Physica status solidi: Volume 18, Number 1 November 1 [Reprint 2021 ed.] 9783112498125, 9783112498118

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plrysica status solidi

V O L U M E 18 • N U M B E R 1 . 1 9 6 6

Contents Review Articlc L.

X'agc Fcldkathodcii mit dünnen Isolatoiseliichten

ECKEUTOVÄ

3

Original I'npcrg G. V.

Dependence of the Geometry of the Formation of X - L ! a y DifCraction Linos on the Instrumental Factors in X - l i a y Investigations of Folyerystals

11

LÜCK

Zur Temperatur- und Fcldabliängigkeit der galvanomagnctisehen .Eigeiiseliaften von Aluminium und Indium

19

11. L Ü C K

Die Temperatur- und Feldabhängigkeit der galvanomagnctisehen Eigenschaften von Blei und der Einfluß der endlichen Probendiekc hierauf

ü'J

It.

DAVYDOV

Li. BliHuxKR u n d W . LAXUE

Diffusion von Sn und Zn in Sn imd in einer homogenen Sn-Zn-Lcgiorung . . .

D. IjEliUXEl! und W. L A N G E Volumen- und Korngrenzendiffusion von Sil und Zn in heterogenen Sn-ZnLegierungen G.

13.

S. 13.

Solvent Diffusivity in a Dilute li.C'.C. Binary Alloy

GIDUS

T.

L.

L. J.

HU

GRAHAM

Edge Dislocation Core Structure and the PeicrJs Barrier in Body-Centered Cubic Iron

V. liAmiAKlilSlIXAN Tunelling in a Two-Band Model Superconductor 1 j . K . SHARMA,

W . DREYHRODT

Localized 2 S 1 ( 2 -States Centres in ZnS It.

and

125

FUSSGAEXGER

Zum linearen Anklingen der Pliotoleitfähigkeit in Halbleitern

145

Die Wärmeleitfähigkeit des gelben PbO zwischen

151

A.

0

und

500

°C

MERLIXI

A Study of the (222) "Forbidden" Reflection in Germanium and Silicon. . . . 157

LANDUYT, It. GEVERS,

and

S.

AMELINCKX

On the Determination of the Nature of Stacking Faults in F.C.C. Metals from the Bright Field Image 1G7

K . L. MERKLF, A. SEEGER u n d

119

Interpretation of the Temperature Dependence of Ag + Dipole Strengths in Alkali llalides 133

EGOROV

J . VAX

113

S . PAT.

SCHNEIDER

and

W . NEUMANN 11. C O L E L L A

and

Temperature Variation of the Grüneisen Parameter of Copper

A. ItÄUBEK and J .

V. D.

99

Forward Bias Current-Voltage Characteristics for a Ileterojunction in which Tunelling Dominates 105

TAXSLEY

l t . 1'. G U I T A ,

75 85

Twinning Deformation in Iron at 1 °Iv

MCKICKAKIJ

I i . C'IIAXÜ a n d

(17

Radiation-Induced Point Defect .Clusters in Copper and Gold (I) G.

I I . BILDER

WOBSEE

Stapelfehlerdipole in kubiseli-fläehenzentrierten Metallen

173 189

Temperatur- und Verformungsabhängigkeit der Koerzitivfeldstärke von EisciiEinkristallen (II) 207

I I . C. BOLTON,

L. A. BURSILL,

T . I t . DUNCAN

and

A. C. MCLAREN,

and

lt. G. TURXER

On the Origin of the Colour of Labradorite D.

221

KUIILMANN-WILSDORE

The Stress Fields of Dislocation Tilt Boundaries in Anisotropic Cubic and Diamond-Type Crystals 231

E . STEixuEiSij

"Wandbeweglichkeitsmessungen an Fcrritcn mit reehteckförniiger Hystereseschleife 211

II.

