

tr t♦rt② ♣rt s ♦ s sr② t♦ ♦♥ tt t r♦t ♦

♥ r♥ ♦ ♣♦st♦♥ t♠♣rtr ♦♥s t♦ t P❩ ss

①

♣tr

♥tr♦t♦♥

♠♦♥t♦r t♥ ♠s r t ss ♦ ♦r ♦♣t♦tr♦♥ t♥♦♦② ♥

♥ ♦♥ r②r ❬❪ rt ♣rt ♦ t rr♥t t♥♠ st s t

♠ t♥♦♦② ♥t♠ t s ♥ s s♣♣♦rt t ♦♣♥ ♦

s♦♣stt r♦t t♥qs s ♦r ♠ ♣t①② ♦t ❲ ♣t①②

❲ ♠ ❱♣♦r ♣♦st♦♥ ❱ ♥ ♦trs ❬❪ ♥ t♥ t qt②

♥ ♦♥tr♦ ♦♥ ♦♣♥ t♥ss ♠str② ♦♠♣♦st♦♥ ♥ strtr ♦ ♠s ♠

s♦ rt tt t rsts ♥ t♦♥ rt② ♦r ♦♠♣trs t♠

s♠r ♥ ♠♦r ♣♦r ♥♥ ♣♦♥s ♥ ♦trs ♠♦ s s②st♠s ♦

♦ ♣♦st♦♥ ♦③t♦♥ P t♦ ♣♦♣t♦♥ srs ♦ sr ♥ts

♠ q♣♠♥ts ♦ s♦ ♠♣t s ♠♥t rs♦♥♥ ♣♦str♦♥tr♦♥

t♦♠♦r♣② ♥ s♦ ♦rt

♠♦♥ t ♠♦st ♣r♦♠♥♥t ♦♠♣♦♥s ♥t♥ ♦r t♥ ♠ ♣r♦t♦♥s

st♥s ♦t t ♠♠r ♦♥ t♦ ts st s♠♦♥t♦r ♣r♦♣

rts s s rt ♥r② ♣ Egap = 1.53 ❱ t♦ 300 ♥ ♦♣t s♦r♣t♦♥

♦♥t ≈ 5 × 105 ♠−1 ❬❪ ♣♣t♦♥s ♦♥r♥ ♦♥ ♠s s♣♥ t

rt♦♥ ♦ s♦r s ♦ ♥② ❬❪ s s ♦ t ❳r② γr② ♥ ♥rr

tt♦rs ❬❪ r r ①t♥s sts ♦♥ r♦t ♥ r♥s s rs s

t ♦♥tr♦ r♦t ♦ ttr♣♦r♥ ♥♥♦r②sts ❬❪ sss♠② ♦ q♥t♠

♦ts ❬❪ ♥♥♦rs ❬❪ ♠r♦ts ♦ trst ♦♣t rs♣♦♥ss ❬❪ t r

tss tr r st ♦rs ♦♥ r♦t ②♥♠s ts ❬❪ sss♥

♥tr♦t♦♥

sss s ♥t r♦♥♥ r♦t s②♠♠trs s s ♠♦r♣♦♦ s♣ts

r♥ s③ r♥ s♣ ♥ tr ♣♥♥② ♦♥ ♣♦st♦♥ t♠♣rtr ♠♦r

① ♥ t♥ss ♥ t ts r t ♠♦st ♠♣♦rt♥t strtrs ♥ ♣r♠trs

t♥ st♦♠♥ ♥ tr ♠ trs ♥ ♦♥sq♥t② t ♥②

♦ s t ♣ r♦♠ t♠ ❬❪

r♦♠ t t♦rt s t sr ♦r ♥tr r♦t s ♥ ♥trst♥

♦t♦qr♠ ttst ♥s st s ♦ s ♥r♥ ♥ ❯♥

rst② ♠r♥ s ♦rs ♥ tr♠ tt♦♥s ♦ qr♠ s②st♠s t rt

t② ❬❪ s♥ ♥r♥ ♠♣s s♥ ♦ ♥② rtrst ♥t ♥ t

s②st♠ ②♦♥ s②st♠ s③ ts ♥ tr♥ t ❯♥rst② ♦♥♣t ♠♥s tt s②s

t♠s ♦ r♥t ♠r♦s♦♣ ♥tr ♥ ①t t s♠ r s ♦r s♥

t② r r ② ♥trt♦♥s sr♥ ♠♥s♦♥t② s②♠♠trs ♥ ♦♥srt♦♥

s ❬❪ t② t ❯♥rst② ♦s ②♦♥ ♦ ♥tr sts ♥ s ♥

♦♥ ♥ ♦trs rr♦♠qr♠ ♦♥t①ts s s ♣♠ s♣r♥ ♥ r♥♦♠

♥t♦rs ❬❪ r♥ ♥♦s ❬❪ ♥ s♦ ②♥♠s ❬❪ ♦r ♠♦st ♦rts

♥ ♦s♥ ♥ sr r♦t t♦ ts qt♦s♥ss ♥ t ♥tr t ①♠♣s

r♥♥ r♦♠ P②ss ♠str② ♦♦② t♦ ♣♣ t♠ts ♥ ♦♦② ❬❪

r rs s♥♦ ♥ ♦♥ t ♥♦ ♥ ♦r♠♥ r♦ ♥tr ①trtr♦♠ ❬❪

♦r ♥st♥ ♦♥ ♥ ♦sr t ♣r♦ss ♦ s♥♦s ♥ ♦♥ t ♥♦

s♥ ♦♥ tr♦ t ♥ st♥ ♦♥ t rst rt t② ♠t t♦s tr♠ rrs t♦ t t♠♣rtr ♦ t sstrt

♥tr♦t♦♥

t ♦ s②st♠ s ♣rt ♦ ♦r ② ①♣r♥ ♥ s♠s t♦ qt s♠♣ t

♥rt ♥tr s ♥♦t② s♥t♥ s ♥ ♥ ♥♦t t ♣rs♥

♦ r ♦s r♥s ♥ r♦ s♣t

♦♠♣① ♥trs ♥ t s♥s ♦ ♦r♠t♦♥ r♦t ♥ ②♥♠ r ♦♥ ♥

t ♥tr r♦♠ t tr r♦t s♣r♥ ♦ ♠ r♦♥ts ♣r♦♣t♦♥ ♦ s

♥ ♣♦r♦s ♠trs t s s srs ♠ ♥ t t♦s r♦♠

tr♦♦♥t♦♥ ♥ ♦tr t♥♠ ♣♦st♦♥ t♥qs ❬❪ r s♣②s

♥♥♥ tr♦♥ r♦s♦♣② sr ♠ ♦ ♠ tr♦♠② r♦♥

♦♥ sstrt s sr r♦♥ ♦♥ t♦♠♥s♦♥ sstrt s ♦♥trst

t tt ♦♥ ♦r♠ ② s♥♦s ♥ ♣rs♥ts r♦r s♣t rs♠s

t♦ r♥ s ♥ ①♠♣ ♦ ♠r♦s♦♣ ♦t ①t♥ rt ♣r♦♣rts ♥

♥tr r♦t s ♦s t♦♣ t♦ t rt ♦♠tr② ② ♥r♦t ❬❪ ♥

♦♥s ♥ t♥ t♠ s sts ♥ t ♣tr

r ♠ ♦ t♥ ♠ r♦♥ ♦♥ ② tr♦♣♦st♦♥ t♥q ♦rts②♦ ♣r♦ ♥ê r♦♠ ❯♥rs r ❱ç♦s r③

♥ ♦rr t♦ sr t ②♥♠ ♦ r♦♥ srs r r♥ ② ♦

♣r♦sss ♦♥t♥♠ r♦t qt♦♥s ♥ ♣r♦♣♦s s qt♦♥s ♦s ♥♦t t

♥t♦ ♦♥t t ♠r♦s♦♣ ♥tr ♦ s②st♠s t ♦♥② t ♥r②♥ s②♠♠trs ♥

r♥t r①t♦♥ ♣r♦sss tt r t ②♥♠ t ♦rs♥♥r♥ ♥

♥ ②r♦②♥♠ ♠t ❯♥r ♣♣r♦♣rt s tr♥s♦r♠t♦♥s s♦♠ sttst

q♥tts r ♣t ♥♥ ♥ ♥ ts ② rt ①♣♦♥♥ts ♥ sssts r ♦♥ ♥ t ♣tr

♥tr♦t♦♥

sr♣t♦♥ t st ♦ ts ①♣♦♥♥ts s rt t♦ ♥ ❯♥rst② ss ❯ ♥

s②st♠s sr♥ ♠♥s♦♥t② ♥ r s ♦r ♥ r♦♣

♠♦♥ ❯s t♦rt② ♣rt t ♠♦st r♠r ♦♥ s tt

♦ rrPrs❩♥ P❩ ❬❪ t s ♠ rt ❯ s ts

♦♥t♥♠ r♦t qt♦♥ s ♥♦♥♥r ♥ ♠♣♣ ♥ ss q

r♠ ♣r♦♠s ♦ t ttst P②ss ❬❪ ♥ ♥ sr ♠t♠t ♠♦tt

♦♥s ❬❪ ♥ ♠♥s♦♥s ♠♣♦rt♥t P❩ ♠♦s s t ♥t♣

♠♦ ❬❪ ♥ t P♦②r r♦t ♠♦ ❬❪ ♦ ♥♦r♥ ♣r

t♥ t rs t strt♦♥ ♦ P❩ ♥trs ♥ t ♠♦s r②

❲♦ strt♦♥s ♠r♥ r♦♠ t ♥♦♠ tr① ♦r② ❬❪ ♥ ②rs

tr ♥②t trt♠♥ts ♦♥ t ss ♦ t ♠♦s rt P♦②♠r ♥ ♥♦♠

♠ P ♠♦ ❬❪ ♦ t♦ s♦ t P❩P qt♦♥ ❬❪

♥ t ♠♥t♠ tt ①♣r♠♥ts rr ♦t ② ♥ ♥♦ ❬❪ r

r♥ t r♦t ♦ tr♥t q r②sts ♦♥r♠ t ♥②t ♥♥s

♥ sst ♥ ♥rs P❩ trs Pr♦s ①♣r♠♥ts ♦♥r♥♥ ♦♥ t

♦ ♦♠st♦♥ ♦ ♣♣r ❬❪ ♥ r♥t ①♣r♠♥t r③t♦♥ ♦♥ t ♣♦st

♦ ♦♦ ♣rts t ♦ tr r♦♣s ❬❪ s♦ ♥♦r♠♦s② ♦♥trt t♦

♥♦ t P❩d=1+1 ♣r♠ ♦rt② tr ♥ ♥♦ ①♣r♠♥ts ❬❪

♥♠r ♠♦s ♦♥♥ t♦ t P❩ ss s♣♣♦rt ♥ ♥ ♦♥

②♦♥ qr②st rsts ❬❪ ♥ ♥② t st ♣s t♦r ♦♥sst♥t

P❩d=1+1 tr♠rt

♥ t P❩ stt♦♥ ♦r s r② r♥t r♦♠ ts ♦r

♠♥s♦♥ ♦♥tr♣rt r r ♥♦t ♥②t rsts ♥ ♠♦st ♦♥ ♥♦s ♦t

t ♠♦st ♠♣♦rt♥t ♠♥s♦♥ ♦r t♥♦♦ ♣♣t♦♥s ♦♠s r♦♠ ♥♠r r

sts t s♥ ①♣♦♥♥ts ❬❪ ♥ t t ❬❪ sqr ♦ r♦♥ss ❬❪

♥ ♠①♠ rt t strt♦♥s ❬❪ ♥ t st②stt r ♦r ①♠♣s

♥t② t ♥rst② ♦r t strt♦♥s ♥ t r♦t r♠ s ♥ r

tr♦ rs s♠t♦♥s ❬❪ r♦♠ t ①♣r♠♥t s ♥s ♦

P❩d=2+1 s②st♠s r ①tr♠② rr ❬❪ ♥ ♦♥ qst t♦ ♥ ♦t ♦♥ r♦st

s ♥♦tt♦♥ ♠♥s ♦♥ sstrt t♦♣♦♦ ♠♥s♦♥ ds ♣s ♦♥ r♦t rt♦♥ ❬❪

♥tr♦t♦♥

r③t♦♥ ♦♥r♠♥ t P❩ ♥rst② ②♦♥ s♥ ①♣♦♥♥ts s ♣rsst

♥t t ♥♥♥ ♦ ❬❪ ♥ t ts ♣♦rt② ♦ ①♣r♠♥t ♥s s

t♦♥ ❯s r ♦rs s ♣tr ♥ t ❬❪ ♥ st♦♥ ♥ t

❬❪ ♥ r♣♦rt♥ ①♣♦♥♥t s tt ♦ ♥♦t ♠t t ♥②♦♥ ♥♦♥

❯ ♥t② t s rt ♣ ♥s♥ ♥ t r sst♥ tt t t♦

rt r♠♦r s ②t s ♥♦t ♦♠♣t ♥s s t ♥♦♠♦s s♥ ❬❪

s♦♥ tt s♦♠ s②st♠s r r ② r♥t ①♣♦♥♥ts t ♦ ♥ ♦

ss ♥ tt t ♥②ss ♦ ♥tr tt♦♥s ♥ t ♦rr s♣ r ss♥ts

♦r ss♥ t tr ♦r♠ ♦ t ♦ s♥ ❬❪ ♦r ♥ ts stt♦♥s

rr ♥♠r ♦ ①♣♦♥♥ts ♠st ♦♥ ♥ ♦rr t♦ ss② t ♥tr ②♥♠

t ♠s t ♦r st rr

r♦♠ ♥ ①♣r♠♥t ♣♦♥t ♦ tr r s♦♠ s s♣ts ♠♣r♥

t ss♦t♦♥ t♥ t♦♠♥s♦♥ r♦t ♥ ts ❯ ② r st ♥ t

♦♦♥

t② ♦r ♠♥ srs ♥ ♦r r♦♥ ♠s t ♦♥ t♠s

❯♥ ♦♥♠♥s♦♥ r♦♥ ♥trs t s r r♦r♥ t t♠ ♦t♦♥

♦ t♦♠♥s♦♥ srs r♥ t r♦t s ♥ ♥r ①st ♣r♦

♠r♦s♦♣ t♥qs r s ♦r ♠♥ t ♥tr t srt r♦t t♠s

r♦t t♠ ♦rrs♣♦♥s t♦ ♥ ♦♥ st♥t s♠♣ ♣r♦ ♥ t

♣♥♥ ♦♥ t r♦t t♥q ♥ ♦♥ t r♦t ♣r♠trs ♥

t s♦♠ ♦rs ♦r ♥ ②s t♦ ♦♠ r② s ♦ ts ♣r♦r t♦

♠♥s♦♥ r♦ts r s② ♦♥t♦♥ t♦ t ♥tt♠ r♦t ♥st t♦

t s②♠♣t♦tt♠ r♦t r t s♥ r♠ ♦ t tr ❯ s ①♣t

t♦ ♠r

♥tt♠ ts r♥ t r♦t t ♥tr ②♥♠ ♥ sr

tr♥st♦♥ r♦ss♦r ♠♣♦r r♦ss♦rs ♥ ♥t ♠sr♥ t

t♠ ♦t♦♥ ♦ s♦♠ ♦rrt♦♥ ♥t♦♥ t ♥ ♥ s♠t♦♥s ts s ♥♦t

♥ s② ♣r♦r ❬❪ ♦r ♥st♥ st②♥ t r♦t ♦ 2 ♦♥

② r t♦r♦② ♥ ♥ t st♦♥

♥tr♦t♦♥

r ①♠♣ ♦ r♦ss♦rs ♦rr♥ ♥ t t♦♠♥s♦♥ r♦t ♦ 2 ♦♥ sstrts ② ❱ r♦ss♦rs r ♥t ② r♥t s♦♣s t♦ r ♥ r t♦ t t ❲t s♥ ♥ ♦ × ♦ ♣♦t s♦rt ♥♠r ♦ s♠♣s ♠s r t♦st♥s r♥t r♠s rrs t♦ t s♦♣ ♦ t r ♥ srt t♦ t ss②♠♣t♦t s♥ r♠ r ①trt ♥ t r♦♠ ❬❪

② ❱ t ❬❪ ♦♥ t tr s②♠♣t♦t s♥ r♠ ♦♥②

♦r s♠♣s r♦♥ t t r♥ ♦ ∼ 102 t♦ 103 ♠♥ tr t♦ ♥t r♦ss♦rs

s r♦♠ ts ①♠♣ t s r tt t t r♦♠ ♥ ①♣r♠♥t

♦rrs♣♦♥s t♦ r♦ss♦r r♦♥ r② t t♦ tt t ①♣♦♥♥t

①trt r♦♠ tr ♦ ♥♦t ♠t t ♥② ♥rs s♠

♥ ♦r ♥ tr♥s♥t ♥♦♠♦s s♥ ts ♣ ❬❪ ♦t ts r s♦

rs♦♥s ♦ ② s♦ ♠♥② ①♣r♠♥t ♦rs ♦♥ ①♣♦♥♥ts r ♥♦t

t♦rt② ①♣t

♣rs♥ ♦ ♠♦r♣♦♦ ♥stts ❲♥ t sr ♥

♦♠♣♦s ♥t♦ ♥ rr② ♦ ♦s r♥s ♠♦♥s t ♦♥ s②s tr s t

♣rs♥ ♦ ♠♦r♣♦♦ ♥stts ❬❪ s tr ♥s rtrst

♥t ♥ t s②st♠ ζ ♦ s♥ ♥r♥ s t t♦ r♦♥

♣rs♥ ♦ ζ ♠♦s ♦ s♥ ♦ ♦rrt♦♥ ♥t♦♥s ♥ s

t♦ ♠♥② q♦t ss♦t♦♥s t♥ ①♣r♠♥t r♦ts ♥ ❯s ♥

♣rtr ♥♦♥♥rs ♦ ①♣♦♥♥ts ♥ ♦♥s t rt ♦♥s

s r② ①♣♥ ② r ♥ rã♦ s ❬❪ s sts r

rss ♥ t st♦♥

♥tr♦t♦♥

♠♥s♦♥ rt② ♦ t P❩ qt♦♥ ♦♥r♥♥ ♦♥ t P❩ ss

s t♦rt② sss ♥ t r ❬❪ s♠ ①♣r♠♥t P❩d=1+1 s②st♠

♥ r♦♥ ♥ ♠♥s♦♥s ♠t s ts ①♣♦♥♥ts t r♦♠

t♦s ①♣t ♦r t P❩ ss s ♦rs t♦ t ♣rs♥ ♦ ♠♦r♣♦♦

♥stts ♥ ♥♦♥♦t② r ♥tr♦ ② ♥ s ♣rtrt♦♥

tr♠ ♥ t ♥♦♥♥r tr♠ ♦ P❩ qt♦♥ ♦ t ♥ ♥

tt♥t♦♥ ♦r ts ♠♥s♦♥ rt② s rtr ♦st ♥ r♦♥t ♦ ①♣r♠♥

t ♦♥r♠t♦♥s ♦ P❩d=2+1 r♦t s ①t♥s♦♥ ♦♥ ♥ ♦♥r tr

ts rt② s r② ♣rtr tr ♦ t P❩ qt♦♥ ♦r ♥ t ♣

♥ ♦trs ♥♦♥♥r r♦t qt♦♥s

② ts ①♣r♥ t t♦♠♥s♦♥ ①♣r♠♥t s②st♠s s

s t ♣ ♥♦ ♦t t P❩d=1+1 tr♠rt ♦♥ ♦t t

rst r♦st ①♣r♠♥t ♦♥r♠t♦♥ ♦ t P❩ ss ♥ t♦♠♥s♦♥ s②st♠s

♦♥ ②♦♥ t st♥r ♦♠♣rs♦♥ t ①♣♦♥♥ts ❬❪ r♦ r② s②st♠t

s♠ ♦r ♥stt♥ t ❯ ♦ r♦♥ ♠s ♥s ♦r t ♦♠♣rs♦♥ ♦

s♥ ①♣♦♥♥ts ♥ ♣r♦ r ② ♦r r♠♥t ♦t ♣rt② t ♦sts

♦♥r♠ tt t r♦t ♦ t♥ ♠s ♦♥ ♦r T = 250 C

♦♥s t♦ t P❩ ss s ♠♦♥strt ② s♥ ①♣♦♥♥ts ♥ ♥rs t

strt♦♥s ♥rs sqr ♦ r♦♥ss strt♦♥s ♥ ♥rs ♠①♠

rt t strt♦♥s ❬❪ ♦r♦r ts s t rst t♠ tt t s ♥

①♣r♠♥t② ♠♦♥strt t ♥rst② ♦ s strt♦♥s ❬❪

s t ♦ ♦ t ♣rs♥t ♦r s t♦♦ t♦rt② ♥ ①♣r♠♥t②

♠♦tt s♦ st t t ♦ t ♣♦st♦♥ t♠♣rtr ♦♥ ts ♥♦

t♦♠♥s♦♥ rrPrs❩♥ s②st♠ ❲ s♦♥ tt ♥ t ♣rs♥ ♦

♥tt♠ ts r♦ss♦rs ♥ ♥♦♠♦s s♥ s♥ ①♣♦♥♥ts r ♥♦t

t♦ ♣♦♥t ♥ ♦♠♣♥ ② t ❯ ss ♦ t s②st♠ ♠♥② t♦ ♥①♦r

①♣r♠♥t ♦sts ♦r s♦♥ tt ts ♥ r♠♥t ②

♥②s♥ strt♦♥s ♦ s t♦ ♣r♦ tt tt♦♥s ♦ ♥tr r

sr s②♠♣t♦t② ② t rrPrs❩♥ qt♦♥ ♥ r♦ r♥ ♦ ♣tr

♥tr♦t♦♥

t♠♣rtr ❬❪ ♥② ♦♥ ♠♦♥strts tt s ♣♦ss t♥♥♥ ①♣r♠♥t②

t P❩ ♥♦♥♥rt② tr♦ t ♣♦st♦♥ t♠♣rtr ❬❪

rst ♦ ts ssrtt♦♥ s ♦r♥③ s ♦♦s ♣tr ♠s ♥

t♥ rts ♥ sr tt♦♥s ♥tr♦♥ s sttst t♦♦s ❯♣t♦

t rsts ♦t t s♣ sts ♦ st♦♥ ♥ ♦♥ t ♦♥

t t①t ♥ ♥ ♥t ♥ t ♣♣♥① st♦♥s ♣tr s sr

♦r s♦rt r ♦♥ t rrPrs❩♥ qt♦♥ ♥s t ♠♥②

♥r♥ts ♥ ♥s t♥ t P❩ t♦r② ♥ t ♣tr ①♣r♠♥t

♣r♦rs s r♥ ts ♦r r r② sr s s t t♥qs ♥

r♦t ♣r♠trs ♣tr ♣rs♥ts t rst rsts ♦ ts ♦r ♥ ①ts

rtr♦ ♥ t ♦♥t①t ♦ t t♦♠♥s♦♥ P❩ ♣r♠ ♥ t sq♥

♥ ♥ t ♣tr ♦♥ sts t t ♦ t ♣♦st♦♥ t♠♣rtr ♦♥ t

r♦t ②♥♠ ♥ ♦♥ ①♣♦rs t r♥♥s ♠r♥ r♦♠ t s②st♠

♥② ♥ t ♣tr ♦♥s♦♥s ♥ ♣rs♣ts r r♥ t ♦♠♣r♥s

♦r ♦ ts ♦r st♥♥ ♦t ts ♠♥② ♦♥trt♦♥s t♦ t ♦♥t①t ♦ ♥t

r♦♥♥ srs ♣♣♥① st♦♥s ♦r t♦♣s s t r♥♦♠ r♦t qt♦♥

t ❱♥sr♠ ❱ ss t t ♦r♥ ♥ ♦♣♠♥t ♦ ♥♦♠

♦s s♥ ♥ ♥② ♥ ♥tr♦t♦r② ♠tr ♦r ②♦♥ st♥ts ♦♥r♥♥ ♦♥ t

s ♣②ss ♦ r②st r♦t

♣tr

rts ♥r♥ ♥

❯♥rst② ♥ ♥tr r♦t

♥ ts ♣tr ♥ t♥ r♦♥ srs ♥ tr sr♣t♦♥s t ♦♥

t ♥♦② ♦ t♦♦s r♦♠ ttst♥s ♦ qr♠ ♣♥♦♠♥ t t rt

t② s ♥ ♦♥t♥♠ r♦t qt♦♥s r sss ♥ tr rt ①♣♦♥♥ts

r ①trt ❯♥rst② s sss ♦♥ t t♦rt ♥♠r ♥ ①♣r♠♥t

♣♣r♦s

r♦♠ rtt② t♦ t ♠②❱s ♥sät③

rt s ♦♠♣① ♥ rrr ♦t ♥r ♥ ♣♣r♦♣rt s

tr♥s♦r♠t♦♥ ♥②♦♥ ♦ ts ♣rts r♣rs♥t t s t ♦ ♦♥t♥s ♦s tr

s ♥ ♦st♥s r s♦♠ ①♠♣s ♦ rts ♦rt♥ t ♥tr ❬❪ ♥

♠t♠t s♥s rt s s tr♠♥st ③♦♦♠ ♥ t s②st♠ ②s

r♣r♦s ①t② t ♦ ♦t ♦r ♥ t ♥tr ♦♥ ♥s sttst

rts ♦r ①♠♣ ♦♥ ♦♠♣rs t♦ s♥♣s♦ts r♦♠ ♠♦♥t♥ t r♥t

♠♥t♦♥s t② ♦ ♥♦t ♦r♣ t ♥♦♥tss tr sttst ♣r♦♣rts r t

s♠ ❬❪

rt ♦t ♣rs♥ts tt♦♥ s②♠♠tr② ♦r ♦♠♦♥t② ♣r♦♣rt② ❬❪

ttst ♦♠♦♥t② s sr ② t ♦♥t♦♥

♥ t s♥s tt ♥ ♦♠tr② ♥ ♥♦t sr t

rts ♥r♥ ♥ ❯♥rst② ♥ ♦♥qr♠Pr♦sss

f(ςα1x1, ςα2x2, ς

α3x3, ...) = ςαf(x1, x2, x3, ...),

r t ♦t s ♦r♠ ② st ♦ ♣♦♥ts f = f(①) ς s s t♦r ♥ α s

t ör ①♣♦♥♥t ❬❪

α1 = α2 = ... = α ♠♥♥ ♥ s♦tr♦♣ s tr♥s♦r♠t♦♥ stss t

q s♦ ♦♥ s ss♠rt② trs ♦r ♥s♦tr♦♣ tr♥s♦r♠t♦♥s ♦♥ s

s♥t② ❬❪

t rt ♣♦♥t ♦ ♣s tr♥st♦♥ ♦rrt♦♥ ♥t♦♥s ①t②

s q ❬❪ ♥ t ttst ♥s ♥ t ♠♣s tt t ♦rrt♦♥

♥t ξ rs ♥ tr s ♥♦ rtrst ♥t ♥ t s②st♠ ②♦♥ s②st♠

s③ ts ❬❪ ♦r ♥♦♥qr♠ ♣r♦sss s s sr r♦t t s ♥ s♦♥

tt ♥tr tt♦♥s s♦ s t♦s t rt ♣♦♥t ♦ ♣s tr♥st♦♥

❬❪ ♥ ♦tr ♦rs t②♣② ♥trs ♣r♦ ② t ♥tr r rts ♥

t sttst s♥s t h(x, t) t t ♦ ♥ ♦♥♠♥s♦♥ ♥tr t t

♣♦st♦♥ x t t♠ t ss♠♥ ♦♠♦♥t② s ♥ q t♥

h(x, t) = ς−αh(ςx, ςzt),

r ③ s t ②♥♠ ①♣♦♥♥t ♥ α ♥ t sr r♦t ♦♥t①t s ♥♠

r♦♥ss ①♣♦♥♥t ❬❪ ♦t tt t t♦♥ s②♠♠tr② s s♦ ♥ ss♠ ♦♥

t t♠♣♦r ①s s t♦ t ③ ♥t♦♥ ♥ α ♥ z r rtrst

♦ s②st♠s s♣②♥ s♣ ♥ t♠♣♦r s♥ ♥r♥

sr ♦♥ ♥ t♠ s ♥♦t tr♠♥st ♣r♦ss t h(x) t

sr s st♦st r ♥ st ♦ ♣♦ss ♦t♦♠s h1, h2, h3... strt r♦♠ ♥ ♣r♦t② P (h) ♥ ♥r t ♣r♦t② ♦r ♥ t ♦

t ♥tr t t h s ♥ s t rt♦ t♥ t ♥♠r ♦ ♦rr♥s

Nh ♦ ts s♣ ♥t ♥ t t♦t ♥♠r N ♦ ts s♠♣

♠sr♠♥t ♦ ♦ ♦ tt♦♥s ♥ ♦♥ ♣rt ♦ t s②st♠ ts t♦s ♥ t ♦tr ♦♥s② r ♥ ①♣t② ♥ t ♥①t st♦♥

♥t ♦r t ♦♥ ♦ ♣rt ♦ t s②st♠ ts t ♦tr ♦♥s


P (h) = limN→∞

Nh/N.

♣r♦t② ♣♥s ♦♥ t r♥ dh ♥ t♥ h ♥

h + dh s ♠sr ♦ s ♠♦r ♥trst♥ t♦ ♥ t ♣r♦t② ♥st② ♥t♦♥

♣ p(h) ≡ dP (h)/dh ❬❪ ♠st sts② t ♥♦r♠③t♦♥ rqr♠♥t

∫ ∞

−∞

dh p(h) = 1.

♣ s t ♣ r♦♠ ts ♠♦♠♥ts ♦r ♠♥ts ♥t♠♦♠♥t mn ♦

♣ s

mn ≡ 〈hn〉 ≡∫ ∞

−∞

dh p(h) (hn),

r t rts ♠♥ t♦ t t ①♣tt♦♥ ♦ t r hn

♠♦♠♥ts r ♥rt tr♦ t rtrst ♥t♦♥ ♣(k) s

t ♦rr tr♥s♦r♠ ♦ p(h) ❬❪ r♦♠ t ♦rt♠ ♦ ♣(k) ♦♥ ♥s t ♥t

♠♥t 〈hn〉c

ln ♣(k) =∞∑

n=1

(−ik)nn!〈hn〉c.

② ①♣♥♥ t ♦rt♠ ♥ t ♣(k) ♥t♦♥ ♥ ♦♠♣r♥ tr♠ ②

tr♠ t t ①♣♥s♦♥ ♦ ♣(k) ♥ ♣♦rs ♦ ♦♥ rs t♦ t rt♦♥ t♥

t ♠♦♠♥ts ♥ t ♠♥ts ❬❪ ♦r t rst ♦r ♠♥ts ♦♥ s q

〈h〉c = 〈h〉〈h2〉c = 〈h2〉 − 〈h〉2〈h3〉c = 〈h3〉 − 3〈h2〉〈h〉+ 2〈h〉3〈h4〉c = 〈h4〉 − 4〈h3〉〈h〉 − 3〈h2〉2 + 12〈h2〉〈h〉2 − 6〈h〉4

rst ♠♥t s ♠♥ ♥ t s♦♥ ♦♥ s r♥

tr ♠♣♦rt♥t q♥tts r ♠♥s♦♥ss ♠♥t rt♦s ♥ ♣rtr

t ♥ss q s ♥ ♥t ♦ ♣ s②♠♠tr② ❬❪ t ①♣tt♦♥


x

p(x

)

S = K = 0

S > 0 and K > 0

S = 0 and K < 0

r ♦♠♣rs♦♥ t♥ r♥t strt♦♥s ♥ t s♠ ①♣tt♦♥ s ♥t ② t s♦ r ♥ r ♣rs♥t♥ ♥ ♦t♥ ♣r♦r♠♥ ♣rt② tr♥s♦r♠t♦♥ ♦♥ t r♥ r

s rr t♥ t ♠♦r ♣r♦ ♦♥ t♥ S > 0 ♦trs S < 0 rt♦ss

q ♣r♦ ♥♦r♠t♦♥ ♦t t t ♦ ♣ ts s②♠♠tr ♣ t

K > 0 ♣rs♥ts ♣ sr♣r t♥ tt r♦♠ t ss♥ ♥ ts ts t ♦♥r

t♦ ♦♥ ♦♣♣♦st ♦rs ♦r K < 0 ② ♥t♦♥ t ss♥ ♣ s

S = K = 0 r♣rs♥tt♦♥ ♦♠♣r♥ strt♦♥s t t s♠ ①♣tt♦♥

♦r r♥t ♥ s s s♦♥ ♥ t

S = 〈h3〉c/[〈h2〉c]3/2 ♥ K = 〈h4〉c/[〈h2〉c]2.

♦ sqr r♦♥ss w2 ♦ ♥ ♥tr s ♥ s t r♥ ♦

ts ♦♠♣♦s♥ t ♥trst♥② t r♦♥ss ♦r t s r② ♠♣♦rt♥t r

r♦♠ ♦t ①♣r♠♥t ♥ t♦rt ♣♦♥t ♦ tr ♦♥tt②

♦ s♦♠ t♥ ♠s ♦r ①♠♣ s str♦♥② r s r♦r s tr sr ❬❪

♠♣♦rt♥ r♦♠ t t♦rt s r s♦♦♥ ♦♥sr ♥ ♦♥♠♥s♦♥

♥tr ♦ s③ L r♦♥ss ♦ s ♥tr t t♠ t rs

♥ srs t tr♥st♦♥ s②♠♠tr② ts s s♦ t r♥ ♦ p(h)


w(L, t) = [〈h(x, t)2〉 − 〈h(x, t)〉2]1/2,

r t 〈...〉 rrs t♦ s♣t r ♦r t ♦ s②st♠ ♦ s③ L

♦ ♥ rtr♥ t♦ t ②♣♦tss ♠ ♥ q ♥srt♥ ts qt♦♥

♥ t q ♥ ♣r♦r♠♥ r ♠♥♣t♦♥s ♦♥ ♥s

w(L, t) = tα/zf(L/t1/z),

ts qt♦♥ ♣r ♥♦♥ ♥sät③ ♥ t ♥t r♦♥♥ t♦r② ♥♥ t

r♦t ①♣♦♥♥t β ≡ α/z t ♠②❱s ❱ ②♥♠ s♥ ♥sät③ ♣rts

tt f s②♠♣t♦t② s s f(u) ∼ uα ♦r u ≪ 1 ♥ ∼ const ♦r u ≫ 1 ❬❪

♥ t ♥sät③ ♦r q ♥② ♦♥ ♦t♥s

w(L, t) ∼

tβ, ♦r t1/z ≪ L,

Lα, ♦r t1/z ≫ L.

♦♥ss ♦r s st ♥ t s②st♠ r♦♥s s tβ

t ♦rrt♦♥s r s♣r♥ tr♦ t s s r♦t r♠ t t t♠ tx

t ♦rrt♦♥ ♥t ♦♠s ♦ t s♠ ♦rr ♦ L ♥ st♥t r♠ s r

r♦♥ss st♦♣s r♦♥ ♥ t♠ ♥ tr♥s t♦ ♣♥ s♦② ♦♥ L s s t st②

stt ♦r strt♦♥ r♠ ♦♠♣r♥ t q t s t s r t♦ ss♦t

ξ|| ∼ t1/z r ξ|| s t ♣r ♦rrt♦♥ ♥t ♥ ♥st ♦ t ❱ ♥sät③

s tst s♦♥ r♠r ♦♣s ♦r t rs ❲ ♥♦t tt t r♦♥ss

♣rs♥ts ♣♦r ♣♥♥ ♥ s♣ ♥ t♠ s ♦rs t t ♦rrt♦♥

♥t♦♥s ♥ qr♠ rt ♣♥♦♠♥ ♣r ♦s rtr ♥ ♥ t

rt ①♣♦♥♥ts α ♥ z ♦ ♥♦t ♣♥ ♦♥ ♠r♦s♦♣ ts ♦ t s②st♠ ♥r

♥stt♦♥ tr s ♥rst② ♥ tt♦♥s ♦ r♦♥ ♥trs

♥ t strt♦♥ r♠ ♣♥s ♦♥ t1/z ♦ t s♠ ♦rr ♦ L ♥ ①

♣r♠♥t stt♦♥s r L s ♠ rr t♥ t rtrst s③ ♦ ♣rts

♦♥sttt♥ t ♥tr t t♠ rqr t♦ t s②st♠ ts ♥t♦ t st② stt s

♥ ♠st ss♠ ςzt = 1 s ♥ ♦♥ s ς s st ♥ rtrr② t♦r ❲ ♥ ♥ts ♥t ♦t♥ t rst ςzt = 1


t

w (

L,t

)

10-2

100

102

L/t1/z

w/t

β

β

tx(16L)t

x(L)

α

const.

r ②♣ t♠ r♦♥ss ♣♦t ♦ × ♦ r♥ ♥ ♥tr r♦t ♦r r♥tsstrt s③s rs rr t♦ sstrt ♦ s③ rs ♦r♥ t tr♥s ♣ tr♥s r♥ ♠♦♥s ♥ r sqrs tx ♥ts trtrst t♠ ♥ ξ|| ≈ sstrt s③ ♥st ♦♥r♠s t ❱ ♥sät③ sr ♥q s♥ t ♣♣r♦♣rt s ♦r β ♥ z

r② s r s ♥♦ ♦♥② t ♦t♦♥ ♦ ♣r♦s ②

② s♦ ♦♠st♦♥ ♦ ♣♣r ❬❪ ♥ t r♦t ♦ 2 ♠s ② ❱

tr ②s ♦ ♣♦st♦♥ ①♣r♠♥t② t stt♦♥r② stt ❬❪

❱ ♥sät③ s ♥ ♦♥r♠ ♥ sr ①♠♣s ♦ sr r♦t s

s t ♣r♦♣t♦♥ ♦ ♦ ♥ ♣♦r♦s ♠♠ ❬❪ ♣♣r tt♥ ❬❪ t r♦t

♦ tr♥t q r②sts ❬❪ t s♦ ♦♠st♦♥ ♦ ♣♣r sts ❬❪ ♥ s♦

♦♥ ❬❪ ♦r t ❱ s♥ s ♥♦t t ♠♦st ♥r ♦♥ ♥ ❱ ♥sät③ s

♦r sr♥ ♦ s♥ ♦ r♦t ♣r♦ss t ♥♦♠♦s r♦♥♥ t

sr♣t♦♥ ♦ ♥♦♠♦s s♥ s t t♦ t ♣♣♥① st♦♥

♦rrt♦♥ ♥t♦♥s

♦rrt♦♥ ♥t♦♥s ♣② ♥tr r♦ ♥ s②st♠s ①t♥ rtt②

♦t♦ ♦r t qr♠ s t② ♣r♦ ♠sr♠♥t ♦ t ♦rrt♦♥ ♥t

② r ♦r♠ ② ♥ ♦♣rt♦♥ s♠ ♣r♦t t ♥♦♥ st q♥tts


sr♥ t ♠r♦s♦♣ stt ♦ t s②st♠ r s♣rt ② st♥ ♦ l

♥ ♥ ♥tr ♦ ♥r s③ ♦r ♥st♥ ♦♥ ♥ sr t ② ts t

hL ♦r ♥ ② ts s♦♣ ∇hL ♦ ♦rs ♥ ♥ ♣♣r♦♣rt ♦rs♥♥ r♥

♦r ♥ ♥tr r♦♥ ♥ ♥ rtt♥ s

Ch(|l|, t) = 〈[h(①, t)− h(①+ l, t)]2〉.

t s②st♠ ♣rs♥ts tr♥st♦♥ s②♠♠tr② ♥② ♣♥ s♦② ♦♥ t

♠♥t ♦ l ❬❪ t s ♥♦t tr ♦r ♥s♦tr♦♣ s②st♠s s ❬❪ ♥ r tr♥

rtr♠♦r s sss t t ♥♥♥ ♦ ts ♣tr s♣②s t♦♥

s②♠♠tr② t t rtt② ♥ ♥srt♥ q ♥ q ♥ ♣r♦r♠♥

r ♠♥♣t♦♥s ♦♥ rs t♦

Ch(l, t) = t2α/zf(l/t1/z),

r f(u) s t s♥ ♥t♦♥ tt ♦r ♦②s t ❱ ♥sät③ Ch s ♦t♥

tr♥ ♦rrt♦♥ ♥t♦♥

♦t tt ts s t s♠ s♥ ♦r♠ ♦t♥ ♦r t r♦♥ss ♥ t q

t β r♣ ② 2β ♥ L ② l ♦ t rst ♥♦t t ①♠♣ s♦ ♥ t

s sr ② Ch(t) t t t r♦t r♠ ♦♥ s t2β ♥ t s♠ ②

♦♥ ♥ r♣ L ② l ♥ t q t♦ ♦t♥ s♥ ♦r♠ ♦r t ♦ r♦♥ss

❬wloc(l, t)❪ s q② ♦rrt♦♥ ♥t♦♥ r♦♠ ♥ ①♣r♠♥t ♣♦♥t ♦

t s ♥♣rt ♥♥ L ♦ s②st♠ ♥ ♦rr t♦ ①trt α tr ♦♥ ♦t♥

ss ♦ ♠sr♠♥ts s♣♥♥♥ ♦①s ♦ tr ♥t l ♥ t ♥tr [0, L] ♥

♦t♥♥ t ①♣♦♥♥t r♦♠ t ②♣♦tss C1/2h (l, t) ∼ wloc(l, t) ∼ lα s♦s

t②♣ ♦r ♦ wloc ♦r ♥trs ♥ t r♦t r♠

tr ♠♣♦rt♥t s t s♣t ♦r♥ ♦ ts Cs ♥ s

Cs(||, t) = 〈[h(①, t)h(①+ l, t)]〉 − 〈h〉2,

①♠♣s r t ♦ ♠♥t③t♦♥ ♦r s♣♥ tt t ♦ t ♦r ♥ ♥tr t♥ t s♥ ♦ ♥♦♠♦s r♦♥♥ t loc s♥ ♦♦s t ❱ ♥sät③ s ♥ t q


l

wlo

c (l,t)

10-3

100

103

lz/t

wlo

c/lα

α

ξ(t) ξ(16t)

const.

