Physica status solidi / A.: Volume 86, Number 1 November 16 [Reprint 2021 ed.] 9783112501481, 9783112501474

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plrysica status solidi (a)

ISSN 0031-8965 * VOL. 86 • NO. 1 . NOVEMBER 1984

Classification Scheme 1. Structure of Crystalline Solids 1.1 Perfectly Periodic Structure 1.2 Solid-State Phase Transformations 1.3 AlloysrMetallurgy 1.4 Microstructure (Magnetic Domains See 18; Ferroelectric Domains See 14.4.1) 1.5 Films 1.6 Surfaces 2. Non-Crystalline State 3. Crystal Growth 4. Bonding Properties 5. Mossbauer Spectroscopy 6. Lattice Dynamics. Phonons 7. Acoustic Properties 8. Thermal Properties 9. Diffusion 10. Defect Properties (Irradiation Defects See 11) 10.1 Metals 10.2 Non-Metals 11. Irradiation Effects (X-Ray Diffraction Investigations See 1 and 10) 12. Mechanical Properties (Plastic Deformations See 10) 12.1 Metals 12.2 Non-Metals 13. Electron States 13.1 Band Structure 13.2 Fermi Surfaces 13.3 Surface and Interface States 13.4 I m p u r i t y and Defect States 13.5 Elementary Excitations (Phonons See 6) 13.5.1 Excitons 13.5.2 Plasmons 13.5.3 Polarons 13.5.4 Magnons 14. Eleetrical Properties. Transport Phenomena 14.1 Metals. Semi-Metals 14.2 Superconductivity. Superconducting Materials and Devices 14.3 Semiconductors 14.3.1 Films 14.3.2 Surfaces a n d Interfaces 14.3.3 Devices. Junctions (Contact Problems See 14.3.4) 14.3.4 High-Field Phenomena, Space-Charge Effects, Inhomogeneities, Injected Carriers (Electroluminescence See 20.3; Junctions See 14.3.3) 14.4 Dielectrics 14.4.1 Ferroelectrics 15. Thermoelectric and Thermomagnetic Properties 16. Photoconductivity. Photovoltaic Effects 17. Emission of Electrons and Ions 17.1 Field Emission Microscope Investigations 18. Magnetic Properties 18.1 Paramagnetic Properties 18.2 Ferromagnetic Properties 18.2.1 Ferromagnetic Films 18.3 Ferrimagnetic Properties 18.4 Antiferromagnetic Properties (Continued on cover three)

physica status solidi (a) applied research

B o a r d of E d i t o r s

S. A M E L I N C K X , Mol-Donk, J . A U T H , Berlin, H. B E T H G E , Halle, K. W. B Ö E R , Newark, P. G Ö R L I C H , Jena, G. M. H A T O Y A M A , Tokyo, C. H I L S U M , Malvern, B. T. K O L O M I E T S , Leningrad, W. J . M E R Z , Ziirich, A. S E E G E R , Stuttgart, C. M. V A N V L I E T , Montréal Editor-in-Chief

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Volume 86 • Number 1 • Pages 1 to 460, K1 to K86, and A1 to A10 November 16,1984 PSSA 86(1) 1-460, K 1 - K 8 6 , A1-A10 (1984) ISSN 0031-8965

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phys. stat. sol. (a) 85 (1984)

Author Index G . A . ADEGBO YEGA M . ADOLF S . C. AGARWAL J . M . ALAMEDA J . A . ALONSO S . A . ALTEBOVITZ V . M . ALYABIEV S . AMELINCKX K . P . ABEFIEV D . A . ABONOV A . K . AROBA

V . G . ARTYUSHENKO H . ATMANI V . ATZRODT M . AUBIN S. A . AZIMOV

215 K81 297 511 423 69 11,315 29 K31 K159 491

167 399 K9 649 K159

Y . H A R I BABU I . BAKEB D . BALTBÜNAS A . BABBULESCU U . BABKOW A . BABTL J . H . BASSOK B . A . BAUM G . BEBGEB H . BEBGEB K . - H . BEBTHEL P . A . BESIBGANYAN V . A . BESPALOV U . BIRKHOLZ G . BLASEK M . G . BLAZHA R . BLINC H . BOOYENS S. BREHME W . BRUNNER G . H . BU-ABBUD W . BURCKHABDT P . BUSSEMER

265 481 K97 K129 K77 257 243, 449 497 K9 45 KL75 349 K73 K81 473 553 409 243, 449 627 257 69 97 97

J . CABRERA-CANO J . CASTAING J . (IEBMÄK A . R . CHELYADINSKII V . CHERNYI V . G . CHUDINOV M . C . CONTBERAS J . W . CORBETT M . COUZI M . R . CZERNIAK

445 445 173 K43 627 105,435 51 1 KL09 359 235

Y U . N . DALUDA M . DAVID

575 375

G . A . DENISENKO I . DIACONU U . DIETRICH R . DIETSCH K . DINI A . DOMINGUEZ-RODRIGUEZ A . F . DUNAEV R . A . DUNLAP J . DURAN A . V . DVURECHENSKII V . P . DYMONT S . A . DZUBA D. A. V. N. C. R.

D . ELEY A . EL-SHARKAWY V . EMTSEV P . ESINA ESNOUF ETHIBAJ

G . FANTOZZI G . M . FILARETOVA H . - J . FITTING Z . FRAIT J . FRÖHNER Y . FUKUDA M . GABEB R . T S . GABRIEL YAN V . I . GATALSKAYA P . GAWOBZEWSKI R . GEVERS M . H . GHONIEM S. GHOSH H . GLAEFEKE H . - C H B . GOLINSKY E . M . GOLOLOBOV E . S . R . GOPAL I . K . GOPALAKRISHNAN B . N . GOSHCHITSKII G.GÖT Z I . N . GBINCHESHEN R . GBOETZSCHEL J . GBONKOWSKI A . K . GROVER V . I . GUBSKAYA A . I . GUSEV V . S . GUSHCHIN P . HARGKAVES F . HASHIMOTO H . HASHIMOTO W . HERGERT B . HERMONEIT

553 283 97 473 K101 445 591 K101 291 K39 K69 257 283 429 575 655 463 K27 463 655 195 179 257 K141 195 349 K51 133, 5 7 5 375 429 535 K149 569 K51 K165 K89 435 KL K85 K35 389 K89 585 159 497 K101 227 335 641 553

Author Index

662 W . HERREMANS R . HERRMANN H . HESS E . HILD D . HILDEBRANDT O. HÖLZER K . HOSHINO

375 K183 543 133 K35 K43 K5

K . IKEZAKI T . ISHII M. IWATA

615 615 K105

K.JACOBS Z. JANÁCEK K . N. JOG T.JUNG K . JÜRGENS

627 77 417 K15 K77

S . KACIULIS A . A. KAMINSKII Y . KAMIURA S . A . KARANDASHEV B . P . KASHNIKOV F . R . KESSLER S . KHAN E . KIBICKAS F . - G . KIRSCHT V . F . KISELEV J . KLIKORKA N . M. L . KÖCHE H . - J . KÖHLER 0 . KOHMOTO V . G. KOHN M . KOIZUMI L . S . KORNIENKO E . KOT AI W . KRAAK A . R . KRASNAYA H . KRAUSE W . KRECH P . Y . KUCHINSKII Y U . A. KULYUPIN A . KUMAR G . S . KUMAR N . KURAMOCHI L . A . KUROCHKIN M . F . KUZNETSOV J . LAUZIER H . M . LEDBETTER H . LEMKE M. L . LEVITAN P . LILLEY J . L . LINDSTRÖM 1. S . LISITSKII D . C. LIU V . M . LOMAKO V . A . LOMONOV J . M. LÓPEZ

K179 553 227 655 K39 K77 535 K179 133 273, K 7 3 K123 K65 K175 K155 349 523 167 K35 K183 K145 569 K175 585 45 297 K27 121 K51 K31 463 89 K133, K137 51 235 K109 167 69 585, K 5 7 553 423