Contribution to the Theory of Nonlinear Itcsponsc

STOLZ

(Continued

on cover

three)

251

physica status solidi B o a r d of E d i t o r s P. A I G R A I N , Paris, S. A M E L I N C K X , Mol-Donk, W. D E K E Y S E R , Gent, W. F R A N Z , Münster, P. G Ö R L I C H , Jena, E. G R I L L O T , Paris, R. K A I S C H E W , Sofia, P. T. L A N D S B E R G , Cardiff, L. N É E L , Grenoble, A. P I E K A R A , Poznan, A. S E E G E R , Stuttgart, 0. S T A S I W , Berlin, M. S T E E N B E C K , Jena, F. S T Ö C K M A N N , Karlsruhe, G. S Z I G E T I , Budapest, J. T A U C , Praha Editor-in-Chief P. G Ö R L I C H Advisory Board M. B A L K A N S K I , Paris, P. C. B A N B U R Y , Reading, M. B E R N A R D , Paris, W. B R A U E R , Berlin, W. C O C H R A N , Edinburgh, R. C O E L H O , Fontenay-aux-Roses, H.-D. D I E T Z E , Aachen, J. D. E S H E L B Y , Cambridge, G. J A C O B S , Gent, J. J A U M A N N , Köln, E. K L I E R , Praha, E. K R O E N E R , Clausthal-Zellerfeld, M. MATYÀS, Praha, H. D. M E G A W , Cambridge, T. S. MOSS, Camberley, E. N A G Y , Budapest, E. A. N I E K I S C H , Jülich, L. P A L , Budapest, M. R O D O T , Bellevue/Seine, B. Y. R O L L I N , Oxford, H. M. R O S E N B E R G , Oxford, R. V A U T I E R , Bellevue/Seine

Volume 18 1966

A K A D E M I E - V E R L A G

-

B E R L I N

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Contents Review Article L.

ECKERTOVX

Page Feldkathoden mit dünnen Isolatorschichten

3

Original Papers G. V.

DAVYDOV

R. LÜCK R . LÜCK

D . BERGNEB u n d W .

Dependence of the Geometry of the Formation of X - R a y Diffraction Lines on the Instrumental Factors in X - R a y Investigations of Poly crystals Zur Temperatur -und Feldabhängigkeit der galvanomagnetischen Eigenschaften von Aluminium und Indium Die Temperatur- und Fcldabhängigkeit der galvanomagnetischen Eigenschaften von Blei und der Einfluß der endlichen Probendicke hierauf

41 49

59

LANGE

Diffusion von Sn und Zn in Sn und in einer homogenen Sn-ZnLegierung

67

D . BERGNER u n d W . LANGE

G. S.

B. GIBBS B. M C R I C K A R D

Volumen- und Korngrenzendiffusion von Sn und Zn in heterogenen Sn-Zn-Legierungen Solvent Diffusivity in a Dilute B.C.C. Binary Alloy Twinning Deformation in Iron a t 4 °K

75 85 89

R . CHANG a n d L . J . GRAHAM

T.

L . TANSLEY

V.

RADHAKRISHNAN

Edge Dislocation Core Structure and t h e Peierls Barrier in BodyCentered Cubic Iron Forward Bias Current-Voltage Characteristics for a Heterojunction in which Tunelling Dominates Tunelling in a Two-Band Model Superconductor

99 105 113

R . P . G U P T A , P . K . SHARMA, a n d S . P A L

Temperature Variation of the Grüneisen Parameter of Copper . . A . R Ä U B E R a n d J . SCHNEIDER2

Localizad S 1/2 -State Centres in ZnS

119 125

W . DREYBRODT a n d K . FUSSGAENGER

V. D. EGOROV W. N E U M A N N

Interpretation of the Temperature Dependence of Ag + Dipole Strengths in Alkali Halides Zum linearen Anklingen der Photoleitfähigkeit in Halbleitern. . Die Wärmeleitfähigkeit des gelben P b O zwischen 0 und 5 0 0 ° C .

133 145 151

R . COLELLA a n d A . M E R L I N I

A Study of the (222) "Forbidden" Reflection in Germanium and Silicon

157

J . VAN L A N D U Y T , R . G E V E R S , a n d S . A M E L I N C K X

K . L.

MERKLE

On the Determination of the Nature of Stacking Faults in F.C.C. Metals from the Bright Field Image Radiation-Induced Point Defect Clusters in Copper and Gold (I).

167 173

A . SEEGER u n d G . WOBSER H . BILGER

Stapelfehlerdipole in kubisch-flächenzentrierten Metallen. . . . Temperatur- und Verformungsabhängigkeit der Koerzitivfeldstärke von Eisen-Einkristallen (II)

189 207

H . C. BOLTON, L . A . BURSILL, A . C. MCLAREN, a n d R . G . T U R N E R

On the Origin of the Colour of Labradorite

221

IV

Contents

T . R . DUNCAN a n d D . KUHLMANN-WILSDORF

E.