−β

r ②♣ ♦ r♦♥ss ♣♦t ♦ × ♦ ♦r ♥ ♥tr r♦♥ t r♥tt♠s rs rr t♦ r♦t t♠s rs t ♦r♥ t tr♥s t ♣tr♥s t r♥ ♠♦♥s t ♥ r sqrs t ❱rt s ♥s ♥t tξ|| ♠sr ♥st ♦♥r♠s t ❱ ♥sät③ sr ♥ q t r♣ ② l ♥s♥ t ♣♣r♦♣rt s ♦r α ♥ z

r t s strt♦rr t♦ s♦ tt Ch = 2w2loc − 2Cs

♦r♥ Cs(l, t) s ♥ ♦♠♣t ♥ ♣rtr ♦r ♦♥♠♥s♦♥

♠♦s ♦♥♥ t♦ t rrPrs❩♥ P❩ ❯♥rst② ss ❯ s

t② r ♥rs ♥ ♥ ② t ♦r♥ ♦ r② ♣r♦sss ❬❪ s

♥ t r ❬❪ ❱r② r♥t② ♣♥② ♥ P③♥t③s ♦♥

♥♠r② ♥ ♦♥r♠ ①♣r♠♥t② t ♥rst② ♦ t rs Cs(KPZ) s♦

♥ ♠♥s♦♥s ❬❪

s ♦♥ ♥ ♥ s♦♣s♦♣ ♦rrt♦♥ ♥t♦♥s s♥ ∇ ♥st ♦

♥ t q ♥ q ♥ sr ①♣r♠♥t sts ❬❪

s t s♦♣s♦♣ ♦r♥ q ♥ ♦rr t♦ ♦t♥ ♥ st♠t ♦ ξ||

Γ(||, t) = 〈[∇h(①, t)∇h(①+ l, t)]〉.

♥ st♠t ♦ ξ|| ♥ ♦♥ ♠sr♥ tr t rst ③r♦ ♦r t rst

♠♥♠♠ ♦ t r ❬❪ r s♦s ♣♦t ♦ t②♣ ♦r ♦ t s♦♣


s♦♣ ♦r♥ s ♥t♦♥ ♦ t st♥ ♥ t♠ t ♣r♦r s ♦r

st♠t♥ ξ|| s s♦ ♥t s s t ♣♦t ♦ ξ|| s ♥t♦♥ ♦ t r♦t t♠

r♦♠ r t ①♣♦♥♥t 1/z ♥ ♦♥

l

0Γ(l,t)/

Γ(0,t)

ξ

ξ(t)

ξ(16t)

t 2t 4t 8t 16t

r ②♣ ♥♦r♠③ s♦♣s♦♣ ♦r♥ s ♥t♦♥ ♦ t st♥ ♥ tt♠ t ♥ ♥r × ♥r ♣♦t r ♦♦rs rr t♦ r♦t t♠s t ♦r♥ t t r♥ t ♥ r t ♥t s ♥ sts Γ(||, t)/Γ(0, t) q t♦ ❱rts ♥s ♥ts t ♣♦st♦♥ ♦ t st♠t ♦rrt♦♥ ♥t ♥st s♦s ξ|| s♥t♦♥ ♦ t♠ ♥ ♦ × ♦ ♣♦t r♦♠ r t ①♣♦♥♥t 1/z ♥ ①trt

♦♥t♥♠ qt♦♥s ♥ ❯♥rst② sss

t ♥ ♥tr sr ② ts t ④①t0⑥ ♥ ♣♣r♦♣rt

♦rs♥♥ r♥ ♥ ♦♥sr tr s ♥♦♥ r♥ ♦r tr s r

r♦♠qr♠ stt♦♥ ♥ ♠♣♦rt♥t qst♦♥ ♦♥r♥♥ ♦♥ t ♦ ♦ ♦s

♦♥ ♥ sr t t ♦t♦♥ ♦ t ♥tr ♣r♦r t ♥sr ♦s

♥♦t ♣♣r t♦ r s t s♠s tt tr s ♦ ♥♦r♠t♦♥s s

s t ♥ ♦ ♥tr ♦♦ ♣②s ♠ rt♦♥ r♦♥t t

s ♥ t ♥ t s♣ ♥trt♦♥s r♥ t ②♥♠s t t ♠r♦s♦♣


♥ ♥ ♣♣r♦ t♦ t qst♦♥ r♦♠ ts ♣♦♥t ♦ ♠s t ♣r♦♠

♣rt② ♥trt rtss ♦♦♥ t t ♠r♦s♦♣ ss ♦ ♦♥

♥trs ♦♥ ♥ ♠♣s tt t ♦t ♦r ♦ t♠ srs ♠♥② ♥r②♥

s♠rts ♦s ♥♦t ♣♥ ♥ t ♦ ♥ ♥ s ♦♥ ts ♦♥ ♦

♣r♦r♠ ♦♥♥t sr♣t♦♥ ♦ t s②st♠ ♥♥ ♥ ts t♦ts ♦♥

♥ r tt ②♦♥ ♦ t ♦♥♥t ②♣♦tss t ♦♥t♠ r♠ s s♦

♥ssr② ♦♥t♦♥ s♥ tr♥s♥t ♥tt♠ ♦rs s♦ ♦ s

♣rtr ♠t ♦ ♦♥ ♥ts ♥ ♦♥ t♠s s ②r♦②♥♠ ♠t ②

♥♦② t♦ t rt♦s qt♦♥ ♦r ♦ ♣rts ❬❪

② ts s ♦♥ ♥ ♥r qt♦♥s ♦r sr♥ t t

♦t♦♥ ♦ r♦♥ srs t t ♦r♠

∂th(①, t) = F +Θ(①, h, t) + η,

t ♥ t r ♥♠r ♦ ♣rts ♣r ♥t t♠ rr♥ t t ♥tr

t r♥ ♦r ♦ ♦♥ ♠st ♦♥t tt ts rr♥ ♣r♦ss s st♦st

♥ t tr♠ η s ♥srt ♥ t qt♦♥ ♦r ♣tr♥ ts tr r♥ η

♦♥ s

• t t ③♦♥ ♥tr r♦♥t ♥s ♦♥t♦ ♥ ♥♦♠♦♥♦s ♠♠ s

♣♦r♦s sstrt ♦r ♣♣r st t r♥t ♥♦s ♥ t ♣r♦ss s stt

♥ ♥s ♣♦♥t t♦ ♣♦♥t ♥ t ♠♠ s s ♦ q♥ ♥♦s ♦♥

η = η(①, h) ♦♥② t q♥ ♥♦s s ♣rs♥t t r♦t ♦ t ♥tr s

tr♠♥st ♦t♦♥ ♦r t tr♠ ♥♦s s ②s ♣rs♥t

♥ ①♣r♠♥ts str♦②s ts tr♠♥s♠

• ②♦♥ ♦ tr♠ ♥♦s ♥ ♥ ♥tr r♦s ② r♥ ♣rts r♦♠ ♥

①tr♥ ① t s♦ s♦t ♥♦s s s♦ ♣rs♥t ♥ ♣②s t r r♦

♦♥ t ②♥♠ s ♥♦s ♦♠s r♦♠ t ♥r♥t r♥♦♠♥ss ♦rr♥ ♥

t ♣♦st♦♥ ♣r♦ss ss♠♥ tr s ♥♦ ♣rr♥t r ♦♥t♦ t sstrt

♦r r♥ ♠♦r ① ♦♥ s tt 〈η(①, t)〉 = 0 t t s♣tt♠♣♦r

ss♠ t ♣rt s③ ♥ q t ♥t ♥ ♦rr t♦ sts② ♠♥s♦♥ ♥②ss


♦r♥ ♥ ②

〈η(①, t)η(①, t′)〉 = 2Dδds(①− ①)δ(t− t′),

r√D s t ♠♣t ♦ t t ♥♦s

♦ tr♥ t tt♥t♦♥ t♦ t ♦r♠s t ♥t♦♥ Θ(①, h, t) ♥

ss♠ s♣♣♦s♥ tt t ②♥♠ ♦ t ♥tr s r ② ♦ ♣r♦sss s

♥ ♥r②♥ ②♣♦tss sr♣t♦♥s ♦ ♣②s t ♥ ♥♦t ♣♥ ♦♥ t ♦r♥

♦ ts ♦srt♦♥←→S = (t0,①, h0) s s♣t ♥ t♠♣♦r tr♥st♦♥s ♠st

sts ② t q t rs ♦t r♦♠ t ♥t♦♥ ①♣t tr♠s ♥♦♥ tn,①m

♦r hm r m ∈ R∗ ♦r♦r s t r♦t ♦s ♥♦t ♠ st♥t♦♥ t♥

rt ♥ t♥ t q ♠st s♦ ♥r♥t ♥r s♣t ♣rt②

tr♥s♦r♠t♦♥s t rs♣t t♦ t ① ①s s ②♣♦tss r t ♦ tr♠s

♥ Θ t♦ ♦♠♥t♦♥s ♦ ♥ rts s s (∇2nh)(∇h)2p t n, p ∈ N

♥ ♥r ♣tr ♦r ♦♥srt♦♥s ♥t r s t♦ ♦♥sr

∂th(①, t) = F + a1∇2h+ a2∇4h+ ...+ b1(∇h)2 + b2(∇h)4 + ...

+c11(∇2h)(∇h)2 + ...+ cnp(∇2nh)(∇h)2p + η,

r ai bi ♥ cnp r ♣♣r♦♣rt q♥tts ♠♥ t q ♠♥s♦♥② ♦♥sst♥t

s r ♥trst ♥ t ②r♦②♥♠ ♠t rts ♦ r ♦rr r

rr♥t t♦ t s②♠♣t♦t s♥ ♦r s r ❬❪ ♣ s t s♠♣st

♥r qt♦♥ ♥♦♥ ts tr♠s rs

∂th(①, t) = F + a1∇2h+ b1(∇h)2 + c11(∇2h)(∇h)2 + η,

s s s ♠♣♦rt♥t ♦♥t♥♠ r♦t qt♦♥s r ♥♦ ♥ t

qt♦♥ rtss tr r s♦ ♦trs ♠♣♦rt♥t r♦t qt♦♥s tt

♦r ♣②s②♠♦tt tr♠ ♠s ♣♣r r rts ♦ h ♥ t

♦♦♥ st♦♥ sss ts st ♥ ts


rs❲♥s♦♥ ♥ t ♥r qt♦♥

rs❲♥s♦♥ ❲ qt♦♥ q s ♣r♦♣♦s ♥ ♦r

sr♥ t s♠♥tt♦♥ ♦ r♥r ♣rts ❬❪ qt♦♥ ♣rsrs ♣rt②

s②♠♠tr② ♥ t r♦t rt♦♥ t rs♣t t♦ t ♠♥ t ♥ t t

♣ t♦ s S = 0 s ♣rtr s②♠♠tr② rs ♦t ♥ ♣♦rs ♦ h s s (∇h)2n

♥ ♥ ♦r♥ t t ♥r q t rs

∂th(①, t) = ν∇2h+ η(①, t),

r ν ≡ a1 s q♥tt② ♦ ♠♥s♦♥ [m2/s] ♥ t ♦♥♥t♦♥

r♦t r ♦t② ♦ t ♥tr v ≡ 〈∂th〉 = Ft ♦♥ 〈∇nh〉 ♥ss ♦r ♣r♦ ♦♥ ♦♥t♦♥s q ♦ s rtt♥ t t rr♥t ♦ t

♠♥ t ♦♥ t ①♣t ♣♥♥ ♦♥ t ① tr♠ ♥ stt♥ v = 0

P②s② t ♣♥ tr♠ ts s ♦♥srt s♠♦♦t♥ ♠♥s♠ r

strt♥ t rrrts ♦♥ t ♥tr ♠♥t♥♥ t r t

♥♥ s ♣ ♥ t r ❬❪ ♦r ♦♠tr ♥tr♣rtt♦♥

t♦ t ♥r rtr ♦ ts qt♦♥ rt ①♣♦♥♥ts ♥ ♦♥

strt♦rr② ② rs♥ ♦r ② ♦rr tr♥s♦r♠ ♠t♦s s ♣r♦t♦t②♣ s

s ♥ ①♣♦♥♥t s ② ♣♣②♥ rs♥ t♦♦s ♦♥ ♥r ♥r qt♦♥

♦t♥ r♣♥ 2 ② 2n ♦♥ t ♥ ♦♣rt♦r ♥ t q ♠t♦ ②

♦rr tr♥s♦r♠ ♥ ♦♥ ♥ ts ♥ t ♣ ♦ t ❬❪

s♥ ♣♣♦s rs♥ s ①→ ς① t→ ςzt ♥ h→ ςαh ♥srt♥ ♥

t q t 2 r♣ ② 2n ♦♥ t ♥ ♦♥ s

ςα−z∂th = νςα−2n∇2nh+ η(ς①, ςzt),

r s♥ t ♥♦s ♦r♥ ♥t♦♥ q ♥ t t ♥t♦♥ ♣r♦♣rts

δds(ς①) = ς−dsδ(①) ♦♥ ♥ rrt t rs ♥♦s ♥ t ①♣rss♦♥ ♦ ②

ς(−z−ds)/2η(①, t) ♦ ss♠♥ s ♥r♥ ♦♥ ♥s t rt ①♣♦♥♥ts

♥ ♥ s♦ tt t tr♠ (∇2h)(∇h)2 ♥r r♥♦r♠③t♦♥ s rr♥t ♦♠♣r t (∇2h) ❬❪ ♣


♦r n ≥ 1

α =2n− ds

2and z = 2n.

❲ q s r♦r ♦r n = 1 ②s α = (2 − ds)/2 ♥ z = 2

♥ ♣rtr ♦r ds = 2 ♦♥ ♦t♥s α = 0 ♥ β = 0 ♠♥♥ tt ♥ t q

t r♦♥ss ①t ♦rt♠ ♣♥♥ ♦♥ t ♥ ♦♥ L st ♦ ①♣♦♥♥ts

α = α(ds) ♥ z = 2 ♦♠♣♦s t s♦ rs❲♥s♦♥ ❯♥rst② ss

sr ♥ t ②r♦②♥♠ ♠t ② t ❲ qt♦♥ ♠♦s ♥♠r ♠♦

♦♥♥ t♦ t ❲ ss s t ♥♦♠ ♣♦st♦♥ t r ①t♦♥

♣r♦♣♦s ♥ ② ♠② ❬❪ r t ♣♦st ♣rts r ♦ t♦ s

t sr ♥t r♥ t ♦ ♦st t ①♣r♠♥t ♥s ♦ srs

♦♥♥ t♦ t ❲ ss ♥ tr♥ r r② rr s r s r ♦♥r♥ t ❲

♥rst② s ♦♥② ♥ ♦♥ ♥ t r♦t ♦ ❲ ♠t②rs ♦♥ ② ♠♥tr♦♥

s♣ttr♥ ❬❪ ♥ ♥ t s♠♥tt♦♥ ♣r♦ss ♦ 2 ♥♥♦s♣rs ❬❪

tr♥♥ t♦ q ♦♥ sts n = 2 ♦♥ ♦t♥s t ①♣♦♥♥ts ♦r t

♠♦s r♦t qt♦♥ ♥♦♥ s ♥r qt♦♥ q rst② ♣r♦♣♦s ②

❲♦ ♥ ❱♥ ❬❪ ♥ s r♠ ♥ ♠♦r♥ ❬❪ ♦r sr♥ t r♦t

♦ srs ♥ s♦♥ s t r♥t r♦t ♠♥s♠

∂th(①, t) = −Kd∇4h+ η(①, t),

Kd ♦♥ts ♦r t str♥t ♦ t s♦♥ ♥ s ♠♥s♦♥ ♦ ❬m4/s❪

rt♦♥ ♦ q r♦♠ ♦♥srt♦♥ s t t♦ t ♣♣♥① st♦♥

♥ t tr♠♥st ♦r♠ ♦ q s ♥♦♥ ♥ s s♦ s♥

♦♥ t♠ ♦ ② rr♥ ❬❪ ♥ ♥s ❬❪ ♦♥sr♥ t t ♦ t s

♦♥ s♥tr♥ ♣♥♦♠♥ ♥ t ♦♣♠♥t ♦ tr♠ r♦♦s rs♣t②

ts ♦t♥ t st♦st r♦t qt♦♥ s ♥srr♥ qt♦♥ ♥st

♥r ♦r ♥ ts ssrtt♦♥ rr t♦ t q s ♥r

qt♦♥ ts rt ♥rst② ss s ♥ ♥srr♥ ss

rt ①♣♦♥♥ts ♦♥sttt♥ t ss r r♦♠ q t n = 2


α =4− ds

2and z = 4.

ss ①♣♦♥♥ts r rst② t ② ❲♦ ♥ ❱♥ s ♥t♥t♦♥ ♦

sr♥ t ❲❱ ♠♦ ❬❪ t♦ s♦rt② tr s ♦♥ tt t② t

♠♦ ♦s ♥♦t ♦♥ t♦ t ss ❬❪ ♠r ♠♦s ♣tr♥ t s♦♥

♠♥s♠ ♥ tr② ♦♥♥ t♦ t ss rrs t ♥♠ ♦ s r♠ ♠

♦r♥ ss ♥ ♠ ❬❪ ss r ♣♣r ♦♥ t ♠♦s ♥

♦♥ ♥ t r ❬❪ sss♦♥ ♦♥ t ❲❱ ♠♦ s sr ♥ t ♣tr

♦ t r ❬❪

♥ t ①♣r♠♥t s ♥s ♦ ss ♥ ♦♥ ♥ t r♦t

♦ ♦♥ ② ♦r T = 275 C ♥ ♣♦st♦♥ rt ♦ ②rs♠♥ ❬❪

❬s♥ s ♦t♥ tr♦ ♠s❪ tr♠ ♣♦rt♦♥ ♦ ♠♦r♣♦s

♦♥ sstrts t ♣♦st♦♥ rt ♦ 0.8 ± 0.2 /s ❬❪ ❬❪ t s♣ttr

♣♦st♦♥ r♦t ♦ Pt ♦♥ ss t ♣♦st♦♥ rt ♦ /s ♥ t t ♥♦r♠

sstrt ♥ ♦t 45 t t trt sr ♥♦r♠ ❬❪ ❬❪ ♥ tt♦♥s

♦ ♥trr♥ ♦♠♥s ♦ ♦ tr♦♣♦sts r♦♥ t nm/s ❬❪ ❬❪ ♥ ♥

♥trr♥ tt♦♥s ♦ LiCoOx t♥ ♠s r♦♥ ② r s♣ttr♥ tr ♥♥♥

♣r♦ss ❬❪ ❬❪ s s ♥ t rr r♦t ♦ tt r♥ t♠♦r ❬❪

❬♦♣t ♠r♦s♦♣❪

♥r ♣♦st♦♥s♦r♣t♦♥s♦♥ qt♦♥

♥ r♥ ♦s r♦♠ r ♥ s

t ts ♣♦♥t ♦ ♥srt t♦tr t s♦♥ ♥ t s♦r♣t♦♥ ♠

♥s♠s ♥ ♦♥t♥♠ r♦t qt♦♥ s ♠s s♥s ♦♥ s♣ ♦♥t♦♥s

r② ♣♦st♦♥ t♠♣rtr ♥♦r ♦ s♣rstrt♦♥ ♥ t r♦t t♦

♣♥ s♥st② ♦♥ ♦t ♣r♦sss ♣♦st♦♥ ♦rs s♦② tr s s♣r

strt♦♥ r♥ t ♣♦rt♦s♦ ♣s tr♥st♦♥ t ♠♥s tt t r♥ ♦

♠ ♣♦t♥t t♥ t ♣♦r µv ♥ ♦ t s♦ ♣s µ(①, t) s ♣♦s

t ♦r ♥ t s♣rstrt♦♥ s ♦ ❬(µv − µ) & 0❪ tr♠ tt♦♥s


♠ ♣♦ss r♥ rtrst t♠ ♣rt t♦ s♣ r♦♠ t ♥tr ♥

t♦ ♠♦ t♦r t ♣♦r ♣s ♥ ts ♦♥t♦♥s ♥ ♥♥ s♦♥ ♦♥ s

∂th(①, t) = F − B[(µv − µ)]−Kd∇4h+ η(①, t),

t ♥ ♠♥s♦♥ ♦ ♥rs ♦ ♥r ♠♦♠♥t♠

♥srt♥ µ(x, t) ∝ −∇2h ♦♥ ♥s t ♥r ♣♦st♦♥s♦r♣t♦♥s♦♥

qt♦♥

∂th(①, t) = F + ν∇2h−Kd∇4h+Bµv + η(①, t),

r t rt♦ (Kd/ν)1/2 s ♠♥s♦♥ ♦ ♥t ♥ ts ♥ rtrst

♥t ♥ t s②st♠ ζ

tr rs♥ ♥ t q ♣r♦s

ςα−z∂th = νςα−2∇2h−Kdςα−4∇4h+Bµv + ς−z−dsη(①, t),

r ♦r ♦♥♥t tt♦♥s ς → ∞ l ≫ ζ t ♣♥ ♦♠♥ts t

r♦t ♥ t ①♣♦♥♥ts r ♦♥sst♥t t t ❲ ss t s♦rt♥t ss

ς → 0 l ≪ ζ ♦r s♦♥ ♠♥s♠ ♦r♦♠s t ♣♥ t ♥ t

r♦t s tt ② t ♥r qt♦♥

s ♦r s ♥ ♦♥r♠ ♦r ♥st♥ ♥ ♦♣♣r tr♦♣♦st♦♥ ♥

t ♣rs♥ ♦ t②t♦r ♥ ♦r♥ t t ♦♥♥trt♦♥ x 0.3 ≤x ≤ 0.4 mM ♦r ♦ rr♥t ♥st② j = 0.02 cm−2 ❬❪ ♥ ts ♦r t

t♦rs ♦♥ α ♥ β ①♣♦♥♥ts ♦r ♦t r♠s str♦♥② ♦♥r♠♥ t

❲ r♦ss♦r ♦ttst♥♥ ♥ ♠♥② sts ♦♥ t r♦t ♦ t♥ ♠s t ♦

r♦♥ss s t ♦♥② r ♥②③ r♦♠ r α s s② ①trt ♥ ♣rtr

♥ α ≈ 1.00 ♥rs♦♥♦♠♥t ②♥♠ s ♦t♥ sst t r

♥ s ♠♦♥strt tt ts ♣r♦r s ♥ ♠♦st ♦ ss ❬❪

t②♥ t t ♦ r♥s t sr ♦♥ t ♦ r♦♥ss s r ♥

s s♦♥ tt t ♥r wloc(r, t) ♣♦t ①ts t♦ r♦ss♦rs s ♥ ♦♦

❲ r ♣rsr♥ t ♦r♥ ♥♦tt♦♥ ♦ t ♣♣r ❬❪


t rst r♠ ♣♣♥s ♦r r ≪ rc r rc s t r r♥ s③ ♥

s tt ② t ①♣♦♥♥t α1 ♥t ss t♥ r♥s rc ≪ r ≪ ξ ♥ t

s♦♥ r♠ r♦♥ s wloc ∼ rα2 s♦♥ r♦ss♦r s♣rts ♦rrt r♦♠

♥♦♥♦rrt r♦♥s ♦ t r♥② sr

r r♥ ♦ s♠♣t ♦♥ ♥ ♣②r♠ s♣s ♥r♦ r♦♥ss r ♦ × ♦ ♦r r♥② srs ①trt ♥ t r♦♠ r♥ s ❬❪

♥trst♥② t s ♥ ♠♦♥strt tt α1 s ♣♥♥ ♦♥ t r♥

♦♠tr s♣ ♦♥ t ♦ ♦rrt♦♥ ♥t♦♥ ♥ ♦♥ t ♣r♦r t♦ t

r♦♦t♠♥sqr rs t ♠♣s tt α1 ♥ ♥♦t rt t♦ rt ①♣♦♥♥t

♥ t s♥s ♦ ♣tr♥ ♥rs tt♦♥s t sr ♦♠tr ♥tr♣rtt♦♥

s ♦rr♦♦rt t t ♦ r♦♥ss r r t α1 ♥ ♥ r♦♠

0.75 t♦ 1.00 ♦r srs ♦♠♣♦s ② ♣②r♠ ♥ t t♦♣ r♥s rs♣t②

α2 ①♣♦♥♥t ♦r ♣s ♦♥st♥t t t ①♣t ❯ rt t♦ t ♠♦

❲ r♦♠♠♥ t rr t ♦♦ t t ♦r♥ ♣♣rs r♦♠ r ♥ s

❬❪ ♦♠♣rs♦♥ t ①♣r♠♥ts s ♥ rr ♦t ♥ st t♦ s♦♠

①♣r♠♥t rsts ♠t♥ t r ♥ s ♣rt♦♥s ♦♥ ♥s t s♣r②

♣②r♦②ss r♦t ♦ ❩♥ ♠s ❬❪ s 0.94 ≪ α1 ≪ 0.97 ♦r ♦

rts t tr♦♣♦st♦♥ ♦ ♦♦♣r ② ♥③ t ❬❪ ♣rs♥t♥ α1 = 0.87 ±0.06 t s♣ttr♥ ♦ ♥q ♦① ♠s α1 = 0.70 ❬❪ t r♦t ♦ ②rs

♦ ♣♦②②♠♥ ②r♦♦r ♥ s♥ssttt ③♦♥③♥ ♦♣♦②♠r


♦ tr ♣♦st♦♥ ♦ ♦r ②rs ♥ α1 ≈ 0.80 ❬❪ ♥ t

srs ♦ ♥♠r♦tt ♠s ♦ ♣♦②♥♥ ♥ ♥tr ♣♦s♣♥ rt♥♠

♦♠♣① ②s 0.66≪ α1 ≪ 0.82 ❬❪

♣tr

rrPrs❩♥ ❯♥rst②

ss r st♦r ♥

tt♦trt

s ♣tr s ♥tr sq♥ ♦ t ♣r♦s ♦♥ t t♦ t r♥ss

♦ t rrPrs❩♥ qt♦♥ ♥ ♥tr ♣tr s sr t♦ t sss♦♥s

tt ♦♥ ♦♦s s ♥rrt ♦♥ t ♠♦tt♦♥ ♦♣♠♥t ♥ rr♥t stts

♦ ts ♣r♠t ♦♥t♥♠ r♦t qt♦♥ s ♣rs♥t ❲ ♦ ♥♦t t

♥t♥t♦♥ t♦ ♥♦ ♥ ts t①t t t qt♦♥ s ♥ ♠♦♥strt♥ t♦ ♦r

♥st ♦s ♦♥ ts s♥ ♣r♦♣rts ♦♥ ♠♣♦rt♥t ♠♣♣♥s ♥ ♦♥ ts ♥rs

strt♦♥s

♥ ♣♣♥s t strt♦♥s

rrPrs❩♥ P❩ qt♦♥ s ♣r♦♣♦s ♥ ♦r sr♥

r♦♥ ♥trs ♥ r♦t ♥ t ♦ ♥♦r♠ rt♦♥ ♣②s s r♦ ♥

t s②♠♣t♦t ②♥♠s ❬❪ s ♦♥ ♥rst② s s s ♦♥ ①st♥ r♦t

♠♦s s t ♥ ♠♦ ♦r ♦♦♥② ♦r♠t♦♥ ❬❪ ♥ t st ♣♦st♦♥

♠♦ ♦r ♦♦ rts ❬❪ P❩ ♣r♦♣♦s t s♠♣st ♥♦♥♥r st♦st

qt♦♥ ♦r sr♥ t t ②♥♠ ♦ s ♥trs


∂h(①, t)∂t

= ν∇2h+λ

2(∇h)2 + η(①, t),

ν r♣rs♥ts sr t♥s♦♥ t♦ t ♦♠tr ♥tr♣rtt♦♥ ♦ ♣♥ tr♠

s r ❬❪ rs t ♥♦♥♥r tr♠ ♦♥ts ♦r t r♦t ♥ t ♦ ♥♦r♠

rt♦♥ ♥ η s t ♥♦s

r ss ♣r♦s ♦r tr♠♥st P❩ r♦t t r♦ ♥t ♦♥t♦♥♥st ♥ts ♦ ♦ ♥♦r♠ r♦t ♦rs ①trt ♥ t r♦♠ ❬❪

s ♥t ♥ t ♥st ♦ ♥ ♥ ♥tr r♦s tr② t

♥r♠♥t ♦♥ t h ①s δh ♥ t ♦ ♥♦r♠ ♦t② v r rt ② t

P②t♦rs t♦r♠ δh = [(vδt)2 + (vδt∇h)2]1/2 ♥srt♥ t s♠ s♦♣ ♣♣r♦①

♠t♦♥ ∇h ≪ 1 ♦♥ ♥ ♣r♦r♠ ♥ ①♣♥s♦♥ t t rt s ♦ t qt♦♥

♥ t♦ ∂h∂t≃ v+(v/2)(∇h)2+ ... q s ♦t♥ tr tr♥s♦r♠t♦♥ t♦ t

♠♦♥ r♠ h→ h+ vt ♥ tr t r①t♦♥ ♠♥s♠ s ♥srt ♦♥♥r

r♦rr tr♠s r s♦♥sr s t② ♦ t♦ ③r♦ str t♥ (∇h)2 ♥ t

②r♦②♥♠ ♠t s r② sss ♥ ♣tr

❯♥ ♥r r♦t qt♦♥s t ♠♥ ♥tr ♦t② ♦ t P❩ qt♦♥

ts t ♦r♠ v = Ft + λ2

∫ L

0ddsx〈(∇h)2〉 s r♥t r♦♠ ③r♦ ♥ ♥

F = 0 λ ♦♥ts ♦r ts ①ss ♦ ♦t② ♥ ♣rtr λ = 0 ♦♥ ♦t♥s

t ❲ qt♦♥ q ♥ t♦♥② ν = 0 ♦♥ ♥s t ♥♦♠ r♦t

qt♦♥ s ♣♣♥① st♦♥ s ♣♣r♦①♠t♦♥ s ♥♦t rqr ♥ ♦trs rt♦♥s ♦s r s ♠t t ♦r♠ ♦ t

P❩ qt♦♥ ❬❪


P❩ qt♦♥ ♥ s② s♦ ♥ ts tr♠♥st ♦r♠ η(①, t) = 0

s t♦ ♥trs ♦♠♣♦s ② ♣r♦♦ s♠♥ts rs♠♥ ♥rts ❬❪

s ♦r t②♣ ♦♥♠♥s♦♥ r♦t ♣ttr♥ ♥ t ♦tr ♥ t

st♦st qt♦♥ q s ♥ rsst♥ t♦ ♦♠♣t ♥②t ♥♥

rt ♣rt ♦ t ♥s ♦♥ t s♦t♦♥ ♦ t P❩ qt♦♥ s ♠♦ ♦♥ ♦♥

t♦ t rs♥ ♦ ♣♦r t♦♦s tt ♠t♠t♥s s r♦t t♦ t ♦r ♥

st♥ ♥②t ♠♦s tt ♦♥ t♦ t P❩ qt♦♥ s t ♥t♣ ♠♦ ❬❪

♥ t P♦②rr♦t P ♠♦ ❬❪ t♦tr t t ♦t②

s②♠♠tr ①s♦♥ Pr♦ss P ❬❪ ♥ t rt P♦②♠r ♥ ♥♦♠

P♦t♥t P ❬❪ ♥ ♠♣♣ ♦♥t♦ t t ♦ P❩ ♥tr

♣② r r♦ t♥ t P❩ t♦r② ♦r ts st t t ♥♥♥ ♦ ts

♥②t trt♠♥ts ♥ ♦r ♠♥s♦♥s ♦r r

♥t ♦♠trs ♥ t sr r♦t ♦♥t①t rrs t♦ r ❬❪ t ❬❪

♥ stt♦♥r② ❬❪ r♦t ♥t ♦♥t♦♥s ♦t♦♥s ♦ r ♠♥s♦♥s

♦r r ♥ ♦ ♦ ♥②t ♦♣s

r♥ t♦ t P❩ s♥ ♦♥ ♣♣s ♥② t tr rs♥ s

♥ t st♦♥ ♦♥ ♦t♥s tr s♥♦♥sst♥t s♥ rt♦♥s r♥ tt

t ♥♦♥♥r tr♠ s♦ ♦♠♥t t r♦t ♥ t ②r♦②♥♠ ♠t ♥ ♣

♣②♥ t rs♥ ♥ t♦t t ♣♥ tr♠ ♦♥ ♥s α = (2 − ds)/3

♥ β = (2 − ds)/(4 + ds) ♦s ♣rt♦♥s r qt r♥t r♦♠ ♥♠r r

sts ❬❪ ♥ ts ♣r♦r s r♦♥ s tr♠s ν, λ ♥

D ♦ ♥♦t r♥♦r♠③ ♥♣♥♥t② ♥ ♦♣ t♦ ♦tr ❬❪ ♦rrt

♣rt♦♥ ♦r P❩ ①♣♦♥♥ts ♥ s♥ ♥♦r♠③t♦♥r♦♣ t♥qs

♥ ♠♣♣♥s t♦ ♦tr ♣r♦♠s r♦ t u = −∂xh tr♥s♦r♠t♦♥ t P❩

qt♦♥ t λ = 1 s ♠♣♣ ♥ t rrs qt♦♥ t ♥♦s ❬❪ sr♥

t ♦rtt②r ♦t② ♦ strr t♦♥② t t tt λ = 1

♥ t rrs qt♦♥ ♣♦♥ts ♦t tt ♠st ♣rsr t s ♥r♥ ♦♥ t

♥♦♥♥r tr♠ ♥ t P❩ ♦♥t①t s rs♦♥♥ ♣r♦s t ②♣rs♥ rt♦♥

α + z = 2.


s rt♦♥ s ♦♥sq♥ ♦ t rr qt♦♥ t♦ ♥r♥t ♥r ♥

tr♥s♦r♠t♦♥ u(x, t)→ u0 + u(x− v0t, t) ♥ tr♥ s t P❩ qt♦♥ t♦

♥r♥t ♥r tt♥ tr♥s♦r♠t♦♥ ② ♥ ♥ ǫ

h′ → h+ ǫx; x′ → x− ǫλt; t′ → t.

tt♦♥ss♣t♦♥ t♦r♠ ❬❪ rs tt t t st② stt ♦r

d = 1 + 1 ∇h ♦♦s ss♥ strt♦♥ s t ♣♦st♦♥ ♦ ♣rt ♥

r♦♥♥ ♠♦t♦♥ ♥ r♥s ♦t α = 1/2 ♦tr t q ts ♠♣s z = 3/2

♥ β = 1/3 ♥ r♠♥t t ♣r♦s ❬❪ ♥ r♥t ❬❪ ♥♠r

♠♦s ♦♥♥ t♦ t P❩ ♥rst② ss

♦♥♣t ♦ ♥rst② ②♦♥ ①♣♦♥♥ts s ♥tt ♥ t ss

♣♣r r♦♠ r ♥ ♥ ♣♥② t♦ s ❬❪ ♥ tt ♦r ②

s♥ t s♦ r♥ t♦♦♦① ❬❪ t s ♣♦ss t♦ rt ♥rs

♠♣ts ♥ tr♠s ♦ ♠♦ ♣r♠trs ♥♠② A ≡ f(D/ν) ♥ λ ♥

s② ♦t♥ ♥ s♠t♦♥s ❬❪ ♠♦st ②rs tr ts ♠♣♦r♥t

rst ♦♥ss♦♥ st ♠♦ ♦r sr ♣r♦st ♥tr♣rtt♦♥s ♥

♥ ❬❪ ♠♦♥ t♠ ♥ ♥tr♣rtt♦♥ ♥ tr♠s ♦ t ♦♥♠♥s♦♥ P ♥

tr♥ ♠♣♣ ♦♥t♦ t ♥tr tt♦♥s ♦ t ♥t♣ ♠♦ ♦ t♦ s♦

tt t ♣ ♦ r♥♦♠ ♠♣t χ rt t♦ t t ♦ t ♥t♣

♠♦ q s t rt r②❲♦♠ ❲ strt♦♥ ❬❪ ♠r♥

r♦♠ t ♥♦♠ tr① ♦r② ♦♥t①t

h(t) ≃ v∞t+ sign(λ)(Γt)βχ.