F. P. P. T. C.

C. LOVE Y L . LOYZANCE F . LUGAKOV A . LUKASHEVICH W . LUNG

G. I . MAKOVETSKII S . A . MAKSIMENKO J . MÁLEK

C. MALGRANGE Y . I . MALTSEV R . MARQUEZ SH. R . MASTOV B . A . MEN M. MESSAOUDI H . - G . MEYER P . MICHEL P . MICHON G. MICOCCI C. MINIER D . S . MISRA T . MIURA E . MIZERA H . MIZUBAYASHI K . MOCHIZUKI P . C. MORÁIS N . V . MOSEEV V . G . KRISHNA MURTHY

29 359 441 441 K L 13 K69 K23 K 1 2 3

389 529 445 K31 . 51 291 K175 KL 399 609 463 297 615 83 121 249 K65 435 K27

E . NEBAUER S . A . NEPIJKO S . NESPÛREK M . NEVRIVA J . E . NICHOLLS D . O. NORTHWOOD P . NOVAK R . NOVAK

K169 45 619 173 235 149 173 173

G . S . OEHRLEIN G . OELGART H . OESTERREICHER K . OESTERREICHER T . OHKAWA S . OKUDA C. K . ONG L . N . OSTER

K109 205 K61 K61 335 121 199 K145

J . PARSELIÜNAS G . PABTHASARATHY L . PASEMANN F . PASZTI V . A . PAVLOV A . P A WE LEK R.PÉREZ Y U . E . PERLIN R . PICKENHAIN G . S . PLOTNIKOV A . D . POGREBNYAK

K179 K165 641 K35 11, 3 1 5 K L 17 113 553 627 273, K 7 3 K31

Author Index G . P . POKHII

K39

M . SuEZAWA

V . P . POPOV

K39

A . SUKIENNICKI

N . S. POPOVICH

K85

K . SUMINO

E . D . POZHIDAEV

591

M . SuNDBERG

.

K . SUZUKI

.

Y . I . PROTASOV J. PULTORAK

L . PUST S. V . R A K I T I N

105, 435 K93

.

.

.

B . G . SVENSSON

179 K31

H . TAGUCHI .

.

N . TAKEUCHI

.

I . H . RASHED

429

H . S. T A N

B . RAUSCHENBACH

473

G . VAN TENDELOO

P . REICHE

553

A . TEPORE

M . RIEDEL

K175

A . RIZZO J . ROSENZWEIG L . ROTH V . M . RUBINOV H . RUBIO G . RUDLOF

609 K81 K15 K159 511 K149

.

.

.

.

N . T . THUC H I E N L . TICHY

.

. .

V . D . TKACHEV R . TOGNATO . , A . TOMITA A . H . TONEYAN .

.

V . V . RYABCHENKOV

553

B . TOPIC

M. TUICHIEV.

.

.

A . 0 . RYBALTOVSKII

167

A . V . TULINOV

V . S. SAENKO

591

A . A . TURINGE A . M . SALETSKII B . SANDOW

K73

B. P. TYAGI .

K169

S. P . S A N Y A L

417

0 . UEMURA

S . E . SARKISOV

553

K . UHLMANN

A . SASAKI T . SATOW U . SCHALLER F . SCHAUER

K105

619 569

K . SCHMALZ

575

M . SCHMIDT

K19

P . SCHNEIDER

455

E . M . SCHULSON

481

D . SCHULTZE

553

P . SEIDEL W . SEIFERT K . SEN

45 K175 627 603

V . I . SHALAEV

11, 315

R . V . SHARMA

39

M . SHIMADA A . A . SHTANOV

523 K85

L . A . SHUVALOV

409

V . E . SIDOROV

497

E . H . SIN

199

L . SINCAN

.

. .

S. UNTERRICKER

K5 K183

M . SCHELL

N . N . SEDOV

.

A . P . TYUTNEV

D . M . VANDERWALKER A . V . VANNIKOV C. V A U T I E R

.

.

L . VECSERNYÉS S . V . VLNTSENTS Y . V . VOITSEKHOVSKII G. VÖLKEL

.

.

V . G . VOLOGIN . S. A . VOROBIEV F. WALZ

.

.

.

T . WATANABE

.

B. WEBER

.

.

P . WERNER

.

.

U . WERNER

. .

W . WESCH

.

.

M. R . WILLIS

.

J . A . WOOLLAM

K129

R . K . SINGH

417

J. V . YAKHMI

P . SIRCAB

649

V . Y A . YASKOLKO

K . SKEFF NETO

.

K65

I . O . SMITH

149

M . S. ZAGHLOUL

K . SOMAIAH

265

K . 2Î>ÂNSKY.

.

P . Y A . STAROSTIN

K57

J . J . ZEBROWSKI

H . STRUSNY

K35

O . ZMESKAL .

.

S. S U D H A K A R

K27

N . V . ZOTOVA

.

Contents Review Article C.

BOULESTEIX

A Survey of Domains and Domain Walls Generated by Crystallographic Phase Transitions Causing a Change of the Lattice

11

Original Papers and Short Notes Structure R . V . G . RAO a n d S. KUMAB P A L

A New Method of Evaluation of the Structure of Molten Sodium Chloride

43

N . I . BOBGABDT, S . K . MAKSIMOV, a n d D . I . PISKUNOV

Phase Information in Transmission Electron-Microscope Investigation of Defects. Dynamic Images with a Weak Amplitude Contrast

55

M . TAKEDA a n d H . HASHIMOTO

A High Resolution Electron Microscopy Study of Domain Structure in CuAul

67

M . H I D A K A , H . F U J I I , B . J . GARRARD, a n d B . M . W A S K L Y S

Structural Phase Transitions of RbVF 4

75

P . MALÉTKAS e t P . DUVAL

Evolution comparée d'alliages Cu-Al trempés revenus a v a n t et après amincissement

85

N . V . R U M A K , V . V . K H A T K O , a n d V . N . PLOTNIKOV

Structure and Properties of Silicon Dioxide Thermal Films (I)

93

M . G . BLANCHIN a n d L . A . BURSILL

Extended versus Small Defect Equilibria in Non-Stoichiometric Rutile (I)

101

G . SORGE, J . J A N I C H , L . A . SHUVALOV, a n d N . M . S H C H A G I N A

The Influence of an Electric DC Field on the New Intermediate P h a s e between the a- and ^-Phases of NaH 3 (Se0 3 ) 2 Crystals

Ill

E . Z S C H E C H , W . BLATJ, H . V E G A , K . K L E I N S T U C K , S . M A G E R , M . A . K O Z L O V , a n d M . A . SHEROMOV

E X A F S and X - R a y Diffractional Investigation of the Heusler-Type Alloys Co2MnSi and Fe2.4Mno.eAl. Determination of the Ordering Probabilities

117

D . FASOLD, K . H E H L , a n d S. J E T S C H K E

Optical Reflection and Transmission through a F l a t System with a Thick Fluctuating Layer

125

V . V . A R I S T O V , L . S . K O K H A N C H I K , a n d YTR. I . VORONOVSKII

Voltage Contrast of Ferroelectric Domains of Lithium Niobate in SEM . . l*

133

Contents

4 P . A . ALEKSANDROV, A . M . AFANASIEV, a n d S . A . STEPANOV

Bragg-Laue Diffraction in Inclined Geometry

143

A . M . A F A N A S I E V , S . M . A F A N A S I E V , P . A . A L E K S A N D R O V , R . M . IMAMOV, a n d E . M . P A S H A E V

P . KOSTIAL

Grazing Bragg-Laue Diffraction for Studying the Crystal Structure of Thin Films

K1

The Influence of Heating on the Attenuation of Surface Waves in Electrographic Se Layer

K7

A . A . R E M P E L , S . Z . NAZAROVA, a n d A . I . G U S E V

Effect of Atomic Ordering on Heat Capacity of Non-Stoichiometric Niobium Carbide Kll N . K A M E S W A R I , K . V . R A M A N U J A C H A R I , A . M E E N A K S H I STJNDAHAM, a n d C . S . S W A M Y

Synthesis and Structural Characteristics of La 2 Cui_a;Co x 0 4 (x = 0.2, 0.4, and 0.5) K15