STEINBEISS

H . STOLZ

The Stress Fields of Dislocation Tilt Boundaries in Anisotropic Cubic and Diamond-Type Crystals Wandbeweglichkeitsmessungen an Ferriten mit rechteckförmiger Hystereseschleife Contribution to t h e Theory of Nonlinear Response

R . C. LINCOLN, K . M . KOLIWAD, a n d P . B . GHATE

Elastic Constants of Some NaCl Type Alkali Halides

231 241 251 265

W . DEGRIECK a n d G . JACOBS

H J . MATZKE

G.JUNGK

Properties of t h e B-Centre in Additively Coloured Ag + Doped KCl Crystals Radiation Damage and Xenon Release in Quartz and Fused Silica Following Ion Bombardment Zur Anregungs-Rekombinationsstatistik in Halbleitern

279 285 297

P . BRÄUNLICH a n d A . SCHARMANN

H J . MATZKE

Approximate Solution of Schön's Balance Equations for the Thermoluminescence and the Thermally Stimulated Conductivity of Inorganic Photoconducting Crystals Diffusion of Tritium from Lithium Fluoride

307 317

R . G E V E R S , J . VAN L A N D U Y T , a n d S . A M E L I N C K X

On a Simple Derivation of the Amplitudes of the Electron Beams Transmitted and Scattered b y a Crystal Containing Planar Interfaces — Images of Subgrain Boundaries

325

R . G E V E R S , J . VAN L A N D U Y T , a n d S . A M E L I N C K X

The Fine Structure of Spots in Electron Diffraction Resulting from the Presence of Planar Interfaces and Dislocations (I). . .

343

J . VAN L A N D U Y T , R . G E V E R S , a n d S . A M E L I N C K X

The Fine Structure of Spots in Electron Diffraction Resulting from the Presence of Planar Interfaces and Dislocations (II). . .

363

S. A . MOSKALENKO a n d P . I . K H A D S H I

K . TOMPA

H . FRANK

Infrared Absorption b y Excitons due to Photoionization and I n t r a b a n d Lattice Scattering Nuclear Magnetic Resonance of 63 Cu and 65 Cu in Copper . . . . Beitrag zur Widerstandsmessung von nn + - u n d pp + -Epitaxialschichten nach dem Durchbruchsverfahren

379 391

401

H . FLIETNER u n d N . KEMPE

Untersuchung der Schwingungseigenschaften des Oszillistors . .

415

K . ISEBECK, R . M Ü L L E R , W . SCHILLING, a n d H . W E N Z L

Reaction Kinetics of Stage I I I Recovery in Aluminium after Neutron Irradiation

427

K . M. KUNZ, A. K . GREEN, a n d E . BAUER

W. FRANK

On the Formation of Single Crystal Films of F.C.C. Metals on Alkali Halide Cleavage Planes in Ultrahigh Vacuum Die kritische Schubspannung kubischer Kristalle mit Fehlstellen tetragonaler Symmetrie (III)

441 459

W . A . BRANTLEY a n d CH. L . BAUER

The Geometry of Charged Dislocations in the NaCl Structure. .

465

V . A . K U T A S O V u n d I . A . SMIRNOV

Der Einfluß von überschüssigem antistrukturellem Bi auf die Wärmeleitfähigkeit des Kristallgitters von Bi 2 Te 3 , Bi 2 Te 3 -Sb 2 Te 3 und Bi 2 Te 3 -Bi 2 S 3 mit Jodbeimischung

479

V

Contents K . C. A . BLASDALE, R . K I N G , a n d K . E . PUTTICK

The Microstructure of Tensile K i n k s in Cadmium Crystals

. . .

491

Electrical Resistivity a n d Longitudinal Electrical Magnetoresistivity of Palladium-Rich P a l l a d i u m - E r b i u m Alloys a t Low Temperatures

505

S. ARAJS, G. R . DUNMYRE, a n d S. J . DECHTER

P . P E T R E S C U a n d V . GITEORDANESCU

Anisotropy of t h e Photoemission f r o m Some Coloured Alkali Halides G. C. T. L i u a n d J . C. M. L i E n e r g y of Polygonal and Elliptical Dislocation Loops

511 517

G. C. T. L i u a n d J . C. M. L i E n e r g y of Hexagonal Dislocation Loops

527

B. TUCK

541

Photoluminescence of GaAs 0 7 P 0 3

0 . HAUSER u n d M . SCHENK

CH. SCHWINK

U.