r v∞ s t s②♠♣t♦t ♦t② ♦ t ♥tr ❬v∞ ≡ limt,L→∞〈∂th〉❪ sign(λ) st s♥ ♥t♦♥ Γ ≡ aA1/α|λ| t a ♥ ♦♥st♥t ♥ A ≡ f(D/ν) ♥

♠♥s♦♥s s ❬❪

♦rt② tr Prä♦r ♥ ♣♦♥ ❬❪ ♥ ♣♥♥ ♦ ♣χ ♦♥

t ♦♠tr② ♦ t r♦t ② s♥ t ♠♣♣♥ ♦ t P ♠♦ ♦♥t♦

s rt♦♥ s ♦♥② ♦r ♠♥s♦♥s ♥ r ♠♥s♦♥s ♥ s♦ ♥t♦♥♦ λ ❬❪


r ❯♥rs strt♦♥s ♦t♥ ② Prä♦r ♥ ♣♦♥ ♥ t ❬❪ r♦♠t t♦ rt ♥rs ♣s s♦ ♥s ♦r χ ♥r r t ♥ stt♦♥r② ss♠rr♦t rs♣t② ②♠♦s rrs t♦ s♠t♦♥s ♦ t P ♠♦ r ①trt r♦♠t r ❬❪

r♥♦♠ ♣r♠tt♦♥s t② s♦ tt h(x, t) ♦rrs♣♦♥s t♦ t ♥t ♦ t ♦♥st

♥rs♥ ssq♥ ♦ s ♣r♠tt♦♥ ♥ tr♥ s strt ♦r♥ t♦

t ss♥ rt♦♦♥ ♥s♠ ❲ strt♦♥ ♦r t t

ss♥ ❯♥tr② ♥s♠ ❯ ❲ t r♦t strts r♦♠ s ♥ ♦♣s

r ♥tr ♥ t ♥s F0 ♠t♥ strt♦♥ ❬❪ ♦r st②

stt r s♦s ♣χ ♦r r♥t r♦t ♦♠trs ♥ ♦r stt♦♥r② ♥t

♦♥rt♦♥ t r♦♠ t P ♠♦ ♥ ♦♠♣r t♦ t rs♣ts ❯

♥ F0 strt♦♥s ❬❪

tr t rsts ♦ Prä♦r ♥ ♣♦♥ ❬❪ ♥②t s♦t♦♥s ♦♥

t ♦♥♠♥s♦♥ P❩P qt♦♥ r② t ♦ ♦♥r♠ t

♠t♥ strt♦♥s ♦ χ s ❯ ♦r r r♦t ❬❪ ♦r t r♦t ❬❪

♥ F0 ♦r t stt♦♥r② ♥t ♦♥t♦♥ ❬❪ ♦r♦r ♥tt♠ ♦rrt♦♥s r

♦♥ sst♥ ♥ ♥rs P❩ trs s s st ♥ t ♠♥ ♦♥r♥

t♦ ❯ s s t−β t s s♦ ♠♦♥strt tt t ♠t♥ ♣r♦sss

♥ t ♦♥♠♥s♦♥ t ♣r♦s ♦ P❩ ♥tr r tt ② t

s ♠♥s t strt♦♥ ♦ t rst ♥s ♦ ♦rt♦♦♥ ♠tr ♥s♠s ♦s♠♥ts r strt r♦♠ ss♥

♠ s ♦r t ♥♦ t ♠trs r ♥tr②


r②1 ❬❪ ♥ r②2 ❬❪ ♣r♦sss ♦r t ♥ r r♦♥ rs♣t②

r sts ♦r tr♥t q r②sts ①trt ♥ t r♦♠ ❬❪ st♦r♠ ♦ rs ♦ t χ ≡ h− v∞t/(Γt)1/3 ♥ r s♦ s②♠♦s s♦t st♦r♠s ♦r t rr ♥ t ♥trs rs♣t② s ♥ ♦tt rss♦ ❯ ♥ ❲ strt♦♥s ♦ χ ♠♥ts ♣♣r♦ t♦ t ❯ s♥st s♦s t ♠♥ st ②♥ s t−1/3 s♠ ♦r s sr ② t t s♥♦t s♦♥ r

♠t♥♦s② ①♣r♠♥ts ♦ ♥♣r♥t sttsts ♦♥ tr♥t q r②s

ts ♣r♦r♠ ② ♥ ♥♦ ❬❪ ♦♥r♠ r② rt ♣rt

♦ t ♣rt♦♥s t ♦ ♥ ♥ t♦ t P❩d=1+1 t♦r② r rt②

②♦♥ ♠t♠t ♥♠r ♥ ①♣r♠♥t r③t♦♥s ♦♥str♥ t♦ ①

♣♦♥♥t rsts ❬❪ ② s♥ t r♥ t♦♦♦① ♦r ♥rt♥ ♥♦♥♥rs

♣r♠trs A ♥ λ ♥ ♥♦ t t ♣ ♦ t χ r

♥ ♦r♥ t♦ t P❩ ♥sät③ q s χ ≡ (h − v∞t)/(Γt)1/3 ♥ t

♦♥ ♥ s ♦♠♣rs♦♥ t♥ t ①♣r♠♥t ♣χ ♦t♥ ♥ tt ♦r

♦r ♦t r ♥ t ss ♥ t ❯ ♥ strt♦♥s ❬❪ ❲♦♥r②

rt ♦r♥ s ♦t♥ ♣rt r♦♠ st st t ♠♥ ♦ t strt♦♥s

♥ t ♥ t r t st t t ♠♥ s s♦ t sst♥ tt t

②s s ∼ t−1/3 t st ♦r r ♠♥ts ♥ss q②

♠r ①♣r♠♥ts ♦♥ ♦♦ ♣rts ♣♦st t t ♦ ♣♦

rt♥ r♦♣s r s♦ t♦ ♦♥r♠ t P❩ ♥rst② ②♦♥ ①♣♦♥♥ts r

r ♣♦sts ♦ ♣rts st② ♥s♦tr♦♣ ♦♦♥ t ❯❲ strt♦♥


r ♦♥ ❬❪ ♥②t ♥ ①♣r♠♥t rsts ♥ r♥♦r ② ♥♠r

♦♥s ❬❪ t st ♣s t♦r r♦st ♥ ♦♥sst♥t P❩d=1+1

tr♠rt

P❩d=2+1 stt♦♥ s r② ♦♥trst♥ t ts ♦♥♠♥s♦♥ ♦♥tr♣rt

♥ ♠♦st ♦♥ ♥♦s ♦t t ♠♦st ♠♣♦rt♥t ♠♥s♦♥ ♦r ♣♣t♦♥s s ♦♠

r♦♠ s♠t♦♥s ♥ r② r♥t② r♦♠ s♦♠ r♠r ①♣r♠♥t ♦rts ❬❪

st st♠ts ♦r s♥ ①♣♦♥♥ts ♥t tt ❬❪

α ≈ 0.393 β ≈ 0.242 1/z ≈ 1.607.

♥rst② ♦ ♠♥s♦♥ss ♠♥t rt♦s ♦ h ♥♠② t s♥ss

♥ rt♦ss q r rst② t ♥♠r② t t stt♦♥r② stt

r tr s r ♣r♦ t♦ ♥rs s ❬❪ ♥ rr♥s tr♥

♥rst② ♥ t r♦t r♠ t♦ ♠♣s ♥ t ❬❪ ♦♥② s

♦♥♥♥② ♠♦♥strt ♥ t ②r ♦ r♦ rs s♠t♦♥s

♣♥② ❬❪ ♥ r t ❬❪ ♥♦r t ①st♥ ♦

♦♠tr②♣♥♥t P❩ ♥rs t strt♦♥s t t r♦t r♠ r

♠♥s♦♥ ♥ ❯❲ ♦♥tr♣rts ②♥ ♥ t rt ♦ P❩d=2+1 ♥r

st② t♦ t ①t ♦r♠s ♦ ts strt♦♥s r ♥♦t ♥♦♥ r t

♠♦♥strt tt rs t ♣s ♥ tt ② ♥r③ ♠

strt♦♥s ❬❪ s t ♥t♦♥ ♥ t st♦♥ t ♣r♠trs ♠ ♥

♠ ♦r t ♥ r ss rs♣t② ♦r♦r s ♥ ♠♥s♦♥s

rsts r♦♠ t r t st② ❬❪ s♣♣♦rt ♥r③t♦♥ ♦

t P❩d=2+1 ♥sät③ q ♥srt♥ ♣♣r♦♣rt ♥tt♠ ♦rrt♦♥s ♥

P❩ ♥sät③ rs

h(t) = v∞t+ sigλ(Γt)βχ+ ηp + ζpt

−γp + ...,

r ηp ζp ♥ γp r ♥♦♥♥rs ♣r♠trs ❬❪ s ♦r ts ♠♦

♣♥♥t ♣r♠trs ♥ ♦♥ ♥ t rr♥s ❬❪ ♦r

♥ ❬❪ ♦r sr♥ ♠ ♠♦r tt♥t♦♥ t ♥rsP❩ s


♦r t ♠♥ts ♦ t strt♦♥s r r♦♣ ♥ t ts ♥

❯ ♥s〈χ〉c 〈χ2〉c ⑤⑤

❯♥rs P❩ s ♦r ♠♥ts ♦ t strt♦♥s ♥ ❬❪

t r tr♦♥ r♦♦ tt♦♥r②〈χ〉c 〈χ2〉c ⑤⑤

❯♥rs P❩ s ♦r ♠♥ts ♦ t strt♦♥s ♥ ♠♥s♦♥s ♦r t s ❬❪ r ❬❪ tr♦♥ ❬❪ r♦♦ ♥t ♦♥t♦♥ ❬❪ ♥tt♦♥r② stt ❬❪

s ♥ r♠r ♦♥ P❩ t strt♦♥s ♣♦♥t ♦t tt st②♥ P❩

r♦t ♦♥ ♥r♥ t sstrts rrs♦ t r tt t r②

❲♦♠ strt♦♥s ♥ t r② ♣r♦sss s s tr ♠♥s♦♥

♥♦s ♦ ♥♦t ♣♥ ♦♥ t ♥tr ♠r♦s♦♣ rtr t t② ♦♥ t

♥t♦♥ ♦ t tt ♠tr ♦♥ t t ③♦♥ ❬❪ rt♠♦r ♥ t sr ♦r ♥

♣♣r rt ♠♥s♦♥ ♥ P❩ ss s t ❬❪ ♦♥r♠ tt t P❩

♥sät③ s ♣ t♦ ♠♥s♦♥s t st ♦r t ♠♦strt③ ♠♦ ❬❪

❯♥rs qr ♦♥ss strt♦♥s

♥ ♥stt♥ t ♦ ♣r♦♠ ♦ r♥♦♠ ♥trs ót♥ t

❬❪ s♦ tt t sqr r♦♥ss ♣ ❬P (w2)❪ ♦ s ♥tr t st② stt

s s

P (w2) =1

〈w2〉Φ(w2/〈w2〉)

♦t tt t t st② stt t sr r♦♥ss tts r♦♥ ts strt ♥P (w2) s t ♣ ss♦t t♦ ts tt♦♥s


r w2 s t sqr ♦ r♦♥ss q ♦ t ♥tr Φ(u) s ♦s

♦r♠ ♥ s s♦ ♥ ♥rs s♥ ♥t♦♥ ♥ ts ♥rst② s ♦♥r♠

② ♥♠r s♠t♦♥s ♦ ❲ ♥ P❩ ♠♦s ♥ ♠♣s③ t ♣♦r ♦

tt strt♦♥ ♦r ss♥ t ❯ ♦ ♥ r♦t ♣r♦ss ♦♥ t ♣rs♥ts

♣♥♥ ♦♥ ♥ts③ ♦rrt♦♥s

♥ t s♠ ②r Ps t ❬❪ ①t♥ tss sts ♦r rtr

r♥ ♥trs ♥ r♥♦r t t② ♦ t q s s t ♥rst②

♦ Φ ♥ ts s s r♥t ♦r♠ r♦♠ tt ♦r ss♥ ♥trs ♦♦♥

t s♠ á③ t ❬❪ t ♥♠r② P (w2) ♦r sr ❯s ♥

♠♥s♦♥s ♥ ①♣♦s tt ♥ t ♦r♠♥s♦♥ s ΦEW ♥ ΦKPZ

r r♥t ♥ t rst ss♥ ♥ t st ♦♥ ♠r ② s♦ ② ♥ t

rt t ΦMH ♥ ΦV LDS s♦ ①t r♠r r♥s t♥ ♦tr

♥ t s♠ ♣♣r ♦♥ ♥s r♣ ♦ ♦ ♦♠♣r♥ Φ ♦r ♥ ❯ t t

♦♥s ♦t♥ ♥ ①♣r♠♥ts r♣ ♦♥ssts ♥ t sr ♥t♦ ♦①s ♦

tr s③ l ≪ ξ|| ♥s w2 s♦ t t♦ ② r ♥s♠ ♥

t ♦tr ♥ l ♠st s♦ rr t♥ rtrst s③s t sr s s r♥s

♠♦♥s t

qr ♦ ♦♥ss strt♦♥s s ♥ s ♥ s♥

♥②ss ❬❪ ♥ ♥ sss ♦t t ♣♣r rt P❩ ♠♥s♦♥ ❬❪ s t②

r ♦♥sr ♦♥ ♦ t ♠♦st st ②s ♦r ss♥ t ❯ ♦ r♦t ♦r

♥st♥ ♥ r② ♥trst♥ st② ❬❪ rã♦ s ♥②③ t rs strt♦♥

Φ t ♠♥ ♥ ♥ ♥tr② r♥ t Ψ ♥ s ♥ t q ♦r ♦♥ ♥

t♦♠♥s♦♥ ♠♦s ♦♥♥ t♦ t P❩ ♥ ❱ sss

P (w2) =1

〈w2〉1/2c

Ψ(w2 − 〈w2〉〈w2〉1/2c

)

.

rã♦ s ♦♥r♠ tt ♥ ΨKPZ ♣rs♥ts strt

①♣♦♥♥t ♥ t rt t s ♣♣r♦①♠t② exp(−x0.8) ❬❪ s r s♦r♠ s ♦♥trst t t♦s r♦♠ ΨEW s ss♥ ♥ t t s♠♣

①♣♦♥♥t ② ♦ ΨV LDS ♥ ΨMH ♥ t st ♦♥s ♥ s② st♥s

♥ t ΦV LDS ♥ ΦMH s♥ s ①♠♣ ♥ t ♥st ♦


ssq♥t ♦r r♦♠ P & rã♦ s ❬❪ r♦t ♦t ΨKPZ ♥ t r♦t

r♠ ♣rs♥t♥ s♠r ② t t rt t s s♦♥ ♥ t s

tr s ♥ sst s ♥ ♥rs ♥ st♥t P❩ ♥♠r ❬❪ ♥ s

♥ ♦♥r♠ ①♣r♠♥t② ♥ t r♦t ♦ ❬❪ ♥ ♦♦♠r ❬❪ t♥ ♠s

r ♠♥ ♣♦t ①ts t ♦♥trst♥ ♦r♠ ♦ ♥rs Ψ strt♦♥s♦r t ❱ ♥ P❩ sss t r♦♠ t L = 64 L = 256 ♥ L = 128 ♠♦s t t stt♦♥r② stt ♦r d = 2+1 ♦r♥ t♦ t r ❬❪ ♥srt♦♥ r♥♦rs t r♥ t♥ ΨV LDS ♥ ΨKPZ ♥r ♦ t ♣ rs♥srts s♦s t ♦♥strst t♥ ΦV LDS ♥ ΦMH ♦♠♣rs♦♥ t♥ ΨKPZ

t r♦♠ t st② stt s♦ ♥ ♥ r♦♠ t♦♠♥s♦♥ ♥ ♠♦s♥ t r♦t r♠ ♦r ♦① ♦ tr s③ r = 64 sts ♥ ♥ ♥ ♥ ♦♥♥ t rr♥s ❬❪ ♥ ❬❪ rs♣t② ss rsts r ♥② ♣r♦ ② Pr♦rã♦ s

❯♥rs ①♠ t t strt♦♥s

①tr♠ sttsts ❱ ♣② ♥ ♠♣♦rt♥t r♦ ♥ s②st♠s r rr

♥ts rst ♦♥sq♥s s s ♦♦s ♥tr♥t rs st♦ ♠rt rss

s s ♥ ♠♣♦rt♥t t♥♦♦ ♣♣t♦♥s ❬❪ ♦r ♥st♥ t

♦♥st ♦ r♦♥ ♦ ♦rr♦ srs s tr♠♥ ② tr ♣st ♦r st

♣♦♥t rs ♥ ttrs t st ♣♦♥t ♦ t ♠t sr r♥ t ♦♣♣♦st

♠t sr s rs♣♦♥s ② t ♥♥♥ ♦ s♦rtrt ❬❪ ♥♦♥

♣r♦t② ♥t♦♥ ♥ t ❱ ♦♥t①t s t ♠s rst s②♠♣t♦t ♥


t strt♦♥ ♦ t ♥t ♠♦♥N ♥♣♥♥t ♥♦rrt r♥♦♠ rs

❬❪ ♠ ♣ G(X;m) ♦ t r X s ♥ s

G(X;m) =mmb

Γf (m)exp[−m(zX + exp(−zX))],

r m s ♣r♠tr b =√

ψ1(m)/〈X2〉c zX = b(〈X〉 − X + s) s = [ln(m) −ψ0(m)]/b,Γf (X) s t ♠♠ ♥t♦♥ ♥ ψk(X) s t ♣♦②♠♠ ♥t♦♥ ♦r

♦rr ❬❪

r② sts ♦♥ ❱ ♣♣ ♦♥ r♦♥ ♥trs ♦s ♦♥ t st② stt

♦ ♥r ❲ qt♦♥s ❬❪ r♦ ♠①♠ rt t m∗ ♥②ss

♥ s t r♥ t♥ t rst t ♠♥s t r t ♦ t

sr t s s♦♥ tt ♥ t stt♦♥r② r♠ m∗ ss s t ♦ r♦

♥ss ts rst s ♦♥② ♦r t ♦♥♠♥s♦♥ s ❬❪ ♥ tt ts ♥rs

strt♦♥ P (m∗) = L−1/2f(m∗L−1/2) s tt ② t r② strt♦♥ ♥t♦♥

f(x) tr ♣r♦ ♦♥r② ♦♥t♦♥s r s ❬❪

♥trst♥② ♥ ♦r str♦♥② ♦rrt s②st♠s s s t♦♠♥s♦♥ ❲

♥trs t st② stt t s ♥ s♦♥ tt P (m∗) ♥ r② tt ② t

♠ t ♥♦♥♥tr ♥ ♥ ts s ❬❪ ♥♠r ♦r s

r♦t tss sss♦♥s ♦r P❩ ♥ ❱ t♦♠♥s♦♥ srs ❬❪ t s

♦♥r♠ t ♥rst② ♦r tr t s ♠①♠ ♦r ♠♥♠rt t

strt♦♥s s ♣♥♥ ♦♥ t r♥t ♥♦♥♥r tr♠ s♥ ♦r♦r

t r♠r ♦♥trst t♥ t rt t ② r♦♠ KPZ s♠♣ ①♣♦♥♥

t ♥ V LDS ss♥ ♥ t r② ♥ts③ ts t♥ ts

strt♦♥s sst ♥ tr♥t ② ♦r ss♥ t ❯ ♦ ♥ r♦t

r ①♣r♠♥t rsts ♥ t rst ♦♥r♠t♦♥ ♦ s ♥rst② ♦ ts

strt♦♥ ♥ t P❩d=2+1 ♦♥t①t

s♦ t r t rsts ❬❪ ♦t t rt♦♥s♣ t♥ t ♠ strt♦♥♥ P❩ sd=2+1 ② r r② sss ♥ t st♦♥

♣tr


♥ ts ♣tr s♦rt r ♦♥ t ①♣r♠♥t t♥qs s ♦♥ ts

♦r s ♥ ts ♦♥ r♦t ♣r♠trs ♥ ♦♥ t ①♣r♠♥t ♠t♦♦♦② s

sss ♥ ts

♦t ❲ ♥q

♦t ❲ ♣t①② s sts t♥q s ♦♥ tr♠ ♣♦rt♦♥

♦♣ t t ♥ ♦ t s s ♥ s ♦r r♦♥ qt② ♠s

r♦♠ ❱ ❱❱ ♥ s♦ ❱ ♦♠♣♦♥s ❬❪ rt r♥ t♥

s♠♣ tr♠ ♣♦rt♦♥ s②st♠ ♥ ❲ ♦♥ ♦♥ssts t ♣rs♥ ♦ t

♥r ♦t s ♥r srs s ♦r t ♣♦r ♠ t♦ ♦ r♦♠ t s♦r

t♦rs t sstrt ♥sr♥ ♦ ♠tr ♦ss ♥ r♦t ♦♥t♦♥s s ♥r s

♣♦ss ♦ t tr♠♦②♥♠ qr♠

❲ s②st♠s sr t s♠ s strtr s t t② ♥

♠♦ ♣♥♥ ♦♥ t ♣rtr r♦t ♥ssts s ❬❪ ♦r r♥t ❲

♦r♠s ♥ ♥r tr r tr rsst♥s ♥♥s t qrt③ t t♦ t t

sstrt t s♦r ♥ t ♥♣♥♥t② t r♥ts tt t t♠♣rtr

s♣② t t ♦♥tr♦r s ②s ♥ ♠sr t t s♠ rr♥t ②♦♥

♦ ♠♥ t s②st♠s rt♦♥ t♦ r② r ❲ ♦rs ♥ ♠s

t s ♥♦t t s ♦r ♦tr s②st♠s r t♠♣rtr ♠sr♠♥ts ♠st r♦t♥② rts ♦rs ♥ s②st♠s


≈ 10−7 ♦rr ♣r♦♥ rt② ♥ ♥r♦♥♠♥t ♦r t r♦t ❬❪

strt ♣♦st♦♥ ♥ ♦r s ♣ ♦r t ♦r ♥ st② ♥srt ♦ t

♥ sttr s

s r② s♠♣ t♥q ❲ s ♥st ♦r ♦♣♥ ♦r r♦♥ ♦♠

♣♦♥s t r② r♥t ♣♦r ♣rssrs ♥ t s♦ ♦s ♥♦t ♦ ♥st ♠

sr♠♥ts ♥ ts s♥ts tr♥ t ❲ s ♠♥♠③ ♦r ♥

stt♦♥s r ♥st ♠sr♠♥ts r ♥♦t ss♥t r t ♦♠♣♦♥s ♣♦

rt ♦♥r♥t② ♥ ♦ ♥♦t t♠♣rtr ♦ s♠t♦♥ ❲ s s♦t②

♦♥ ♦ t ♠♦st st t♥qs t♦ ♠♣♦② t♦ ts r♣r♦t②

r♦t rt r♥ s ❬❪ ♥ ts rt ♦ ♦st ♦♥rs♣ ♥

♠♥t♥♥ s s ♣rtr② t s ♦r t r♦t ♦ ❬❪

r ♦t ❲ s②st♠ ♦ ♣t①② ♦rt♦r② P②ss ♣rt♠♥t ♦ t ❯♥rs r ❱ç♦s r③ ♥ ♥ t r s♦s t♦ r♥s sstrt ♥s♦r tt ♦r ♥♣♥♥t② ♥ sttr ♦s strt♥♥s♥ t r♦t ♣rssr r♥ t r♦t ♥ r ∼ 10−7 ♦rr


t♦♠ ♦r r♦s♦♣②

tr t ②r ♥♥t♦♥ ❬❪ s ♥♦r♠♦s② ssst t sr

st② ♥ ♠r♦♠tr s ②♦♥ ♦ ♣♥ sr sts ♦ t t♥♠ ♣r♦t♦♥

❬❪ s ♥ ♠♥ ♠st s s ♥rs t♦ t♦♣♦r♣ rt♦♥s r♥ r

tr t ♦♥ssts ♥ t♦ ♦t♥ ♠♦r♣♦♦ t r♦♠ ♥trt♦♥s t♥

r② t♥ t♣ ♦rr ♦ 10−8m r ♥ t s♠♣ sr t♦ ② tr♠♥s♦♥

♠ q♣♠♥t ♠srs t♦♠ ♦rs s♣r♥ ♦r♠t♦♥ ♠♦r ♥♦♥

s ♥tr ♥r t t♣ s ♦♣ ♥ t ♦tr ♥tr sr ♠

s ♦s ♥ rt t♦ ♣♦t♦♦ ♥ ♦rr t♦ s♥ tr s♥s ss♦t

t t ♥trs t♦♥ t♦ t ♦♥tr♦r s♠♣ ♦ ♠♦st ♠♣♦rt♥t

♦♠♣♦♥♥ts s s♦♥ t

♥ t tr s♥s rr t ♦♥tr♦r t② r tr♥st ♥

s♥t t♦ t ♦♠♣tr t♦ ② ♠♥s♦♥ ♣r♦ st ② t t♣ ♥

♠ ♥ ① ♣①s ♠♥s tt tr r ♠♥s♦♥

♣r♦s ♦♥ ♦ t♠ ♥ ♦♠♣♦s ② q② s♣ ♣♦♥ts ♥ s♦♠

r s s♠ ♦ ♦r♥ ♣r♥♣s ♦ ♥ t♦♠ ♦r r♦s♦♣


s②st♠s t s♠♣ s s♣♣♦rt ② ♣③♦tr r♠ srs

s♥♥♦♠tr ♦r♠t♦♥s t♦ ♠♦ t s♠♣ ♥ rt♦♥ t♦ t t♣

r r t♦ s ♠sr♠♥t ♠♦s ♥♠② ♦♥tt ♥ t♣♣♥ ♥

t ♦r♠r t t♣ ♣♣r♦s t♦ t sr ♥t sr r♣s♦♥ ♦r ss

t t♣ t♦ ♥ ♣ ♥ t t♣♣♥ ♠♦ ♦r t t♣ s st t♦ ♦st ♥r t♦ ts

♥tr rs♦♥♥ rq♥② ♥ ts ♦s t♦ t sr ♥t t ♦st♦♥ ♠♣t

♦♠s r t♦ s♠r rr♥t ts ♦rs t st♥ t♥ r♦♥s

♦ r♣s♦♥ ♥ ttrt♦♥ ♦r ♦t ♠♦s t ♣③♦ sts tr♦

♠♥s♠ t t ①s ③ t♦ ♦ ♦r t ♦r ♦♥tt ♦r t ♠♣t t♣♣♥

♦♥st♥t ♥ sr ♠ s ♦t♥ r♦♠ t♦s ③ s r♦r ② t

♣③♦ ♦ ♦ ♠t♦ s♦ ♠♣♦② ♣♥s ♦♥ t s♣ts ♦ t

st ♠tr s♣ strtr ♥tr ♥ t ♥ ♦ rsts tt ♥t t♦

♦♥ ♦ rt♦♥ ♦♥ts ♣ss r♥ ♦♥t♥ rt♦♥s t ♦r

♥st♥ ♦♦ s♠♣s r ♠♦st st ♦r t♣♣♥ ♠♦ ♦♥ t ♦s ♠

s♠♣ ♥ rt♦♥ ♦rs ♦♥tt s ♠♦st st ♦r t♥ ♠s ♥ r②sts

♥ ♥r

♦②s s ♦♥ ♦ t ♠♦st s t♥qs ♦r st②♥ srs t t

s♠r♦♠tr ♥ ♣rtr sr ♦rs s ♠s t♦ ♣r♦r♠

s♥ ♥②ss ♦ ♥trs s ①♠♣ ② ♦rs ♥♦♥ t r♦t ♦ SiO2

② ❱ ❬❪ t ss♦t♦♥ ♦ ♣♦②r②st♥ ♣r r♦♥ ❬❪ t ♠♦r♣♦s ②

tr♠ ♣♦rt♦♥ ❬❪ ♥ Pt s♣ttr ♦♥ ss ❬❪

sr ♥♥

strt ♥♥ s t ② ♥t st♣ ♥ t♥ ♠ ♥ ♣t① r♦t ♥

♦rr t♦ ♦t♥ t ♥ ♦♥t♠♥t♦♥r srs sr ♠ ♥♦r tr♠

trt♠♥ts ♥ ♣r♦♣♦s s♥ s ❬❪ s② t ♠ s t♦ r♠♦

t s♦♥ ♥t ♦① ②r ≈ 0.7 ♥♠ ♦ t♥ss ♥ t ②r♦r♦♥ ♦♥t♠♥♥t

②r ≈ 0.2 ♥♠ ♦r♠r s s② r♠♦ ② ♦♦♥♥trt♦♥ q♦s

s♦t♦♥ ❬%❪ s♥ ♥ tr ♦ ♣rt② ♣r♦ ♥ st


♠♦♥♦②r tr♠♥t sr ♣♥♥ ♦♥ t q♦s ♣ ♦♥♥trt♦♥ ❬

❪ ♦r♦r q♦s s♦t♦♥ ♦s ♥♦t t t r sr ts ♣rsr♥

s♠♦♦t ♠♦r♣♦♦② t②♣② ≈ 0.2 ♥♠ ♦ r♦♥ss ❬❪ ❲rs s♦♠

♦rs strss t ♥sst② ♦ r♠♦♥ r♠♥s♥t ②r♦r♦♥ ♠♣rts ❬❪

❳r② ♣♦t♦tr♦♥ s♣tr♦s♦♣② ❳P ♠sr♠♥ts s♦ tt trt

srs ♣rs♥t r② ♦ ♦♥♥trt♦♥ ♦ O F ♥ C ❬❪ ♦r♦r t

tr♠♥t sr ♦t♥ ② ts ♣r♦ss ♣r♦ t♦ r② st ♥st t

♦①t♦♥ ♥ r ♥ ♣rtr ts ♥♥ ♣r♦r s ♥ s ♦r r♦♥

♣t① s ♦♥ ❬❪ ♥♦tr ♣♣r♦s s t rs♥ ② t♦♥

t♥♦ ♥ ♦♥③ tr ♦♦ ② r♣t② ♦♥ ♥ 3 ♣♣♥ ♥

r♥s t tr ♥ r t 2 s♦ r ♦♠♠♦♥② s ❬❪

t♥ ♠s ♥♥ r♦t ♥ r

tr③t♦♥

♥ ts ♦r ♣t②♣ sstrts ♦ ♠♥s♦♥s 10.0 ♠♠ × 10.0 ♠♠ × 0.3

♠♠ r ♣♣ ♠♥ts ♥t♦ ♥ q♦s 2% s♦t♦♥ ♣r♣r t tr

s t♠ s ♥ t♦ ♦ r♠♦♥ ♦♠♣t② t ♥t ♦①

②r ❬❪ t t sq♥ srs r ①♣♦s t♦ N2 r t st t♦ r♠♦ ♥②

r♠♥s♥t r♦♣t ♦♥ t sr ♦♥ tr trt♠♥t t sr ♦♠s r②

tr♠♥t trt srs r ♠♠t② ♥srt ♥t♦ t ❲ ♠r

♥s ♠ s ♣r♦r♠ r s ♥♦ ♥② t♥ trt♠♥t ♦r

♦r tr t r♦t

❲ s②st♠ s ♥ ts ♦r s ♦♠♣♦s ② t♦ ♥♣♥♥t r♥s

s♦r ♥ sstrt s♣rt ② sttr ♥ ♦ ♠ s s♦♥ ♥ t

♣♦st♦♥ ♦rs t ♣rssrs ≈ 10−7 ♦rr ♦t♥ ② s♦♥ ♣♠♣

s②st♠ t s♦r t♠♣rtr ♥ ♦♥tr♦ r♦♠ 400 t♦ 520 C ♣r♦♥

r♦t rts t♥ ≈ 0.01 ♥ 2.5 s t♠♣rtr ♦ t sstrt ♠♦r

♥♦♥ s t ♣♦st♦♥ t♠♣rtr ♥ r r♦♠ ♣ t♦ 550 C

♦ s ♥ s s s♦r ♠tr t♠♣rtr ♦ t


s♦r s ① t 520 C ②♥ ♣♦st♦♥ rt F = 2.2± 0.3 s ♦r ♣♦st♦♥

t♠♣rtrs st t ♥ 300 C ♦r ♣♦st♦♥ t♠♣rtr t

r♦t t♠ t s r r♦♠ t♦ ♠♥ ♥ ♦♠tr ♣r♦rss♦♥ sq♥ ♦

rt♦ 2 ♣r♦♥ t♥sss th r♦♠ ♣♣r♦①♠t② 0.20 ♣ t♦ 3.5 µm th ♥

t r♦t rt F r tr♠♥ ♣♦str♦t s♥ ❳P ❮ ♦♥tt

♣r♦♦♠tr ♥ ♦♥t♦r ❯ ♦♣t ♣r♦♦♠tr

r rtr③t♦♥ s ♣r♦r♠ ♥ r ② ①st ❲ s ♥

tr Pr♠ P ♦r♥ ♥ ♦♥tt ♠♦ r♥t ♥s ♦ t♣s r s ♥

♦rr t♦ t rt② ♦ t ♦ t♠ r s ♥ t sttst ♥②ss

rq♥② ♦ t s♥ s ♣t ♥r ♦ ♥ss r♥ qst♦♥ ♦ t

♠s t s♦ ♦♥r♠ tt t rq♥② ♦s ♥♦t t t rsts s

r t rq♥② s ♥♦t st t r② s s & ♥ss r t♦♣♦r♣② ♦

t♦ r♥t r♦♥s ♥r ♦ t ♥tr ♠ ♦ s♠♣ s s♥♥ ♣r♦♥

♠s ♦ 10 µm × 10 µm t 1024 × 1024 ♣①s s s③ s ♦s♥ s♦ tt

♠♦r♣♦♦ ♣r♦♣rts ♥ ♦♠♥s s♠r ♥ rr t♥ t r r♥ s③

♦ s♠t♥♦s② ♥stt ♦r r♥t s③s ♦ s♥♥♥ r rr

♦t ♥♠② 1 µm × 1 µm 30 µm × 30 µm ♥ 100 µm × 100 µm t♦ r♥t t

t② ♦ t ♥ r♥t ss ♦r ♠s tt♥s ♦rrt♦♥ ♦ s♦♥ ♦rr

s ♣r♦r♠ t♦ ♦rrt t s♠♣ ♠s♥♠♥t ♥ t ♣③♦ s♥♥r rr♦r

s s s ♥ t ♣trs ♥ t r♦t ♦ ♥ ts ♦♥t♦♥s r ♥♦♥♦♥srts ♠♣s tt t r♦t rt s ♥♦t ♦♥st♥t ♥ t♠

♣tr

❯ ❯♥♦r♥ t P❩

❯♥rst② ♥ ♥ ♠s

rsts ♦♥t♥ ♥ ts ♣tr r rt t♦ ♠s r♦♥ t ♣rtr

♣♦st♦♥ t♠♣rtr ♥♠② 250 C r ♦♣ ♥♦ ♣r♦r t♦

st t ❯♥rst② ss ❯ ♦ ♥ r♦t ♦♥ssts ♥ t♦ ♣r♦r♠

s ♥stt♦♥ ♦ t ♥trs tt♦♥s ♦s♥ ♦♥ t ②♥♠ ♦

s♣r strtrs ♦s ♥t s rtrst rt t t♦ ♣ ♦ s♥

♥②ss ♦♥ t rsts ♦♠♥ r♦♠ t ♦ s♥ ♥ ♥② t♦ ♥

♥♦r t♦ ♦♥r♠ t ❯ ♦ t r♦t ② s♥ ♥rs strt♦♥s s

s♥ ♦ tr♦ ts s♠ r t♦ ♥ t ❯ ♦ t♥ ♠s

♥ t ♠♥t♠ tt ♦♥ s ♥ ①♣r♠♥t② ♠♦♥strt t ♥rst② ♦

P❩d=2+1 strt♦♥s

♠♥ttt ♦r♣♦♦ ♥②ss

t rst t s ♠♣♦rt♥t t♦ ♦sr ♦ t ♠♦r♣♦♦② ♦s s ♥t♦♥ ♦

t r♦t t♠ ♥ s♠q♥ttt s♦♥ r s♦s 10 × 10 µm

♠s ♦r t♠s ♠s ♣t r t♦s ttr r♣rs♥t

t②♣ ♦r ♦r r♦♥s s♥♥

t ♥t t♠s rs ♥ t sr s ♦♠♥t ② r

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

r ♠s ❬10×10 µm t s ♥ ♥♥♦♠trs❪ ♦r t♥ ♠s r♦♥t 250 C ♥trs r♦♥ ② ♥ ♠♥ rs♣t② ②♣ r♥s s♣ ♣rs♥t t t ♥tr ♦r t ♠♥ s♦ ♥s ♥ t ♠♥s ♥s

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

♥♠r ♦ r♥s t ♥ sr♣ s♣ ♣rs♥t r② s♣t

rt♦ Ω ♥ s t rt♦ t♥ t r②stt t ♥ t rtrst

♥t ♦ ts s t r r♥ t ζ ♦② ♦♥ ♦srs ζ ≪ 0.5

µm ♦r t = 15 ♥ 30 ♠♥ st t rt r rss r♦♠ 80 nm t♦ 60 nm

♦♥sq♥t② Ω s♠s t♦ rs ♥ ts ♥tr t t ssq♥t t♠ ♦♥ ♥

s sr ♦♠♣♦s ② r t rr r♥s ζ ≈ 0.5 µm srr♦♥ ②

r♦ ♦ s♠r sr♣ r♥s t s ♥♦t t t♦ s tt Ω s rs ♦♥

♥ ♠r② ♦r t st t♦ srs s♦ ♥ t rs ♥ ♦♥

♥♦ts t ♣rs♥ ♦ r♥s t rr ss ♥ ♣rtr ζ . 0.8 µm ♥ ♥

s♠♦♦tr t♦♣ ♦♥trst♥ t t♦s ♣r♦s ♦♥ r♥s ♦♥sr♥ tt

t sr st ♣rs♥ts r♦♥ ♦ s♠ r②stts t s♣t rt♦ ♥ t

♥tr ♦ 120 − 240 ♠♥ ♣s ♥r t♦ ♦♥st♥t ♦♥ tt r♦♥ s

r♦♥ ♣ ♦t ♥ t ♥ t ♥♦tr ♠♣♦rt♥t ♥ st♥s tr

r♥ srs r♦♠ s t qt② ♦ ♠t♣ r♥s t t

♥tr ♠s ♣♦♥t ♦t tt ♦s♥ ♣r♦sss ② ts

strtrs t st ♦r t♠s rr t♥ t = 60 ♠♥ ♥ t ♦♥ ♦♠♣rs

t②♣ r♥s ♣r♦s t t sr ♦r t = 60 ♠♥ ♥ t = 240 ♠♥ t s r tt

s♥♣sr♣♦♥ strtrs ♦ ♦r rr ♠t♣ ♠♦♥s t

ttr t♦♣ ♥ ♥ s♠r s♣t rt♦

♦r♥ ♦s tss rsts ♥ ♥rst♦♦ s ♦♦s t♦ t r

♠s♠t ♦ tt ♣r♠trs t♥ ♥ ≈ 20.0% ❬❪ t r♦t ♦

♦♥ r② ♣r♦s ♣t①② ①♣t ♥ ♣rtr ♦♥t♦♥s r♣♦rt ②

rrr t ♦♥t♦ sstrts ❬❪ trs ②rs r s② ♣♦②

r②st♥ ♣rs♥t♥ str♦♥ ❬❪ t①tr ❬❪ ♥ ♦♥r♠ ts

② ♣r♦r♠♥ strtr ♥②ss ♦ t ♠s ❯s♥

❱ ❯ ❳② rt♦♠tr ♥ t θ − 2θ ♦♣ ♠♦ rt♦♥

λCuKα= 0.154056 nm t ❳ s♣tr ♦ ♠s r t s s♦♥ ♥ t

♣rs♥ ♦ ♠♦r t♥ ♦♥ r②st♦r♣ ♦r♥tt♦♥ s s♥t ♦♥t♦♥ t♦

ss♥ t ♣♦②r②st♥ ♥tr ♦ ♠ rtss s t ♣♦st♦♥ ♣r♦s t

rtr ♦♥sr tt t r♥ s sr♣ s♣ ♥r ❬♥♠❪ × ❬µ♠❪ s s t②♣ s ♦r ♥②③♥ t r♥② ♠♦r♣♦♦② ❬❪

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

20 30 40 50 60

2θ (°)

102

103

104

105

106

I (a

.u.)