Lattice

properties

SHTJ Z H E N a n d G . J . D A V I E S

Calculation of Potential Energy Parameters for Some Intermetallic Compounds

155

P . C. MORAIS, A . L . TRONCONI, a n d K . S K E F F N E T O

Photoacoustic Signal Behavior near the Ferromagnetic Resonance Field

163

M . A . ALDZHANOV, D . A . GUSEINOV, a n d R . K . V E L I E V

Low-Temperature H e a t Capacity of CdGa 2 S 4

K19

W . S. TSE a n d C. C. CHEN

Lattice Vibrations of Crystalline Silicon and Germanium Tetrachlorides K23

Defects,

atomistic

aspects

H . C. G O N Z A L E Z Hardening of Single Crystals of Magnesium by Low Neutron Doses at 77 K

169

S . V . SOKOLSKII, F . F . LAVRENTEV, a n d O . P . SALITA

Jump-Like Deformation and Stability of the Hardened and Structural States of Zinc Single Crystals with Forest Dislocations in the Temperature Range 293 to 4.2 K

177

V. V . K A L I N I N a n d N . N . GERASIMENKO

On the Effect of Doping Impurities on the Formation of Rod-Like Defects in Silicon

185

A . D . POGREBNYAK

Positron and Positronium States in Semiconductors Irradiated by Supercurrent Beams of Charged Particles

191

5

Contents V . V . KIRSANOV, S . B . KISLITSIN, a n d E . M . KISLITSINA

Interstitial Migration in a Stress Gradient

199

H . M . EISSA a n d M . M . EL-OKER

The Interaction of Straight Dislocations with Dislocation Loops in an Anisotropic Material (Zinc)

207

CH. LUSHCHIK, J . KOLK, A . LUSHCHIK, a n d N . LUSHCHIK

Radiational Creation of Frenkel Defects in KC1-T1

219

H . - R . H O C H E a n d J . SCHREIBER

Anisotropic Deformation Behaviour of GaAs

229

H . YAMAGUCHI, T . TSUCHIDA, I . HASHIMOTO, Y . OHTANI, a n d S . O H T A K I

H . BENDER

Stage I I Recovery of Defects in Cold-Worked Aluminum

237

Investigation of the Oxygen-Related Lattice Defects in Czochralski Silicon by Means of Electron Microscopy Techniques

245

A . KOZANECKI a n d H . RZEWUSKI

Distant DAP Bands in Electron-Irradiated and Ion-Implanted CdSe . . A.

PAWELEK

263

On the Thermodynamic Criterion for the Unstable Motion of a Source Generated Dislocation Group K27

A . A . EVTUSHENKO, Y . F . PETRENKO, a n d I . A . R Y Z H K I N

R.

TOGNATO

Electric Polarization of Ice at Nonuniform Elastic Strains

K31

Overheating in Semiconductors with a Diamond-Like Structure . . . . .

K35

Magnetism A . A . WAFIK a n d S. A . MAZEN

Barkhausen Jumps and Resistivity of Cu-Cd Ferrites

271

M . B . SHAH a n d M . S. C. BOSE

Magnetic NDT Technique to Evaluate Fatigue Damage

275

G . P . R R A M A B a n d Y A . I . PANOVA

The Permeability Dispersion in Ferrites at Different Levels of Direct Current Magnetic Biasing

283

M . TAKAHASHI, Y . SASAKI, a n d S . ISHIO

Magnetic Freezing in Solution-Quenched Dilute Fe-Cu Alloys

289

D . GIGNOUX, J . C. GOMEZ-SAL, a n d J . RODRIGUEZ F E R N A N D E Z

Magnetic Structure of the Er 2 Pt Antiferromagnet

295

6

Contents

Localized

electronic

states

and

transitions

D . I . ALADASHVILI, L . KONCZEWICZ, a n d S . POROWSKI

Studies of the Deep Levels in p-Type InSb under Pressure

301

P . P . LITGAKOV a n d V . V . S H U S H A

A Model of Nonequilibrium Charge-Carrier Recombination in Semiconductors Containing Nonuniformities

309

A . V . D V U R E C H E N S K I I , B . P . K A S H N I K O V , a n d Y . V . STJPRUNCHIK

New B P R Defects in Si

313

I . TALE a n d J . ROSA

Fractional Glow Technique Spectroscopy of Traps in Heavily Doped A1N:0

319

S . C. SABHABWAL, S . P . K A T H U R I A , a n d B . GHOSH

Kinetics of Thermoluminescence in y-Irradiated Nal(Tl)

327

R . CAPELLETTI a n d M . MANFREDI

The Role of E u + + Aggregation on Optical Absorption, Emission, Luminescence Time Decay, and ITC Plots in N a C l : E u + +

333

A . A . K A M I N S K I I , E . L . B E L O K O N E V A , B . V . M I L L , YTT. V . P I S A R E V S K I I , S . E . SARKISOV, I . M . SILVESTROVA, A . V . B U T A S H I N , a n d G . G . K H O D Z H A B A G Y A N

Pure and Nd 3 + Doped Ca 3 Ga 2 Ge 4 0 14 and Sr 3 Ga 2 Ge 4 0 14 Single Crystals, Their Structure, Optical, Spectral Luminescence, Electromechanical Properties, and Stimulated Emission

345

Über die mögliche Existenz „repulsiver" Zentren mit großen Einfangquerschnitten in Silizium K39

H . LEMKE

T . NGOC, V . T . B I C H , T . K . A N H , N . N . D I N H , a n d P . L O N G

Kinetics of Luminescence Quenching in Neodymium-Doped Fluorophosphate Glasses K45

Electric

transport

A . P . T Y U T N E V , V . S . S A E N K O , A . V . V A N N I K O V , a n d V . E . OSKIST

I.

P.

ZVYAGIN

Radiation Induced Dielectric Effect in Polymers

363

Planar Conduction in Inhomogeneous Films. Application to Amorphous Silicon

375

B . T . KOLOMIETS, V . M . L Y U B I N , M . A . TAGUYRDZHANOV, a n d G . ZENTAI

A Comparative Study of Photoconductivity in Vitreous As 2 Se 3 and a-Si: H Thin Films

383

N . E . ALEKSEEVSKII, A . V . M I T I N , E . P . KHLYBOV, C. BAZAN, B . G R E N , a n d A . GILEWSKI

The Influence of Alloying on the Properties of Superconducting Molybdenum Sulphides

389

7

Contents E . HARTMANN, L . KOVACS, a n d J . PAITZ

Electrical Conductivity of Gadolinium-Gallium Garnet (GGG) Crystals

401

U . BRAATZ a n d D . ZAPPE

Investigation of Space Charge Limited, Poole-Frenkel, and Isothermal Currents on Hgl 2 Single Crystals

407

M . T O P I C , A . MOGUS-MILANKOVI(5, a n d Z . KATOVKS

Thermally Stimulated Depolarization Current Study in Novolac PhenolFormaldehyde Resin

413

DANG TRAN QUAN

Electrical Properties and Optical Absorption of SnSe Evaporated Thin Films

0 . A . GOLIKOVA, N . E . S O L O V E V , Y A . A . U G A I , a n d V . A . F E I G E L M A N

Electrophysical Properties of a-Rhombohedral Boron

A . M . KONIN a n d A . P . SASCIUK

Thermogradient Magnetoconcentration Effect

421

K51

K55

V . E . A R K H I P O V , V . I . V O R O N I N , B . N . GOSHCHXTSKII, A . E . K A R K I N , V . N . KOZHANOV, A . V . M I R M E L S H T E I N , a n d E . P . ROMANOV

The Structure, Superconducting Properties and Electrical Resistivity of V2Zr and V 2 Hf Irradiated with Fast Neutrons K59

P . I . BARANSKII, T . G . GRISHCHENKO, V . V . SAVYAK, a n d Y u . V . SIMONENKO

Temperature Dependence of the Thermoelectric Characteristic of n-Si Elastically Strained in the [100] Direction K63

S . SHIGETOMI, T . I K A R I , Y . KOGA, a n d S . SHIGETOMI

Annealing Behaviour of Electrical Properties of n-InSe Single Crystals . . K69

Device-related

phenomena

W . BRATJNE, M . SCHNURER, N . K U B I C K I , a n d R . HERRMANN

C-U Measurements of Anodized InSb MOS Structures at 77 and 4.2 K . .