HEINECKE

D . H . HOWLING

Strahleninduzierte P h a s e n u m w a n d l u n g e n einiger Substanzen des Perowskit-Gittertyps u n d ihre t h e r m o d y n a m i s c h e Behandlung . Ü b e r den Begriff „aktives Gleitvolumen" u n d dessen B e d e u t u n g Ü b e r ein Modell zur Berechnung der Entmagnetisierungskurven von H a r t f e r r i t e n Identification a n d Morphology of Oxide Particles in Internally Oxidized Cu-Mn Alloys

547 557 569 579

L . B E N - D O R , A . G L A S N E R , a n d S . ZOLOTOV

Colour Centres in KCl Doped with Zn+ 2 , Cd+ 2 , a n d Hg+ 2

. . . .

593

G . B . STREET a n d W . D . GILL

P h o t o c o n d u c t i v i t y a n d D r i f t Mobilities in Single Crystal Realgar (AS4S4)

601

F . HÄUSSERMANN u n d M . W I L K E N S

Bestimmung der Stapelfehlerenergie kubisch-flächenzentrierter Metalle aus der Analyse des elektronenmikroskopischen Beugungskontrastes von Stapelfehlerdipolen

609

Measurements of t h e Anisotropy of t h e Dislocation Resistivity in Au, Ag, a n d Cu

625

H . BILGER

Elektronenmikroskopische Oberflächen- u n d Durchstrahlungsuntersuchungen von Eisen-Einkristallen sowie der Einfluß der Orientierung auf deren Verfestigungsverhalten

637

D. SCHMID

D e r j - F a k t o r von F-Zentren in Alkalihalogenid-Kristallen.

. . .

653

Q u a n t u m Theory of t h e Electrical Conductivity of Semiconductors w i t h a Non-Standard Energy B a n d

667

H.

YOSHINAGA

B . M . ASKEROV a n d F . M . GASHIMZADE

1 . P . M O L O D Y A N , D . N . N A S L E D O V , S . I . R A D A U T S A N , a n d V . G . SIDOROV

The Effective Mass of Electrons in (InSb)^ • ( I n T e ) ^ Crystals

.

677

Study of E n e r g y Transfer from 3d t o 4f Electrons in Antiferromagnetic MnE 2 : Eu 3 + Crystals

683

Thermally Activated Motion of Screw Dislocations in B.C.C. Metals

687

V . V . EREMENKO, E . V . MATYUSHKIN, a n d S. V . PETROV

V . VITEK

V . VÎTEK a n d E . KROUPA

Dislocation Theory of Slip Geometry a n d Temperature Dependence of Flow Stress in B.C.C. Metals

703

VI

Contents

B. HARRIS

Solution Hardening in Niobium

715

N . P . GUPTA a n d B . DAYAL

L. I. VAN M. JUNG

TORNE

Effect of Zero-Point Energy on the Specific Heats of Solidified Neon and Xenon Point Defect Electron Scattering in Beryllium Pseudo-abrupte legierte p-n-Übergänge in G a P

731 737 743

M . ASCHE, O . G. SARBEJ u n d V . M . VASETSKII

Piezowiderstand von p-Germanium

749

J . H . C. HOGG a n d W . J . DUFFTN

Twinning in InS

755

B . C . VAN Z O R G E , W . J . C A S P E R S , a n d A . J . D E K K E R

Note on Relaxation Effects in Mössbauer Spectra

761

I . SPINULESCU-CARNARU

The Crystalline Structure of ZnTe Thin Films

769

H . MAYER u n d J . HÖLZL

D.

SCHWARZ

Experimentelle Bestimmung der maximalen Austrittstiefen monoenergetischer Sekundärelektronen Die Ausläuferabsorption chalkogendotierter AgBr-Kristalle bei variabler Fremdkationenkonzentration

779 787

B . DORNER u n d H . H . STILLER

Die innere Dynamik der Tieftemperaturphasen des Molekülkristalls CH 4

795

E . NEMBACH, H . FREYHARDT u n d P . HAASEN

Abhängigkeit des Supraleitverhaltens tordierter Niobeinkristalle vom Probendurchmesser

807

H . TRÄUBLE u n d U . ESSMANN

G . SCHMID

Ein hochauflösendes Verfahren zur Untersuchung magnetischer Strukturen von Supraleitern Kriechexperimente an hochreinen Kupfer-Einkristallen

813 829

S. M . KLOTSMAN, A . N . TIMOFEEV, a n d I . SH. TRAKHTENBERG

On the Mechanism of Lattice Electromigration in M e t a l s . . . .