15 min

120 min

240 min

0 60 120 180 240

t (min)

0.88

0.92

0.96

1.00

Pro

b. (1

11)

250 ºC

111

311

400

331

220

r ❳② s♣tr ♦ ♣♦②r②st♥ ②rs s♥ λCuK−α ♣tr♠ ♦r②rs r♦♥ ② ♠♥ r ♠♥ ♥ ♠♥ t T = 250 C ♥♥♦ts t ♣rs♥ ♦ r♥s t sr r②st♦r♣ ♦r♥tt♦♥s ♥ rr♥ t♦ tsstrt ♥♦r♠ ♥♠② ♥ ♦r s t t♠ ♦st ♣ ♦♠s r ♥st s♦s t ♣r♦t② ♦ ♥♥ r♥ ♦s ❬❪rt♦♥ ♦♥s t t ♥♦r♠ t♦r ♦ t sstrt sr

♣ ♦♠s r t t s♠ t♠ tt t ♦tr ♦♥s r r

♣r♦t② ♦ ♥♥ r②stt ❬❪ ♦r♥t tr t ♠♥ ♦ r♦t ♣ t s

♥ s

p(111, t) ≡ [I(t)111/Aθ−2θ(θ111)]∑

hkl I(t)hkl/Aθ−2θ(θhkl)

r Ihkl,t s t ♥t♥st② ♦ t ♣ t t t♠ t ♥ Aθ−2θ(θhkl) s t

s♦r♣t♦♥ t♦r ♦r t θ − 2θ ♦♠tr② ♣♥♥t ♦ t θhkl ♥ ❬❪

♥ t ♥st ♦ t ♦♥ s♦s t ♣r♦t② ♦ ♥♥ r♥

♥ t ♠ s ♦♥ ♥ ♦♥r♠ r② str♦♥ ❬❪ t①tr ♥ ②rs r♦♥

♦♥ sstrts t T = 250 C r ♦r r② r♦t t♠s r②sts

r② ♦♠♣♦s ♠♦st 90% ♦ t t ②r

♥ rr t♦ ♥t r♠ ♦ r♦t ♣r♦s sts s♦♥ tt t

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

r♦t ♦ ♦♥ tr♠♥t srs ♦♦s t ❱♦♠r❲r ❱❲ r♦t

♠♦ r tr♠♥s♦♥ s♥s ♥t rt② ♦♥t♦ t sr t♦t ♥

♥t tt♥ ②r ❬❪ s rst s ♥ ♦♥ ♥ t r♦t ♦♥ ♦t

❬❪ ♥ s ❬❪ ② s♥ s♦ ♥♦r r ♠♦r ①

r♥ t ①♣r♠♥t t♦ t ❱❲ r♦t ♠♦ ♦♥ r♥s r ♦r♠

♦♥t♦ t t♠♣t ♦ s♠♦♥♦②r s♥s ♥ r♦ ② ♥rs♥ tr s♣t rt♦

s♥ t rt ♦♥tt t t sstrt s ♥♦r ♥rt② Prts ♦ t

r♠♥s ①♣♦s ♥t t s♥s strt ♦s♥ t♦ ♦r♠ ♦♥t♥♦s ②r s t

♥r♦♥♠♥t s ♣♦②r②st♥ r♥ ♦♥rs ♦ ♦ ♥♦r♥ r♥s

r ♥♦r r♦♥s ♦r ♣♦st♦♥ ♥ s♦♥ ♦ ♣rts s♥ ♥ ts ♣s

♦♥ ♥s r ♥♠r ♦ ts ❬❪ ts r ② ♠♥② t

strss ♦t♦♥ ♥② sts ❬❪ s♦♥ tt t strss s ♦♠♣rss ♣r♦r

t♦ ♦s♥ ♦♠s t♥s r♥ t ♦s♥ s ♦♥sq♥ ♦ ttrt

♦rs t♥ ♥♦r♥ r♥s ♥ rs ♦♥st♥t s ♣♦st♦♥ ♣r♦s

♥ ♥ t ♣rs♥ ♦ t str♦♥ t①tr t sr s ♦ t♦ r②sts s♦

② ts s t r♥s s② r♥t r♦tt♦♥ ♦r♥tt♦♥s s

♥ ♦ t ss sr strss t♦rq t♥s t♦ r♦r♥t t r②stts ❬❪

♦r ♦♥ r♦t t♠s ♠t②r r♠ t sstrt♠ ♥trt♦♥ s

①♣t t♦ ♦st ❬❪ ♥ t r♦t ②♥♠ ♦♠s tt ② t r♦t ♦

♦♥ t ♦♠♦♣t① r♦t ❬❪ s t sstrt t♠♣rtr

s r♥ t r♦t t rt s♦♥ s ①♣t t♦ ❬❪ s t

sts t ♥ r♦♥ t s ♦ ♦ ♥♦r♥ r♥s r ♥t② ♦r

② s♦♥ ♥ ♣♦st♦♥ ♦ ♣rts ♥ ♦ r①t♦♥ ♣r♦ss ♦r♥ t ♥

r♦♥ ts r♦♥s ♦s ♦s♥ ♣r♦sss ♦♠♥ ♠♦r ♦♣rt

s ♦♥sq♥ ♦ ts r①t♦♥ t s♣t rt♦ Ω ♦ r♥s rs ♥ t♠ ♥

t t♦♣ ♦ t♠ ♦♠s s♠♦♦tr t♦ t ♥ ♣r♦ss ♦ ♣rts t ♦

♠♥♠ s ♦rr♦♦rt ② t ♠s rs

♥ t ♣♣♥① st♦♥ s♦ st♦♥ ♥ t ♣♣♥①

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

♦ tt♦♥s ♦♥❯♥rs ♥ ❯♥rs

♥ ①♣♦♥♥ts

♦ tr♥ ♦r tt♥t♦♥ t♦ ♦ tt♦♥s t t ♥tr r

s♦s t ♦ r♦♥ss ♥ ♥ t st♦♥ t ♦r srs r♦♥ ②

r♥t t♠s

10-1

100

101

l (µm)

100

101

wlo

c (nm

)

15 min

30 min

60 min

120 min

240 min

101

102

t (min)

10-1

⟨(∇

h)2

⟩

(a)

101

102

t (min)

0.6

0.7

0.8

0.9

α1

(b)

r ♦ r♦♥ss s♥ ♦r t♥ ♠s r♦♥ t T = 250 C ② ♠♥t tr♥s ♠♥ ♣ tr♥s ♠♥ ♠♦♥s ♠♥ sqrs ♥ ♠♥rs ♥st s♦s t r ♦ ♥♥t♦♥ ♥ s♦s ♥ ts t ♥♦♥♥rs♦♠tr s♥ ①♣♦♥♥t s ♥t♦♥ ♦ t r♦t t♠ ♦ ♥s ♥ t ♠♥ ♣♦t ♦ r♣rs♥t t ts s t♦ ①trt α1 ♥

t s♦rt ♥t ss l . 10−1 µm ♥ r② t♠s ♦♥ ♥ ♦sr wloc(l, t)

rs♥ ♥ t♠ s ♥ ts stt♦♥ wloc ♠srs t ♥trr♥ r♦♥ss

♦ r♥s ♦♥ r♦♠ sr♣ t♦ r♦♥ s♣ r♦♠ rr t♦ s♠r t

tt♦♥s ♦r ♦♠♣rs♦♥ ts s t ♦♣♣♦st ♦rr♥ ♥ s②st♠s tt ①ts

♥♦♠♦s s♥ s ♥ s♥ ♥ t ♣♣♥① st♦♥ ♥ ♥r t ♣rs♥

♦r s♥ ♦ ♥♦♠♦s r♦♥♥ ♥ t ② t♥ t ♦t♦♥ ♦

sqr s♦♣s t t sr 〈(∇h)2〉 s ①♣t t♦ s s ❬❪

〈(∇h)2〉 ∼ t2κ.

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

P♦st s ♦ κ ♥t ♥♦♠♦s s♥ ♥t ♦r ♥ s

♦♥r♠ tt t ♠②❱s s♥ q srs t ②♥♠ ♦ ♦ t

tt♦♥s ❬❪

♥st ♦ t s♦s t ♦r ♦ 〈(∇h)2〉 ♥ t♠ s t

s♠ ♦ wloc ♦r ① ♦① s③s l∗ t l∗ . 10−1 µm ♥ ♣rtr ♦♥ ♥ ♥♦t tt

♥ ♥t rs♥ r♠ s ♦♦ ② strt ♦♥ ♣♦r q ♥

ts tr♥s♥t r♠ ♣r♦s κ = −0.15(5) s s r② ♦s t♦ tt ♣rt

② ó♣③ ❬❪ ♥♠② κ = −1/6 ♦r t♦♠♥s♦♥ P❩ s②st♠s ♦r♦r t

s ♥♦♥ tt wloc(l∗, t) ∼ Ω ❬❪ ♥ ts rsts r ♥ r♠♥t t

♦r ♣r♦s ♥②ss s ♦♥ ♠s ♦r♦r t r♦ss♦r ♥ wloc ♦rs ♥

l ≈ ζ ♦♠♣r t ♥ ♦r♥ t ♥♠r rsts ❬❪ t t ♦tr

♥ ♦♥♥t tt♦♥s l ≫ 10−6 µm ♥rs ♥ t♠ s ①♣t ❬❪

s♥ ①♣♦♥♥t α1 ♥ s w(l, t) ∼ lα1 s ts ♥♥ ♥ t♠

r♦♠ t♦ rs♣t② s s♦♥ ♥ t s s ♦r s♦

♥♦t t♥ s r♣rs♥tt ♦ ♥rs tt♦♥s t t ♥tr s ♦♥ ♥

♣r♦s sts ❬❪ tr t♥ s ①♣♥ ② r ♥ s ❬❪ s

st♦♥ α1 s r rt t♦ t r♥ ♠♦r♣♦♦② α1 ≈ 0.6 ♥ts tt

t r♥② ♠♦r♣♦♦② s sr♣♦♥ ♦r♠ ♥ s rr s t α1 s♠♦♦tr

s t t♦♣ ♦ t②♣ r♥s s♣ ♦t tt ts ♥♥s r ♥ r♠♥t t

t ♠s tr ♥rs r♦♥ss ①♣♦♥♥t α ♦r s♦

♦♥ ♥ r♦♥ r ζ ≪ l ≪ ξ ❬❪

♥ ♦rr t♦ ♥rt t ②♥♠ ①♣♦♥♥t s t s♦♣s♦♣ ♦r

rt♦♥ ♥t♦♥ ♥ ♥ t q ♥ sss ♥ t st♦♥ r

s♦s ♥♦r♠③ Γ(l) ♥t♦♥s t ♦r t♠s st r♦♠ t rst ③r♦

♥♦r r♦♠ t rst ♠♥♠♠ rm ♦ ♥♦r♠③ Γ(l) rs ♦♥ ♥ ♠sr t

♦rs♥♥ ①♣♦♥♥t ncoar ♥ s rm ∼ tncoar ❬❪ ♦r s♦rt r♦t t♠s

♦r t ♣♣r♥ ♦ ♠t♣ strtrs t sr ♦♥ s tt ncoar = 1/z

s sst ② ♠s ts r♠ ♥ ♦r ①♣r♠♥t stt♦♥ ♦rs ♦r

t . 60 ♠♥ ♥ ♣r♦ ncoar = 1/z = 0.62(2) s t s ♥ ♥ t ♥st ♦ t

s s ♥ ①♥t r♠♥t t tt ①♣t ♦r t♦♠♥s♦♥

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

0 0.2 0.4 0.6 0.8l (µm)

-0.2

0

0.2

0.4

0.6

0.8

1

Γ(l

)/Γ

(0)

15 min30 min60 min120 min240 min

101

102

t (min)

10-1

r m (µ

m)

First minimun

1/z =

0.6

2(2)

r ♦r♠③ s♦♣s♦♣ ♦rrt♦♥ ♥t♦♥ ♦r t♥ ♠s r♦♥ t T =250 C ② ♠♥ t tr♥s ♠♥ ♣ tr♥s ♠♥ ♠♦♥s ♠♥ sqrs♥ ♠♥ rs ♥st s♦s t rst ♠♥♠♠ rs ①trt r♦♠ Γ(l)/Γ(0) rss ♥t♦♥ ♦ r♦t t♠

P❩ s②st♠s ❬❪ s q ♥ tr♥ t♥ ♦♥st r♦t

t♠s ♦♥ ♦t♥s ncoar ≈ 0.34 s♦ ♥♦t ♥tr♣rt s r♣rs♥tt ♦

t♠♣♦r ♥rs tt♦♥s s r ♣s ♦ rs♦♥ s③ ♣♣r♥ ♦♥

t t♦♣ ♦ ♠♦♥ ♥ ♥ ♥rst♠t ♦r t r r♥ s③ ♠sr ②

t s♦♣s♦♣ ♥t♦♥ q s r ♠sr♠♥t ♦r rm s ♥♦t

♥ ts stt♦♥ ♥ ♥ ♦♥ ♥ ♥♦t t t rt♦♥ ncoar = 1/z

♦ ♥

♦♦♥ t ♠②❱s ♥sät③ q t ♦ r♦♥ss s s

tβ ♥ t ♦♥ ♥ s t w ♣♦t ♥ s ♣♦r t ①♣♦♥♥t

β = 0.24(4) s s r② str♦♥ ♥ ♦ P❩d=2+1 r♦t ♦♥ βKPZ ≈ 0.24

s q

t ts ♣♦♥t ♦♥ 1/z = 0.62(2) β = 0.24(4) ♥ κ = −0.15(5) ss r str♦♥ ♥ts ♦ t♦♠♥s♦♥ P❩ r♦t ♥ ♠s ts

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

101

102

t (min)

101

wglo

b (n

m)

250 ºC

β = 0.24(4)

r ♦ r♦♥ss s ♥t♦♥ ♦ t♠ ♦r t♥ ♠s r♦♥ t T = 250 C r♦♥ss r♦s s ♣♦r t β = 0.24(4)

s r② tr s♦ ♦t♥ ♥rs strt♦♥s ♦ ts ♦ r♦♥ss ♥

♠①♠ ts ♠t♥ t tt ♦♥s ♥♠r② ♣rt ♦r t P❩ ss s

t ♥ st♦♥s ♥

s r ♥ t ♥♠r ♦ r♦t t♠s r ♥♦t

t♦ ♥ t s②♠♣t♦t ♦t② v∞ ♦ t sr r♦t ❬❪ ②♦♥ tt

② ♦♦♥ t r♥ t♦♦♦① ❬❪ t ♦ ♥tr tt♦♥s

♦♥ ♥ t♠ ♥t r♥ t stt♦♥r② r♠ ♦r r♥t ♥t ♠st ♥s

♦ sstrts s♦ ss s r ♥♦t ♥ ♣♦st♦♥ t♦ t

t ts t♠ t t♦♠♥s♦♥ P❩ ♥sät③ q ♦r ♥ ♣r♦r♠

♦♠♣rs♦♥ t♥ rs t strt♦♥s ♥ t strt♦♥ ♦ χ t

s♣t r ♦♥ ♦t ss ♦ t q ♥t♥ t ♦rrt♦♥ tr♠s ♦♥ s

〈h〉 = v∞t+ sign(λ)(Γt)β〈χ〉.

♥ ♥ ♥s t tr♠ v∞t strt♥ q r♦♠ q rst s

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

h− 〈h〉 = sign(λ)(Γt)β[χ− 〈χ〉].

♦ t s② t♦ s♦ tt 〈h2〉c = (Γt)2β[〈χ2〉 − 〈χ〉2] ♥ ♥ q

② t ♦ r♦♥ss w = σh = 〈h2〉1/2c ♦♥ ♥s

(h− 〈h〉)σh

= sign(λ)(χ− 〈χ〉)

σχ,

r σχ 〈χ2〉1/2c

qt♦♥ ♠♥s tt t♦ ♦♠♣r rs t strt♦♥s t ♠♥

♥ ♥ ♥tr② r♥ s q♥t t♦ ♦♠♣r♥ t χ tt♦♥s rs ♥

t s♠ ② s ♦s s t♦ ♦♠♣r ts strt♦♥s ♥ t♦t ♥♦♥ t

♥♦♥♥rs P❩ ♣r♠trs r ♣ts t rs t strt♦♥s ♦r

0 5[h - <h>]/σ

h

10-4

10-2

100

σh p

(h)

Gumbel, m = 6

-4 -2 0 2 4

[h - <h>]/σh

0

0.1

0.2

0.3

0.4

0.5

σh p

(h)

0 60 120 180 240

t (min)

1

2

3SK(B)

(A)

r s t strt♦♥s t ♠♥ ♥ ♥ ♥tr② r♥ r♦♠ srr♦♥s s②♠♦s ♦ t tst s♠♣ r♦♥ t T = 250 C ♦♠♣r t t ♥♠rP❩ r s♦ ♠♥t ♥ ♥st ts t r② ♦♦ ♦♣s t♥ ①♣r♠♥t ♥ ♥♠r t ♦s t♦ t ♣ ♥ ♥r × ♦ ♣♦t ♥srt♦♥ ①tst ①♣r♠♥t ♥ ♦♥ ♥ t♠ ♦ ♥ s ♠♥t ♥s rrs t♦ t♥♠r② ①♣t P❩d=2+1 ♥ s rs♣t②

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

sr r♦♥s ♦ t tst s♠♣ ♠r② ♦♥ ♥♦ts r② ♦♦ r♠♥t

t♥ t ①♣r♠♥t t ♥ t ♠ ♣ t ♠ s q ♦r ♦r

s r♦♥ t ♣ s st♦♥s rst s r♥♦r ② t t ♦♣s ♦s

t♦ t ♣ s t ♥ t t ♥srt♦♥ ♦ t st t♦♥

♦sr ♥ t t t ♦ t♦ t t tt t♣ ♦s ♥♦t s♥ rt②

♣ ②s s s t ♦s ♦r r ts

♦♠♣r♥ t ♥rs P❩d=2+1 ♠♥s♦♥ss ♠♥t rt♦s ♦r t t

s t ♦♥ ♦r t tst ♠ St=240 = 0.34(1) ♥ Kt=240 =

0.3(1) ♥♠r s r S = 0.423(7) ♥ K = 0.344(9) ❬❪ ♦r

s s♦♥ ♥ t ♥st ♦ t t ①♣r♠♥t S ♥ K s ♣♣r♦ t♦

t P❩d=2+1 ♦♥s ♦♥② ♦r t = 240 ♠♥ ♥ ♣r♥♣ ts rsts ♦ st

tt♦♥ ♠♥ t r♠♥t ♦ s ♥ t ♠r ♦♥♥ ♥

ts s ♥♦t t s r s ♦r t♦ s t ♥rs strt♦♥ rt

①♣♦♥♥ts r② ♣♦♥t ♦t t♦ P❩ r♦t ♦r♦r t s ♥♦ tt

①♣r♠♥t S ♥ K s t ♦♥ t♠ t♦ ♦♥r t♦ t s②♠♣t♦t s

t ♥♠r② ❬❪ s t s♦ str♦♥r ♥ ♠♥s♦♥

s②st♠s ♥ t ♥rst♠t S ♥ ♦♥② ♥tt♠ t ♥ rsts

♠r♥ r♦♠ t ♣r♦ t rst s♦t ①♣r♠♥t ♦♥r♠t♦♥ ♦ t

P❩d=2+1 ♥rst② ②♦♥ t ①♣♦♥♥ts

♥ ♦rr t♦ ♣r♦ t ♥rst② ♦ strt♦♥s sr ♥ t st♦♥s

♥ ♥ s♦ t♦ ♥ rtr ♥ ♦ P❩ r♦t t t

♥ strt♦♥s ♦♥str♥ t♦ t ♦① s③ l t♥ t ♥tr [10µm/1024]≪l ≪ ξ r s♦s t s ♦r r♦t t♠s rs ♥

t rs♣t S ♥ K s s ♥t♦♥ ♦ l r ♣t st♦♥s♥② r②

♦♦ ♦♣s s s♥ ♥ ♠♦st ♦r s r♦♥ t ♣ ♠♦♥ t ①♣r♠♥t

♥ ♥♠r P❩ s ♦r♦r t strt ①♣♦♥♥t ② t t rt

t s ♣rs♥t ♥ ♦♥ ♠♦r str♦♥ ♥ ♦ P❩ r♦t

♠r rsts r ♦♥ ♦r t s rs ♥ r t ♥

♦♣s t♥ t ①♣r♠♥t ♥ ♥♠r t s ♥t rs s♦

♥ t ♠♥ ♣♦t r t♦s ♦s S ♥ K s ttr r♣rs♥t t ♠♥ ♦r t l

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

0 5[w

2 - <w

2>]/σ

w2

10-4

10-2

100

σw

2 p

(w2)

SLRD KPZ

15 min

30 min

60 min

240 min

-2 0 2 4 6

[w2 - <w

2>]/σ

w2

0

0.3

0.6

σw

2

p(w

2)

l15−60 min

= 0.342 µm

l240 min

= 0.469 µm

(a)

2

4

SKewness S

KPZ

3 6 9

1/l (µm-1)

5

10

15

20

25

Kurtosis

KKPZ

(b)

r s sqr ♦ r♦♥ss strt♦♥s ♦r s♠♣s r♦♥ t T =250 C ② ♠♥ ♦r♥ t tr♥s ♠♥ ♣ tr♥s ♠♥ r♥ ♠♦♥s♥ ♠♥ rs rs s♦♥ r t♦s ♦s t ♦① s③ s ♥t ♦t ♥ ② r r♥t ♦r t s t ♣♣r♦♣rt ♥tr ζ ≪ l≪ ξ ♣♥s♦♥ t ♥st s♦s t ♦♦ ♦♣s r♦♥ t ♣ ♦r t t♥♥st ♥ tst ♠r♦♥ ♥ss ♥ rt♦ss s s ♥t♦♥ ♦ t ♦① s③ l

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

0 5

[m - <m>]/σm

10-4

10-2

100

σm p

(m)

MRHD KPZ

15 min

30 min

60 min

240 min

-2 0 2 4

[m - <m>]/σm

0

0.1

0.2

0.3

0.4

0.5

σm p

(m)

l15-60min

= 0.195 µm

l240min

= 0.342 µm

(a)

0.8

1.2

1.6

SKewness S

KPZ

5 10 15

1/l (µm-1)

0

2

4

Kurtosis

KKPZ

(b)

r s ♠①♠ rt t strt♦♥s ♦r s♠♣s r♦♥ t T =250 C ② ♠♥ ♦r♥ t tr♥s ♠♥ ♣ tr♥s ♠♥ r♥ ♠♦♥s♥ ♠♥ rs rs s♦♥ r t♦s ♦s t ♦① s③ s ♥t ♦t ♥ ② r r♥t ♦r t s t ♣♣r♦♣rt ♥tr ζ ≪ l≪ ξ ♣♥s♦♥ t ♥st s♦s t ♦♦ ♦♣s r♦♥ t ♣ ♦r t t♥♥st ♥ tst ♠r♦♥ ♥ss ♥ rt♦ss s s ♥t♦♥ ♦ t ♦① s③ l

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

sr ♥ t r♣

t s r② ♠♣♦rt♥t ♠♥t♦♥ ② t S ♥ K s ♥ t l s

♣♣♥s s ♦♥ ♦s ♥♦t ♥♦ ♣r♦r t r t l′s sts②♥ t ♦♥t♦♥

ζ ≪ l ≪ ξ r♠♠r tr s rtrst ♥t t t sr s ♥ t

s②st♠ ζ ≈ ξ t ♣♣r♦♣rt ♥tr ♦r l s r② s♦rt ♥ s♦

♥t s ♦♥r♥ ♦ S ♥ K ♦r s♦♠ s♣ ♥ t s s♦♥

♥ t rs ♥ ts ♦♥r♥t s r t t♦s ♥♠r②

t ♦r t ♥ ♦ P❩d=2+1 ♠♦s ❬❪

♦r♥ ♦ t P❩ ♠♥s♠ ♥ ♦♥

s♦♥s

rsts ♣rs♥t ♥ t ♣r♦s st♦♥s ♣♦♥t ♦t tt t ♦♥♥t

sr tt♦♥s t st ♦r ♠ r♦♥ t T = 250 C ♥ F ≈ 2.2 s

♦ ♦r♥ t♦ t P❩ qt♦♥ q ♦r♥ ♦ t P❩ s♥ ♥

ts ①♣r♠♥t s②st♠ ♥ ♥rst♦♦ ♥♥ t ♦♠♣① ②♥♠ ♦ r♥s

s tt ② ♥ ♥tr♣② t♥ ♥trr♥ sr ♥rts ♦♥str♥ts

② ② t ♦s♥ ♦ ♥♦r♥ r♥s ♥ ♥ ♣r♦sss ♦r♥ t ♥

r♦♥ t s ♦ ♦ ♥♦r♥ r♥s s s t♦ ♦♠♣① ♣♥ ♦

r②st♥ r♥s s♠♣ strt♦♥ s ♣r♦ ② t r♥ ♣♦st♦♥ ♠♦ ♥

t ❬❪ ♥ t ♠♦ ♥ r♥ s r♠② tt t♦ t ♦♥r② ♦

t r♥s ♦ t t ♦s ♥♦t s♣ ♥ tr ♥♦r♦♦ s

ts s♣ s ♦♥str♥ t♦ s rt♦♥ ♠♥s♠ s t s♠ t

♦ t tr rt♦♥ s ♥ t st ♠♦ ❬❪ t ♥rts ①ss ♦t②

s t ♥♠r ♦ P❩ s♥

♥ ♦♥s♦♥ ♦♥ t rst r♦st ①♣r♠♥t KPZd=2+1 s②st♠

♥ t ss ♦ s♠q♥ttt ♠♦r♣♦♦ ♥ ♦♥♥t tt♦♥

sts ①trt rt ①♣♦♥♥ts r♥ t tt ①♣t ♦r t P❩

ss ♦r♦r t ♥rst② ♦ P❩ strt♦♥s s s ♥ s s

♦♥ ②♦♥ ♦ t st♥r ♦♠♣rs♦♥ t ①♣♦♥♥ts

❯♥♦r♥ t P❩ ♥rst② ♥ t♥ ♠s

♥ ①♣r♠♥t② ♠♦♥strt ❲ ♥♦t t P❩d=2+1 ♥sät③ q

rt② t♦ ♥r♥t t♦♠♥s♦♥ ①♣r♠♥t ♦sts r rsts r

♣s ♥ t P②s ❬❪

♦rt② tr t ♣t♦♥ ♦ ♦r ♦r ❬❪ ♣♥② ♥ Ps

♥t③s ❬❪ s ts ♥♦ s♠ ♣r♦♣♦s ② s t♦ ♦♥r♠ t P❩ ♥rst②

♥ ♦♦♠r t♥ ♠s s r② ♦rr♦♦rts tt ♥ ♥ ♣rs♣t

♦r ♥♥ ♥ ♦♥r♠♥ t♦♠♥s♦♥ r♦♥ srs ♦♥♥ t♦ t P❩

♥rst② ss ❬❪

♣tr

❯ t ♦ ♠♣rtr

♦♥ r♦t ②♥♠

♥ ts ♣tr ♦♥ sts t t ♦ t ♣♦st♦♥ t♠♣rtr ♥ t r♥

♦ ❬ ❪ C ♦♥ t ♠♦♥ ♦t♦♥ ♥ ♦♥ t ♦♥♥t tt♦♥s ♦

srs r♦♥ ♦♥ sstrts

♥trr♥ ♠♦r♣♦♦② ♥ ♦ tt♦♥s

r s♦s t②♣ ♠s ♦r srs r♦♥ t r♥t T ♦r t

t♦ rst r♦t t♠ t = 120 ♠♥ ♥ 240 ♠♥ ♦r T = 150 C

♥ ♦♥ ♥ s r♥s t ♥ sr♣ s♣ ♦♠♥t♥ t

sr t t s♠ t♠ tt Ω s♠s t♦ ♥rs t t s sst ② t rt

r ♦r ♠s r♦♥ t T = 200 C t s♥r♦ s qt s♠r t♦ tt ♦r T = 150 C

①♣t ♦r t rst r♦t t♠ r Ω s rs ♥ t ♥tr

♦ t ∈ ❬ ❪♠♥ ❯♥ t ♠♦r♣♦♦② ♦ srs r♦♥ t T = 250 C

♠t♣ strtrs r rr ♥ t = 240 ♠♥ ♥ r♥s t ♥ s♣

st r t ♠♦rt② ♥ t②♣ ♠♦♥ ♣r♦s r s♦♥ ♥

♦rr t♦ ♠ r t♦s ♥♦ts ♥ t ♦tr ♥ t r t♠♣rtrs ♥♠②

T = 300 C t srs ♣rs♥t ♠ ♠♦r ♦♠♣① strtrs s ♥ ♥♦t

r♦♠ t t rs ♥ ♥ t ts ♠s r ♦♠♣♦s ② ♠① ♦

♦♥ r♥s ♥ r ♠♦♥s ♦r♠ ② ♦s r♥s t ζ ≈ 2 µm s t

♠♣rtr t ♦♥ r♦t ②♥♠

t♠ ♦s ζ ♥rss str t♥ t ♠♦♥ t t st t♥ t ♥tr t ∈❬ ❪♠♥ t♦ ♦r♠ rr ♠♦♥s

r ♠s 10× 10 µm t s ♥ nm ♦ t♥ ♠s r♦♥ t T 150 C ♥ T 200 C ♥ ♥ T 300 C ♥ ② t = 120 ♠♥♥ 240 ♠♥ rs♣t②

♦r♥tt♦♥ ♦ r♥s ♦s r②st♦r♣ ♣♥s r ♣r t♦ t s

strt sr s tr♠♥ ② θ − 2θ ❳ ♠sr♠♥ts s r② ♦♥ ♦r

♠s r♦♥ t T = 250 C str♦♥ ❬❪ t①tr s ♥ ♦sr s♦ ♦r ♦tr

t♠♣rtrs ♦r♦r t s♠ ♣s ♥ s♥ ♥ t ❳ s♣tr r

❳ s♣tr♠ s ♥♦t s♦♥ s t s s♠r t♦ tt r♣rs♥t ♥ t

♠♣rtr t ♦♥ r♦t ②♥♠

r ②♣ r♥♠♦♥ ♣r♦s t t sr ♦r T = 200 C ♥ T =300 C ♦r r♥t r♦t t♠s

0 60 120 180 240

t (min)

0.9

0.95

1

Pro

b. (1

11)

150ºC

200ºC

300ºC

r Pr♦t② ♦ ♥♥ r♥s t rs♣t t♦ t sstrt ♥♦r♠ ♥ ②r r♦♥ t T = 150 C rs 200 C sqrs ♥ 300 C ♠♦♥s

s♦s t ♣r♦t② ♦ ♥♥ r♥s q ♥ t ②r s ♥t♦♥

♦ t r♦t t♠ ♥ ♦r r♥t t♠♣rtrs rsts sst tt ts t①

tr s st② ♥♥ ② t ♣♦st♦♥ t♠♣rtr ♥ ts st t♦♥

♣r♦② t♦ t t tt s r s T rr s t t ♦ r♥s ❬❪ ♥

♥ t♦s r♦♥ ♥ r♥t ♦r♥t♦♥s rts ♠♦r t ❳r② ♥♥t

♥ ♥ ♠♥s p(111, t, T ) ♥♣♥♥t ♦ t sstrt ♦♥ s♠r ❬❪

t①tr s ♥ ♦♥ ② r♦ t ❬❪ ♥ rrr t ❬❪ ♥ t r♦t

♠♣rtr t ♦♥ r♦t ②♥♠

♦ ♦♥ ss sstrts ♥ s♦ ② ♣♦r♥ t s♥ sstrts ❬❪

♥②② t ♠♣♦rt♥t ♦r t r♦t ②♥♠ s tt ♠♦st ♦s♥ ♣r♦ss ♣

♣♥s t♥ r♥s ♥ ♥ ts ② t② rs r ♥♠r ♦ ts

t t s ♦ ♦ ♥♦r♥ r♥s t♦ s② t r♥ t♥ t

r♦tt♦♥ ♦r♥tt♦♥ ♦ ♥♦r♥ r②stts

10-1

100

101

l (µm)

100

101

wloc

(nm

)

15 min

30 min

60 min

120 min

240 min

(a)

10-1

100

101

l (µm)

100

101

wlo

c (nm

)10

110

2

t (min)

10-2

100

⟨(∇

h)2

⟩

150ºC

200ºC

300ºC

(b)

10-1

100

101

l (µm)

100

101

wlo

c (nm

)

0 60 120 180 240

t (min)

0.4

0.6

0.8

1

α1

(c)

r ♦ r♦♥ss ♦r ♠s r♦♥ t T = 150 C 200 C ♥ 300 C ② ♠♥ t tr♥s ♠♥ ♣ tr♥s ♠♥ ♠♦♥s ♠♥ sqrs ♥ ♠♥ rs ♥srt♦♥ ♦ s♦s t sqr ♦ ♥♥t♦♥s s ♥t♦♥ ♦ t r♦tt♠ ♦r T = 150 C rs T = 200 C sqrs T = 300 C ♠♦♥s s♠ ♥s s ♥ t ♥st ♦ t r ♣ts t ♦♠tr α1 ①♣♦♥♥t s ♥t♦♥♦ t ①trt r♦♠ t s♦ ♥s s♦♥ ♥ t ♠♥ ♣♦ts