427

J . W . TOMM, K A R I N H E R R M A N N , C . B A R T H E L , a n d U . B A R T H E L

On the Dispersion of the Refractive Index in Active Layers of Lead-Salt Injection Lasers

433

A . J . MUHAMMAD, D . V . MORGAN, a n d A . E . G U I L E

The Effect of Overlap of Ohmic Contacts on the Characteristics of a MIM Structure Having a Single Discrete Shallow Trap Level

W . TORBICZ, D . SOBCZYNSKA, A . OLSZYNA, G . FORTUNATO, a n d A . D'AMICO

A Stable Pd-BN-Si 3 N 4 -Si0 2 -Si F E T for Hydrogen Detection

439

453

8 Tran Chot

Contents Effect of Vacuum Annealing on Electrical Characteristics of Ni-Si Schottky Diodes K73

D. Ray Chatjdhuki, J . B. Roy, and P. K. Basu Density of the 2D Electron Gas in Modulation Doped GalnAs/AlInAs Layers from a Charge Control Analysis K79 S. A. Gbusha, A. M. Evstigneev, K. A. Ismailov, R. V. Konakova, A. N. Kbasiko, I. K. Sinishohuk, Ytj. A. Tkhoeik, and V. I. Fainbekg Electric Field Effect on Radiation-Induced Gettering of Defects in n + - n - n + + -GaAs K83

Erratum M. J. Makcinkowski Erratum to: The Mechanism of Cell Wall Formation

460

Pre-Printed Titles of papers to be published in the next issues of physica status solidi (a) and physica status solidi (b)

Al

Contents Systematic List Subject classification: 1.1 1.2 1.3 1.4 1.5 1.6 2

Corresponding papers begin on the following pages (pages given italics refer to the principle subject classification): 143, 345, K15 11, 75, 111, Kll, K59 85, 101, 117, 155 11, 55, 67, 245 93, K7 143, K1 43,93, 375, 383, K7

3

93, K15

4

43, 93, 155

6

K l l , K23

7

163, K7

8

Kll,

9

K19

199

10 10.1 10.2

55 169, 177, 199, 207, 237, 275, K27 101, 185, 219, 229, 245, 263, 401, Kl, K31, K35

11 12.1 12.2 13.3 13.4 14.1 14.2 14.3 14.3.1 14.3.2 14.3.3 14.3.4 14.4 14.4.1

169, 185, 191, 263, 313, 327, 363, K83 K27 Ill, 345, K31 427 301, 309, 313, 319, 327, 333, 407, K39, K69 237, K59 389, K59 271, 301, K39, K51, K55, K63, K69 383, 421 375, 439, K79 427, 453, K79, K83 309, 407, 439, K73 345, 363, 401, 413, K31 11, 111, 133

15

K51, K63

16

383

18 18.1 18.2 18.3 18.4 19

117 289 275 271, 283 295 163, 219, 313

20.1 20.2 20.3

125, 219, 319, 333, 345, 421, 433, K23 345, 433 219, 263, 319, 327, 333, 345, K45

10 21 21.1 21.1.1 21.3 21.4 21.6 22 22.1 22.1. 1 22.1.2 22.1. 3 22.2 22.2.1 22.2. 2 22.2.3 22.2. 4 22.3 22.4.2 22.5 22.5.2 22.5.4 22.6 22.6.1 22.7 22.8 22.8.1 22.8.2 22.9

Contents 85, 237, 275, K59 67, 85, 117, 177, 207, 289, K59 117, 199, 289 169 295 67 11, 383, 421, 433, K l l , K23, K31, K35 K51 K55 143, 185, 191, 245, 309, 313. 375, 383, 453, K39, K63, K73 K7 319 191, 229, K l , K83 Kl 301, 427, K55 K79 K69 263 407 43, 219, 327, 333 75 11, 93, 101 K15 155 11, 111, K19 133, 345, K45 163, 271, 283, 401 363, 413

Contents of Volume 86 Continued on Page 463

Review

Article

phys. stat. sol. (a) 86, 11 (1984) Subject classification: 1.2 and 1.4; 14.4.1; 22; 22.6; 22.8 Laboratoire de Microscopic Electronique Appliquée (EU A 545), Faculté des Sciences Saint-Jérôme, Marseille1)

A Survey of Domains and Domain Walls Generated by Crystallographic Phase Transitions Causing a Change of the Lattice By C. BOULESTEIX

Contents 1.

Introduction

2. Ferroelastic

and ferroic

crystals

2 . 1 Ferroelastic crystals 2 . 2 Other ferroic crystals 2 . 3 Common properties and differences between ferroic crystals 3. Theoretical

study of domains and domain

walls

3.1 T h e classical point of view 3.2 Space group formulation 3.3 Small disorientations and epitaxial growth 4. Experimental

study of domains

and domain

walls

4.1 Observation of domains and domain walls b y diffraction contrast electron microscopy 4 . 2 Observation of interfaces b y high resolution electron microscopy 5. Examples 5.1 5.2 5.3 5.4 5.5 5.6

of domain

structures

BaTi03 NbTe2 Pb3(P04)2 V 2 0 3 and Y2-XCTX03 Gd 2 (Mo0 4 ) 3 Si02

5.7 W 0 3 , a t o p o t a c t i c reconstructive phase transformation 6. Discussion

and

conclusion

References 1)

rue H. Poincaré, 13397 Marseille Cédex 13, France.

12

C. Boulesteix 1. Introduction

I n this paper we deal mainly with crystallographic phase transitions which occur by temperature lowering, with formation of a new lattice obtained by a slight deformation of the high-temperature lattice. Such transformations give rise to the formation of domains of different orientations which, in some cases, will be shown to be related to the high-temperature structure by epitaxy. Such transformations are cooperative phase transitions: displacive or shear transformations or even differential dilatation transformations [1,2], In special cases reconstructive transformations can give rise to similar phenomena [3], We will only consider the case of transformations with conservation of the total volume and obtained without straining the crystal. The new phase must exactly occupy the volume of the parent phase t h a t has been transformed or grow outside the crystal on its surface. I n the first case if a slight deformation of the lattice occurs, the conservation of the volume transformed into the new structure can be entirely achieved only thanks to the formation of different orientation domains related to each other by accomodation twins. When these domains are related to the high-temperature structure by the same kind of epitaxy they will be said to be equivalent, but different kinds of epitaxies can occur giving rise to non-equivalent orientation domains [4, 5]. Such transformations are generally ferroelastic ones, so t h a t before the study of domains and domain walls connected with the phase transition, we will briefly recall the properties of ferroelastic and, more generally, of ferroic crystals. Then, we will study domains and domain walls between ferroelastic domains and in one case between prototypic and ferroelastic structures (theoretical and experimental results) and then give some examples. 2. Ferroelastic and Ferroic Crystals Ferroelastic crystals and, from a more general point of view, ferroic crystals have properties rather similar to the well-known properties of ferroelectric crystals (see for example [6 to 8]). We will first recall these properties. 2.1 Ferroelastic

crystals

The concept of ferroelasticity and ferroelastic crystals has been introduced by Aizu [9 to 14]. A crystal is ferroelastic if it has two or more domains called "orientation states" (or orientation domains) t h a t can be transformed one into the other by means of a mechanical stress [9 to 12]. This definition is rather similar to t h a t of ferromagnetic and ferroelectric crystals where domains can be transformed one into the other by electric or magnetic fields. A ferroelectric crystal can be regarded as resulting from the slight deformation of a certain non-ferroelastic ideal crystal which is referred to as the "prototypic structure" [10]. I t has been shown t h a t the condition for a crystallographic phase transformation to be ferroelastic is t h a t the high-symmetry and lowsymmetry structures belong to different crystal systems [15, 16], the point group of the ferroelastic structure being a subgroup of the prototypic one (this point will be developed later on). Domains due to crystallographic phase transitions occur when the prototypic structure is transformed into the ferroelastic one. I n the deformation of the lattice one or more symmetry operations are lost which relate different orientation domains. They are said to be " F " operations [10 to 12],2) If for two different transformations the high-symmetry structures have the same point groups, and if the low-symmetry structures have also the same point groups, the F operations relating different