847

A . F . LUBCHENKO a n d S. I . DUDKIN

Effect of Concentration on Light Absorption and Dispersion b y Impurity Centres in the Case of Weak Electron-Phonon Coupling

853

V . I . S T A R T S E V , V . P . SOLDATOV, a n d M . M . B R O D S K Y

The R a t e of Twin Layer Growth in Bismuth Single Crystals. . .

863

G . A . S M O L E N S K I I , V . P . Z H U Z E , V . E . A D A M Y A N , a n d G . M . LOGINOV

Magnetic Properties of Ce, Pr, and Nd Monochalcogenides a t 4.2 to 1300 °K

873

V . F . MOISEEV a n d V . I . TREFILOV

Change of t h e Deformation Mechanism (Slip Poly crystalline a - I r o n

Twinning) in 881

K . K . SHVARTS, A . Y A . VITOL, A . V . P O D I N , D . O . K A L N I N , a n d Y u . A . E K M A N I S

Radiation Effects in Pile-Irradiated LiF Crystals

897

G . D . GUSEINOV a n d A . I . RASULOV R.

I.

ANISHCHENKO

H e a t Conductivity Study of GaSe Monocrystals Calculation of the Mean Inner Potential of a Crystal in t h e Statistical Theory

911 923

G . D . GUSEINOV, M . Z . ISMAILOV, a n d A . G . TALYBOV

On the New Semiconducting Compound HgTlS 2

929

VII

Contents Short Notes K . F . L I D E R , B . V . NOVIKOY, a n d S . A . PERMOGOROV

Application of Bound-Exciton Optical Spectra in the Study of Radiation Damage in Crystals

Kl

H . FLIETNER u n d W . GRÄFE

I.

LICEA

Großsignal-Feldeffekt zur Oberflächentermbestimmung bei Silizium On the Variation of the Electron Temperature in Polar Semiconductors by an Electric Field

K5 K9

B . F . ROTHENSTEIN, C. ANGHEL, a n d M . LUPULESCU

Influence of Hydrogen on the Magnetomechanical Damping in Nickel K13 K . S . ALEKSANDROV, L . M . RESHCHIKOVA, a n d B . V . BEZNOSIKOV

Behaviour of the Elastic Constants of KMnF 3 Single Crystals near the Transition of Puckering K17 H . L . BROWN a n d C. P . K E M P T E R

H.

WEVER

Elastic Properties of Zirconium Carbide K21 Eine Methode zur Bestimmung partieller Diffusionskoeffizienten aus dem Kirkendall-Effekt im gesamten Konzentrationsbereich . K25

G . B . ABDTTLLAEV, A . U . MALSAGOV, a n d V . M . GLAZOV

E.

PRAVECZKI

Thermoelectric Power of A ' B 1 1 1 ^ 1 Type Compounds in the Solid and Liquid State K29 Decoupling of Green's Functions in the Theory of Ferromagnetism K33

B . S. RAZBIRIN, I . FILINSKI, a n d B . WOJTOWICZ-NATANSON

H o t Excitons in Cadmium Selenide

K37

W . HASE u n d W . MEISEL

Mössbauer-Effekt von Fe 57 in Sesquioxiden des D 5 3 -Strukturtyps K41

U . GERHARDT a n d E . MÖHLER

Piezooptical Experiments on the Excitons in KBr and K I . . . .

K45

C. VAROTTO a n d A . E . VIDOZ

R.

KRISHNAN

On the Recovery and Ordering Processes of Cold Worked N i - F e Alloys K49 Influence of Tetrahedral Ni 2+ Ions on the Magnetic Properties of YIG K53

B . P . KOZYREV a n d T . BOTILA

H. REHME V. V. PARANJAPE

E. KAMIENIECKI

Some Photoelectrical Properties of KRS-5 Single Crystals . . . K57 Elektronenmikroskopischer Nachweis von Sperrschichten in Bariumtitanat-Kaltleiterkeramik K101 Distribution Function of Hot Electrons K103 Remark on Solving Boltzmann Equation for H o t Carriers . . . K105

W . H E Y E a n d G . WASSERMANN O. HENKEL

Mechanical Twinning in Cold-Rolled Silver Single Crystals . . . K107 Winkelabhängigkeit der magnetischen Eigenschaften bei normalerweise positiv wechselwirkenden Teilchenketten K l 13

C. P . KEMPTER

Vegard's " L a w "

K117

G. ALBRECHT u n d G . WOLE

Spezifische Elektronenwärme und A-Anomalie von CrH 0

84

. . . K l 19

G . S. NIKOLSKII a n d V . V . EREMENKO

The Effect of a Magnetic Field on the Heat Conductivity of Er Polycrystals K123

VIII P. V. K.