♥ rr t♦ sr tt♦♥s t ♦ ss ♣rs♥ts t ♦ r♦

♥ss t s ♥t♦♥ ♦ t ♥ T rst ♥♦t t♦ t ♦♦ t s tt ♥ t

♠♣rtr t ♦♥ r♦t ②♥♠

t rs s♦t t♦ ♣ s t t♠ ♦s ♥ ♦t s♦rt l . 10−1 µm

♥ r♥ts l ≫ 10−1 µm ♥ ♣♣r♥t② ♥♦ ♦♥ r①t♦♥ s ♦sr

♥ ts wloc ♥rs♥ t s♦rt♥t ss s t ♥♠r ♦ t ♥♦♠♦s

r♦♥♥ sss ♥ t ♣♣♥① st♦♥ r t rst st rr ♥♠r

♦ rt ①♣♦♥♥ts ♠st t♦ ♦♥ ♦r tr♠♥♥ t ❯ κ ①♣♦♥♥t ♥

♥ t qt♦♥s ♥ s ♥ t ♦r t ♠s r♦♥ t T = 150 C

s ♥srt♦♥ ♦ t ♦t♥ s κ = 0.5(1) t♦ ts s

♣♦st ♥t♥ ♥♦♠♦s s♥ ts ♣rtr s ♦♥sst♥t t tt ♦♥

♦t♥ r♦♠ ♥ ♥♦rrt ♥tr r t sqr♦s♦♣ tt♦♥s ♥

rs ♥ t♠ t ♥t ♣♦r ♥ ♥♦tr ♦rs t r♥♦♠ r♦t s ♥tr♥s②

♥♦♠♦s ♥ ts s♥s

♦r srs r♦♥ t T = 200 C t stt♦♥ s ♠♦r ♦♠♣① t♥ tt ♦r

t ♦st T s s♦♥ ♥ t r ♦♥ ♥ s tt t r ♦r t = 240

♠♥ s s♦t ♦♥ t s♦rt♥t ss ♥st ♦ s♦t♥ ♣ s ♥ t

♥t tr♥ ❱r② ♥trst♥ ts s ♥ t s♠ ♦r ♣rs♥t ② Ω s

s ♥ t s ♥s♣t♦♥ ♦ ♠s sqr ♦ s♦♣s

s♦ s♦ ♦r ♦ t♦ r♠s s ♥st ♦ t r ♥ ♥t

♥♦♠♦s r♠ rtr③ ② κ = 0.19(5) s ♦♦ ② s♦♥ ♦♥ ♥

t ♠②❱s s♥ s s②♠♣t♦t② r♦r ❬❪ t κ ≈ −0.7♦r t ♣♦st♦♥ t♠♣rtr T = 300 C t s♠ qtt rst ♦ t♦

r♠s s ♦sr s tr♥s♥t ♥♦♠♦s s♥ s ♣ t♦

♥♦r♠ ♦♥ t t . 60 ♠♥ s ♦rr♦♦rt ② t ♦ r♦♥ss t s♦rt♥t

ss ♥ ② 〈(∇h)2〉 s ♥t♦♥ ♦ t s ♦r κ rtr③♥ ts t♦ r♠s

r rs♣t② 0.2(2) ♥ −0.56(3) ♦t tt t rst ♥ ♥♦t r♥t

t ♣rs♥ ♦ ♥♦♠② s♥ κ = 0 s ♣♦ss t♥ t rr♦r r

♦ rr♥ t♦ t ♦♠tr s♥ ①♣♦♥♥t α1 ♥ s wloc ∼ lα1

st♦♥ ts s ♥t♦♥ ♦ t r♦t t♠ s ♣t ♥ t ♥st ♦ t

♦r r♥t t♠♣rtrs α1 s ♦s t♦ 0.6 ♦r 150 C ♥ts

t ♣rs♥ ♦ r② sr♣ r♥s t sr t t♠s ❬❪ ♦r♥ t t

s ♥s♣t♦♥ ♦ ♠s ♥ tr♥ ♦r T = 200 C ♥ 300 C α1 ♥s

♠♣rtr t ♦♥ r♦t ②♥♠

r♦♠ 0.6 t♦ 0.8 ♥ r♦♠ t♦ rs♣t② s rs tt sr♣ r♥s

♣ t♦ s♠♦♦tr strtrs t ♦♥ t♠s s rsts r t♦t② ♦♥sst♥t

t t rs ♥ s♠r ♦r ♦r α1 ♦rs ♦r ① t s T

♥rss s ♦♥ ts rsts ♦♥ ♥ s♦ ♦♥ tt t ♦r ♦ wloc t

s♦rt♥t ss l ≪ 10−1 µm s ♦r♥ ② t♦ ♦♥trt♦♥s t rst s t

r♥ s♣ rt t♦ α1 ♥ ♣rs♥t r sr♣ r♥s ♦r s♠ t t♦♣

r♥s tt♦♥s s♦♥ ♦♥ ♦♠s r♦♠ t s♣t rt♦ s♥ ♦r ③ ♦①

s③ l∗ ♦♥ s wloc(l∗, t) ∼ 〈(∇h)2〉 ❬❪

♦r♥ ♦ t rsts ♦ ♥ ♥rst♦♦ s ♦♦s ♥t②

♠s ♦ ♦r♥ t♦ t ❱♦♠r❲r r♦t ♠♦ ❬❪ ♥ tr t

♦♠s rr s r s t ♣♦st♦♥ t♠♣rtr ❬❪ s s t

r♥s t ♥ s♣ ♥r tr② t② ♦ ♦r♠♥ ♦♥t♥♦s ♠

♥ ♥ ♦r♥ t♦ r♥s ♦♥rs s r ♥♦♥r②st♥ ts r♦♥s

♠r ❬❪ s s r♦♠ ② ♦ ♥♦r♥ r♥s r ♦r♠ ♠♥②

t t ♥tr t♥ r♥s ♥ r♥t r♦tt♦♥ ♦r♥tt♦♥s s③

♦ ts t r♦♥s s rr s r s T s♥ r r t r♥ ♣r♠trs

r♠♠r tt ♥t② ζ ♥rss t T ❬❪ t ts s ♥ ♥ t♦♥

♥r② rrr EGB rss t♥s t♦ r♣ t s♦♥ ♥ t ♣♦st♦♥ ♦

♣rts t ♥ ♥r t♦ ts sts ❬❪ s♠r rrr s s♦ ♥ sst

r♥t② ♥ t r♦t ♦ ♠s ❬❪ ♦r ♦ T r sr s♦♥ s

s♦ s♠ ♥♠r ♦ ♠♦s ♥ ♦r♦♠ t EGB rrr ♥ ♠♦st ♦ t♠

rts ♥s t r♥ tt t② rr s ♦♠♣s t r♥ t t♦

♥rs str t♥ ts t ♥ Ω ♥ 〈(∇h)2〉 t♦ ♥rs s t t♠ ♦s

s s ♥ ♥st ♦ ♦r t ♦♥ t♠s r①t♦♥

♣♣♥s s ♣rts r ♥t② ♣♦st t ♥ r♦♥ t s ♦ ♦

♥♦r♥ r♥s s ♠♥ss t ♥♠r ♦ t sts ♥ t t ③♦♥ ♥

♦♥sq♥t② ♥♥s s♦♥ t♦rs ts r♦♥s r♦♠ ts ♠♦♠♥t ♦s♥

♣r♦sss ♦♠ ♠♦r ♦♣rt ♥ r♥s t ♥ s♣ ♣ t♦ r

♠♦♥s s♦ tt Ω ♥ 〈(∇h)2〉 rs ♥ t♠ ♥ ♣rtr ♦r T = 150 C ts

r♠ s ♥♦t ♦sr t♦ t ①♣r♠♥t ♦♥t♦♥ ♦ ♥tt♠ r♦t ♦r

♠♣rtr t ♦♥ r♦t ②♥♠

r T ♦r t s♦♥ r♠ s r② ♥♦t ♥ ♣♣rs s♦ rr s r

s T s ♥ ♥ ♥sts ♦ t s ♥ ♥ ♥ t ♥st ♦

♥t ♦♥t r♦ ♦

rt② ♦ t ♦ rs♦♥♥ s strt ♥ ♥ ♦♥♠♥s♦♥ t♦♠

st r♦t ♠♦ ♥ ♦r ♥trst s t ♦s♥ ♣r♦ss t r♦t strts ♦♥

♣r♦ rr② ♦ ♣②r♠ r♥s t t s♠ t ζ ♥ t H ♦r s♠♣t②

♥t t♠ s st t♦ ♦rrs♣♦♥ t♦ t ♣♦st♦♥ ♦ ♦♥ ♠♦♥♦②r ♦ ♣rts

r♥ ♣♦st♦♥ ♥t st s r♥♦♠② st ♥ ♣rt s ♦

s♥ t sr strt♥ r♦♠ t st ♥t ♥♥ st j r t rts

♣r♠♥♥t② tr t ♦♥str♥t |hj − hj±1| ≤ 1 s sts s ♥s t r

trst r♥ rt♦♥ ♦♦s t ♦♥srt rstrt s♦♦♥s♦

r ❬❪ t t s ♦ ♦ ♥♦r♥ r♥s ♦r ♥ ♥r② rrr EGB

s ♣rs♥t s♦ tt ♣rt ss t♦r t♠ t ♣r♦t② PD = e−EGB/kBT

♥ ♣rt rts t ♥ i t rrr EGB t i ♦♠s ♥ t

♣r♦t② PR = e−ER/kBT ♦t tt ts s♠♣ ♠♥s♠ ♣trs t ♥r②♥

tr ♦ t r①t♦♥ ♣r♦ss t t s ♦ ♦ ♥♦r♥ r♥s

r s♦s t②♣ sr ♦t♦♥s ♦r T = 150 ♥ 200 C ♥ s

♦♥sr EGB = 0.10 ❱ ER = 0.30 ❱ ζ = 64 ♥H = 8 ♥ ♦r T = 150

♥ 300 C rs♣t② ♦r T = 150 C ♦♥ ♦srs r♥s t ♠♦st ① t

♥ ♥rs♥ t s♠r ♦r s ♦sr t s♦rt t♠s ♦r T = 200 C

t ♦r r t r ♠♦♥s ♦r♠ ② ♦s r♥s ♣♣rs s♠ ♦rs

♦r rr T s qtt r♠♥t t t ①♣r♠♥t s ♦rr♦♦rt ② t

♦t♦♥ ♦ t sqr ♦ s♦♣s s♣② ♥ t ♦♠♣r♥ ts

rsts t t ①♣r♠♥t ♦♥s ♥st ♦ ♦♥ ♥ ♦♥r♠ tt t

♥tr♣② t♥ t r①t♦♥ ♣r♦ss t t s ♥ ♥t ♦♥t♦♥s ❱♦♠r

❲r r♦t ♠♦ ♥ t rr ♦r r T ♥ t ①♣♥s t

♠♦♥ ♦t♦♥ rtss t s ♠♣♦rt♥t ♠♥t♦♥ tt ts s♠♣ ♠♦ ♦s

♥♦t r♣r♦ ♥ ♦ ♥♦t ts ♥t♥t♦♥ t ♦♠♣① ②♥♠ t♥ ♣ ♥ t

♠♣rtr t ♦♥ r♦t ②♥♠

0 128 256 384 512x

0

20

40

60

0

20

40

60

h -

m

(a)

100

101

102

t (min)

⟨(∇

h)2

⟩

150 ºC200 ºC300 ºC

(b)

r sts r♦♠ ♥ ♦♥♠♥s♦♥ ♥t ♦♥t r♦ ♠♦ ♥ ♦♥ ♣tst t ♣r♦s ♦r T = 150 C t♦♣ ♥ 200 C ♦tt♦♠ ♦r t = 10, 100 ♥ ♥st ② m = 10, 80 ♥ rs♣t② s ♥s r♣rs♥t t ♥t t s ♦♦ ♥♦r♥ r♥s rsqr♦ s♦♣ 〈(∇h)2〉 rss t♠ ♦r srsr♦♥ t T = 150 C rs T = 200 C sqrs ♥ T = 300 C ♠♦♥s

r♦t ♦ ♠s t t s ♥sts ♦t ♥r②♥ trs

r t ♦s♥ ♣r♦ss

❯♥rs ①♣♦♥♥ts

♦ ♦♥ ♦t ♥ ①♣♥t♦♥ ♦r t s ♦♥ ♦♥ t ♦ ss ♥

t ♣♦st♦♥ t♠♣rtr s ♥ r ♣r♣r t♦ tr♥ ♦r tt♥t♦♥ t♦ t

♥②ss ♦ ♦♥♥t tt♦♥s ♥ t♦ t ❯♥rst② ss ❯ ♦ t

r♦t

r s♦s t ♦ r♦♥ss s ♥t♦♥ ♦ t r♦t t♠ ♦r

r♥t t♠♣rtrs t rst ♦♥ ♥♦ts tt w(T ) ♥rss t T ♥ t s

t ②r s ♣♦②r②st♥ t ♦♠♣① ♦♠♣tt♦♥ t♥ r♥s s rs

t♦ ♥trr♥ tt♦♥s tt r rr s r s t t♠♣rtr s t

r♥s t♠ss r rr ♥ t ②s s♣rt♥ ts strtrs r ♣r

s r s T s ♦sr ♥ t ♠r rsts ♥ ♦♥ ♥ t r♦t

♦ r♦♥ ♦♥ ss sstrts ♦r ② ♦r♥ ♦♣ t t♥ ♦① ❬❪

r♥ t♦ t r♦t ①♣♦♥♥t ♦r T = 150 C ♦♥ ♦t♥s β = 0.51(4)

♠♣rtr t ♦♥ r♦t ②♥♠

101

102

t (min)

100

101

102

wg

lob

(nm

)

150ºC

200ºC

300ºC

100 200 300

T (ºC)

0.2

0.4

β

βKPZ

(a)

~t0.5

0 0.1 0.2 0.3 0.4 0.5

l (µm)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Γ(l

)/Γ

(0)

15 min

30 min

60 min

120 min

240 min

101

102

t (min)

101

102

r m (n

m)

150ºC

200ºC

300ºC

(b)

r ♦ r♦♥ss s ♥t♦♥ ♦ t r♦t t♠ ♦r r♥t t♠♣rtrs♥♠② T = 150 C rs T = 200 C sqrs ♥ T = 300 C ♠♦♥s ♦ ♥sr t♦ ②s r♦♠ t β ①♣♦♥♥t s ♥ ♦♥ ♥st ♦ s♦s ♦ tr♦t ①♣♦♥♥t ♣♥s ♦♥ T s♦ ♥ ♠rs t s ♦♥sst♥tt KPZd=2+1 ❬❪ ♦♣♦♣ ♦rrt♦♥ ♥t♦♥ ♦r t♥ ♠s r♦♥ tT = 200 C ♦st♦r② ♦r s ♦sr ♦r t♠♣rtrs ♥ t r s♦♥ s t②♣ ♦♥ ♦r t♠♣rtrs ♥②③ ♥srt♦♥ ♥ ♣ts t rst ③r♦ rm♦ t ♦♣♦♣ ♦rrt♦♥ ♥t♦♥ s ♥t♦♥ ♦ t♠ ♦r sr T ′s ♥ s ts♠ ①t ♥ t ♥st ♦ s ♥ s♦ ♥s r t♦ t ②s r♦♠ rt ncoar ♥ 1/z ①♣♦♥♥ts ♥ tr♠♥ ①♣♥ ♥ t t①t

♠♣rtr t ♦♥ r♦t ②♥♠

s ♥st ♦ ♥ ss♦t♦♥ t κ = 0.5(1) s str♦♥ ♥

tt ♦♥ s P♦ss♦♥♥ r♦t t ts ♦ t♠♣rtr ♦r T = 200 C ♦r

♦♥ ♥ ♥♦t tt ♦r ♥t t♠s t . 60 ♠♥ ♦ r♦♥ss ♥rss t

βinitial ≈ 0.5 t ts r♠ s ♦♦ ② st♥t ♦♥ r r② ♦♥ ♥s

t♥♥② ♦ β ♦♠♥ s♠r ♦♠♣r ♦r C ♥ C

s♣t ts t ♦♥ ♥s β = 0.41(5) ♦r T = 200 C t s ♦rt ♠♥t♦♥ tt

ts ♦s ♥♦t ♠t t ♥②♦♥ ♥♦♥ ❯ ❬❪ ♥ ♣rtr ts

s ♥ ♦♥ ♥ ♦♠♦♣t① r♦t ♦ t ❬❪ ♦ ♥

t ❬❪ ♥ s♦ ♥ t r♦t ♦ ♦♥ ss sstrts ② ❲

t T = 250 C ❬❪

♥② ♦r ♠s r♦♥ t T = 300 C ①♣r♠♥t rsts ♣♦♥t ♦t β =

0.21(5) ♦s t♦ t P❩ ♦♥ q ❬❪ ♥ s♦ t♦ t ♥ ❱

sss s st♦♥ ♥ ♣♣♥① st♦♥ rs♣t② ♥ s♦♥

s♦ ♦♠♥t t r♦t t r② r T ♦r♦♠♥ ♦tr ♠♥s♠s ♦r

t ♦♥② ts ♥ ♥ ♥♦t tr♠♥ r♦t qt♦♥ srs t

sr tt♦♥s ♦ t ♥tr

r s♦s t s♦♣s♦♣ ♦rrt♦♥ ♥t♦♥ q ♦r t♥

♠s r♦♥ t T = 200 C s♠ ♦st♦r② ♦r s ♦♥ ♦r s♠♣s r♦♥

t T = 150 C ♥ 300 C ♥♦t s♦♥ r s sss ♥ t st♦♥ r♦♠

t rst ♠♥♠♠ ♦r r♦♠ rst ③r♦ ♦ Γ(l, t) ♦♥ ♥ ①trt t r ♠♦♥ s③

rm ss s rm ∼ tncoar ❬❪ ♥ ♥ ♣♣r♦♣rt r♥ ♦ t♠ ♦♥

s ncoar = 1/z ♥srt♦♥ ♥ t s♦s t rst ③r♦ rm s ♥t♦♥ ♦

t r♦t t♠ ♦r r♥t t♠♣rtrs ♦r T = 150 C ♠t♣ strtrs

♦ ♥♦t ♣♣r t t sr ♥ t qt② ncoar = 1/z ♠st ♦r t

♥ ♥s 1/z ≈ 0.07 s ♦♥ ♠♦r ♣r♦ tt t r♦t s ♥♦rrt t

ts T ❬❪ ♦r T = 200 C t♦ r♠s ♥ rs♦♥② s♥ t② r ♥♥t

s s ♥ s♦ ♥s ♥ t ♥st ♦ r♦♠ t rst r♠ ♦♥ s

1/z ≈ −0.02 ♥ r♦♠ t s♦♥ ♦♥ ♦t♥s 1/z = 0.6(1) rst s ♦♥sst♥t

t r♥♦♠ r♦t ❬❪ st t s♦♥ ♦♥ r t KPZd=2+1 t♥ t rr♦r

r q ❬❪ ♥② ♦r T = 300 C t♦ s♦rt ♥ ♦♥t♠ r♠s r

♠♣rtr t ♦♥ r♦t ②♥♠

s♦ ♦sr ♥ ♣r♦ ncoar = 0.32(5) ♥ ncoar = 0.7(1) rs♣t② rst

r♠ s ♦♥sst♥t t t ③ ♦r t ❱ ss ❬❪ ♥ t s♦♥ s ♥r

♦ t ①♣t ③ ♦r t P❩ ss ♦t s r t ♥♠r ♦♥s

t♥ t rr♦r r rtss t ts t♠♣rtr ♠st r ♥ ss♠

ncoar = 1/z ♦♥ ♠t♣ strtrs ♣♣r s♥ ♥t r♦t t♠s ♦r t♠s

t . 60 ♠♥ ♦ ♣s ♦♥ t t♦♣ ♦ t ♠♦♥s s♦ ♦♥trt t♦ ♥rst♠t

t r s③ ♦ t strtr ♦r t r r♦t t♠s ♦ ♣s ♦♠

s♦rtr ♦♠♣r t t ss ♦ r② r ♠♦♥s t t sr s

♥ ♥ t② s♦ ♥♦t str♦♥② ♥♥ ♥ t ♠sr♠♥t ♦ rm s t

♠♦st r t♦♥ s♦ st ncoar = 1/z ♦♥② ♦r ♦♥ r♦t t♠s t st ♦r

♠s r♦♥ t ♣♦st♦♥ t♠♣rtrs s T = 300 C

Prt ♦♥srt♦♥s

t ts ♣♦♥t ♣r♦r♠ s♥ ♥②ss ♦ sr t

t♦♥s s s♦ ♥ ♦ trs ♦ t r♦t r♦♠ ♦ r♦♥ss rs tt

tr♥s♥t ♥♦♠♦s s♥ ♥ r♦ss♦r ts r♥ t ②♥♠ ♥

rt t♦ t ♠r♥ ♦ ♥ ♥r② rrr t t s ♦ ♦ ♥♦r♥ r♥s

♥ t♦ t r①t♦♥ ♣r♦ss ♥t ♦♥t r♦ ♠♦ s s♣♣♦rt ♦r r

s♦♥♥s r♣r♦♥ qtt② t ①♣r♠♥t rsts r♦♠ t ♦ r♦♥ss

rs t s ♥♦t ♣♦ss t♦ ♥rt t α ①♣♦♥♥t s t s♦♥ ♥rs

r♠ s ♥♦t ♥ ♦sr ♦r t s ♦♥ ♦r α1 r ♦♥sst♥t t t

♣rt♦♥s ♥ t rs ❬❪ s ♦rr♦♦rt ② t ♠s s t ♥

t sq♥ ♦♥ s ♦♥ β ♥ 1/z s ♥t♦♥ ♦ t ♣♦st♦♥ t♠♣rtr ♦r

T = 150 C rsts ♥t P♦ss♦♥ r♦t ♦r T = 200 C r♦ss♦r r♦♠

♥♦rrtt♦P❩ r♦t s♠s t♦ ♦r β = 0.41(5) ♦s ♥♦t

♠t t ♥② ♥♦♥ ❯ r♥♦rs ts ♦r t st t♠♣rtr st

♦♥ s β = 0.21(5) ♥ 1/z = 0.7(1) β s ♦s t♦ t ❱ ♥

P❩ sss ❬❪ ♥ ♣r♥ts r st♥t♦♥ ♦ r♦t qt♦♥ srs

t sr tt♦♥s t ts t♠♣rtr ♦r t ♦♥ ♦r 1/z t

♠♣rtr t ♦♥ r♦t ②♥♠

♦♥ r♦t t♠s ♣♦♥ts t♦ P❩ r♦t ♥ t ts r② ♣♦st♦♥ t♠♣r

tr ♦r♦r strss tt s ♦♥② ♦♥ rsts ♦♠♥ r♦♠ ①♣♦♥♥ts ♦♥ ♥

♥♦t ♠ r t ❯ t tt♦♥s ♦ srs t 200 C ♥ 300 C

♦♥

s ♦ ①♣♦♥♥ts ♦♥ ♥t r ♥ s♥ ♥ t ts

♥

t ♠♥ α1(T = 150 C) α1(T = 200 C) α1(T = 250 C) α1(T = 300 C)

❱s ♦r t ♥♦♥♥rs ①♣♦♥♥t α1(t, T ) ♦♠♥ r♦♠ srs r♦♥♦♥ sstrts ② ❲ t F ≈ 2.2 s ♥ ♣♦st♦♥ t♠♣rtrs t 150 C200 C 250 C ♥ 300 C

T ( C) κ ≈ −0.7 ≈ −0.05

ncoar ≈ 0.07

❱s ♦r t ♥♦♥♥rs ①♣♦♥♥ts κ(t, T ) ♥ ncoar(t, T ) ♦♠♥ r♦♠ srs r♦♥ ♦♥ sstrts ② ❲ t F ≈ 2.2 s ♥ ♣♦st♦♥ t♠♣rtrst 150 C 200 C 250 C ♥ 300 C ❱s s♣rt ② t s②♠♦ / r ♥ ♣♣r♦♣rt r♥ ♦ t♠ s r♥s ♦ t♠ r ♥t s s♦s ♥s ♥ t♥srt♦♥ ♦ ♦r T = 150 ♥ 300 C ♥ s ss♦ ♥s ♥ t ♥st ♦ ♦r T = 250 C

T ( C) β 1/z ≈ 0.07 0.7(1)

❱s ♦r t ♥rs ①♣♦♥♥ts β(T ) ♥ 1/z(T ) ♦r sr tt♦♥s ♦ r♦♥ ♦♥ sstrts ② ❲ t F ≈ 2.2 s t♥ t r♥ ♦ 15 t♦ 240♠♥ ♥ ♦r sr ♣♦st♦♥ t♠♣rtrs

❯♥rs strt♦♥s

♥ ts st♦♥ s♣♣♠♥t ♦r ♣r♦s sts t ♣ ♥②ss ♦

t sqr r♦♥ss ♦ ♥ ♠①♠ rt t strt♦♥s s s

♠♣rtr t ♦♥ r♦t ②♥♠

♠♦♥strt ts ♥②ss rtr t♥ ♦♠♣♠♥tr② ♦♥ s ♥ s♦♠ ss ss♥

t ♦r ♥♥ t ❯♥rst② ss ♦ ♥ r♦t

sr tt♦♥s t T = 150 C P♦ss♦♥♥

r♦t

r s♦s t rs t strt♦♥s ♦r r♥t r♦t t♠s

♦r srs r♦♥ t T = 150 C s r ♦♠♣r t♦ t ss♥

strt♦♥ ①♣t ♦r ♥♦rrt r♦♥ srs ♥ ♥♦ts tt ①♣r♠♥t

t ♦♣s r② t t ss♥ ♥r ♦ t ♣ ♥st ♦ t

s s t t ts ♥ t st ♦r s r♦♥ t ♣

0 5

[h - <h>]/σh

10-4

10-2

100

σh p

(h)

Gaussian

15 min

30 min

60 min

120 min

240 min

-2 0 2

[h - <h>]/σh

0

0.2

0.4

σh p

(h)

0 60 120 180 240

t (min)

-0.2

0

0.2

0.4

SK

(A)

(B)

r s t strt♦♥s t ♠♥ ♥ ♥ ♥tr② r♥ ♦r r♥tr♦t t♠s s②♠♦s r♦♠ srs r♦♥ t T = 150 C ①♣r♠♥t tr ♦♠♣r t♦ t ss♥ s♦ ②♥ ♥ ♥st ts t r② ♦♦ ♦♣st♥ ①♣r♠♥t t ♥ t ss♥ ♦s t♦ t ♣ ♥ ♥r × ♦ ♣♦t ♥srt♦♥ ①ts t ①♣r♠♥t ♥ ♦♥ ♥ t♠ ♦s ②♥ ♥ ♠♥t ♥srrs t♦ t s ♦r t ss♥ ♥ P❩ rs♣t②

①♣r♠♥t s ♦r t ♥ss S ♥ rt♦ss K r s♣② ♥ t

♥srt♦♥ ♦ t r tr s s♠ t♦ tt r♦♥ ③r♦ s ①

♣t r♦♠ ♥trs ♥ ss♥ ♥ t♦ ts rst t♦s ♦♠♥ r♦♠

♠♣rtr t ♦♥ r♦t ②♥♠

t s♥ r♦♥ss ♥②ss s t s s t♦s r♦♠ t ♦ ②♥♠

♦♥ ♥ ♦♥ tt t ts ♦ t♠♣rtr t tt♦♥s ♦

♠s r P♦ss♦♥♥ ❬❪

1

2

3

SKewness

15 min

30 min

60 min

120 min

240 min

5 10 15 20 25

1/l (µm-1)

3

6

9

12

Kurtosis

(a)

0.5

1

1.5

SKewness

5 10 15 20 25 30

1/l (µm-1)

0

0.5

1

1.5

2

2.5

Kurtosis

(b)

r S ♥ K s ♦r s ♥ s r♦♠ srs r♦♥t T = 150 C ② ♠♥ ♦r♥ t tr♥s ♠♥ ♣ tr♥s ♠♥ r♥♠♦♥s ♠♥ r sqrs ♥ ♠♥ rs ♦ ♦♥ ♦♥r♥ ♦r ♥ rs r ♦sr s ①♣t ♦r ♥♦rrt r♦♥ srs

S ♥K s s ♥t♦♥ ♦ t ♦① s③ l ♦r s ♥ s s♦ ♦

♥ ♦r t♦ ♥♦rrt r♦t s s ♥ s sss ♥ t st♦♥s

♥ ts ♦① strt♦♥s ♠st t ♥ t r♥ ♦ ζ ≪ l ≪ ξ

❲♥ l stss ts ♠t S ♥ K s ♦♠ ♥♣♥♥t ♦ l ♥ ♦♥r t♦

♥rs s ❬❪ ♦r ♥ t sr s ♥♦rrt ξ ♣s

r② s♠ ♥ ♦ t s♠ ♦rr ♦ ζ ♥ tr r r♥s t t sr s

♥♦ ♦♥ ♦♥r♥ ♦ S ♥ K s♦ ♦sr s ♥ ♦♥r♠ ♥ t rs

♥

♥♦♠t♦P❩ r♦ss♦r ♥ r tt♦♥s

t T = 200 C

s ♦r ♠s r♦♥ t T = 200 C ♥ s♥ ♥ t ❱r②

♥trst♥ ♦♥ ♥♦ts tt t strt♦♥s t ♥t r♦t t♠s r r② r r♦♠

t P❩ ♦♥ ♥st t② r ♦sr t♦ ss♥ s s ♦♥r♠ ② S(t) ♥

♠♣rtr t ♦♥ r♦t ②♥♠

-6 -4 -2 0 2 4 6

[h - <h>]/σh

10-6

10-4

10-2

100

σh p

(h)

Gaussian

KPZ

15 min

30 min

60 min

120 min

240 min

240 min

-2 0 2

[h - <h>]/σh

0

0.2

0.4

σh p

(h)

0 60 120 180 240

t (min)

-0.4

0

0.4

0.8

1.2

SK

(A)

(B)

r s t strt♦♥s t ♠♥ ♥ ♥ ♥tr② r♥ ♦r r♥tr♦t t♠s s②♠♦s r♦♠ srs r♦♥ t T = 200 C ①♣r♠♥t t r♦♠♣r t♦ t ss♥ s♦ ②♥ ♥ ♥ t♦ t ♥♠r P❩ r s♦ ♠♥t♥ ♥st ts t r② ♦♦ ♦♣s t♥ ①♣r♠♥t ♥ ♥♠r t♦s t♦ t ♣ ♥ ♥r × ♦ ♣♦t ♥srt♦♥ ①ts t ①♣r♠♥t S ♥ K♦♥ ♥ t♠ ♦s ②♥ ♥s rrs t♦ t ss♥ s rs t♠♥t ♥ ♥ts t P❩ S/K s

K(t) r ♦sr t♦ ③r♦ t♥ t♦ t P❩ s s ♥st ♦ t t

♦♥ t♠s ♦r S ♥K s s♠ t♦ ♦♥r t♦ t P❩ ♦♥s ♥ r② ♦♦

r♠♥t t t ♥rs P❩ s ♦sr s sr♠♥tr♠♥t t

P❩ s r♥♦r t♥ ♥t♦ ♦♥t t ♦♣s ♥r ♦ t ♣ s ♥st

♦ t ♦r srs r♦♥ t t = 15 ♠♥ ♥ t = 240 ♠♥ rs♣t②

s♣t ♦ r rr♦r rs rt t♦ ♥ s st♠ r♦♠ ♦

sttsts t r♠♥t t♥ t ♦♥st r♦t t♠ ♥ t ①♣t

P❩ s s t s r♠r ♥♠② t=240 = 0.43(5) ♥ t=240 = 0.5(2)

rs P❩ s r S = 0.42(1) ♥ K = 0.34(2) s ♦♥ ♦r ♣r♦s ♦

♥ r♦♥♥ s♥ st② ♥ r tt t s ♦ s♦ r♦ss♦r

♠♣rtr t ♦♥ r♦t ②♥♠

♥ t♠ t♦r t P❩ r♠ ♥ ♣rtr ts ♥ ♦ r♦ss♦r s ②

st ♥♠r② s ❬❪ ♥ r tr♥ ♥ ts ①♣r♠♥t s r s

♥♦ s t rst ♥ ♥ ①♣r♠♥t ♥ ♦ s r♥♦♠t♦P❩ r♦t ♥

d = 2 + 1

0 5 10[w

2 - <w

2>]/σ

w2

10-6

10-4

10-2

100

σw

2 p

(w2)

SLRD KPZ

15 min

30 min

60 min

120 min

240 min

-2 0 2 4 6

[w2 - <w

2>]/σ

w2

0

0.3

0.6

σw

2

p(w

2)

l60-120 min

= 0.146 µm

l15-30 min

= 0.117 µm

l240 min

= 0.195 µm

(a)

2

3

SKewness S

KPZ

5 10 15 20 25

1/l (µm-1)

0

4

8

12

16

Kurtosis

KKPZ

(b)

r s sqr ♦ r♦♥ss strt♦♥s ♦r s♠♣s r♦♥ t T =200 C ② ♠♥ ♦r♥ t tr♥s ♠♥ ♣ tr♥s ♠♥ r♥ ♠♦♥s ♠♥ r sqrs ♥ ♠♥ rs rs s♦♥ r t♦s ♦s t♦① s③ s ♥t ♦ t ♥ ② r r♥t ♦r t s t ♣♣r♦♣rt♥tr ζ ≪ l ≪ ξ ♣♥s ♦♥ t ♥st s♦s t ♣♦♦r♥ ♦♣s r♦♥ t ♣♦r t t♥♥sttst ♠ r♦♥ ♥ss ♥ rt♦ss s s ♥t♦♥ ♦ t ♦①s③ l

s ♥ s s♦ s♣♣♦rt ts ♥♥s s ①t ♥ t rs

♥ t ♥t t♠s s rs ♦ ♥♦t ♣rs♥t t strt ①♣♦♥♥t

② t t rt t s ♥♦♥ t♦ P❩ ♥♠r ❬❪ ♦r♦r S ♥

K s ♦r ts t♠s ♦ ♥♦t ♦♥r t♦ t P❩ ♦♥ s t ♦① s③ s ♥rs

s t ♦♠♣rs♦♥ t♥ t ♦r srs r♦♥ t t = 15

♠♥ ♥ t P❩ r♦♥ t ♣ s ♠♦r ♥s ♦ s sr♣♥②

♥st ♦ rtss t stt♦♥ s t♦t② r♥t t ♦♥ t♠s r

t r♠♥t t♥ t ①♣r♠♥t t ♥ t t♦rt ♦♥ rs ♠♦r

t♥ ♦r s r♦♥ t ♣ t S ♥ K s ♦♥r♥ t♦ t P❩ s

s t ♦♥t♥♦s ♠t s ♣♣r♦ ♦♥str♥ t♦ ζ ≪ l ≪ ξ sts ♦♠♥ r♦♠

t s r ♥♦t s♦ rt s t♦s ♠r♥ r♦♠ s t t②

♠♣rtr t ♦♥ r♦t ②♥♠

0 5

[m - <m>]/σm

10-6

10-4

10-2

100

σm p

(m)

MRHD KPZ

15 min

30 min

60 min

120 min

240 min

-2 0 2 4 6

[m - <m>]/σm

0

0.2

0.4

σm p

(m)

l15-30 min

= 0.127 µm

l240 min

= 0.195 µm

l60-120 min

= 0.146 µm(a)

0.6

0.9

1.2

SKewness S

KPZ

5 10 15

1/l (µm-1)

0.5

1

1.5

Kurtosis

KKPZ

(b)

r s ♠①♠ rt t strt♦♥s ♦r s♠♣s r♦♥ t T =200 C ② ♠♥ ♦r♥ t tr♥s ♠♥ ♣ tr♥s ♠♥ r♥ ♠♦♥s ♠♥ r sqrs ♥ ♠♥ rs rs s♦♥ r t♦s ♦s t♦① s③ s ♥t ♥①t t♦ t ♥ ② r r♥t ♦r t s t ♣♣r♦♣rt♥tr ζ ≪ l ≪ ξ ♣♥s ♦♥ t ♥st s♦s t ♦♦ ♦♣s r♦♥ t ♣ ♦r tt♥♥st ♥ t tst ♠ r♦♥ ♥ss ♥ rt♦ss s s ♥t♦♥ ♦ t ♦①s③ l

s♦ ♦rr♦♦rt tt ♦r t♥♠s r♦♥ t T = 200 C ♥♦♠t♦P❩

r♦ss♦r ts ♣

P❩ r♦t t ♣♦st♦♥ s r

tt♦♥s t T = 300 C

♥② t ♥②ss ♦ ♠s r♦♥ t T = 300 C ♥♦s ts st♦♥ ♥ r♥

t♦ ♦t t r ❯ ♦ t♥ ♠s r♦♥ t t♠♣rtrs r ♣ts

t ①♣r♠♥t s ♥ ♦♠♣r t♦ ♥♠r P❩ ♥r qt♦♥

♥ ❱ ♥♦♥♥r qt♦♥ rs

st t♦ ♠♥t♦♥ t ♥rst② ♦r t s ♦ ts s♦♥

♦♠♥t sss ② ♥trt♥ t ♥r qt♦♥ s s s♠t♥ sr s

rt ♠♦s s ❬❪ ♥ r tr♥ tt ♦♥ t♦ t ♥ ❱ sss

s ♥ t♦♥ ♥♦t t P❩ ♣♦tt s tt ♦♥ ♦r λ < 0 s t s♠

♣♦tt ♥ t ♣r♦s rs t rt r♦♥ t ♦r♥ ♥②② ♦♥ ♥

♦sr tt s ♦r r② r♦t t♠s ♦ ♥♦t r t strt♦♥s ♦ ♥② ss

♠♣rtr t ♦♥ r♦t ②♥♠

-5 0 5[h - <h>]/σ

h

10-4

10-2

100

σh p

(h)

KPZ, λ < 0

MH

VLDS

15 min

30 min

60 min

120 min

240 min

-4 -2 0 2 4

[h - <h>]/σh

0

0.2

0.4

σh p

(h)

0 100 200 300

t (min)

0

1

2SK

(A)

(B)

r s t strt♦♥s t ♠♥ ♥ ♥ ♥tr② r♥ ♦r r♥tr♦t t♠s s②♠♦s r♦♠ srs r♦♥ t T = 300 C ①♣r♠♥t t r♦♠♣r t♦ t ♥♠r P❩ r s♦ ♠♥t ♥ ♦♥sr♥ λ < 0 t♦ t ♥rs s♦ ②♥ ♥ ♥ t♦ t ♥rs ❱ s r♥ ♥ ♥st ts t r② ♦♦ ♦♣s t♥ ①♣r♠♥t ♥ P❩ ♥♠r t ♦s t♦ t♣ ♥ ♥r × ♦ ♣♦t ♥srt♦♥ ①ts t ①♣r♠♥t ♥ ♦♥ ♥ t♠♦s ♥s rrs t♦ t ①♣t s

t ♦r ♦♥ t♠s ♦♥ ♥ rs♦♥ r♠♥t t t P❩ s ♦sr

t ts s♠ t♦ ♦♥r str t♥ t rt ♦♥s t♦ t P❩ r s s

♦r t > 30 ♠♥ s ♠♦♥strt ② t ♠s ♥ ♥ ②

t ♦ ①♣♦♥♥t α1 ♥st ♦ t t t♦♣ ♦ r♥s ♦♠ s♠♦♦tr s

t t♠ ♦s t s t rt t ♦ t s t♦ ♠♦ ♥ t t rt♦♥ ♥

♦♥sq♥t② t♦ ♣♣r♦ t♦ t P❩

❯♥ t ①♣♦♥♥ts t ①♣r♠♥t s sr t ♥ ❱ sss

s ♣♦sss ❯ ♦ t r♦t ♦r t rst t♠ t ♥st ♦ t

♣rs♥ts ♥ ♦♣s r♦♥ t ♣ t♥ t ①♣r♠♥t rst ♥

t ♥♠r ♦♥ ♦r♦r t s r s♥ tt S(t) ♥ K(t) s r ♣♣r♦♥

♠♣rtr t ♦♥ r♦t ②♥♠

0 5[w

2 - <w

2>]/σ

w2

10-4

10-2

100

σw

2 p

(w2)

SLRD KPZ

SLRD MH

15 min

30 min

60 min

120 min

240 min

0

[w2 - <w

2>]/σ

w2

0

0.3

0.6

σw

2

p(w

2)

(a)

l120-240 min

= 0.586 µm

l15-60 min

= 0.342 µm2

4

SKewness S

KPZ

SMBE-L

3 6 9

1/l (µm-1)