Survey of Domains and Domain Walls by Crystallographic Phase Transitions

13

orientation domains, are the same. Such transformations are said to be of the same "species". The number of orientation domains has been shown by Aizu [12] to be equal to the ratio of the order of the prototypic point group to the order of the ferroelastic point group. For the same ferroelastic species the number of possible orientation domains is then the same. The same kind of species is indicated by the point groups of the prototypic and of the ferroelastic (or ferroelectric or ferromagnetic) structures and by the letter F : for example m3m F 2/m(p) or m3m F 2/m(s) where m3m is the point group of the prototypic structure, 2/m the point group of the ferroelastic structure and where (p) or (s) indicates that the twofold axis of the ferroelastic structure was the fourfold axis or one of the twofold axes of the prototypic structure. This indication (p) or (s) is only given in case of ambiguity. All the different kinds of species: ferroelastic, ferroelectric, and ferromagnetic have been listed by Aizu [12]. Some crystallographic transformations can correspond to species which are simultaneously ferroelectric and ferroelastic, but this does not always occur. Ferroelectric, nonferroelastic transformations occur for the prototypic and ferroelectric crystals belonging to the same crystal system [8, 15]. I t must be noticed that a ferroelectric structure must be non-centrosymmetric. A ferroelectric crystal is characterized by its spontaneous polarization represented by a polar vector, a ferroelastic crystal by its spontaneous strain represented by a polar tensor of rank two. The simplest case occurs when the order parameters of the Landau theory of second-order phase transitions [16] are polarization and strain, respectively (many results of this theory can also be used in the case of first-order phase transitions). The corresponding transitions are then said to be "proper" transitions. But it must be noticed : (i) that such transitions are generally first-order transitions (it can be predicted by symmetry arguments when a transition is necessarily of first order and when it can be of second order [17, 18]; (ii) that a number of ferroelectric or ferroelastic transitions are not "proper" transitions. "Improper" transitions have been extensively studied mainly in the case of ferroelectric transitions [19 to 21]. It must be noticed that, for instance, a proper ferroelastic transition where the spontaneous strain creates a spontaneous polarization, is also a ferroelectric transition, but an improper one. More generally ferroelectrics and ferroelastics are improper when their main property (polarization or strain) is the consequence of another one that can be described by a tensor of a rank higher than one or two, respectively. It can be shown that ferroelectric and ferroelastic transitions involving a modification of the translation symmetry of the crystal must be classified as improper, so t h a t transitions accompanied by a change of the number of atoms in the unit cell are improper ones [22, 23]. 2.2 Other ferroic

crystals

Some other crystallographic phase transitions have some common features with ferroelectric or ferroelastic ones, but are not of any of these types. They can for example have different point groups and twins can occur in the low-temperature phase. They are said like ferroelectric and ferroelastic transitions to be ferroic transitions [14, 24, 25], These transitions are characterized by polar tensors of rank higher t h a n two which are related to changes of properties like piezoelectric coefficients (rank three) or elastic coefficients (rank four). Because of their common properties with ferro2

) In some papers all the symmetry operations of the high-symmetry structure are said to be "F" operations, these operations transform then one orientation domain of the ferroelastic structure into itself or into another orientation domain of the same structure.

14

C. BOULESTEIX

electric or ferroelastic transitions they are sometimes said to be improper ferroelectric or ferroelastic, though they do not present the main properties of ferroelectric or ferroelastic species and are not indicated as such in Aizu's classification [12], I t has been shown that every ferroelastic transition not only changes the crystal system (by spontaneous strain), but also some other properties like optical index of refraction, electrical conductivity, elastic stiffness, etc. [14]. I n the case of ferroic transitions which are improper ferroelastic transitions these kinds of effects can always occur but without spontaneous strain at the transition. A transition between different orientation states can still occur by straining the crystal, but in this case by a minimization of the total free energy. This is the case of the phase transition in quartz [14] where the orientation states differing in the elastic coefficients can be shifted by straining the crystal like in the case of ferroelastic crystals. 2.3 Common

properties

and differences

between

ferroic

crystals

For all ferroic transitions (ferroelectric, ferroelastic, or other ferroic) the point group of the low-symmetry structure must be a subgroup of the point group of the highsymmetry structure. In the case of ferroelastic transitions, the high-symmetry and low-symmetry structures must belong moreover to different crystal systems [8, 15, 26], the hexagonal and the rombohedral systems being considered as the same crystal system; the low-symmetry structure cannot be centrosymmetric in the case of ferroelectric crystals. Transitions between different crystal systems and with a low-symmetry structure which is not centrosymmetric can be simultaneously ferroelectric and ferroelastic. For instance, 42m F 3 species can be neither ferroelastic nor ferroelectric, though 3 is a subgroup of 32m because both structures are quadratic and because the low-symmetry structure is centrosymmetric, but 32m F m species can be simultaneously ferroelastic and ferroelectric transitions. Each transition being really ferroelastic if orientation domains can be changed by strain. Each transition being really ferroelectric if orientation domains can be changed by an electric field. Ferroic crystals are said to be "full" ferroelastics or "full" ferroelectrics when all the different orientation states are different in spontaneous strain or in spontaneous polarization, respectively; if not, they are said to be "partial". 3. Theoretical Study ol Domains and Domain Walls For any phase transition occurring by a reduction of symmetry, one single crystal of the high-symmetry phase (high-temperature phase) gives, by lowering temperature, domains of different possible orientations. Translation domains can also occur when the number of atoms per unit cell (and then also when the volume of the unit cell) is multiplied by an integer. Orientation domains have been first and extensively investigated in the case of ferroelectric transitions (see for example [6, 7, 10, 12, 27, 28]). They have been studied in the case of ferroelastic transitions by Aizu [9 to 14] and from a general point of view by Newnham and Cross [24, 25, 30], Janovec [31 to 33], Portier, Gratias, Fayard, and Guymont [34 to 39], In these studies the number of domains and their relative positions have been investigated. Only in the last papers a space group formulation has been taken into account; moreover in [4] and [5] small rotations occurring between domains of the same orientation state have been taken into account. We will first consider results deduced by taking into account a point group formulation (the classical point of view) and then consider results deduced from space group formulation, and at last by taking small disorientations into account.

Survey of Domains and Domain Walls by Crystallographic Phase Transitions

15

3.1 The classical point of view

The number of domains as well as their relative orientation is determined in the case of any ferroic crystal by a similar process. For ferroic transitions of the G 0 F G1 species, Gx is a subgroup of G0 [10, 12], where G0 and Gx are the point groups of the high-symmetry and of the low-symmetry structures, and where ga and gx, respectively, will be the order of these point groups. When the g0 symmetry operations are applied to one orientation state of the low-symmetry phase, each orientation state is obtained g1 times (because Gx is a subgroup of G 0 ), so that the number of different orientation states is n = gjg^. The number of translation states is equal to the ratio r of the volume of the primitive unit cell of the low-symmetry phase to the volume of the primitive unit cell of the high-symmetry phase. Orientation states are related one to the other by symmetry operations of the high-symmetry phase which have been lost in the low-symmetry phase. Translation states are related in the same way by translation operations of the high-symmetry phase which have been lost in the low-symmetry phase. The number of orientation and translation domains as well as their relative positions can then be easily deduced from this classical point of view. Orientation states are generally said to be in "twin position", though in this classical theory they are for ferroelastic structures not really related by a twin operation (here symmetry operations about a pseudo-symmetry element of the low-symmetry structure) but by a twin operation plus a small rotation (Fig. 1 to 3) or by a symmetry operation of the high-symmetry structure which has been lost in the phase transition. This point will be developed later on. A procedure to predict the orientation of domain walls in ferroics has been proposed by taking into account the strain compatibility between different orientation domains. It has been first elaborated by Fousek and Janovec [40] in the case of ferroelectric crystals when "piezoelectric effect and electrostriction lead to spontaneous deformations of the paraelectric lattice", i.e. in the case of ferroelectric-ferroelectric structures, but obviously such a procedure is important in the case of any ferroelastics. In this case the strain compatibility has been taken into account by Sapriel [41] to predict the orientation of domain walls. On each side of a wall between two orientation domains the change in length due to spontaneous strain must be the same so as to preserve the continuity at the interface, even at large distances. This is satisfied, if at the origin O of an orthogonal coordinate system being taken at the boundary between two domains S and S', we have (Sy — Sij) x t x] = 0 ,

(1)

where S^ are the components of the spontaneous strain tensor of S. This equation is representative of a cone with its summit at the origin. Owing to the fact that the surface that has been defined must be independent of the origin, the cone must be degenerated into two planes, which occurs for —