Contents SASTRY

STEENBECK

H . BILGER

R . J . FLEMING

On the Photo- and Thermo-Chemical Reactions in Additively Colored Sodium Chloride K127 Zur Magnetisierungsänderung dünner Nickelschichten durch elektrostatische Aufladung K131 On t h e Recovery of Plastically Deformed Iron Single Crystals above Room Temperature K135 Potential Energy Curve for F-Centre Electrons in Calcium Fuoride K 1 3 9

R . L A I H O , P . K E T O L A I N E N , a n d E . M Ä N T YSALO

G.

V . DAVYDOV

G.

RICHTER

Ultrasonic Modulation of F - B a n d in Additively Colored K B r . . K143 Dependence of the Geometry of the Diffractional Line Formation on t h e Instrumental and Other Physical Factors a t X - R a y Investigation of the Polycrystalline Fine Structure K145 Zur Bestimmung von Transportkoeffizienten f ü r Minoritätsträger inlnSb K149

G . B . ABDUIXAEV, F . B . GADIEV, G . M . ALIEV, S. I . MEKHTIEVA, a n d D . SH. ABDINOV

Interaction of Sodium Admixtures and Oxygen in Selenium . . . K153

Author Index D . SH. ABDINOV G . B . ABDULLAEV V . E . ADAMYAN G . ALBRECHT K . S . ALEKSANDROV G . M . ALIEV S . AMELINCKX . . . . C. ANGHEL R . I . ANISHCHENKO S . ARAJS M . ASCHE B . M . ASKEROV CH. L . B A U E R E . BAUER L . BEN-DOR D . BERGNER B . V . BEZNOSIKOV H . BILGER K . C. A . BLASDALE H . C. BOLTON T . BOTILÄ W . A . BRANTLEY P . BRAUNLICH M . M . BRODSKY H . L . BROWN L . A . BURSILL W . J . CASPERS R . CHANG R . COLELLA G . V . DAVYDOV B . DAYAL S . J . DECHTER W . DEGRIECK A . J . DEKKER B . DORNER W . DREYBRODT S . I . DUDKIN W . J . DUFFIN T . R . DUNCAN G . R . DUNMYRE L . ECKERTOVÄ V . D . EGOROY Y U . A . EKMANIS V . V . EREMENKO U . ESSMANN I . FILINSKI R . J . FLEMING H . FLIETNER H . FRANK

K153 K29, K153 873 K119 K17 K153 1 6 7 , 3 2 5 , 343, 3 6 3 K13 923 505 749 667 465 441 593 67, 7 5 K17 207, 637, K 1 3 5 491 221 K57 465 307 863 K21 221 761 99 157 41, K 1 4 5 731 505 279 761 795 133 853 755 231 505 3 145 897 683, K 1 2 3 813 K37 K139 415, K 5 401

W . FRANK H . FREYHARDT . . . . K . FUSSGAENGER . . .

459 807 133

K153 F . B . GADIEV F . M . GASHIMZADE . . 667 K45 U . GERHARDT R . GEVERS 167,325,343, 363 P . B . GHATE 265 V . GHEORDANESCU . . . 511 G . B . GIBBS 85 601 W . D . GILL A . GLASNER 593 K29 V . M . GLAZOV K5 W . GRÄFE L . J . GRAHAM 99 441 A . K . GREEN 731 N . P . GUPTA R . P . GUPTA 119 911,929 G . D . GUSEINOV . . . . . . . . 807 715 K41 547 609 569 K113 K107 755 779 579

P . HAASEN B . HARRIS W . HASE O . HAUSER F . HÄUSSERMANN . . . U . HEINECKE . . . . 0 . HENKEL W. HEYE J . H . C. HOGG J . HÖLZL D . H . HOWLING . . . . K . ISEBECK M . Z . ISMAILOV

427 929

. . . .