0

10

20

Kurtosis

KKPZ

KMBE-L

(b)

r s sqr ♦ r♦♥ss strt♦♥s ♦r s♠♣s r♦♥ t T =300 C ② ♠♥ ♦r♥ t tr♥s ♠♥ ♣ tr♥s ♠♥ r♥ ♠♦♥s ♠♥ r sqrs ♥ ♠♥ rs rs s♦♥ r t♦s ♦s t♦① s③ s ♥t ♦ t ♥ ② r r♥t ♦r t s t ♣♣r♦♣rt♥tr ζ ≪ l ≪ ξ ♣♥s ♦♥ t ♥st s♦s t ♦♦ ♦♣s r♦♥ t ♣ ♦r tt♥♥st ♥ t tst ♠ r♦♥ ♥ss ♥ rt♦ss s s ♥t♦♥ ♦ t ♦①s③ l

t♦ t P❩ ♦♥s t♥ t rr♦r rs ♥st ♦ t Prtr② ♦r t = 240

♠♥ ♦♥ s t=240 = −0.2(2) ♥ t=240 = 0.3(2) r t t P❩ s

t ♦r λ < 0 ♦♥sr♥ t r rr♦r rs

♦ ♥ rtr ♥ ♦ P❩ s♥ t s ♥ s ♥

t ♥ r ♣t ♥ t rs ♥ rs♣t② r♦♠ t t t

♥ r♦♠ t ♣ ♦ t s ♥ ts ♥st ♦♥ ♥♦ts tt ①♣r♠♥t

rsts r r② sr ② t P❩ r rs t ② t t rt t

♥ ♥♦t ♠ r st♥t♦♥ t♥ t P❩ ♥ t ss ♥ t

S s ♦♥r t♦ t P❩ ♦♥s ♥ t ♠♥t♠ tt K s s♦ ♥

♥♦t st♥s r♦♠ tt ①♣t ♦r t P❩ ♥ ♥ ts

stt♦♥ r t ①♣♦♥♥t ♥②ss ♣♦♥t t♦ P❩ ♦r t♦ r♦t t s

r ♥♦t st t♦ ♠ s♣rt♦♥ t♥ ts t♦ sss s s♦ ♥ t

t ①♣t S ♥ K s ♦r t P❩ ♥ t ss r r② ♦s

t♥ ♦tr ♥ r♦♠ t ①♣r♠♥t ♣♦♥t ♦ ♦♥ ♥ ♥♦t st t

❯ s ♦♥② ♦♥ ts rst

♠♣rtr t ♦♥ r♦t ②♥♠

0

1

2

3

SKewness S

KPZ

SMBE-L

5 10 15

1/l (µm-1)

0

2

4

6

Kurtosis

KKPZ

KMBE-L

r S ♥ K s ♥t♦♥ ♦ t ♦① s③ ♦r s ♦r s♠♣s r♦♥ t T =300 C ♦s ♥ ♥ts t ①♣t s ♦r t S ♥K ♦r t P❩ ♠♥t♥ ♥ ♦r t ss ②♥ ♥

s ♥♥s ♣ s t♦ s ♦ r s tr♠♥♥ t ❯ ♦ t t♦

♠♥s♦♥ ①♣r♠♥t ♠s ♥②♦ ② t ♦strtr ♦t♦♥

r♦♥♥ s♥ rsts & strt♦♥s ♥②ss ♥ ♦♥ tt t ♥

tr ♦r T = 300 C ♦s ♦r♥ t P❩ qt♦♥ q t λ < 0 s

♥t t ①st♥ ♦ ♠♥s♠ ♦ rs♥ t ♣♦st♦♥ ♦ ♣rts ♣♥♥

♦♥ t ♦ s♦♣s s ♦rs ♥ t ♠♦s ♠♦ ❬❪ ♣♦ss ①♣♥t♦♥

♦r λ < 0 s tt t st♥ ♦♥t s s♠r ♥ r♦♥s t r② r s♦♣s

t sr ♥ ♦♥ ♥♦ts ♥ t tt 〈(∇h)2〉 ♦r T = 300 C s r②

rr t♥ tt ♦r ♦r T s ♥ ①♣♥ ② ts t ♣♣rs ♦♥② t

T ❬❪

♥ ♠rs

t st② ♦♥ ♦ t ♣♦st♦♥ t♠♣rtr ts t r♦t

②♥♠ ♥ r♥ ♦ t♠♣rtr ❬ ❪ C s ♥ ♣r♦r♠ ♦

♠♣rtr t ♦♥ r♦t ②♥♠

♠♦♥ ♦t♦♥ s ♥ ①♣♥ ♥ tr♠s ♦ ♥ ♥tr♣② t♥ ♦ ♥r②

rrr EGB tt t♥s r♣♥ t s♦♥ ♥ t ♣♦st♦♥ ♦ ♣rts t t

sts ② ② t ♦s♦♥ ♦ s ♦ ♥♦r♥ r♥s ♥ t t♥ss ♦ t

r①t♦♥ ♣r♦ss t ts s t♥s t♦ ♠♥t ts ♦♥ t t ③♦♥ ♦

t ♥tr ♥ tr♥ t st ♠♥s♠ ♣♥s s♥s♥t② ♦♥ t sstrt

t♠♣rtr s ♦r T = 150 C t ♦ s♦♥ t t s ♦ ♦ ♥♦r♥

r♥s ♣r♥ts ♦s♥s ♥ t ♣r♦♣t♦♥ ♦ ♦rrt♦♥s t ♥tr s♦ tt

♥trr♥ tt♦♥s r sr ② P♦ss♥♥ ♣r♦ss s ②♥♠ s♦ ♣♣♥s

t s♦rt t♠s ♦r ♠s r♦♥ t T = 200 C ♦r t r①t♦♥ t ♥ r♦♥ t

s ♦ ♦ ♥♦r♥ r♥s s rs t♦ ♦s♥s ♥ ♥ ♥ s②♠♣t♦t

♦rrt r♦t ♦r r T r s♦♥ rt s r ts ♣r♦sss strt

r② s s t P❩ s♥

♥ tr♠s ♦ P❩ qt♦♥ q t r♥♦♠ r♦t t ♦ T ♠♣s ν ≈ 0

♥ λ ≈ 0 ♦r T = 200 C ♦♥ ①♣ts λ > 0 t s♠ s♦ tt r♦t s ♦♠♥t

② ♥♦s♥♦♥♥r ts ♥t②s②♠♣t♦t② s♥ ♦ r♦ss♦r ♥

T = 250 C ❬❪ ♥ts tt λ s ♣♦st② r s♠ ♥tr♣② s♦ ts ♣

♥ ts t♠♣rtr t s t s♦♥ s st ♠♦r ♦♣rt t♥ tt ♦r ♦r T

♦♥ ♣r♦sss strt r② ♥♥♥ t tt② t r♥♣♥ sss

♥ t ♣r♦s ♣tr ♥ ♥ t str♥t ♦ λ > 0 s λ(T ) s♠s t♦

♣♦st ♥rs♥ ♥t♦♥ ♥ ts r♥ ♦ T ♦r ♦r T = 300 C t

♥s ♦ P❩ s♥ ♥ t ♥t s sP❩ rs λ < 0 s

t②♣ ♦ P❩ s②st♠s r tr s ♣♦st♦♥ rs s ♥ t ♠♦ ❬❪

♣♦ss ①♣♥t♦♥ s tt t t st♥ ♦♥t ♦♠s s♠r ♥ r♦♥s

t r② r s♦♣s t sr s rs♦♥♥ s ♦rr♦♦rt ② 〈(∇h)2〉 ♦r ♠s

r♦♥ T = 300 C r ♦♥ ♦♥r♠s t ♣rs♥ ♦ rr s♦♣s t♥ t♦s ♦r

♦r T s ♥ ①♣♥ ② ts t ♣♣rs ♦♥② t T ❬❪ ♥②②

t s st♦♥s♥ tt s♦ ♦♥trst♥ P❩ ♠♥s♠s ♥ ♠r ♥ t r♦t

s②st♠ ♦♥② st♥ T

♣tr


♥ ts ♦r ♦♥ s ♣r♦r♠ t st② ♦♥ t r♦t ②♥♠ t ♦t

s♦rt ♥ r♥t ss ♦ t♥ ♠s r♦♥ ♦♥ sstrts ②

♦t ❲ ♥q rsts ♣r♦ ♦r t rst t♠ r ♥ r♦st

♥ ♦ s②st♠ ♥ ♠♥s♦♥s ♦♥s t♦ t rrPrs

❩♥ ss r t ♣rs r♦st ♥ ♠♥s tt t ♣r♦ ♦s ②♦♥ t

♦♠♣rs♦♥ t rt ①♣♦♥♥ts s ♦♥r♠ ② t rs t strt♦♥s

sqr ♦ r♦♥ss strt♦♥s ♥ ♠①♠ rt t strt♦♥s ♥ t

♠♥t♠ ts ♦r ♠♦♥strts t ♥rst② ♦ ts P❩ strt♦♥s ♥

t♠ r rt② ②♦♥ ♥♠r s♠t♦♥s

♦♥ t ♥②ss ♦♥ s ♦♥ sr ♣ts ♠♣r♥ t ①trt♦♥ ♦

s②♠♣t♦t s♥ ①♣♦♥♥ts ♦r ♥st♥ tr♦ t ♣♦r ♥ t wloc× ♣♦t

t s ♥♦t ♣♦ss t♦ ♥rt t r♦♥ss ①♣♦♥♥t tr t♥ ♦♥② t ♦

♠tr ①♣♦♥♥t α1 ♦s ♥♦t ♣r♦ ♥② ♥♦r♠t♦♥ ♦t t ❯♥rst②

ss ❯ ♦ t s②st♠ s ♦♥ ♥♦tr t② s t rt♦♥ t♥ t

r ♠♦♥ s③ ζ ♥ t ♦rrt♦♥ ♥t ξ t② ♦ ζ ≈ ξ s♦

t♥ s tr ♦♥② ♦r ♥ ♣♣r♦♣rt r♥ ♦ t♠ ♦r ♦ ♣s ♦♥ t t♦♣

♦ r ♠♦♥s ♦ ♥♦t str♦♥② ♥rst♠t t ♠sr♠♥t ♦ ζ s ♥ ♥

t ♣s ♥s♣t♦♥ ♦ srs ♠ ② s♦♠ ♠r♦s♦♣ t♥q ♦♠s

♠♣♦rt♥t

P❩ ♠♥s♠ ♦rr♥ ♥ ♠s r♦♥ t T = 250 C s rt t♦


t ♦r♠ ♥ ♥ ♣rts tts t♦ r♥ ♦♥rs s ♦ ♦ ♥♦r

♥ r♥s ♦ ♥♦t ♥ s♣ ♥ tr ♥♦r♦♦ ♥ ts ♥rt♥

①ss ♦ ♦t② t P❩ ♥♠r s sss ♦ ♥ ts ts

♣♥ ♠♥s♠ s♦ s♦ ♣rs♥t ♥ ♦tr t♠♣rtrs t♦ ♥①♦r

①♣r♠♥t ♦sts t s ♥♦t ♣♦ss t♦ t t P❩ ♥sät③ ♥

♠♥s♦♥s ♦r ts s ♥tr ①t♥s♦♥ ♦ ts ♦r

t ♦ t ♣♦st♦♥ t♠♣rtr T ♦♥ t r♦t ②♥♠ s s♦

♥ st ♥ r♦ r♥ ♦ T ♥♠② T ∈ ❬ ❪ C rt♦♥ t♥

s♦rt ♥ r♥t ②♥♠s s ♥ sts ♥ s ♥ ♦♥ tt

t ♠♦♥ ♦t♦♥ s tt ② t ♥tr♣② t♥ t ♦r♠t♦♥ ♦ ts

t r♥ ♦♥rs ♦ ♦ ♥♦r♥ r♥s ♥ t r①t♦♥ ♣r♦ss ♥

② s♦♥ ♥ ♣♦st♦♥ ♦ ♣rts t♦r ts r♦♥s s♠♣ ♦♥t r♦

♠♦ ♦rr♦♦rt ts rs♦♥♥ s ♥tr♣② s t♦ r♥t s♥r♦s t r

♥t tt♦♥s s T ♥rss ♦r T = 150 C t ♦ s♦♥ t t

s ♣r♥ts ♦s♥s ♥ t ♣r♦♣t♦♥ ♦ ♦rrt♦♥s t ♥tr s♦ tt

♥trr♥ tt♦♥s r sr ② P♦ss♦♥♥ ♣r♦ss ♥ ①♣♦♥♥ts ♥

strt♦♥s s♣♣♦rt t♦ ts rs♦♥♥s

♦r T = 200 C ♦r ♠♦r ♦♠♣① s♥r♦ s ♥ ♦♥ ♥ t r♥

♦ r♦t t♠ st t r①t♦♥ ♣r♦ss ♦r♦♠s t ts ♦ t rrr t

♥ r♦♥ ♦ t s ♥ s rs t♦ r♦ss♦r ♥ t r♦t ②♥♠

sts r♦♠ ②♥♠ s♥ ♥ strt♦♥s s♦ ♦rr♦♦rt t t ♣rs♥

♦ ts r♦ss♦r ♦♥ss s♥ ♥②ss ♥rtss ♥♦t ♥ t♦

♦♥♥♥② ♣♦♥t t ❯ ♦ t r♦t t ts ♣♦st♦♥ t♠♣rtr

s♣♣♠♥t st② s ♦♥ t strt♦♥s ♥ tr♥ ♣r♦ ts ss♥t

♦r ♥♥ t ❯ ♦ ts s②st♠ ♦♥ r ♥tt♠ ts t♥ tt ♥ t

st♥r r♦♥ss s♥ s ♥ ♦♥ s t s ♣♦ss t♦ ♥ ♥♦♠t♦

P❩ r♦ss♦r t♥ ♣ ♥ t r♦t t t s♠ t♠ tt t rst r♦st

①♣r♠♥t r③t♦♥ ♦ s r♦ss♦r ♥ t♦♠♥s♦♥ s②st♠s s ♠♦♥strt

r♦t ♦ ♠s t T = 300 C rrs s♦ ts st♥t ♠♣♦rt♥

t♦ ②♥♠ s♥ ♥②ss ♦ ♥♦t ♦ ♠♥ r st♥t♦♥ ♠♦♥


s♦♥♦♠♥t qt♦♥s ♥ t P❩ ♦♥ ♦r sr♥ tt♦♥s t

ts ♣♦st♦♥ t♠♣rtr t strt♦♥s sr t ♦r♠rs ♥ ♣♦♥t ♦t

t ♣rs♥ ♦ P❩ r♦t t λ < 0 r♥t② ♦ ♦r t♠♣rtrs s

♥♥t♣t rst s ♥ rt t♦ t rs♥ ♦ t st♥ ♦♥t ♥

r♦♥s t r② r s♦♣s t sr rs♦♥♥ s ♦rr♦♦rt ② t sqr

♦ s♦♣s r ♦t t♠s rr t♥ t♦s ♦r ♦r t♠♣rtrs s

♥ ♥♦t ♥ t t♦♠ ♦r r♦s♦♣ ♠s s ♥ ①♣♥ ② ts P❩

♠♥s♠ ♣♣rs s♦② t t♠♣rtrs

♥ tr♠s ♦ t P❩ qt♦♥ q t rsts r tt ♦r T = 150 C

t sr t♥s♦♥ ν ♥ t ①ss ♦ ♦t② λ r r② ♥r ♦ ③r♦ s♦ tt ♥♦s

♦♠♥ts t r♦t ♦r ♠s r♦♥ t T = 200 C ♦r t ♥♦♠

t♦P❩ r♦ss♦r ♦♥ ♥ts λ > 0 t s♠ s♦ tt ♥♦♥♥r ts ♦r♦♠

t ♥♦s ♦♥② t ♦♥ r♦t t♠s ♥ tr♥ ♥ ♥ ♦ t P❩ s♥ s♥

♥t r♦t t♠s s ♣rs♥t ♥ ♠s r♦♥ t T = 250 C ♥ λ s ♣♦st

♥ r t t ♠♦♠♥t tt ♦♥s♣r t♦ ♦♥tr λ(T ) s ♥ ♥rs♥ ♥t♦♥

♦ T t rsts ♦♠♥ r♦♠ ♠s r♦♥ t T = 300 C s♦ tt λ s ♥t

s ssts tt s ♣♦ss t♦ st T t s♦♠ s♣ TEW t T ∈ ❪ ❬ C

♥ ♦rr t♦ ♦t♥ λ = 0 t♦ r♦t sr ② t rs❲♥s♦♥

qt♦♥ ♥ t♦ st T(EW−KPZ) = TEW ± δT s♦ tt ❲t♦P❩ r♦ss♦rs ♠r

♥t♦ t ②♥♠ ♦ sr tt♦♥s ♥ s♠♠r② t s ♣♦ss t♦ st

t P❩ ♥♦♥♥rt② ♥ t s②st♠ ♦♥② st♥ t ♣♦st♦♥ t♠♣rtr

r s♠♠r③s tss sss♦♥s ♠♥ r r♥t r♠s tt ts

♣ ♥ t r♦t s s ♥rs ♥♠② r♦♥ rrs t♦ t t♠♣rtr ♥tr

♥ P♦ss♦♥♥♥♦♠t♦P❩ r♦ss♦r rs t ②♥♠ s rst r♦♥

s ♦③ t t rt ♦rr ♦ t♠♣rtrs ♦s t♦ 150 C r λ→ 0 ♥ r

t P♦ss♦♥♥♥♦♠ r♦t ♠rs ♥ t r♦♥ ♥ P❩ s♥ ♠rs

♥ ♣rs t t♥ r♠ ♠t ② T = TEW ± δT ♥ s②♠♠tr ♥ rt♦♥

t♦ t TEW ♣♦♥t ♥ t ♣r ❲ r♦t δT = 0 C ♥ t ❲t♦P❩

r♦ss♦r r ①♣t t♦ t ♣ ♥② t t♠♣rtrs t♦ ①♣t

♦ ♥♦t ♠ r t♥ 300 C t P❩ s♥ s r♦r t λ < 0


r ♦♥tr ♦r t ♦r ♦ t ①ss ♦ ♦t② λ s ♥t♦♥ ♦ t♣♦st♦♥ t♠♣rtr T ♥ t s②st♠ rs rr t♦ ①♣r♠♥t ♣♦♥ts♥ s ♥s ♥ts ①♣t r♥t r♠s t♦ ❱ r♥ t ♦rs♥♥r♥②♥♠ × ♣♦♥t ♥♦tts t♠♣rtrs ♦r ♦ tt♦♥s r ①♣tt♦ ♦♥ t♦ ♥♦♠ r♥ × ♥ ❲ × sss

❲ ♥s ts ♦r tr② ♥ tt t ♥♦ ♣r♦r ♦r ♥stt♥ t

❯ ♦ r♦♥ srs ♦♣♥ ♥ ♠♦tt s ①♠♣ ② ♣♥② ♥

Ps♥t③s ❬❪ ♥ ♦♥♥t ♣rs♣t ♥ t s s t ♣♣t♦♥

♦ ts ♠t♦s ♥ ♣r♦s② st s②st♠s r s②st♠ s♦ ♦r t ①♣r♠♥t

♣♦sst② ♦ ♦s st② ♦ ♥♦♠t♦P❩ ♥ ❲t♦P❩ r♦ss♦rs

♣♣♥①

♦r ts ♦t ♦♥t♥♠

r♦t qt♦♥s ♥ ❯♥rst②

sss

♥♦♠ r♦t qt♦♥

♦♥r♥♥ ♦♥ ❲ ♥ ♥r qt♦♥s ♦♥ ♥ s② s t ♣rtr

s r ν ♥ K r ♥s rs♣t② rst♥ ♥ t s♦ ♥♦♠ r♦t

qt♦♥

∂th = η(①, t).

♥ st t ♥♦s rs t r♦t ②♥♠s tr r ♥♦ ♦rrt♦♥s t♦

♣r♦♣t tr♦ t s②st♠ ♥ ♥ r z → ∞ α ①♣♦♥♥t s s♦

♥ s♥ t s②st♠ ♥r rs t stt♦♥r② r♠ rtss st

♥ ♥ s t r♥ ♦ s r♥♦♠ ♥trs ♦s t t ♣♦st♦♥ t♠

rst s ♥t♣t ♦♥ ♦♦s t ts srs s ♥ ♦♠♣♦s ② st ♦

♥♣♥♥t r♦♥♥ ♠♦t♦♥s

♥trt♥ t r♥♦♠ q ♦♥ ♦t♥s∫

∂th(①, t)dt =∫

η(①, t)dt r♥

♦t ss ♦♥ s 〈h(①, t)〉 = 0 t ♥st sqr ♥ r ♦♥ rs

〈h2(①, t)〉 = 2Dt ♦ t r♥ r♦s s ♥ t q ♥♥ t r♦t

♣♣♥① ♦r ts ♦t ♦♥t♥♠ r♦t qt♦♥s ♥❯♥rst② sss

①♣♦♥♥t β = 1/2

w2(t) = 〈h2〉 − 〈h〉2 = 2Dt.

rt♦♥ ♦r t ♥r qt♦♥

t s ♥♦♥ tt ♥ ♥r♦♥♠♥ts ♥r s ♦♥t♦♥s s♦r♣t♦♥

s ♥ ♣r♦ss ♥ ♥ t tr♠ ∇2h ♥ t q s♦ ♥♦t ♣② ♥

♠♣♦rt♥t r♦ ♦r♠t♦♥ ♦ ♦r♥s ♥ ts r ♥♦♠♠♦♥ ♥ t

♣t①② ❬❪ ♥ sr t ♥♦♥♥r (∇h)2 tr♠ r strt♥ ♣♦♥t ♦r ♥

♦♥t♥♠ r♦t qt♦♥ ♦r ♥r♦♥♠♥ts s t ♦♥srt♦♥

∂th = −∇ · + η(①, t),

r r♣rs♥ts ♣r rr♥t ♦ ♣rts s♥ ♦♥t♦ t ♥tr

s ♥trs r♦♥ ♥ ♠rs r r♥ ② ♠ ♦♥ ♥ ♦rr

t♦ ♠♥♠③ tr sr r ♥r② tr♦ s♦♥ s♦ ∝ −∇µ t ts♠♣st ss♠♣t♦♥ ♥ tr♥ ♣♦♥ts ♦t tt t ♠ ♣♦t♥t s ♣r♦♣♦rt♦♥ t♦

−1/R r s t ♦ rtr s ♦tt♦♠ ♦ ② ♣♦st

rtr R > 0 ♦r ♥st♥ s t st r ♣rt ♠♦r ♥♠r ♦

♥♦rs ♥ ♠♦ ② r♦♠ tr t ♦ t ♦tt♦♠ ♦ ② s

♠♥♠♠ ♦r µ(①, t) ♦♣♣♦st ♣♣♥s ♦r t t♦♣st st ♦ s♥ R < 0

r tr s ♦ ♠①♠♠ ♦r µ(①, t) ♥ trr ♣rts t s♠

♦♦r♥t♦♥ ♥♠r ♥ ♦② µ = 0 r②r ♥ ♦♥ ts ♦♥srt♦♥s

♦♥ ♥ r strt♦rr② µ(①, t) ∝ −∇2h ♦♥sq♥t② ∝ ∇(∇2h) ♥ ♦♥

rs t♦ t q tr q

♥ ♦r♥ t ♦r ♣r♦s ♥t♦♥ β ≡ α/z ♦r t r♥♦♠ r♦t s ♣rtrs r ♦♥ ♥ ♥ ♣♦r ♥ t♠ ♦r t r♥ ♥ ♥ t s♣t ♥ t♠♣♦rsttstrtt② ①♣♦♥♥ts r ♥ ♥ ts s♥s t β ①♣♦♥♥t s ♥♠♥t② r♥tr♦♠ t♦s ♣r♦s② t

♥t ts ♥ t ♣♣♥① st♦♥


r ♠t ♦ t ♠ ♣♦t♥t ♣♥♥ ♦♥ t ♦ rtr t t t♦rt ♣♦st rtr s♦♥ ② r tr s ♦ ♠♥♠♠ ♦ µ(①, t) s♣rts st t t ♦tt♦♠ ♦ t ② ♥♦t ♠♦ ② r♦♠ tr ♥ trrR →∞ ♥ µ(①, t) s ♥ r②r t rtr s♦♥ ♥ s♥ r µ(①, t)s ♦ ♠①♠♠ s ♣rts ♦♥ t t♦♣ ♦ t s♥ r ♦ tr ♦♥s♥ ♦♣ t♦ ♦sst ♥♦r st r t r ♥r② ♦ t ♥tr s rs

♦♥♥r qt♦♥ ♥ t ❱

ss

♥ ♠♦s t ♦♥sr ②♥♠ tr s ♥sst② ♦ ♥srt♥ ♥♦♥♥r

tr♠s s♥ α > 1 rs t ②♣♦tss tt ♦♥t♥♠ ♣♣r♦①♠t♦♥s r ♥

t ♠t ♦ s♠ s♦♣s ∇h ≪ Lα−1 ❬❪ s ♣♦♥t ♦t ② ❲♦ ♥ ❱♥ ❬❪

t ♦st ♥♦♥♥r tr♠ ♦②♥ ♦♥sr ②♥♠ s t tr♠ ∇2(∇h)2

♦♠tr ♥tr♣rtt♦♥ s r ❬❪ ♥ts tt ♣rts ♥♥ t st♣s

r rts r① t♦ ♦r st♣s s♠ rts ♦rrs♣♦♥♥ t♦ t

t♠♣rtr r♠ ♥ ♥r♦♥♠♥ts r t♦♠s r tr ♦♥ ♥ s

t rr s t♦ s♠ s♦♣ ♥s ♥♦♥♥r qt♦♥ q s

♣r♦♣♦s ♦r♠② ♥ ② ♥ s r♠ ❬❪

∂th(①, t) = −K∇4h+ λ1∇2(∇h)2 + η(①, t),

λ1 s t s♠ ♠♥s♦♥ ♦ ν ♥ ♦♥ts ♦r t str♥t ♦ s♦♥ ♦ ♣rts

t ♦ s♦♣s t♦rs s♠ rts

rt ①♣♦♥♥ts rt t♦ t ♥♦♥♥r qt♦♥ ♥ ♦♥ ②

s♥ ♥♦r♠③t♦♥r♦♣ ♣♣r♦s ❬❪ ♠t♦ ♣rts


α =4− ds

3; z =

8 + ds3

tr r② ♠♣♦rt♥t rst ♦♠♥ r♦♠ ♥②ss s t ②♣rs♥ r

t♦♥ ♦r ♥② r♦t t ♦♥sr ②♥♠ ♥ ♥♦♥♦♥rs ♥♦s

z − 2α− ds = 0

st ts ①♣♦♥♥ts ♦♠♣♦s t ❱♥sr♠ ❱ ❯♥r

st② ss r r s♠ ♥♠r ♦ srt ♠♦s ♦♥♥ t♦ t ❱ ss

♠♦♥ ♣♦ss② r t ♦♥sr rstrts♦♦♥s♦ ♠♦ ♣r♦

♣♦s ② ♠ t ❬❪ ♥ t s r♠ ♥ ♠♦r♥ ♦♥ ❬❪ r②

♦♦ ♥♠r st② ♥ sr♣t♦♥s ♦ ts ♠♦s ♥ ♦♥ ♥ t ❬❪

♣♣♥①

♥♦♠♦s ♥

♥♦♠♦s s♥ ♦rs ♥r sr s ♥♦t s♥ ❬❪ t ♠♥s

tt ♦ ♥ ♦ tt♦♥s ♦ ♥♦t ♦ ♥ t s♠ ② ♥t② t tr♠

♥♦♠♦s s s t♦ sr t r♦t ♦ ♥trs ♥ t ①♣♦♥♥t α s

♦♥ t♦ rr ♦r q t♥ ❬❪ ♥ ts ♦♠s r♦♠ t t tt

α > 1 tr s r♥ ♦ tt♦♥s ♥ t st② stt ♠♥ t sr

s♣rr♦ s ♥t ② t ♠②❱s ♥sät③ q Wsat/L ∼ Lα−1

rr♥t stt♦♥ ♦r s r② r♥t r♦♠ tt ♦ ♠♦st t♥t②

②rs ♦ ♦②s ♦♥ ♥♦s tt sr s②st♠s s♣② ♥♦♠♦s s♥ ♥

♣♥♥t② ♦ t α s ❬❪ ♥ rr♥s tr♥ ♦r♦r tr r ♠♦r

t♥ ♦♥ ♣♦ss s♥ ♦r♠ t♦ sr t ♦tt♦♥s ②♥♠ ♦ r♦♥

♥tr ♥ t ❱ ♦♥ ♦♥② ♣rtr s r tr s ♥♦ r♥ t♥

t ♦ ♥ ♦ ♦♥ ❬❪ ♥♠r ♦ ①♣r♠♥t sts s♦ ♦r

r♦♦rt ts ♦♥tr ♠♦♥ r t ♦♣♠♥t ♦ ♦♦ rtrs ❬❪

t tr♦♣♦st♦♥ ♦ ❬❪ t ss♦t♦♥ ♦ ♣r r♦♥ ❬❪ ♥ t ♣♦st ♦

♦♦ ♣rts t t ♦ ♣♦rt r♦♣s ❬❪

♥t s ♦ r ♥ t ❱ ②♣♦tss r ♦♥ ♥ t ♦♥

♠♥s♦♥ ❲♦❱♥ ♠♦ ❬❪ ♥ ssq♥t② ♥ ♦tr ♥r ♥ ♥♦♥

♥r r♦t ♠♦s ♦♠♥t s♦♥ ❬❪ r♦♠ tr ♥②t ♥

♥♠r trt♠♥ts r str♦♥② ♦t t♦ ts ss ♥ t♦ t ♦♥s♦♥s

♦♥t♦♥ α < 1 s ♥♦t s♥t t♦ ♣r♥t t s②st♠ ♦ ①t♥ t

♣♣♥① ♥♦♠♦s ♥

♥♦♠♦s s♥ ❬❪

r ①st tr t②♣s ♦ ♥♦♠② s♣rr♦ ♥tr♥s ❬❪ ♥ t ❬❪

t② r ♥ ♦

♦♥♦♥sr r♦t ♠♦s ♦ ♥♦t ①t ♥♦♠♦s s♥ ♥ ♥ ♣rt

r ♥tr♥s s♥ ♥ ♥♦t ♦r ♥ ♦ r♦t ♠♦s ❬❪

t s ♦rt ♠♥t♦♥ tt t ♣rs♥ ♦ t ♥♦♠② ts ♦♥② ♦ ♦rrt♦♥

♥t♦♥s ♣♥ t ♦ r♦♥ss ♦r ♥st♥ s♥ ♦r♥ t♦ t st♥r

❱ s♥ ❬❪ ts s ♣r♦ ♥ t ♦♦♥ ♦r ♥♦ ♦♥sr t ♦rr tr♥s♦r♠

♦ t ♥t♦♥ h(①, t) s

(, t) = L−ds/2∑

①

[h(①, t)− 〈h(t)〉] exp(i · ①),

r r t ♥♠rs sr♥ t♥ t s♣ ♦♥ ② t sstrt

strtr t♦r ❬S(k, t)❪ ♦r ♣♦r s♣tr♠ q s ♦rrt♦♥

♥t♦♥ ♠sr♥ t tt♦♥s ♥ t r♣r♦ s♣

S(k, t) = 〈(, t)(, t)〉 = 〈|(, t)|2〉.

S(k, t) ♥ rt t♦ t ♦ r♦♥ss ♥ t♦ t tt ♦rrt♦♥

♥t♦♥ Ch ② t qs ♥ rs♣t②

W 2(L, t) =1

Lds

∑

k

S(k, t) =

∫

ddsk

(2π)dsS(k, t),

Ch(l, t) ∝∫

[1− cos( · ①)] ddsk

(2π)dsS(k, t),

r t ♥trs r♥ t♥ t ♥tr 2π/L ≤ k ≤ 2π/a ♦r k rt♦♥ t

a ♥ t tt ♣r♠tr

s t♦♦s rqr t♦ r t r♥t ♦r♠s tt ♥♦♠♦s s♥

♥ ♣♣r r rtt♥ ♥ ♦rr ♥ t qt♦♥s ♦ ♥srt♥ t ❱ ♥sät③ q

♥t♦ t rt♦♥ ♦ q t s strt♦rr s♦♥ tt t ♣♦r s♣tr♠

♣♣♥① ♥♦♠♦s ♥

s s

S(k, t) = k−(2α+ds)sFV (kt1/z),

r

sFV (u) ∼

const, if u≫ 1,

u2α+d, if u≪ 1.

t ♠♥s tt ♦r s♦rt♥t ss k ≫ 1/ξ t ♣♦rs♣tr♠ s t♠

♥♣♥♥t ♥ t rs ♦r t♠s s♦ ♣♦r s♥ t k−(2α+ds) s

s t s ❱ s♥ ♥ t r♣r♦ s♣ t ♥♦ ♦♦ t ♣♣♥s ♥ ts

s♥ rt♦♥ s ♥srt ♥t♦ t q ❲♥ ♦♥ sts α > 1 t ♥tr ♥ q

♦♠s r♥t ♥ t ♠t ♦ l ≪ ξ ♦r L → ∞ ♥ a → 0 ❬❪ ♥ t

♠t l ≪ ξ rst ♥ ♣♥ L ♥ a ① ♦♥ ♦t♥s r♥t s♥ rt♦♥ ♦r

Ch

Ch(l, t) ∼

l2t2(α−1)/z, if l ≪ ξ ≪ L,

l2L2(α−1), if l ≪ L ≈ ξ.

♦ ♥♦ r♥t r♦♥ss ①♣♦♥♥t s ♠r ♥ ♦② Ch(l, t) ∼l2(αloc) r αloc 6= α ♥ αloc = 1 ♦r♦r ♣♥♥ ♥ t♠ ♦r Ch t

♥ts l ≪ ξ ♣♣rs ♥ t s②st♠ s♥ t t2κ r

κ ≡ (α− αloc)/z.

r♥st♥ ts s♥ ♦r t ♦♥t①t ♦ t ♦♥ ♦ s t rs ♦r

r♥t r♦t t♠s s♦♥ ♥ tt r st t♦ ♣ s t t♠ ♦s ♦r

l ≪ ξ r s♦s r② r② ts ♥♦♥s ♦r ② t♥ t

sqr♦s♦♣ ♦t♦♥ q ♦r t ♦♥♠♥s♦♥ ❲♦❱♥ ♠♦ ❬❪

〈(∇h)2〉 ∼ Ch(l = 1, t) ∼ t2κ.

♥ t ❲❱ ♠♦ t s ♦♥ κ = 0.19(1) r♥ tr♥s♥t r♠ ♠t ②

♦t ♥ts③ ♥ ♥tt♠ ♦rrt♦♥s ❬❪ t ts tr ♦ ♥♦♥s ♦r

♣♣♥① ♥♦♠♦s ♥

r ①♠♣ ♦ ♥♦♠♦s s♥ ♥ ♦♥ s t ♥t♦♥ t ♦rrs♣♦♥♥t♦ t Ch(1, t) ♥t♦♥ ♦r sr s②st♠s s③s r♦♠ t ♦tt♦♠ ♦ t t♦♣ ♥ rs♣t② ♦s t ♥♦♥s ♦r ♦r t strtrt♦r t r♦♠ t ♦tt♦♠ t♦ t t♦♣ t rs rr t♦ r♦t t♠s t 24 27 210213 216 219 222 rs♣t② r ①trt r♦♠ r ❬❪

♦ ♦rrt♦♥s ♥t♦♥s t ♦ ss s ♥♦t t ♦♠♣t s♥r♠ ♥ t

♥♦♠② ♥ ♠r ♥ ♠♦s ♥♦r♣♦rt♥ r♥♦♠ s♦♥ ❬❪ ♥ s♦♥

♦♠♥t ❬❪ ♣rs♥t ♥♦♥s ♦r ♥♦t ♦♥② ♦r Ch(l, t) t s♦ ♦r t

♣♦r s♣tr♠ ❬k ≫ 1/ξt❪ s t♠ ♣♥♥t s s t ♠

r tt t ❱ ♥sät③ rtt♥ ♥ t r♣r♦ s♣ q ♦ ♥♦t t

♥r s♥ tt♥ ♦ tt♦♥s ♥ t sr r♦t ♦♥t①t

♥r ②♥♠ s♥ ♥ ♥t r♦♥♥ s ♥ ♦♣ ♦♥

♦ ♠♥② ②rs ❬❪ t ♦r♠ t♦♥ s ♥ ♥ t ②r ♦

r ♠s♦ t ❬❪ ♣r♦♣♦s s♥ ♦r s(u) s

s(u) ∼

u2(α−αs), if u≫ 1,

u2α+d, if u≪ 1,

r αs s ♥ ①♣♦♥♥t s♣tr r♦♥ss ①♣♦♥♥t

s t ♣♦rs♣tr♠ ♥r② ss s

S(k, t) ∼

k−(2αs+ds), if k ≫ 1/t1/z,

t2α+ds

z , if k ≪ 1/t1/z.