= 0

(2)

This gives two planar solutions said to be "permissible walls" [41]. One of the solutions of (2) consists of mirror planes of the high-symmetry structure, lost in the phase transition, said to be W walls. The other solution (not so obvious) is a plane containing a symmetry axis of the high-symmetry structure lost in the phase transition, which is generally not a crystallographic plane ( W ' walls) which is perpendicular to the W wall. I t must be noticed that this point of view gives the same results as the theory of mechanical twinning: (see for instance [42 to 44]), W walls corresponding to type I twins and W ' walls corresponding to type I I twins. In the first case the interface (habit plane) is also the mirror plane, in the second case the interface is a plane

16

C. Boulesteix Fig. 1. a) I n a structure with a centre of symmetry located a t the intersection of a plane P of symmetry and of an axis A of symmetry, the symmetry operations about P and A are equivalent, b) I n a slightly deformed structure where P and A are pseudosymmetry elements, b u t where a centre of symmetry exists a t the intersection of A and P, the symmetry operations about P and A are not exactly equivalent. From a domain S they respectively give two different domains Sj and Sji which are related by a rotation of small angle about the axis (dash-dotted line) common to P and P', where P ' is a plane perpendicular to A

containing the rotation axis (of angle 71). In the latter case the orientation of the interface can be determined by Sapriel's calculations, but the small orientation difference between domains Sf and Six related to S by a type I twin (Sj) and by its reciprocal type II twin (Sh) is not taken into account (Fig. l a , b) see [45]. In the case of a type I twin (which is a reflection or mirror twin), classified by Sapriel as a W wall, the relative position of the two twinned domains is such that they have a common lattice plane. In reciprocal space this gives a common row for the two twinned crystals [45], It is said to be an unsplit row (U.S.R.). All the other nodes of the reciprocal

twin. A row of unsplit spots (R.U.S.) occurs. I t is perpendicular to the twinning plane. The other nodes of t h e two reciprocal lattices are shifted in a direction parallel to the R.U.S. b) Relative position of the two reciprocal lattices of twinned crystals related by a type I I twin. A plane of unsplit spots (P.U.S.) occurs. I t is perpendicular to the twinning axis. The other nodes of the two reciprocal lattices are related as indicated on the drawing

Survey of Domains and Domain Walls by Crystallographic Phase Transitions

17

lattice of the two twinned crystals are shifted parallel to this direction (Fig. 2a). I n the case of a type I I twin (which is a rotation twin), classified by Sapriel as a W ' wall, the relative position of the two twinned domains is such that they have a common lattice direction. In reciprocal space this gives a common plane for the two twinned crystals. I t is said to be an unsplit plane (U.S.P.). All the other nodes of the reciprocal lattice of the two twinned crystals are shifted (Fig. 2 b.) Hence, taking into account the reciprocal space, the positions of domains Sf and Sfi related to domain S are very different, though they are only desorientated by a few degrees [45], These two kinds of twins give domains which are considered to be identical in the classical formulation, but, as shown by [45], they can be easily differentiated by electron microscopy and electron diffraction. An important point in Sapriel's paper is that orientation states can occur without any permissible wall. This occurs for four different species: 23 F 222, m3 F mmm, 3 F 1, 3 F 1. I n some other cases some domains can be related by permissible interfaces and some not; that is the case for example for 23 F 2 species where six different orientation domains can occur and where only the antiparallel domains are related by a permissible wall. This is also the case for example for 6 F 1 transition species and so on. Such interfaces without spontaneous strain compatibility (non-permissible interfaces) occur for example between domains t h a t are related by a rotation of angle 2tt/3 about the c-symmetry axis of the prototypic structure and which are not related at the same time by a twin operation (type I or type I I twin). This phenomenon was first shown to be possible by Fousek and Janovec [40] in the case of ferroelectric structures with a related strain (improper ferroelectrics). They also predicted that such non-permissible interfaces could dissociate into two twins separated by a new orientation domain. This has been observed recently as indicated in the experimental studies. We will see now two different alternative points of view taking into account a space group formulation in the first case and small disorientations in the second case.

3.2 Space

group

formulation

Only a space group formulation can take into account simultaneously twin and glide operations relating different domains resulting from a phase transition. Moreover the dimension and symmetry of the order parameter is related to the change of the space group and not only to the change of the point group (for example a change of the number of atoms per unit cell makes a ferroelectric or a ferroelastie transition to be improper). A thermodynamic theory deduced from Landau's theory of phase transitions [16, 18, 46] based on a free energy expansion in terms of two parameters has been elaborated by Levanyuk and Sannikov [47] and applied to the case of gadolinium molybdate. The results of this study have been completed by Dvorak [48], Similar studies have also been made by Dvorak in the case of langbeinites and rubidium trihydrogen selenite [49, 50] and in the case of lead phosphate by Torres [51]. In this latter case an ambiguity occurs, two different irreducible representations being able to give the observed transition, but the free energy expansion is the same in each case. A space group formulation is then necessary for the study of order parameters and also to get an expansion of the free energy. Similar studies including also the possibility of a transition and the symmetry of their order' parameters have been made in the case of improper ferroelectric, non-ferroelastic transitions and in the case of higher-order ferroics by Toledano and Toledano [22, 23], 2

physiea (a) 86/1

18

C. B o u l e s t e i x

A space group formulation has been used in the papers by Portier and coworkers [34 to 39] so as to determine all possible kinds of boundary operations in domain structures. The relative positions of the different domains are obtained by a decomposition of the space group of the high-symmetry structure in the space group of the low-symmetry structure. This method is quite general, it can predict the existence of irreducible translation-twin domains. Such domains are related by a glide and a twin operation which cannot be reduced to only a twin operation, the glide being parallel to the mirror plane or to the rotation axis. Such domains had been predicted by Wondratschek and Jeitschko [52] but they cannot be directly deduced if using the classical point group formulation used for ferroics (as seen previously) or for ordered alloys [53]. Any ferroelastic transition with the loss of a mirror-glide plane or of a rotation-glide axis would give rise to such domain operations. The determinations of the symmetry of the order parameter and of the relative positions of domains are tightly related. But, the symmetry of the order parameter corresponds to the symmetry operations which are preserved at the transition, while the relative positions of domains are related to the lost symmetry operations [32]. The interface between domains can be treated as a two-dimensional residue of the parent phase and described by a two-sided plane group. From this, the existence and the orientations of coherent stress-free domain walls can be deduced [32, 33, 54],

Fig. 3. a) From the classical point of view two orientation domains I c i and II c j occur in a 4mm F mm phase transition. These two domains are related by a rotation of n/2 or by a symmetry operation about one of the diagonal directions of the high-symmetry structure. It must be noticed that these two orientation domains can be related by a twin operation about one of their own diagonal directions plus a rotation of angle 28 where 8 is the angle between the diagonal directions of the highand low-symmetry structures, b) The conservation of the area of the unit cell makes the length of the diagonals to be preserved in a phase transition up to first order. An epitaxial growth is then possible along one of the diagonal directions of the high-symmetry structure. Two kinds of epitaxy can occur for the same diagonal direction giving rise to four different orientation domains, c) Domains I and II (or III and IV) correspond to two different kinds of epitaxy for the same diagonal direction. They are related by a twin operation. Domains I and III (or I I and IV) are related by a rotation of angle 28. Domains I and IV (or II and III) are related by a twin operation plus a rotation of angle 26

Survey of Domains and Domain Walls by Crystallographic Phase Transitions 3.3 Small

disorientations

and epitaxial

19

growth

Small disorientations and epitaxial growth have been taken into account by the author and Yangui [4, 5, 55, 56] in the study of domain structures so as to explain some experimental results observed in the case of monoclinic rare earth sesquioxides, but which are quite general. A two-dimensional transition can give a good example of the importance of these small disorientations and of epitaxial growth. Let us consider the case of a phase transition (without external strain) of the 4mm F mm species: taking into account the classical method two orientation domains occur, the number of orientation domains being equal to the ratio of the order of the point group of the highsymmetry structure (8) to the order of the point group of the low-symmetry structure (4) (Fig. 3a, where the two orientation states are indicated Ii c and IIi c ). These two domains are related one to the other by a rotation over an angle jr/2 or by a symmetry operation about one of the diagonal directions of the high-symmetry structure which are equivalent, but they are not exactly related by a twin operation of the lowsymmetry structure about one of the diagonal directions because the latter are not exactly parallel to the diagonal directions of the high-symmetry structure. The two domains are related by a twin operation about one of the diagonal directions plus a small rotation over an angle 26 where 6 is the angle between diagonal directions of high- and low-symmetry structures (Fig. 3 a). Let us notice that the conservation of the area of the unit cell implies the conservation of the length of the diagonals up to first order [4], Then, the strain compatibility is satisfied at the interface if the diagonals of the two structures are superimposed (Fig. 3 b), these diagonals are then the possible interfaces for epitaxial growth. Two different epitaxies can occur for each diagonal of the high-symmetry structure (Fig. 3 c). The two corresponding domains (I, II) are related by a symmetry operation about one of the diagonals of the high-symmetry structure or by a twin operation of the low-symmetry structure. The two other possible domains in epitaxial growth on the other diagonal (III, IV) are also related by similar operations. Domains I and I I I as well as domains I I and IV are related by a small rotation over an angle 26. The relative positions of these domains, as well as their total number cannot be deduced directly from the classical formulation, but can be deduced from some more sophisticated considerations. The epitaxial conditions being supposed to be well defined for one orientation domain, the orientation of any other one can be deduced by the symmetry operations of the high-symmetry structure lost in the transition; but it is generally not possible to relate orientation domains by simple twin operations of the low-symmetry structure. The number of domains, taking into account small disorientations, can be determined from a quite general formulation by Gratias and Portier [35, 39] deduced from the general Curie laws on symmetry. If the transition is induced by a temperature variation without external strain, the number of orientation domains is equal to the ratio n01 of the order g0 of the point group (that can be used here because we do not take into account the translation domains) of the high-symmetry structure S 0 to the order of the intersection g0 fl gi of g0 and g1 (g, being the point group of the low-symmetry structure Sj), 01 —

order of g0 order o f g o n gi'

If the symmetry elements of

go n gl = 2*

gl

and

are parallel to their homologues of S 0 , oi =

w

order of g0

Qrder pf gi

20

C. Boulesteix

which is the classical result (as indicated in Fig. 3a). If (as in Fig. 3 c) the low-symmetry phase grows epitaxially on the high-symmetry phase, the symmetry elements of S0 and Si are generally not parallel, and in this case, if the structures are centrosymmetric, g 0 Pi gi = i (where i is the inversion) and nn ,

=

order of g0 order of i

order of g0

In the case of our two-dimensional model this formulation would give: n01 = 8/2 = 4 (as in Fig. 3 c) while the classical formulation would have given n 01 = 8/4 = 2 (as in Fig. 3 a). In the case of the monoclinic structure of the rare earth sesquioxides, epitaxy of the monoclinic structure (B) on the hexagonal one (A) occurs so that the symmetry elements of B are not parallel to their homologues of A, hence the number of equivalent orientation domains is order of s n 12 „ «01 = — " 2 ~ - = ~ 2 = 6 ( F i g-

4b

)

while it would have been n01 — 12/4 = 3 if the symmetry elements of B had been parallel to those of A (Fig. 4 a). This means that for any kind of epitaxy of B on A, six equivalent orientation domains occur. Two different kinds of epitaxy have been observed (Fig. 5 a), domains of these two different kinds are not equivalent, they are not related by symmetry operations of the high-symmetry structure. The total number of orientation domains is then 6 X 2 = 12 (Fig. 5b). Since there are moreover three translation variants the total number of variants is then 6 X 2 x 3 = 36.

Fig. 4. a) From the classical point of view three equivalent orientation domains occur for a 5m F 2/m ferroelastic phase transition, b) B y taking into account small rotations due to the epitaxial growth that makes the symmetry elements of the high- and low-symmetry structures not to be parallel, six different equivalent orientation domains really occur

Survey of Domains and Domain Walls by Crystallographic Phase Transitions

21

Fig. 5. a) Monoclinic seeds of Nd 2 0 3 (or Sm 2 0 3 ) growing into the hexagonal structure often contain an accomodation twin. The epitaxy on the hexagonal structure is different for each p a r t of the twinned seed, so t h a t these two parts of the twinned seed are not equivalent orientation domains, b) Six equivalent twinned B seeds can occur. Each of them is made of two M and M' domains. Domains M and N or M' and N' are equivalent orientation domains, but not domains M and N '

4. Experimental Study of Domains and Domain Walls Diffraction contrast and high resolution electron microscopy are proper ways for the study of domains and domain walls. We will first consider results obtained by diffraction contrast electron microscopy (domain structures, interfaces between domains, domain shift, star patterns, and interfaces between prototypic and ferroelastic structures). Second we will consider results obtained b y high resolution electron microscopy (interfaces between orientation domains and between prototypic and ferroelastic structures). 4.1 Observation microscopy

4.1.1 Domain

of domains

and domain

walls

by diffraction

contrast

electron

structure

The general aspect of a domain structure resulting from a ferroelastic phase transition is given in Fig. 6a, b in two different cases: Ta 4 N [57] and Sm 2 0 3 [56, 58]. I n each case a great number of different orientation domains is visible. Though being of very similar species (rhombohedral to monoclinic phase transitions) the general aspect to the two domain structures is quite different. Each domain of Ta 4 N (Fig. 6a) is uniformly dark or clear because of the different orientation of the domains. On the contrary, bend contours are clearly visible in each Sm 2 0 3 domain (Fig. 6b), very probably because stresses occurring at the interfaces can cause bending of the orientation domain (moreover crystals used for electron microscopy studies can be at most 100 n m thick and are generally much thinner). These strong forces are due to a strong S m - 0 bonding inside p a r t l y covalent layers separated b y oxygen ions [56, 59]. I t is quite general t h a t ferroelastic domains can be either very flat, like in the case of Ta 4 N, or strongly bent, like in the case of Sm 2 0 3 , when strong stresses occur at the interfaces.

22

C. Boulesteix

Fig. 6. a) The Ta 4 N domain structure. I t must be noticed t h a t each orientation domain is uniformly dark or clear, showing t h a t domains are f l a t . A great number of star patterns are visible on the micrograph (by courtesy of P. Delavignette and coworker), b) The Sm 2 0 3 domain structure. Bend contours are clearly visible showing t h a t domains are strained by strong forces, c) Sm 2 0 3 domains related by W ' (interfaces) (type I I twins). The interfaces are not so regular as in the case of W interfaces 4.1.2

Interfaces

between

domains

I t is clearly visible in the case of the ferroelastic domains of Sm 2 0 3 (Fig. 6 b) t h a t the images of interfaces can be very thin or have a certain thickness. I n this case all the possible W walls (type I twins) are parallel to the [101] axis [60, 61]. The micrograph has been made with the electron beam parallel to this axis so t h a t all the thin inter-

Survey of Domains and Domain Walls by Crystallographic Phase Transitions

23

Fig. 7. Antiphase boundaries in a Sm 2 0 3 crystal visible between a twin and a grain boundary (the lines visible on each side of the micrograph are surface steps)