♣♣♥① ♥♦♠♦s ♥

♥srt♥ q ♥ q ♦♥ ♣r♦s tt ♦ tt♦♥s r tt

② t st♥r ♠②❱s s♥ q ♥♣♥♥t② ♦ α ♥ αs s

rtss s s♦♥ ♦ t ♦ s♥ s ♠♦ ♣♥♥ ♦♥ t

♦rr ♦ t sr ♠ts ♥♦ ♥ ♦♥ s ♦ t ①♣♦♥♥ts ❬❪ ♥ ♥r

② t♦ ♠♦r ss ♥ st♥s ♥♠② αs < 1 ♥ αs > 1 ♦r t ♦r♠r

♦♥ ♦t♥s t s♠ tt ♥ t q t t L→ l ♥ α→ αs

w(l, t) = tβfαs<1(l/ξ),

t

fαs<1(u) ∼

uαs , if u≪ 1,

const, if u≫ 1,

♠♣②♥ tt αs = αloc ❬❪ s s ♥tr♥s ♥♦♠② ♦♥ s t ♣♦r

s♣tr♠ ♥ w(l, t) ❬Ch(l, t)❪ ♥ ♥♦♥tr② ♠②❱s s♥ s r♦r

♥ α = αloc ❬❪

trs ♦r αs > 1 t ♥tr ♥ t q s r♥t ♥ L → ∞

♣♥ ① ♦♥ ♥s tt

fαs>1(u) ∼

u, if u≪ 1,

const, if u≫ 1.

s ♠♣s tt αloc = 1 ♥♣♥♥t② ♦ αs t♦♥② tr α = αs

s♦ t ♣♦rs♣tr♠ ♦s tr② ♥ t♠ rs w(l, t) ♥ Ch(l, t) ♦ ♥♦t s

s t s♣rr♦♥♥ ♥♦♠② sss t t ♥♥♥ ♦ ts st♦♥ ♦r

♥ t qt② t♥ α ♥ αs s ♥♦t s♦ ♥ ♥♦♠② st♠s ♥♠②

t t s♥ ❬❪ ♠♥ rtrst ♦ t t ♥♦♠② s tt t ♥

tt ♦♥② ② s♥ t ♣♦r s♣tr♠ s tr s ♥♦t ♦♥str♥ t♥ α

♥ αloc ts ①♣♦♥♥ts ♥ q κ = 0 ♦r r♥t κ 6= 0

s♠♠r② ♦ ♦♥t♦♥s ♥ t♦ t r♥t s♥ ♥ ♣♣r

♥ sr r♦t s ♥

♣♣♥① ♥♦♠♦s ♥

If αs < 1 ⇒ αloc = αs

αs = α ⇒ Family − V icsek,αs 6= α ⇒ Intrinsic,

If αs > 1 ⇒ αloc = 1

αs = α⇒ Super − rough,αs 6= α⇒ Faceted.

r♦♠ t ①♣r♠♥t ♣♦♥t ♦ sr sts ♦♥ t r♥t

♥♦♠♦s s♥s ♦r ♥st♥ tr♥st♦♥ r♦♠ ♥tr♥s t♦ t s♥ s ♥

r♣♦rt r♥ ss♦t♦♥ ♦ r♦♥ ❬❪ rs t ②♥♠ s ♥ ♠ ♥

t r♦t ♦ ♦♥ ss sstrts ❬❪ ♥tr♥s ♥♦♠② s♠s t♦ ♣♣r ♥ t

r♦t ♦ t♥ ♠s ② rt s♣ttr♥ ❬❪ ♥ r♥ t ♣♦st ♦ ♦♦

♣rts t ♦ ♣♦rt tr r♦♣s ❬❪ tr sts ♥♦♥ ♥♦♠② ♥

♦♥ ♥ t ❬❪

Pr♦rss s s♦ ♥ ♠ ♥ t s♥s ♦ ss② ♥ ♣♦ss ♥♦♠②

♥ ♣♣r ♣♥♥ ♦♥ t ♦♥t♥♠ qt♦♥ ♦♥sr ó♣③ t ❬❪ s

s tr♥s♦r♠t♦♥ Υ = ∇h t♦ st② t r♦♥ss WΥ = 〈(∇h)2〉 ♦ srs

sr ② s♦♣s s ❲♥ s srs r r♦ t ♠♣s κ > 0 ♥ ♦♥

s ♥♦♠② s♦♥sr♥ t t s r♦♠ ts strt♥ ♣♦♥t ♦♥ ♥ s♦

tt ♥♦♥♦♥sr r♦t ♠♦s s t♦s r ② t rrPrs❩♥ P❩

ss ♥ ♥♦t ①t ♥♦♠② rs ♦ r♦t ♠♦s ♦r s♣② ❱ s♥ ♦r

s♣rr♦♥♥ ② ♥ ♥♦t ①t ♥tr♥s ♥♦♠② ❬❪

♣tr

♣♣♥①

♦♥♣ts ♦r rs ♥ qr♠

♥ ♦r t ♥ ♠ r♦t

r ♥s♦♥ ♥ qr♠ ♣

s②st♠ s ♥ qr♠ ♥ ts ♠r♦s♦♣ ♣r♦♣rts ♦ ♥♦t ♥ ♣

♣r② t t ♥t s ♦ ♠sr♠♥t t♠ ❬❪ ♥ ts r♠ tr s

st st ♦ tr♠♦②♥♠ ♦♦r♥ts sr♥ t s②st♠ ♥s ♥

r③ s♣♠♥ts ① ♥ tr rt ♥r③ ♦rs ❬❪ ①♠♣s

♦r t ♦r♠r r t ♦♠ ❱ ♦r s ♦r t r Af ♦r ♠ rs tr

rt ♦rs r t ♣rssr P ♥ t sr t♥s♦♥ γ rs♣t②

♣♥♥ ♦♥ t ♦♦r♥ts r ① ♥ ♥ ①♣r♠♥t t s②st♠ stt

♥ s♣ ② ♣♣r♦♣rt tr♠♦②♥♠ ♣♦t♥ts ♦r ♥st♥ ♦r ♣r♦sss

♣♣r♦♥ t♦ t qr♠ s♦tr♠② ❬t t♠♣rtr s ♦♥st♥t❪ t♦t

♦r ➒W = 0 t ♠♦t③ r ♥r② FH = FH(T, ①) s ♥ trs ♦♥② ♠ ♦r ➒Wche =

∑

i µidNi r µi s t t♠ ♣♦t♥t ♥

Ni s t ♣rt ♦ t♥tr s s♥t ♦♥ ♥ s t s r ♥r② G =

G(T, ) ♦t r ♠♥♠♠ t t qr♠ ♥ ♦♥t♥s tr♠♦②♥♠

♥♦r♠t♦♥ ss t♦ t s②st♠ ❬❪

♥ ♦♥ ♥ t srs ♦r ♥trs ♦♥ s ♦♥sr tt ♦♥r②

t♦♠s r ♠♦r ♥rt t♥ t♦s ♥ t s t② ♥♥ ♦♥s

s ♦r rt♥ ♥ sr ♦ r dAf ♦♥ ♠st ♣r♦ ♥ ♠♦♥t ♦ ♥r②

♣♣♥① ♦♥♣ts ♦r rs ♥ qr♠ ♥ ♦r t ♥ ♠r♦t

dFH |T,V = γdAf ♥ssr② ♦r r♥ ♠ ♦♥s ♥ ts stt♦♥ γ s s♦

♥tr♣rt s t ①ss ♦ r ♥r② ♣r ♥t ♦ r ♦t♥ s♣r ♥r②

❬❪ trs tr r ♣rt ♠rt♦♥s t♥ t ♥ t sr

♣♣rs tr♠ ♦ ♠ ♦r ♥ ♦♥ ♠st ♣r♦ dΞ|T,V = γdAf −ΣiNidµi t

Ξ ♥ t r♥ ♣♦t♥t

♥②② t ♠♣♦rt♥t s sr ♠♥♠③s ts ♥r② rs♥ ts r ♦r

♥♥ γ rst s s t rs♦♥ ♦r tr r♦♣ ♦s ♥♦t s♣r ♦♥

t ♥ ♦r t♦t rt② ts s♣ s ♣rt② s♣r ♣♣rt②

♣♥♦♠♥ ♥ s♦♣ ②♥♠s r s♦ tt ② t sr t♥s♦♥ ❬❪

②♣② ♠ts r γ t♥ ♦①s ♥ ♦r♥ strtrs ❬❪ ❲tr

q♣♦r ♥trs s γ = 72.94 ♠♠ rs tt ♦♥s ♦r♠ ② ②r♥ ♦r

♠t♥♦ ♥ ♠♠ t 20 C ❬❪ t sr ♥♥♦t rs ts

r γ rt♦♥s s t♦ ♠♦t♦♥s ♥ t t♦♠ rr♥♠♥ts ♥ r rs♣♦♥s

♦r sr r♦♥strt♦♥s ♥ r②sts ❬❪

r ♠ ♦ P r②st t qr♠ r♦♠ t ❬❪ ③♠t ♦r 300 C 320 C ♥ 327 C ♦ t P ♠t♥ ♣♦♥t ①trt r♦♠ ❬❪

s♣ ♦ ♥ ♦t t t qr♠ ♥ ♣rt ② rt♦♥ ♣r♥

♣ ♦ t st tr♠♦②♥♠ ♣♦t♥t ❬❪ ♦r r②st t sr t♥s♦♥

♣♥s ♦♥ t s ♦♥sr ♥ γ = γ(hkl, T ) r hkl r t rs ♥

s s ♥s♦tr♦♣② ♦r♥s t r②st s♣ ♥ s t t♠♣rtr ♥rss t

t♥s t♦ ♠♥♠③ ❬❪ ♦ t ♠t♥ ♣♦♥t t ♥s♦tr♦♣② ♥ss ♥ t

s♣r s♣ s tt ♦♥ tt ♠♥♠③s ♥ r ♦r ♥ ♦♠ s♥ γ s s♦tr♦♣ ❬❪s s ♥tt ♦♥ ♠t sr t♥s t♦ ♦♠ ♦①

♣♣♥① ♦♥♣ts ♦r rs ♥ qr♠ ♥ ♦r t ♥ ♠r♦t

♠tr ♠st r♦r t s♣r ♦r♠ ❬❪ ♥ t ♦♥ ♥ s♥ q

r♠ s♣s ♦r P r②sts t r♥t t♠♣rtrs s t♠♣rtr ♥rss t

♥s♦tr♦♣② rss ♥ r ts r ♦r♠ ♦ t ♠t♥ ♣♦♥t r P

s q t s♣r s♣ s

t♦♥

t♦♥ s ♥♠♥t ♣r♦ss ♣♣♥♥ r♥ ♣s tr♥st♦♥ ❬❪ t

s ♠♣♦rt♥t ♦r t♥ ♠s s t strtrs ♣♣r♥ ♥ t s♠♦♥♦②r r♠

str♦♥② ts t ②♥♠ ♦ t r♦t ❬❪ ♠♣ ♠♦s ♦r ♥t♦♥ r

s ♦♥ t q r♦♣ ♠♦ s♦ ♦rs ♦r s♦s s r♥ s

tt s♦s t♦ t♦♥ trs stt② ♥ ♦♠♠♥srt② ❬❪ s

ts sss ♦♥ ts st♦♥ t♦♥ ♠♦s r ♠♣♦rt♥t st s

qtt s♦♥ ♦♥ ♦♥ s t t♦♠s ♥ ♥ ♦♥t♥♠ sr♣t♦♥

♦♥ tr♠♦②♥♠ ss ♦ t ♣♥♦♠♥♦♥ ♦♠s ♦t ❬❪

r ♠t ♦ ♥t♦♥ ♣r♦sss ♥ st♦s♦ ♣s tr♥st♦♥

♦♥sr ♣♦ss tr♦♥♦s ♥t♦♥ ♦ ♦♥♥s ♠ ♦♥ r②st♥

sstrt s s♠ ♥ t ♣♦st♦♥ ♥ ♦rs s♦② t ♣♦r s

s♣rstrt Ss > 0 t ♠♥s tr ①st ∆P r♥ t t♦♠s t♦r t

sstrt q s♠♠r③s ts ♦♥t♦♥

∆Gv = −KBT

Valn(1 + Ss),

♣♣♥① ♦♥♣ts ♦r rs ♥ qr♠ ♥ ♦r t ♥ ♠r♦t

r ∆Gv s t ♥ ♥ t r ♥r② ♣r ♥t ♦♠ KB s t ♦t③♠♥♥

♦♥st♥t Va t t♦♠ ♦♠ ♥ Ss ≡ PV −PS

PSs t ♣♦r s♣rstrt♦♥ t

PV − PS ♥ t r♥ t♥ ♣rssrs ♦ t ♣♦r ♥ ♦ t s♦

rs♣t②

♠♦s ♦r t♦♠s ♠♣♥ ♦♥t♦ t sstrt ♥ ♣r♦ ♥ r

s♦♥ ♣♣r♦①♠t♦♥ s♣r ♥s ♦ ♠♥ rs r ♥ ∆G ♦r t

sstrt♠ s②st♠ s t♦ tr ♦♥trt♦♥s t rst ss♦t t

♦♥s t♦ rt t ♥s ∼ r3∆Gv s♣r ♠♣♦r tr♠ ♦♥ t

♥ ♥s ∼ r2γfv ♥ t ♦♥trt♦♥ ♠r♥ r♦♠ t r♥ t♥

t ♥r② t t ♦ sstrt♣♦r sr ∼ r2γsv ♥ t t ♥ sstrt♠

♥tr ∼ r2γsf t rs

G = a1r3Gv + a2r

2γfv + a3(r2γfs − r2γsv),

r a1 = π3(2− 3cosθ+ cos3θ) a2 = 2π(1− cosθ) a3 = πsen2θ ♥ θ s t tt♥

♥

♥ t tr s ♣♦t ♦ ∆G s ♥t♦♥ ♦ r t ♥ ♥st qr♠

♣♦♥t d∆Gdr

= 0 tr s rt ♥s s③ r∗ r♦♠ s♥ r♦t ♦♠s

♥rt② ♦r ♥ ♦tr ♦rs t st ♥ ♠♦♥t ♦ ∆G∗ = ∆G(r∗) ♥r②

♠st s♣♣ t♦ t s②st♠ ♦r trr♥ s♣♦♥t♥♦s r♦t r t ♥s

s③ r♦s ♥♥t②

♥ ♠♣♦rt♥t qst♦♥ rss ♣ ♦ s♥s r t st s③ s♥ ♦r

r < r∗ t② t♥ t♦ sr♥ ♥ ♠st t♦ t ♥t♦ ♦♥t tt tss ♣r♦sss

r st♦st ♥ tr♠ tt♦♥s KBT ≈ ∆G∗ r rs♣♦♥s t♦ ②

♥st② N∗ ♦ st s♥s ♣r ♥t t♠ ss♠♥ ♦t③♠♥♥ sttst t♥

N∗ ∼ exp[−∆G∗/KBT ] ♥ ∆G∗ ♠st ♥t♦♥ ♦ ♥ ♦ t ♠♦r ①

❬t♦♠s/cm2s❪ ♦♥ ♥ t t ♥ ♦t ♦ ts ♣r♠trs t ♥t♦♥

♣r♦sss

♥ d2∆G

dr2< 0

♣♣♥① ♦♥♣ts ♦r rs ♥ qr♠ ♥ ♦r t ♥ ♠r♦t

r tt ♥ ♦ t r ♥r② s ♥t♦♥ ♦ t ♠♥ ♥s s③ r♥ st♦s♦ ♣s tr♥st♦♥

t♦♥ ♣♥♥ ♦♥ sstrt t♠♣rtr

t s rs♦♥ tt ♠♦r ① s rt② rt t♦ S s tt ♦♥ ♥

r♣ 1+Ss ② F/Fe ♥ t q t Fe ♥ t ♣♦rt♦♥ rt r♦♠ t ♠

t qr♠ ♥srt♥ ts ♥ ∆G ♥ ①trt♥ r∗ r♦♠ t ♥st qr♠

♣♦♥t ♦♥ ♥ s♦ ♦♥sr♥ ∂γ/∂T ≈ 0 ♠ ❬❪

(∂r∗/∂T )|F ∼ [a2γfv + a3(γfs − γsv)]/T 2.

t ts ♣♦♥t t s r tt t ♦r ♦ t rt s③ s ♥t♦♥ ♦

♣♥s ♦♥ t rt♦♥s♣ t♥ t sr t♥s♦♥s ♥ ts rt♦♥s♣ s

s♦ s t♦ sr r♦t ♠♦s ♦ ♠s s ♥①t st♦♥ ♦r ♥st♥ t♥

t s♠♣st stt♦♥ r γfs ≈ γsv ♦♥ s (∂r∗/∂T )|F > 0 ♥ ♦② ♠♥

rt s③ ♦ t s♥s ♦♠s rr s t♠♣rtr ♥rss t st ♦r r♥

♦ s t rst ∼ 1/T 2 ② s♥ t s♠ ♣r♦r ♦♥ ♥ s♦ ♥

∂(∆G∗)/∂T )|F > 0,

s r∗ t ♥r② rrr ∆G∗ ♥rss s ♥rss s rst ♠♣s tt

♣♣♥① ♦♥♣ts ♦r rs ♥ qr♠ ♥ ♦r t ♥ ♠r♦t

t ♥t♦♥ rt s r ①♣♦♥♥t② ♥ ♦r♥ t t ♦t③♠♥♥ stts

t tt② tss rsts r② ♥ ♦♥ ♥ t ss ♦r ♦♥

s♣r♦♦ q t♥ ♥ tr r♦♣s ♥ t♦ t s ❬❪ r②st③

t♦♥s ♦ s②♥tt r♥t ♥ r♥♦♦rt ❬❪ s s t ♠♦s r♦t ♦ ♦♥

❬❪ r t t♦s rsts ♦r tr r s♦ ①♠♣s ♦♥ ♥ t

♦♣♣♦st rt♦♥ rrr t ❬❪ ①♣♦r rt tr♠♥s♦♥

s♥ ♦r♠t♦♥ ♥ t r♦t ♦ ♦♥ sstrts ♦r s♦♥ tt t

♥t♠♦ts s ♥st② ♥rss t ♥ t r♥ ♦ t♦ 300 C st

t ♠♥ s s③ rss

r t♦♠ ♦r ♠r♦s♦♣② ♠s ♦ ♥t♠♦ts r♦♥ ② ❲ ♦♥ sstrts t 200 C 250 C ♥ 300 C ♦t ♥st② ♥ ♠tr• s ♥t♦♥ ♦ ♦rts② ♦ Pr♦ rrr s rsts ♥ ♦♥ ♥ t ❬❪

t♦♥ ♣♥♥ ♦♥ ♠♦r ①

♥ ♦♥ s r∗ = r∗(T, F ) t s strt♦rr s♦♥ t ♣♥♥ ♦♥ F

t ♦♥st♥t ♦♥sr♥ t s♠ s♠♣st stt♦♥ ♦♥ rs t♦

(∂r∗/∂F )|T < 0 ♥ [∂(∆G∗)/∂F ]|T < 0.

q ts s tt ♥ s③ r s♠r s t ♠♦r ① ♥rss t t

t s♠ t♠ t ♥t♦♥ rt s r s rst r t t ♥tt ♥

tt ♥ ♠♦r ♣rts rr ♦♥t♦ sstrt ♣r ♥t t♠ tr s ♥♦ t♠ ♥♦

♦r t ♣rts ♥ ♥ r② st s♥ r t② ♥ ♣ ♣ ♦♥sq♥t②

♣♣♥① ♦♥♣ts ♦r rs ♥ qr♠ ♥ ♦r t ♥ ♠r♦t

t s ② t② ♦r♠ ♠rs ♥ ♦tr ♥st s♠ s♥s s rsts ♠t

♦r ♥st♥ t t♦s r♣♦rt ♦r t r♦t ♦ ♥ss s ♦♥ s

sstrts ② ❬❪ t T = 490 C ♥ F = 0.016 s−1 t s ♥st②

s ♦t 0.2 × 1011 cm−2 ♥ tr ♠tr s ♥r r♦♠ ♦r s F s

♥rs t♦ 0.094 s−1 t ① t♠♣rtr t s ♥st② s♦ ♥rss t

st t♠s rs t s ♠tr ♦s t♦ ♦t

ts ♦♥r♥ ♥ ts ♥ ♥ rt t♦ t♠ ♥ ♦♥ ♥ t ♣

♦ t ❬❪ s s ♥ t r② t s ❬❪ ♥ s

tr♥

r♦t ♥ strtr ♦ ♠s

r♦t ♦s

♠♥ qr♠ ♦♥t♦♥ q ♣rt ② t r♦♣ q

♠♦ ♣r♦ rs♦♥ ② t♦ ①♣♥ tr s r♦t ♠♦s ♦r s♦s

γsv = γsf + γfv cos θ.

❲♥ t ♠ ts t ♦ sstrt sr ♥t♥ t♦♠♥s♦♥

s♥s ♦♥ t ♥tr t♥ θ ∼= 0 ♥ γsv ≥ γsf + γfv s s t r♥♥

r r ♦r ②r②②r r♦t ♠♦ ♦t♥ ♦sr ♥ ♠t ♣♦st ♦♥

♠t ♥ ♦♠♦♣t① r♦t ❬❪ ♥ ts ♠♦ ② t ssq♥t ②r strts

r♦♥ st tr t ♦♠♣t ♦r♠t♦♥ ♦ t ♣r♥t ②r

trs ♥ tr s ♥♦t t tt♥ ②r ♥ tr♠♥s♦♥ s♥s

r ♥t rt② ♦♥ t sstrt t ❱♦♠r❲r ❱❲ r♦t ♠♦ ts

♣ ♥ ts stt♦♥ ♣rts ♦ t sstrt r♠♥ ①♣♦s ♥ t ♥r② s ♠♥

♠③ r♥ t ♠♣♦r ♥tr s♥ γfv ≫ γsv +γsf r♦♥ ♦♥

♦r ♥st♥ ♦♦s ts ♥ ♦ r♦t ❬❪

st② t tr♥srst♥♦ r♦t ♠♦ s ♥rst♦♦ s tr♥st♦♥

t♥ t♦ ❱❲ ♠♦ ❬❪ t t rst ♠♦♥♦②rs ♦ ♣♦st♦♥ γsv ♦♠♥ts

♣♣♥① ♦♥♣ts ♦r rs ♥ qr♠ ♥ ♦r t ♥ ♠r♦t

r♦t ♠♥ tt♥ tr tt rt tr♠♥s♦♥ s♥s strt t♦ ♥t

♦♥ t ssq♥t ②rs ♥② ♥rt str ts ♠t str♥ t ♥

t ♦r♥ ♦ ts tr♥st♦♥ ❬❪ t②♣ ①♠♣ ♦ r♦t ♠♦ ♦rs

♥ ♠s ♣♦st ♦♥ r ♠tst ♣s ♦♥t♥♥ s♠ strs

♣rs t ♠r♦s♦♣ tr♠♥s♦♥ r♦t t ❬❪

r ♠s r♦♥ ♦♥ tr♦ ♠♦ ♥♥♥ ♥♥♥ r♦s♦♣② ① ♠ ♦ t strs ♦♥ ②rs r♦♥ ♥ t r♦t ♠♦ strs r . ♦♥ ♥ ♥ s tt ts s s♥tr ♣r♥ ♠r♦s♦♣ tr♠♥s♦♥ ♥t♦♥s ♠ ♦ t ssq♥t r♦t ♦♠♥t ② t ♠r♦s♦♣ tr♠♥s♦♥ s♥s s strs r∼ ♥ ♠ rr t♥ t t strs ♥ ♥♦t ♥ ts s♠s ①trt ♥ t r♦♠ ❬❪

t s r② ♠♣♦rt♥t ♠♥t♦♥ tt t r♦t ♠♦ ♣♥s ♥♦t ♦♥② ♦♥ t ♠

trs ♥♦ ♥ t ♣r♦ss t ♠♥② ♦♥ t r♦t ♣r♠trs ♥♥ tr

♠♣rts r ♣rs♥t r♦t ♦ ♦♥ ♥ t r♥ ♦ t♠♣rtr r♦♠

t♦ ♦r ♥st♥ ♥s r② r♦♠ tr♠♥s♦♥ ❱❲ t♦ st♣♦

r♦t ♦r s♠♦♥♦②r ♣♦sts ♦ r ♣rs♥t s♦ t r♦t ♦r

♦♠s ②r②②r ♦r t ♥tr r♥ ♦ t♦ ❬❪

♦♠♠♥srt② ♥ P♦②r②st♥t②

r♥ ♥t♦♥ ♥r r② s♣ ♦♥t♦♥s t ♠ ♥ ♦♣② t

sstrt r②st♥ strtr ❬❪ ♥ ts ss ♦♥ s②s tt t r♦t s ♦♠♠♥

srt ♦r ♣t① ❬❪ r s♠♦♥t♦r s ♣♥s ♦♥ ♣t①② ♠♦st

①♣♥ ♥ t st♦♥

♣♣♥① ♦♥♣ts ♦r rs ♥ qr♠ ♥ ♦r t ♥ ♠r♦t

♦♠♠♦♥s r ♦♣t♦tr♦♥ ♦♥s s s ♥ srs s ♦♥ ♥ ♠trs ♥

t♦s s ♥ s♣ ♠r♦tr♦♥ t rss ♦♠♠♥t♦♥s ♦♠♣♦s

♦ x1−x ❬❪ ②♣② t rst st♣ ♥ ♣t① ♠s s t ♦♠♦♣t① ♦♥

r♦t ② ❱ rs♦♥ s tt t ♣②r s r r♦♠ ts ♣rr t♥ t

r ♥ ♥ ♦♣ ♥♣♥♥t② ♦ t

♣t①② ♣♥s ♦♥ t ♠s♠t t♥ t tt ♣r♠trs |af −as|/as♦ t ♠ af ♥ ♦ t sstrt as ♥♥ tr tr♠ ♣♥♥ t

tt♦♥ ♦♥t ❬❪ t ♦rs tr t rt ♠s♠t s ♦

≈ 10 − 15% ❬❪ rtss ①♦t ss r ♣t①② s t tr♦

♣t① ②rs r♦♥ ♦♥ s♦ ①st ❬❪ ♥ s♣t ♦ ♠♦st 19% ♥

48% ♠s♠t ♦ t tt ♣r♠tr ♥ tr♠ ♦♥t rs♣t② rr

♠s♠t ♠♣s rr ♠t str♥ st ♥r② ♥ rs s

ts ♦r ♥ s♣♣♦rt tr♥st♦♥ r♦♠ t♦ ❱❲ r♦t ♠♦

t s ♦rt ♠♥t♦♥ tt t t ♦ tr♦♣t①② ♥ ♥ ♥ t r♦♣

q ♠♦ r♣♥ t tr♠ ∼ ∆Gv ② [∼ (∆Gv + ∆Gs)] ♥ t q r

∆Gs ♥srts t ①tr ♥r② ♠t ♥ t str♥ ♦r♠ ∆G∗ ♥rss ♥

t ♥t♦♥ rt t♦ ♥ tr ts s ♠♣rts ts ♠ rt♦♥s

s♦ ♥ ♥srt ♥t♦ t ♠♦ ♥ ♦rr t♦ t t ♥ ♦t tr ♦♥sq♥s

♦♥ ♥t♦♥s ❬❪

❯♥ ♣t① s♥r②sts ♣♦②r②st♥ ♠s r ♦♠♣♦s ② ♥ ♦

t♦♥ ♦ r♥s ♦ t♠ ♥ ts ♦♥ r②st♦r♣ ♦r♥tt♦♥ s ♥

r♦♥♠♥ts t♦ ♦♠♣① ♦♠♣tt♦♥ t♥ ♥♦r♥ r♥s r r♦

② t sr t♥s♦♥s ♦♥ γ = γ(hkl) ❬❪ ♥r② ♣rr♥t rt♦♥

♦ r♦t Λ t①tr ❬❪ ♣♣rs ♥ ♦s s ♣♦st♦♥ ♣r♦s

r♥sΛ r♦ str t♥ t ♦trs ♥ ♥t② ♦♠♥t t sr ①tr

ts ♠♥ ♥ tr trs ♦ ♠s ♥♥ tr st ♠♦s ②

str♥t ♠♥t ♣r♠t② t♥ rt s♦♥ rt ♥ ♦trs ❬❪ t♥r

♣r♦r q♥t②♥ t①tr s s ♦♥ ❳r② rt♦♥ t♥qs ❬❪

♣♣♥① ♦♥♣ts ♦r rs ♥ qr♠ ♥ ♦r t ♥ ♠r♦t

♥t P♥♦♠♥ ♥ ♣r trtrs

r②st sr r♦t s rr♦♠qr♠ ♣♥♦♠♥♦♥ r♥

r♦t r♦♠ ♣♦r ♣s t♦♠s ♠♣♥ ♦♥t♦ t sstrt ♥tr r♥ ♥♦r

♦r♠♥ ♦ ♦♥s ♥♥ t ♠♦r♣♦♦② ♥ ♠♣r♥ t s②st♠ t♦ ♠♥♠③

ts r ♥r② ❬❪ s t sstrt s t t ♦sr t♦♠s ♦r ♠♦s

t♦♠s ♠t s ♦♥ t sr ♣r♦r♠♥ r♦♥♥ ♠♦t♦♥s ♦♠

♠♦♥② ts t♦♠s st t ♣♦st♦♥s ♠①♠③ tr ♥♠r ♦ ♦♦r♥

t♦♥ ♥ ♣♦st♦♥ ♦rs ❬❪ r♠ tt♦♥ ♠② s♦ ♥s s♦r♣t♦♥

t♦♠s t sr rtr♥♥ t♦ t ♣♦r ❬❪ ♣♦st♦♥ s♦r♣t♦♥ ♥

rt♦♥ ♠t ② s♦♥ r t s ♠♥s♠s r♥ ♥r r♦t

♦rs s♣ stt♦♥s ①s ♦♥ ♠② t♦ ♦ ts ♣r♦sss t r ♦♥

sr♥ t s s st t♦ ♥ ♦tr t ♦ r tr♠② tt

♥♠② t② ♦r t rts ♥ ② rr♥s s q ❬❪

τ = τ0 exp [−Eτ/KBT ],

r τ s rt ♦ ♣rtr ♥t τ0 ♦♥st♥t ♥ Eτ s t ♥r② ss♦t

t t τ ♣r♦ss

r ♠t ♦ ♥t ♣r♦sss ♦rr♥ r♥ t r♦t

♣♣♥① ♦♥♣ts ♦r rs ♥ qr♠ ♥ ♦r t ♥ ♠r♦t

r②st sr s ♥♦t ♣rt ♣♥ ♥ t ♣rs♥ts r ♣s trrs

s♣rt ② st♣s ♦ t♦♠ t ♦r ♠t♣s ♦ ♥ t♦♠ t s ♥ ♦♦

t ♥ tss st♣s r ♥♦t ♣rt② strt rtr t② ♦♥t♥ strt

♣rts s♣rt ② ♥s ❬❪ Prts rr♥ t t sstrt ♥ ♠♦♥t ♦

♥r② rt t♦ t ♣♦r ♣s ❬❪ ② s ts ♥r② ♦r r♥ ♠ ♦♥s

r② sts t sr ♥♦r ♦r ♦♠♥ t♦♠s t♦♠s s t

rt τdiff s♥ tt ♦r ♦♣ ♥ t♦♠ ♠st ♦r♦♠ t tt ♣♦t♥t

①st♥ t♥ ♥♦r♥ ♣♦st♦♥s s s ♣♦t♥t ♣♥s ♦♥ t

♥tr ♦ ♦♠♣♦♥s ♥♦ ♥ t r♦t ♥ ♦♥ t sstrt

Prr♥t sts ♦r st♥ r t ♦ st♣s ♥ ♥s ♦♥ t②

rr ♥♠r ♦ ♥♥ ♦♥s ❬❪ ♦r t♦♠s r ts ♣♦st♦♥s ♦♥②

t t♠♣rtr s s♥t② ♦r ♣r♦♠♦t♥ ♥t s♦♥ xdiff ♦♠♣r

t♦ t trr s♣rt♦♥ ♥t xterr ❬❪ t♣♦ r♦t s ♥ xdiff ≫xterr ❬❪ ♥trst♥② ♥ t♦♠ ss ♦♥ trr t t♦rs ♥♦tr ♦♥