faces are W walls. The t h i c k i n t e r f a c e s are W ' walls. I t can be d e d u c e d f r o m t h i s image t h a t W ' walls are as u s u a l a s W walls a n d can be planar-like W walls. T h e W walls (type I twins) are planes parallel t o t h e t w i n n i n g plane, a n y d i s o r i e n t a t i o n is v e r y exceptional e x c e p t for t h e case of interfaces n e a r t h e t i p of m i c r o t w i n s which will be s t u d i e d l a t e r . On t h e c o n t r a r y W ' walls (type I I twins) can be in some cases r a t h e r irregular like in t h e case of Fig. 6 c which is a high m a g n i f i c a t i o n of W ' walls b e t w e e n different o r i e n t a t i o n d o m a i n s of S m 2 0 3 . W ' walls c o n t a i n t h e c r y s t a l l o g r a p h i c t w i n ning axis. F o r some special o r i e n t a t i o n of t h e electron b e a m (like in t h e case of Fig. 6 b), or in some cases w h e n t h e y look like r a t h e r irregular i n t e r f a c e s (Fig. 6c) it is e a s y t o distinguish b e t w e e n W a n d W ' walls. B u t generally a special s t u d y of t h e relative orientation of t h e t w o t w i n n e d crystals is necessary t o distinguish between W a n d W ' walls. I t is possible t o distinguish t h e m b y t a k i n g i n t o a c c o u n t t h e d i f f e r e n t p r o p e r t i e s of t y p e I a n d t y p e I I twins. As previously shown t w o d o m a i n s s e p a r a t e d b y a W wall (type I twin) h a v e one c o m m o n plane direction w h i c h m a k e s t h e m h a v e a c o m m o n row of unsplit spots (R.U.S.) in t h e reciprocal lattice. Two d o m a i n s s e p a r a t e d b y a W ' wall (type I I twin) have one c o m m o n lattice direction, which m a k e s t h e m h a v e a c o m m o n plane of unsplit spots (P.U.S.) in t h e reciprocal lattice. T h i s t e c h n i q u e h a s been used in order t o d i f f e r e n t i a t e t y p e I a n d t y p e I I t w i n s (W a n d W ' walls) in d i f f e r e n t cases [45]. A n t i p h a s e b o u n d a r i e s s e p a r a t i n g t r a n s l a t i o n d o m a i n s can also occur (like in t h e case of alloy ordering), w h e n t h e ferroelastic p h a s e t r a n s i t i o n occurs w i t h c h a n g e of t h e volume of t h e p r i m i t i v e u n i t cell. I n t h e case of t h e ferroelastic S m 2 0 3 s t r u c t u r e (Fig. 7) t h e v o l u m e of t h e ferroelastic p r i m i t i v e u n i t cell is t h r e e t i m e s t h e v o l u m e of t h e p r o t o t y p i c one giving rise t o t h r e e t r a n s l a t i o n d o m a i n s a n d t o t w o d i f f e r e n t a n t i p h a s e b o u n d a r i e s . I t m u s t be noticed t h a t t h e existence of such a change in t h e v o l u m e of t h e p r i m i t i v e u n i t cell is possible only f o r i m p r o p e r ferroelastic p h a s e t r a n s i t i o n s . 4.1.3 Domain shift Ferroelastic s t r u c t u r e s are characterized b y t h e possible t r a n s f o r m a t i o n of one orient a t i o n d o m a i n i n t o a n o t h e r b y straining t h e crystal, so t h a t W a n d W ' walls m u s t be mechanical twins. Mechanical t w i n n i n g occurs b y shear. I t is c h a r a c t e r i z e d b y t h e

24

C . BOTJLESTEIX

Fig. 8. The four twinning elements of mechanical twins are: K t the habit plane, K 2 the second undeformed plane, 7)j the shear direction, and r)2 which is the intersection of K 2 and of a plane perpendicular to K j and containing T)x; 2q> is the angle between r tl and v)2

knowledge of the shear modulus and of the four twinning elements: K 1 ; K 2 , r^, T)2 (Fig. 8). Mechanical twins can be either type I or type I I twins. K x the habit plane is also the twinning plane for type I twins. I t contains ra which is the shear direction, -/¡J is also the twinning axis for a type I I twin. K 2 is the second undistorted plane and v]2 is the intersection with K 2 of a plane containing ra and perpendicular to K x . The knowledge of two of these elements (K x and rj2 or K 2 and r a ) is sufficient to determine the two other ones. Generally only two of these elements are crystallographic planes or axes (Kj and ri2 or K 2 and T]x). If 2q> is the angle between K , andK 2 the shear modulus is equal to s = 2 cot {2

^ = T (ff2 — ° l )

with

(6)

(7)

+ ^22)

where

while

W

X22),

S r/Ja, = - X 2 1 ( F n 1 2 i

(3)

°2 > ffl '

(B)

Xij — yij(oij) ,

(9)

Yij = yij{a,-j) ,

(10)

^ij ~ y,j{vrj) >

(11)

=

~12(i]'iï]j)ll2 r Ca(r)

+ /3'eiÊi er

(12)

Here y^ioyj) stands for the w-th derivative evaluated at Cy. Further Cfi(r) = 0«(r) +

P'etei er

(13) 00

72 Fy = — (n'iVi)112 e i S Wi! J* xg xga(x) dx , ai 71 I il 71 and rj'i = — pt.

(14)

Thus the seven unknown coefficients can be solved through the seven equations (1) to (7) which involve quite a bit of numerical calculations. Further some approximations are made with respect to the dielectric constant. Also the Cy(r) and C{+j(r) are given by different expressions inside the core. In fact the Ci^j(r) is given by three different expressions in three different regions [22], Thus a search is felt necessary for numerical computation of partial structure functions S{j(K) with less assumptions and parameters. Preliminary calculations revealed that the primary form of the DCF for the primitive model electrolytes as derived by W L does not change if the ionic diameters are not differing by orders of magnitude. Waisman and Lebowitz perturbed the hard sphere potential by a Coulomb potential outside the core, Oy and obtained the DCF Cf-L{r) inside the core as CfiHr) =

son

|2 -

Oij )

for

r^aj,

(15)

A New Method of Evaluation of the Structure of Molten Sodium Chloride

45

where i and j stand for ionic species; eit e } are the ionic changes and fi' is equal to [k^Ty1 where kB is the Boltzmann constant, T the working temperature, and CgV'T(r) the well-known Percus-Yervick (PY) solution for neutral hard spheres given by [CsWlF =

~ GmMzk X -

= 0

for

X >

(1/2) Tfo&te X 3

1.

for

X ^ l ,

(16) (17)

Further we assume the mean spherical model (MSM) approximation [14] to hold good outside the core and this is given by Oij(r) =

-

- ¡ ^

for

r> ^ ,

(18)

where X = r/oij and 7] is the packing fraction which is given by (tcqo 3 !6). Farther (l + 2y) 2 ^ = ^ S r (i - v)'

(19)

n,

,20)

and £ ± M l

4(1 — r}Y (Vij + 2) 2 4(1 - % ) 4 q is the number density and is given by Q=J

N

(23)

where N is the number of ions, V the volume, and e the dielectric constant of the molten sodium chloride. B is given by U = j [ l + | - ( 1 +

2 | )] 1 ' 2

(24)

with | = Kj,a. K u is the well-known inverse Debye length given by

(25)

We rearrange equation (15) and write it as GJTW = «1 - A ( 7 ) + y ( 7 ) '

for

r=g co o co co co co i o © 1 o o © co ico

© ©

tr-

co

©

ia

. i> Tt* co I Hrtei m (M ©

ta

CO 00 CO co ©

00 00 CO 00 t© r—ti-H —* 1—4 lO !N ph; 4*

oa

© "i CO CÖ eö o

© ©

co ti©-K

51

52

R . V . G . R A O a n d S . KTTMAK P A L

Table 3 we give the structural features of the radial distribution function ( R D F ) in NaCl along with other various values, while in Table 4 we present the compressibility sum rule values. Table 4 Compressibility sum rule values m ,S(0)MSM [22]

S{0)«XP T = 1073 K 0.068

S( 0)(e*P> T = 1148 K

T = 1073 K 0.026

0.079

jS(0)(Present> T = 1148 K 0.038

Pt (exp [26]) ( 1 0 " u m 2 /N)

¡iT (present) (10" 1 1 m 2 /N)

32

45.5

3. Results Using the experimental value of the isothermal compressibility at the working temperature (T) we get a value of 0.079 for S(0). The present extrapolated value is 0.038 which is a reasonable value (deviation from experiment is 5 2 % ) like that obtained by previous workers [22] (deviation from experiment is 6 2 % ) . I t may be pointed out that all the Sij(K) curves in the limit K -» 0 give the same value as expected. The behaviour of S ™ ( K ) in the long-wave limit is related to the isothermal compressibility (f} T ) of the fused sodium chloride through the relation [20] l i m SNN(K)

JT-Î-0 where qt =

= q

t

(47)

JcxTPt

+ ¡j } = 2q = 31.2 nm~ 3 . 1

2i0 150

1

1

1

1

i

-Cua-CI

fKiJ Cno-NO -150-

-300

C 0.2 0.3 r/2.75 Inm) Fig. 5

-450

1

10

1

20

50 30 40 K inm'1/ —

Fig. 6

Fig. 5. Partial radial distribution functions