t t♥s t♦ ♦♠ ♥st t♦ ♠♣ ♦ t trr s s ♥♦♥ s r

♦♦ t ❬❪ ♥ t t t ♦ t rrr s

♥ ss r t rrr ♦♠♥ts t r♦t s Pt r♦♥ ♦♥ Pt

♥ t r ❬❪ strtrs ♦♦♥ ♥ s r ♦r♠ ❬❪ ♠♦♥ sr

②s t♦ ♥ ♥ ①♣rss♦♥ ♦r t s♦♥ ♥r② tt ♥s t rrr ♦♥

s② ♠♣♦② s

Ediff = E0 + nEn + Ees,

r n s t ♥♠r ♦ ♦♦r♥t♦♥ En s t ♥r② ♣r ♥t ♦♥ ♥ Ees s t

ss♦t ♥r② t♦ t rrr ②♣② En ∼ 0.1 ❱ ♥ Ees ∼ 0.01 ❱ ❬❪

♦r t s ♦rt ♣ ♥ ♠♥ tt ts s ♠② ♥ ♣♣r② ♣♥♥

♦♥ t s②st♠ ♦♥sr t♦♥② t q s s♦② s♠♣ ♠♦

♦s ♥♦t t s♦r ♦ ♥rs ♥♦ ♥ r ♣r♦sss

♥ ♥ t♦♠ stss ♦♥s ♣♦st♦♥ ♣♣♥s ❲♥ ♥ t♦♠ ♠ts

♥♦tr t♦♠ t② ♥ ♦r♠ ♠r ♠rs s♦ s t t s ♠ss

♣r♦ t♦ t ♥♥ ♦ Ediff ♦♠t♠s ♠rs r♦ ♥ ♦♠ t♦ ♦r

♣♣♥① ♦♥♣ts ♦r rs ♥ qr♠ ♥ ♦r t ♥ ♠r♦t

r ♠t ♦ t tt ♣♦t♥t

tr♠♥s♦♥ s♥ t♥ t♦ ♣tr t♦♠s r > r∗ t t

s♠ t♠ s♦♠ t♦♠s r ♦ tr ♦♥s ♥ t sr ♦ s♦r♣t♦♥

♦rs s♦r♣t♦♥ s ♥ t ♦♠♣t♥ t ♣♦st♦♥ t♦ ts t t

♠♦r ① ♠t r♥t r♦♠ t ♣♦st♦♥ rt ♦r ♥ s

♥ rt t♥q ♦♥t♦♥s s♦r♣t♦♥ s ♥ ♣r♦ss ❬❪ ♦r

♦♥ s ♦r t r♥ t♦ t s♦r♣t♦♥ ♥r② s Edes ≈ 2.5

❱ ❬❪ ss♠♥ 1/τ0des = 10−14 s ♥ T = 850 ♥ t ♥rs ♦ q ♦♥

♦t♥s tt ♦♥② ♦♥ t♦♠ s s♦r r♦♠ t sr ♥ s ♦♠♣r♥

ts s♦r♣t♦♥ rt t t ♣♦st♦♥ rt s ♥ ∼ 1016−18 t♦♠s cm−2 s−1

t ss t♦ s tt s♦r♣t♦♥ ♣♥♦♠♥♦♥ s ♥♦♣rt ❬❪

♥rs ♦ ♥ rt s t rtrst t♠ ♦ ts ♣r♦ss

st ♦ r♦♥②♠s ♥ ②♠♦s

r♦♥②♠s

r♦♥②♠ sr♣t♦♥

t♦♠ ♦r r♦s♦♣②

st ♣♦st♦♥

♦rrt♦♥ ♥t♦♥

♦♥sr strt ♦♥♦

❱ ♠ ❱♣♦r ♣♦st♦♥

♦♥③

P rt P♦②♠r ♥ ♥♦♠ ♠

sr♠♠♦r♥

❱ ①tr♠❱ ttsts

❲ rs❲♥s♦♥

r♥♥rr

❱ ♠②❱s

s r♥ ♦♥rs

ss♥ rt♦♦♥ ♥s♠

P ♦ P♦st♦♥ ②st♠

❯ ss♥ ❯♥tr② ♥s♠

t ♦ r♥s s s ♣r♠tr ♥ ♥t ♦♥t r♦ ♦

s t strt♦♥s

❲ ♦t ❲ ♣t①②

♥t ♦♥t♦♥s

st ♦ r♦♥②♠s ♥ ②♠♦s

P❩ rrPrs❩♥

s t ♠ss♦♥ ♦s

♦r ♠ ♣t①②

♥srr♥

♦♥♦②r

s ①♠ t t strt♦♥s

♣ ♣r♦t② ♥st② ♥t♦♥

P P♦②rr♦t

s ♥t♠ ♦ts

♥♦♠ ♣♦st♦♥ t r ①t♦♥

strt ♦♥♦

♥♥♥ tr♦♥ r♦s♦♣②

tr♥srst♥♦

s qr ♦ ♦♥ss strt♦♥s

♥♥♥ ♥♥ r♦s♦♣②

P ♦t② s②♠tr ①s♦♥ Pr♦ss

❲ r②❲♦♠

❯ ❯♥rst② ss

❱ ❱♥ssr♠

❱❲ ❱♦♠r❲r

❲❱ ❲♦❱♥

❳P ❳r② P♦t♦tr♦♥ ♣tr♦s♦♣②

❳ ❳② rt♦♥

②♠♦s

t♥ ②♠♦s sr♣t♦♥

af tt ♣r♠tr ♦ t ♠

st ♦ r♦♥②♠s ♥ ②♠♦s

as tt ♣r♠tr ♦ t sstrt

A tt ♠♣t ♥ t P❩ ♦♥t①t

Af r ♦ ♠

Aθ−2θ s♦rt♦♥ t♦r ♦r θ − 2θ ❳ ♦♠tr②

Ch tr♥ ♦rrt♦♥ ♥t♦♥

Cs ♣t ♦r♥

d ♠♥s♦♥ ♦ r♦t ♣r♦ss

ds strt ♠♥s♦♥√D ♠♣t ♦ t t ♥♦s

E0 ♥r② ♦ t tt ♣♦t♥t

Ediff rtrst ♥r② ♦ s♦♥

Ees r♦♦ rrr ♥r②

Egap ♥r② ♣

En ♥r② ♣r ♥t ♦♥

EGB ♥r② rrr t s ♦ ♦ ♥♦r♥ r♥s

ER ①t♦♥ ♥r② rrr t t s ♦ ♦ ♥♦r♥ r♥s

Eτ ♥r② ♦ ♣rtr ♥t t

F ♦r ①

F0 ♥s strt♦♥

Fe ♣♦rt♦♥ rt r♦♠ t ♠ t qr♠

FH ♠♦t③ r ♥r②

G s r ♥r②

Gs s r ♥r② rt t♦ str♥

Gv s r ♥r② ♣r ♥t ♦♠

G∗ rt s r ♥r②

G(X;m) ♠ ♣ ♦ t ❳ r ♠t ♦rr

hi r t t t st

h(x, t) r t t t sstrt ♣♦st♦♥ x t t♠ t

〈hn〉c ♥t ♠♥t ♦

r t ♥ t r♣r♦ s♣

st ♦ r♦♥②♠s ♥ ②♠♦s

I111 ♥t♥st② ♦ t ♣ ♥ ❳ s♣tr♠

j rr♥t ♥st②

J rr♥t ♥st② ♣r t♦ t sr

K rt♦ss ♦♥t

Kd tr♥t ♦ s♦♥

KB ♦t③♠♥ ♦♥st♥t

l∗ rtrst ♦① s③

L tr s③ ♦ t sstrt

mn ♥t♠♦♠♥t ♦ ♣

m∗ ①♠ t rt t♦ t ♠♥ t ♦ ♥ ♥tr

ncoar ♦rs♥♥ ①♣♦♥♥t

Nh ♠r ♦ ♣♦♥ts t t sr t t

N∗ ♥st② ♦ st s♥s ♣r ♥t t♠

p(h) ♥st② ♣r♦t② ♦ t r h

♣(k) rtrst ♥t♦♥

P Prssr

PD Pr♦t② ♦ ♣rt s♥ t♦rs t st

PR Pr♦t② ♦ ♦rr♥ r①t♦♥ ♣r♦ss t s

PS Prssr ♦ ♣♦r t s♦

PV Prssr ♦ t ♣♦r ♣s

P (h) Pr♦t② ♦ t r h

Q ♥r③ tr♠♦②♥♠ ♦r

r ♥ rs ♦ qss♣r ♥s

rc r r♥ s③

rm rst ♠♥♠♠③r♦ ♦ t ♦♣♦♣ s♣t ♦r♥

r∗ rt s③ ♦ ♥ s♥

R ♦ rtr

S ♥ss ♦♥t←→S ♣t♠ rr♥t

S(k, t) trtr t♦r ♦r P♦rs♣tr♠

st ♦ r♦♥②♠s ♥ ②♠♦s

Ss ♣rstrt♦♥

t r♦t t♠

th ♥ss ♦ t♥ ♠

tx r♦ss♦r t♠

T ♠♣rtr ♦ t sstrt

TEW strt t♠♣rtr ♦rrs♣♦♥♥ t♦ ♥ ❲ r♦t

TEW−KPZ strt t♠♣rtr ♦rrs♣♦♥♥ t♦ ♥ ❲t♦P❩ r♦ss♦r

v r r♦t ♦t② ♦ ♥ ♥tr

v∞ s②♠♣t♦t r♦t ♦t② ♦ ♥ ♥tr

V ❱♦♠

Va t♦♠ ♦♠

wloc ♦ r♦♥ss

wsat trt♦♥ ♦r t r♦♥ss

w(L, t) ♦ r♦♥ss

W r♠♦②♥♠ ♦r

xdiff rtrst ♥t ♦r s♦♥

xdiff ♥t ♦ trr

z ②♥♠ ①♣♦♥♥t

r ②♠♦s sr♣t♦♥

α ♦♥ss ①♣♦♥♥t

α1 ♦♠tr ①♣♦♥♥t

αs ♣tr r♦♥ss ①♣♦♥♥t

γ r ♥r② ♦r r t♥s♦♥

γfv r t♥s♦♥ t t ♠♣♦r ♥tr

γp ♦♥♥rs ♣r♠tr ♥ t P❩ s♥ t♦r② ♦♥t①t

γsf r t♥s♦♥ t t sstrt♠ ♥tr

γsv r t♥s♦♥ t t sstrt♣♦r ♥tr

Γ ♦♥♥rs ♣r♠tr ♥ t P❩ s♥ t♦r② ♦♥t①t

st ♦ r♦♥②♠s ♥ ②♠♦s

Γf ♠♠ ♥t♦♥

Γ(l, t) ♦♣♦♣ s♣t ♦r♥

ǫ t♥ ♥

ζ rtrst ♥t ♥ t sr

ζp ♦♥♥rs ♣r♠tr ♥ t P❩ s♥ t♦r② ♦♥t①t

η ♦s

ηp ♦♥♥rs ♣r♠tr ♥ t P❩ s♥ t♦r② ♦♥t①t

θhkl ❲tt♥ ♥

θhkl ♥ ♦ rt♦♥ ♦r t ♣ ♥ ❳ ♠sr♠♥t

Θ ♥t♦♥ ♦ ① ♥ t

κ ♣♣ ①♣♦♥♥t

λ ①ss ♦ ♦t② ♦ ♥ ♥tr

λCuKα❲♥t ♦ t rt♦♥ ♥ α tr♥st♦♥ ♦r t♦♠

Λ Prr♥t rt♦♥ ♦ r♦t ①tr rt♦♥

µ ♠ ♣♦t♥t

µv ♠ ♣♦t♥t ♥ ♣♦r ♣s

ν tr♥t ♦ t sr t♥s♦♥

ξ ♦rrt♦♥ ♥t

ξ|| Pr ♦rrt♦♥ ♥t

Ξ r♥ ♣♦t♥t

σh t♥r t♦♥ ♦

σχ t♥r t♦♥ ♦ χ

ς t♦r

τ t ♦ ♣rtr ♥t t

τ0des t ♦ s♦r♣t♦♥

τdiff t ♦ s♦♥

Φ ❯♥rs s♥ ♥t♦♥

χ ♥♦♠ r

〈χ〉 ♥ ♦ chi

〈χ2〉c ♦♥ ♠♥t ♦ χ

st ♦ r♦♥②♠s ♥ ②♠♦s

ψk P♦②♠♠ ♥t♦♥ ♦ t♦rr

Ψ ❯♥rs s♥ ♥t♦♥

Ω s♣t rt♦ ♦ r♥♠♦♥

♦r♣②

❬❪ r♥ trs s♥ ♦ t♥ ♠s ♠ ♣rss

❬❪ ❱♥s ♥tr♦t♦♥ t♦ sr ♥ t♥ ♠ ♣r♦sss ♠r ❯♥

rst② Prss

❬❪ P♠♣♥ ❱♥ P②ss ♦ r②st r♦t ♠r ❯♥rst② Prss

❬❪ r♦t P rt ♥ t ♦♠♣♦♥s P②ss ts tr♦

♥ ♥♦strtrs r②st r♦t rs ♥ ♣♣t♦♥s sr ♥

❬❪ ♥ ❨ ❲ P♦♣s② P♥♥②♦♦ P ❲ ❨♥

P ①② ♣♥ ss♠ P♥♥②♦♦ ❨ ❨♥ r♥

♦♥r②♥♥ rrr ♦t♦♥ ♥ ♦r s P②s tt

❬❪ ♠s ❱ ♦♥s ❱ ♥õ③ rr ♣②rs ♦r ss

♥ ♦♣t s ♣♣ P②s

❬❪ ♥♥ r♦♥ s r P st♦s ♦♥tr♦

r♦t ♦ ttr♣♦r♥ ♥♦r♥ ♥♥♦r②sts tr trs

❬❪ rrr P ♦♥ts s rtr③t♦♥ ♦

q♥t♠ ♦ts r♦♥ ♦♥ ② ♦t ♣t①② ♣♣ P②s

❬❪ r♦ rrr s ♦♥ts

♦♥t♦r♦ ♠r③ ♦t♦♥ ♦ r②st♥ ♦♠♥ s③ ♥ ♣t① ♦r

♥tt♦♥ ♦ q♥t♠ ♦ts ♣♣ P②s

❬❪ ❲♠s P ② ♥s r♦s r♦strtr ♥

♣♦♥t ts ♥ ♥♥♦rs ♦r ♣♦t♦♦t ♣♣t♦♥s ♥♦t♥♦♦②

❬❪ t ♦ ❱ rr② ♥ré ♥ ♥r

♠♥♥ r ♦♥♦♥♥ t t♠♣rtr

trst ♣♦rt♦♥ ♣r♠tr ♠♣t♦♥ ♥ s♠♦♥t♦r ♠r♦ts

tr ♦♥♦♥

❬❪ rrr r♦ ♥③s♦r♥♦ rrr

s t ♦ t♠♣rtr ♦♥ t rst ♥ r♦t ①♣♦♥♥ts ♦

♣♦②r②st♥ ♠s ♣♣ P②s tt

❬❪ r♦ r rrr P ♦ts ♦

t♠♣rtr r♦t ♦ qt② ♣♦②r②st♥ ②rs P②s ♣♣

P②s

❬❪ t rrr r♦ rrr ♥♦♠♦s s♥

♥ s♣rr♦♥ss ♥ t r♦t ♦ ♣♦②r②st♥ ♠s P②s

❬❪ s♠♥t♦ rrr rrr t ♥♦♠♦s s♥ ♥

t ♣t① r♦t ♦ s♠♦♥t♦r ♠s r♦♣②s tt

❬❪ ss♥r ❱ ③♥② P♦r②rs ♦ r ♦♣♦r♣② ts t♦

trs Pr♦♣rts ♥

❬❪ ♥s♦ ② r ♦r♠ ttst ♥②ss ♦

♦♥ rs ♣♣t♦♥ t♦ t ♦t♦♥ ♦ ❯trt♥ ♦ ♠ ♦r

♣♦♦② t ♣♦st♦♥ ♠♣rtr ♥♠r

❬❪ rr ttst ♣②ss ♦ s ♠r ❯♥rst② Prss

❬❪ rás rt ♠r♥ ♦ s♥ ♥ r♥♦♠ ♥t♦rs ♥

❬❪ P t♥ ♠♥ ②rs r♥ ♥♦s tr

❬❪ st♥♦ ♦rt♥t♦ ❱♦rt♦ ttst ♣②ss ♦ s♦ ②♥♠s

♦ ♦r♥ P②s

❬❪ rás t♥② rt ♦♥♣ts ♥ sr r♦t ♠r

♥rst② ♣rss

❬❪ ♥r♦t rt ♦♠tr② ♦ ♥tr ♠s ♦♦s

❬❪ rr Prs ❨ ❩♥ ②♥♠ s♥ ♦ r♦♥ ♥trs

P②s tt

❬❪ ♣♥② ❨ ❩♥ ♥t r♦♥♥ ♣♥♦♠♥ st♦st

r♦t rt ♣♦②♠rs ♥ tt s♣ts ♦ ♠ts♣♥r② sttst

♠♥s P②s ♣

❬❪ ♦r♥ rrPrs❩♥ qt♦♥ ♥ ❯♥rst② ss ♦r②

♣♣

❬❪ ♦♥ss♦♥ ♣ tt♦♥s ♥ ♥♦♠ trs ♦♠♠ t P②s

❬❪ Prä♦r ♣♦♥ ❯♥rs strt♦♥s ♦r r♦t Pr♦ss ♥

♠♥s♦♥s ♥ ♥♦♠ trs P②s tt

❬❪ r② ❲♦♠ s♣♥ strt♦♥s ♥ t r② r♥ ♦♠

♠♥ t P②s

❬❪ r② ❲♦♠ ♥ ♦rt♦♦♥ ♥ s②♠♣t ♠tr① ♥s♥s ♦♠

♠♥ t P②s

❬❪ rr ♦♠♠♥t ♦♥ t ♦♥♥ ② ♠♣rts t ♥t t♠♣rtrs

P②s tt

❬❪ s♠♦t♦ ♣♦♥ ♥♠♥s♦♥ rrPrs❩♥ qt♦♥ ♥

①t ♦t♦♥ ♥ ts ❯♥rst② P②s tt

❬❪ ♠r ♦r♥ st Pr♦t② strt♦♥ ♦ t r ♥r② ♦ t

♦♥t♥♠ rt r♥♦♠ ♣♦②♠r ♥ ♠♥s♦♥s ♦♠♠♥ Pr ♣♣

t

❬❪ P rs P ♦ss ♦ss♦ r♥r② strt♦♥ ♦ t rt

♣♦②♠r t t♠♣rtr r♦♣②s tt

❬❪ ❱ ♦ts♥♦ t ♥st③ rt♦♥ ♦ t r②❲♦♠ strt♦♥ ♦r ♦♥

♠♥s♦♥ rt ♣♦②♠rs r♦♣②s tt

❬❪ P rs P ♦ss ①t ♦t♦♥ ♦r t rrPrs❩♥ q

t♦♥ t t ♥t ♦♥t♦♥s P②s tt

❬❪ ♠♠r s♠♦t♦ ①t ♦t♦♥ ♦r t tt♦♥r② rrPrs

❩♥ qt♦♥ P②s tt

❬❪ ♥♦ ❯♥rs tt♦♥s ♦ r♦♥ ♥trs

♥ ♥ r♥t q r②sts P②s tt

❬❪ ♥♦ s♠♦t♦ ♣♦♥ r♦♥ ♥trs ♥♦r

♥rs tt♦♥s ♥ s ♥r♥ ♣

❬❪ r♦ss♦r r♦♠ r♦♥ t♦ tt♦♥r② ♥trs ♥ t rr

Prs❩♥ ss P②s tt

❬❪ ♥♦ ♥ ♦r ♦♠tr②♣♥♥t ❯♥rs t

t♦♥s ♦ t rrPrs❩♥ ♥trs ♥ qr②st r♥

tt P②s

❬❪ ♥s ②②s P äö♥♥ ♠♦♥♥ Pr♦ts

♥ ss ♥t ♦♥♥ ♥ ♦ ♦♠st♦♥ ♦ P♣r

P②s tt

❬❪ ②②s ♥s ss ♥ ♠♦♥♥ ♥

♥ ♦s ♥ ♦ ♦♠st♦♥ ♦ P♣r P②s tt

❬❪ ②②s ♥s ss r♦s ♥ ♠♦

♥♥ ♥t r♦♥♥ ♥ s♦ ♦♠st♦♥ ♦ ♣♣r P②s

❬❪ r♦s ♥s ②②s ♥ ♠♦♥♥ ♠♣♦r ♥ ♣t

Prsst♥ ♦ ♦♠st♦♥ r♦♥ts ♥ P♣r P②s tt

❬❪ P ❨♥r ♦r t ♦r♦♥ r♥ ♥ ❨♦

ts ♦ Prt ♣ ♦♥ r♦t ②♥♠s t s ♦ ♣♦rt♥ r♦♣s

♦ ♦♦ s♣♥s♦♥s P②s tt

❬❪ s r rrr ❯♥rs tt♦♥s ♥ r

r♦t ♠♦s ♦♥♥ t♦ t P❩ ♥rst② ss r♦♣②s tt

tt

❬❪ r rrr s ❯♥rs tt♦♥s ♥ rr

Prs❩♥ r♦t ♦♥ ♦♥♠♥s♦♥ t sstrts P②s

❬❪ Pr♦ ♣♦♥ t strt♦♥ ♦ t rrPrs❩♥ qt♦♥

t sr♣ ♥t ♦♥t♦♥ ♠r t♦♥s P②s

❬❪ ♣♥② ❨ ♥ ❯♥rs s♣ts ♦ r t ♥ stt♦♥r②stt

rrPrs❩♥ sttsts P②s

❬❪ ♥ Ó♦r ①tr♠② rs s♠t♦♥ ♦ rrPrs❩♥

♠♦ s♥ r♣s rs P②s

❬❪ r♥r P♥♥ Prs rt ①♣♦♥♥ts ♦ t P❩ qt♦♥

♠tsr ♦♥ ♥♠r s♠t♦♥s P②s t ♥

❬❪ rã♦ s ❯♥rst② ♥ t♦♠♥s♦♥ rrPrs❩♥

r♦t P②s

❬❪ ❩ á③ Ps ❲t strt♦♥ ♦r ♠♥s♦♥ r♦t ♥

♣♦st♦♥ ♣r♦sss P②s

❬❪ r♥r P♥♥ Prs ❩ á③ ❲t strt♦♥s ♥ t ♣♣r

rt ♠♥s♦♥ ♦ rrPrs❩♥ ♥trs P②s

❬❪ r rã♦ s ①♠ ♥ ♠♥♠t strt♦♥s

♦ tt♥ ♥trs P②s

❬❪ ♣♥② ♠♥s♦♥ rt P♦②♠r ♥ ♥♦♠ ♠

♥ P♥♦♠♥ ♥ ❯♥rs strt♦♥s P②s tt

❬❪ ♣♥② ①tr♠ ♣ts t st♦st t qt♦♥ ♥ t tr

♠♥s♦♥ rrPrs❩♥ P②s

❬❪ r s rrr rrPrs❩♥ ♥rst② ss

♥ ♠♥s♦♥s ❯♥rs ♦♠tr②♣♥♥t strt♦♥s ♥ ♥tt♠

♦rrt♦♥s P②s

❬❪ rrs♦ rrr r ♥tr

tt♦♥s ♦r ♣♦st♦♥ ♦♥ ♥r♥ t sstrts P②s

❬❪ r♥♦ r③③ ♥ ❱á③q③ ②♥♠s ♦ ♦ ♥tr

s ♥ ♠ ❱♣♦r ♣♦st♦♥ ①♣r♠♥ts ♥ ♦ ♦r P②s

tt

❬❪ t rr♥s ❬❪ ♥ t P②s tt

❬❪ ♠ rrr r rã♦ s ❯♥rs

tt♦♥s ♥ t r♦t ♦ s♠♦♥t♦r t♥ ♠s P②s

❬❪ ♣♥② Ps♥t③s ❯♥rs ♦rrt♦rs & strt♦♥s s ①♣r

♠♥t s♥trs ♦ ♠♥s♦♥ rrPrs❩♥ r♦t r♦♣②s

tt

❬❪ r♥♦ ❱á③q③ ❯♥rst② sss ♥ sr ♥t r♦♥♥ ♦ t♥

s♦ ♠s r❳ ♣r♣r♥t ♦♥♠t

❬❪ r♦r rt ❲♦ ♦r ♥ Ps ♥ ♦

r♦♥ rs t r ♦ ♦♣s r♦♣②s tt

❬❪ ó♣③ ♦rí③ ♦ s♥t② ♥ ♥♦♠♦s r♦♥♥

♥ r♦t ♣r♦sss P②s

❬❪ ó♣③ ♦rí③ r♥♦ ♣rr♦♥♥ rss ♥tr♥s

♥♦♠♦s s♥ ♦ srs P②s

❬❪ ó♣③ ♥ ♣♣r♦ t♦ t rt ①♣♦♥♥ts ♥ ♥♦♠♦s

r ♦♥♥ P②s tt

❬❪ ♠s♦ ó♣③ ♦rí③ ♥r ②♥♠ ♥ ♥

♥t ♦♥♥ P②s tt

❬❪ ó♣③ ♥ ♦ ♦ ♦♣s ♦♥srt♦♥ s ♥ ♥♦♠♦s ♦

♥♥ ♥ r r♦t P②s tt

❬❪ ❲♦ ❱♥ r♦t t sr s♦♥ r♦♣②s tt

❬❪ P ♠r ♦tr r♦ss♦r ts ♥ t ❲♦❱♥ ♠♦ ♦ ♣t①

r♦t ♥ ♥ ♠♥s♦♥s P②s

❬❪ ❩ ❳♥ ♥ ♥ ❳ ♦ ❨ s②♠♣t♦t ②♥♠ s♥

♦r ♦ t ♠♥s♦♥ ❲♦❱♥ ♠♦ P②s

❬❪ ♥ ♥ ♥tr♦t♦♥ t♦ s♣ts ♦ r♠♦②♥♠s ♥ ♥ts

♥t t♦ tr ♥ sr ♥ ♦♦② ♦♦s

❬❪ r rã♦ s ts ♦ r♥s trs ♥ sr r♦

♥ss s♥ ♣♣ P②s

❬❪ r rã♦ s ♦♥ss ①♣♦♥♥ts ♥ r♥ s♣s

P②s

❬❪ ♦ r♥♦ str♦ ♠♥s♦♥ rt② ♦ t rrPrs

❩♥ ♥rst② ss tt ♦r ①♣ P

❬❪ ♠ rrr r♦ ♥ r ♠♣r

tr t ♦♥ ①♣r♠♥t rrPrs❩♥ r♦t ❬t♦ ♣s❪

r♦♣②s tt

❬❪ rr ttst ♣②ss ♦ ♣rts ♠r ❯♥rst② Prss

❬❪ ♥ r♠♦②♥♠s ♥ ♥ ♥tr♦t♦♥ t♦ tr♠♦sttsts ♦♥

❲② ♦♥s

❬❪ ♠② ❱s ♥ ♦ t t ③♦♥ ♥ t ♥ ♣r♦ss ♦♥ ♣r♦t♦♥

♥t♦rs ♥ t st ♣♦st♦♥ ♠♦ P②s

❬❪ r r♥s ♦ s ♥r♥ ♥ r♦t ♣r♦sss P②s

❬❪ ❱♦ ♥r ♥r♥ ♥ ♦♥t♥♠ ♦s ♦ ♦♠♥s♦♥

rs ♦t♦rt ss ♥ P②ss ❯♥rs r♦s r ♣♥

❬❪ P ór♦♦rrs sqt st♦s P ♦r ♦♠♣① ②

♥♠s r♥ t ss♦t♦♥ r♦♠ ♥tr♥s t♦ t ♥♦♠♦s ♥

P②s tt

❬❪ rs ❲♥s♦♥ sr sttsts ♦ r♥r rt

Pr♦ ♦② ♦ ♦♥♦♥ r

❬❪ ♠② ♥ ♦ r♦ srs ts ♦ sr s♦♥ P②s

❬❪ tt t③r Ps ♥t ♦♥ss ♦ ♠♦r♣♦s t②rs

t ② s ❳② ttr♥ P②s tt

❬❪ tt t③r r♥t ❯ ♠rt Ps tr♠♥t♦♥ ♦

t stt s♥ ①♣♦♥♥t ♦ s♥ ♥trs ② ♥♦♥s♣r ①r② sttr♥

P②s

❬❪ r③③ ❱á③q③ í③ ②♦r ó♣③ sr

r❲♥s♦♥ ♦r ♦ r②st rs r♦♥ ② ♠♥t♦♥ ♦ 2

♥♦s♣rs P②s tt

❬❪ s r♠ P ♠♦r♥ ♥ ♥rst② ss ♦r ♥t r♦t ♥

♠♥s♦♥ ♠♦r♠ ♣t①② P②s tt

❬❪ rr♥ s♦♥ s♦st② ♦ ♣♦②r②st♥ s♦ ♣♣ P②s

❬❪ ♥s ❲ ❲ ♦r② ♦ tr♠ r♦♦♥ ♣♣ P②s

❬❪ s r♠ ❱ ss ♦♦♥♦ rs ♥ ♠♦s ♦r ♥♦♥qr♠

r♦t ♥ ♠♥s♦♥s P②s tt

❬❪ ♠ s r♠ srt ♠♦s ♦r ♦♥sr r♦t qt♦♥s P②s

tt

❬❪ s r♠ ♥③② ♦t②r ❱ ss ♥r♥ ♥

②♥♠ ♦rrt♦♥s ♥ r♦t ♠♦s ♦ ♠♦r ♠ ♣t①② P②s

❬❪ ❨♥ ❲♥ ♥stt② ♥ ♦♠♣rtr ♦r

♠ ♣t①② r♦t ♦ P②s tt

❬❪ ❨♥ ❨P ❩♦ ❲♥ ♦s♥ ♦♥♥ ♦

t♦♥ ♦ ♠♦r♣♦s ♠s r♦♥ ② r♠ ♣♦rt♦♥ P②s tt

❬❪ rs ❩♦ r ♥stt② ♦ ♥t ♦♥♥ ♥

♣ttr♣♦st♦♥ r♦t ♦ Pt ♦♥ ss P②s tt

❬❪ ❱á③q③ r③③ P rrst P ó♥ ❱r r

②♥♠s♥ ①♣♦♥♥ts ♥ t r♦♥♥ ♥ts ♦ ♦ tr♦♣♦sts

P②s

❬❪ ❯ ♥ ♦s P ♦♥s ♦r♥st♥ ♥ s ♥ ♥♥

♦x ♠s ♣♣ P②s tt

❬❪ rú Pst♦r r♥ rú r♥r ♣r♦

②♥♠s ♦♥ ♠♦r r♦t P②s tt

❬❪ ❱á③q③ r③③ r ❱t② ♦ t ♥r r♦t q

t♦♥ ♦r ♥tr ♦t♦♥ ♦r ♦♣♣r tr♦♣♦st♦♥ ♥ t Prs♥ ♦ r

♥ ts P②s tt

❬❪ ♦té ♦ P❱tr♦t ♦ ♦ rt ♥ ♥tr r♦

♥ss ♦ ③♥ ♦① t♥ ♠s ♣♦st ② s♣r② ♣②r♦②ss t♥q ♦ ♣♣

P②s

❬❪ ♥③ ♥rs♥ P r r♦ ❱á③q③

r③③ r ②♥♠ ♥ ①♣♦♥♥ts ♦ ♦♣♣r tr♦♣♦sts r♦♠

♥♥♥ ♦r r♦s♦♣② ♠♥ ♥♥ ♦ ♦r t ♦♥ t

♥ts ♦ ♦♥♥ ♥ rt♥♥ ♥♠r

❬❪ r③ ❯ ♥ ♦r♥st♥ ♥ ♦ ♦ ♥ ♦ s♥

r♠s ♥ t♥ ♠s ♣♦st ② s♣ttr♥ ♥ t♦♠ ♦r ♠r♦s♦♣② ♥

tr♦♠ st② ♣♣ P②s tt

❬❪ ♦③ ❱ ❩♦♦tt♦ ♥t♦s ♥t♦s r

♦ r r ♦♠tt ♦r♣♦♦② rtr③t♦♥ ♦

②r②②r ♠s r♦♠ P♦ t r♦ ♦ ♠ t♥ss ♦

♦ ♥tr

❬❪ ♦③ Prr ♣♦s♦ r

♦♠tt r r ♦r♣♦♦② rtr③t♦♥ ♦ ♥♠r

♦tt ♠s r♦♠ ♣♦②♥♥ ♥ rt♥♠ ♦♠♣① ♣② t ♥♥

♦ rt ♦♥♥trt♦♥ ♦ ♣② ♥♦t♥♦♦②

❬❪ ♥ ♥ Pr♦♥s ♦ t ♦rt r② ②♠♣♦s♠ ♦♥ t♠t

ttsts ♥ Pr♦t② ❱♦♠ ❱ ♦♦② ♥ Pr♦♠s ♦ t t ②

②♠♥ ❯♥rst② ♦ ♦r♥ Prss r②

❬❪ ❱♦ ♥♠r ♣♣r♦ t♦ t ♣r♦♠ ♦ s♠♥t ♦♠ ♦

❬❪ Prä♦r ♣♦♥ ttst ss♠rt② ♦ ♦♥♠♥s♦♥ r♦t ♣r♦

ss P②s

❬❪ P rrr ♣♦♥ ♥ ♠t ♦r t s♣t♠ ♦r♥ ♦ t stt♦♥

r② t♦tt② s②♠♠tr s♠♣s ①s♦♥ ♣r♦ss ♦♠♠♥ t P②s

❬❪ r P ♥ ♣♥② ♠♣t ♥rst② ♦r r♥ ♥tr

s ♥ rt ♣♦②♠rs ♥ r♥♦♠ ♠ P②s

❬❪ s r rrr ♦♥♥rs ♣r♠trs ♦rrt♦♥s

♥ ♥rst② ♥ rrPrs❩♥ r♦t ♦ tt P

❬❪ rrs ♦♥♥r s♦♥ qt♦♥ ♦st♦♥

❬❪ s rrr r ♥ t ♦r♥s ♦ s♥ ♦rrt♦♥s

♥ st r♦t ♠♦s P②s

❬❪ r P ♥ r♦strtr ♥ sr s♥ ♥ st ♣♦st♦♥ t

♦q ♥ P②s

❬❪ r P ♥ ❯♥rs ♥ts③ ts ♥ t rt ♦ r♦t ♣r♦sss

P②s

❬❪ ♥s ♠t♥ strt♦♥s ♦r ♣♦②♥r r♦t ♠♦ t

①tr♥ s♦rs tt P②s

❬❪ s♠♦t♦ ♣t ♦rrt♦♥s ♦ t P❩ sr ♦♥ t sstrt

P②s t ♦r

❬❪ ♦♥ss♦♥ srt ♣♦②♥r r♦t ♥ tr♠♥♥t ♣r♦sss ♦♠

♠♥ t P②s

❬❪ ♦r♦♥ P rrr s♠♦t♦ r♥st♦♥ t♥ r②1 ♥ r②2 ♣r♦

sss ♥ P tt♦♥s ♦♠♠♥ ♦♥ Pr ♥ ♣♣ t

❬❪ Prä♦r ♣♦♥ ♥r♥ ♦ t P r♦♣t ♥ t r② ♣r♦

ss tt P②s

❬❪ rr♥s ❬❪ ♥ ♥ ♥♦ P②s tt

♦r ♥s ♥ ♦♥ ♠♥s♦♥

❬❪ ♠ ttsts ♦ ①tr♠s ♦♠ ❯♥rst② Prss ❨♦r

❬❪ P rrr r♥s ♥t t♠ ♦rrt♦♥s ♥ P❩ r♦t ♠♦s

tt P②s

❬❪ s r rrr ❯♥rst② ♦ tt♦♥s ♥ t

rrPrs❩♥ ss ♥ ♠♥s♦♥s ♥ ts ♣♣r rt ♠♥s♦♥

P②s

❬❪ ♠ ♦strt③ r♦t ♥ rstrt s♦♦♥s♦ ♠♦ P②s

tt

❬❪ ♦t♥ r♥ ❩ á③ ❲♦r♠♥ P ❩ ❲t strt♦♥

♦r r♥♦♠ ♥trs P②s

❬❪ Ps ❩ á③ ❲t strt♦♥ ♦ rtrr♥ ♥trs st②

♦ ♥rst② P②s

❬❪ ♥t r♦③ ②ör② ❩ á③ ♦♥ss strt♦♥s ♦r 1/fα s

♥s P②s

❬❪ rã♦ s ♠r st② ♦ r♦♥ss strt♦♥s ♥ ♥♦♥♥r

♠♦s ♦ ♥tr r♦t P②s

❬❪ P rã♦ s t ♥ r♦♥ss strt♦♥s ♥ t♥ ♠s

t rrPrs❩♥ s♥ r

❬❪ ②r r♥st♦♥ Pr③②② ❨ ♣r ①♠ t ♥

♦ ♥t② r♦♥ rs P②s tt

❬❪ ②ör② P ❲ ♦s♦rt P♦rt ❩ á③ ttsts ♦ ①tr♠

♥t♥sts ♦r ss♥ ♥trs P②s

❬❪ ♠r ♦♠tt ①t ①♠ t strt♦♥ ♦ tt♥

♥trs P②s tt

❬❪ strt♦♥ ♦ ①tr♠s ♥ t tt♦♥s ♦ ♦♠♥s♦♥

qr♠ ♥trs P②s tt

❬❪ tr♦ ♦t ❲ ♣t①② ♥ ♦ ♠s

❬❪ ❲♥ s♠♦t♦ ss ♦t ♣t①② ♦

qt② ♦ r②st r♦t

❬❪ ❲♥ ❲ ♠ s♠♦t♦ ss rt

r♦t ♦ ♣②rs ♦♥ sstrt ② ♦t ♣t①② ♣♣

r

❬❪ rrr r r P ♦ts r♠♦

rtr③t♦♥ ♦ ♥ ♠s r♦♥ ♦♥ ss ② ♦t ❲ ♣t①②

r③ ♦ P②s

❬❪ ♥♥ t rr t♦♠ ♦r r♦s♦♣ P②s tt

❬❪ s③ ❨ r ♦ ♠♣rtr r ♥♥ ♦ ♦♥ ♥ ts

♣♣t♦♥ t♦ ♦♥ tr♦♠ ♦

❬❪ st♥ r♦ ❨ s ♦r♠t♦♥ ♦

②r♦♥ ♣sst s♦♥ s♥r②st srs s♥ tr♦t ♥♥ ♥

t♥ ♣♣ P②s

❬❪ ❨ t ts ♦ srs trt ♥ q♦s ♦r♥

s s♦t♦♥s ♣♣ r

❬❪ rst ♥♦st ♦ ts ♦ sr ②r♦♥

♦♥ t r ♦①t♦♥ t r♦♦♠ t♠♣rtr ♦ trt srs ♣♣

P②s tt

❬❪ ❨ s r ❱ rr♦s ♥rr s♣

tr♦s♦♣② ♦ ♥ srs tr trt♠♥t ②r♦♥ tr♠♥

t♦♥ ♥ sr ♠♦r♣♦♦② ❱

❬❪ s ❨ ❲ rs r ♣♣ P②s tt

❬❪ ♣♦r♥ ♥♥t♥ ♦♥r♦② ♦r

P r ♦r ♠ ♣t① r♦t ♦ ♥ ♦♥

♣♣ P②s tt

❬❪ r♦③ ♥ ♠ ♥②ss ② ❳② ttr♥ ♦♥ ❲② ♦♥s

❬❪ ♦ ♦r s♦♥ ♥ ❲ ♦♥ ♥t ♦ ♦ trss

♦t♦♥ r♥ ♦s♥ ♥ r♦t ♦ P♦②r②st♥ ♥ ♠s P②s

tt

❬❪ ♦♥③á③♦♥③á③ P♦♦♣ ❱s♦ P♦st♦s♥ ♦t♦♥ ♦ r♦t

trss ♥ P♦②r②st♥ ♠s P②s tt

❬❪ r s s♣♥ rrr ♦♥strt ♦♥

t♦r♥ ♦ t ♦t♦♥ ♥ ♣t① ♥ ②rs r♦♥ ♦♥

♦ P②s ♠

❬❪ ♦♥ ❨ ♠ ♠r ♦♠♣♥ r♥ r♦t ♥♦♠♦s

s♥ ♥ r♥ ♦♥r② r♦♦♥ ♥ ♣♦②r②st♥ t♥ ♠s ♣♣

P②s

❬❪ r♠♥ ❲ tr ttr ♣t①② P②s Pr♥♣s ♥ ♥

♠♣♠♥tt♦♥ ♣r♥r

❬❪ r♠♥ ♠t ♦r ♠ ♣t①② ♥♠♥ts ♥ rr♥t

stts r♥ ♣r♥r❱r

❬❪ r t♦s rã♦s ♥ ♣rí ♥ s ♦s str

ssrtt♦♥ ♥ P②ss ❯♥rs r ♠♥♥s r③

❬❪ ♦ ❲ r③r ♥♦♠♦s ♥ ♦ t r ❲t r♥

tr♦♣♦st♦♥ P②s tt ♦rss P r

♥ ❲ r③r ♥♦♠♦s ♥ ♦r tr♦♣♦st ♠s

P②s tt

❬❪ ❲ ♥s P rtt ♦r♣♦♦ ♦t♦♥ r♥ ♣t①

t♥ ♠ r♦t ♦r♠t♦♥ ♦ s♥s ♥ ♠♦♥s r ♣

❬❪ ❨ ♠ Pr ♠ ♦♥sr r♦t ♥ rstrt s♦♦♥s♦

♠♦ P②s

❬❪ ♦t③ P P ❲ t♣♥s ♠♣rtr ♣♥♥ ♦ sr

r♦♥♥ r♥ ♦♠♦♣t① r♦t ♦♥ P②s

❬❪ ❲ ♦tt P s P ❲ t♣♥s ♠♣rtr ♥ ♦r♥t

t♦♥ ♣♥♥ ♦ ♥t r♦♥♥ r♥ ♦♠♦♣t①② q♥ttt ①r②

sttr♥ st② ♦ P②s

❬❪ r♦ r ♥③ r ♥ ♣r♦♣rts ♦

st ♣♦st♦♥ ♠♦s t ♦♥ r♥ P②s

❬❪ ♦♦s ♦♦t♥② P ❱r♠ ❯♥rs s♥ ♥ ♠①♥ ♦rr

t r♦t t r♥♦♠♥ss P②s

❬❪ ♦♦s ♦♦t♥② ♦♥♥rs ts ♥ ♠①♥ ♦rrtr♦t

♣r♦sss t r♥♦♠♥ss ♥tr♣② t♥ ♠♦r♣♦♦② ♥ sr r♦

♥♥ P②s

❬❪ rã♦ s ♠r st② ♦ srt ♠♦s ♥ t ss ♦ t

♥♦♥♥r ♠♦r ♠ ♣t①② qt♦♥ P②s

❬❪ ❨P ❩♦ r♦tr ❲♥ ♦r♣♦♦② r♥st♦♥ r♥

♦Prssr ♠ ❱♣♦r ♣♦st♦♥ P②s tt

❬❪ ❩ ❲ s r♠ ♥t r♦t t sr r①t♦♥ ♦♥t♥♠

rss t♦♠st ♠♦s P②s tt

❬❪ s♦r♥ ♥ ♦♠♠♥t ♦♥ st tr♥ ♥ ♥♦♠ P♦t♥t

P②s tt

❬❪ ♠r P ♠ ♠② r♦♦ ♥stts ♥ sr r♦t t

s♦♥ P②s

❬❪ s r♠ ❱ ss ♠ ♥t s♣rr♦♥♥ ♥ ♥♦♠

♦s ②♥♠ s♥ ♥ ♥♦♥qr♠ r♦t ♠♦s P②s

❬❪ ó♣③ ♠tt ♥♦♠♦s s♥ ♦ rtr sr P②s

❬❪ ♦ r♥♦ str♦ ♦♠♠♥t ♦♥ ts ♦ Prt ♣ ♦♥

r♦t ②♥♠s t s ♦ ♣♦rt♥ r♦♣s ♦ ♦♦ s♣♥s♦♥s P②s

tt

❬❪ Ps ♦r r♦r rt ❲♦ ♦♠♠♥t ♦

♦♥♦ s ♥ ♦s ♦r ♦♥qr♠ r♦t ♥ ♠♥s♦♥s

P②s tt

❬❪ r ❱á③q③ r♥♦ str♦ r á♥③ ♥tr♥

s ♥♦♠♦s sr r♦♥♥ ♦ ♠s ♣♦st ② rt s♣ttr♥

P②s

❬❪ ❲ ♠s♦♥ P st P②s ♠str② ♦ srs

❬❪ ❱♦♥♥t ❱rt♦♥ t t♠♣rtr ♦ t ♥t♦♥ rt ♦ s♣r♦♦

q t♥ ♥ tr r♦♣s ♦r♥ ♦ ♦♦ ♥

❬❪ ♥s♦♥ t♦♥ ♦ ♥t♦♥ ♥ r②str♦t rt t♦ t ♦♣♠♥t

♦ r♥t t①trs ♠r♥ ♥r♦st

❬❪ ② t♦♠ ♠♦t♦♥ ♦♥ srs P②ss ♦②

❬❪ r t♦♠ ❱ ♦ r s♦♥ ♥st♥ ♦♥

♥st♥ ♠ P②s ♦ ♣s② t♣

♠♦t♦♥ ♦♥ r②st srs ♣♣ P②s

❬❪ ♦♠ ♦r♠çã♦ strtrs r♠♥s♦♥s ♠ rs

♠♥t♦ ♣t① ♦t♦r ss ♥ P②ss ❯♥rs r ❱ç♦s

r③

❬❪ P ♦② r③②③s ♦♥s s

rr② t ♦ r♦t rt ♦♥ t s③ ♦♠♣♦st♦♥ ♥ ♦♣t ♣r♦♣rts ♦

♥ss q♥t♠ ♦ts r♦♥ ② ♠♦r♠ ♣t①② P②s

❬❪ ❨❲ ♦ rt③♥trr ② ♥t ♣t②

♥ tr♥srst♥♦ r♦t ♦ ♦♥ P②s tt

❬❪ ♥ r ❱t ♥ P♥①tr♥ ♦♠r ❱

♦r♥t♦♥ rt♥t♥ ②r②②r r♦t ♦ ♦♥ P②s

tt

❬❪ r♦③ ♥s ❲ s trtr♥t♦♥ rt♦♥s♣

t♥ ♣rrr ♦r♥tt♦♥ ♦ r②stts ♥ tr rsstt② ♥ t♥ ♣♦②

r②st♥ ❩♥ ♠s P②s

Documents

KARDAR-PARISI-ZHANG UNIVERSALITY, ANOMALOUS SCALING … completo.pdf · RENAN AUGUSTO LISBÔA ALMEIDA KARDAR-PARISI-ZHANG UNIVERSALITY, ANOMALOUS SCALING AND CROSSOVER EFFECTS IN