279 743 297

G . JACOBS M . JUNG G . JUNGK D . O . KALNIN E . KAMIENIECKI . . . N . KEMPE C. P . KEMPTER . . . . . P . KETOLAINEN . . . . P . I . KHADSHI . . . . R.KING S . M . KLOTSMAN . . . K . M . KOLIWAD . . . . B . P . KOZYREV . . . . R . KRISHNAN F . KROUPA D . KUHLMANN-WILSDORF K . M . KUNZ Y . A . KUTASOV

.

897 K105 415 . K21, K117 K143 379 491 847 265 K57 K53 703 231 441 479

X R . LAIHO J . VAN LANDUYT . W . LANGE J . C. M . L I I . LICEA K . F . LIDER R . C. LINCOLN G . C. T . L I U G . M . LOGINOV A . F . LUBCHENKO R . LÜCK M . LUPULESCU A . U . MALSAGOV E . MÄNTYSALO E . V . MATYUSHKIN H J . MATZKE H . MAYER A . C. MCLAREN S . B . MCRICKARD W . MEISEL S. I . MEKHTIEVA K . L . MERKLE A . MERLINI E . MÖHLER V . F . MOISEEV I . P . MOLODYAN S . A . MOSKALENKO R . MÜLLER

Author Index

.

.

K143 167,325, 343,363 67, 7 5 517, 527 K9 KL 265 517, 527 873 853 49, 59 K13 K29 K143 683 285, 317 779 221 89 K41 K153 173 157 K45 881 677 379 427

D . N . NASLEDOV E . NEMBACH W . NEUMANN G . S. NIKOLSKII B . V . NOVIKOV

677 807 151 K123 KL

S. PAL V . V . PARANJAPE S . A . PERMOGOROV P . PETRESCU S. V . PETROV A . V . PODIN E . PRAYECZKI K . E . PUTTICK

119 K103 KL 511 683 897 K33 491

S . I . RADAUTSAN V . RADHAKRISHNAN A . I . RASULOV A . RÄUBER B . S. RAZBIRIN H . REHME L . M . RESHCHIKOVA G . RICHTER B . F . ROTHENSTEIN

677 113 911 125 K37 K101 K17 K149 K13

0 . G . SARBEJ P . V . SASTRY A . SCHARMANN M . SCHENK W . SCHILLING D . SCHMID G . SCHMID J . SCHNEIDER D . SCHWARZ CH. SCHWINK A . SEEGER P . K . SHARMA K . K . SHVARTS V . G . SIDOROV 1. A . SMIRNOV G . A . SMOLENSKII V . P . SOLDATOV I . SPINULESCU-CARNARU V . I . STARTSEV K . STEENBECK E . STEINBEISS H . H . STILLER H . STOLZ G . B . STREET A . G . TALYBOV T . L . TANSLEY A . N . TIMOFEEV K . TOMPA L . I . VANTORNE I . SH. TRAKHTENBERG H . TRÄUBLE V . I . TREFILOV B . TUCK R.G.TURNER C. V. A. V. A.

VAROTTO M . VASETSKII E . VIDOZ VITEK Y A . VITOL

G. H. H. M. G. B. G.

WASSERMANN WENZL WEVER WILKENS WOBSER WOJTOWICZ-NATANSON WOLF

749 K127 307 547 427 653 829 125 787 557 189 119 897 677 479 873 863 769 863 K131 241 795 251 601 929 105 847 391 737 847 813 881 541 221 K49 749 K49 687,703 897 K107 427 K25 609 189 K37 K L 19

H . YOSHINAGA

625

V . P . ZHUZE S . ZOLOTOY B . C . VAN ZORGE

873 593 761

Review

Article

phys. stat. sol. 18, 3 (1966) Lehrstuhl jür Elektronik und Vakuumphysik, Mathematisch-Physikalische Fakultät der Karlsuniversität,

Prag

Feldkathoden mit dünnen Isolatorschichten Von L . ECKERTOVA

Inhaltsübersicht 1.

Einleitung

2. Der Malter-Effekt 3. Spitze, 4.

Isolierschichten

Sekundärelektronenemission

bedeckte

Kathoden

Sandwich-Kathoden

4.1 4.2 4.3 4.4

Konstruktion und Materialien Kathoden mit Dielektrikumsdicken d < 100 A Kathoden mit Dielektrikumsdicken d zwischen pa 100 A und 1 ¡im Kathoden mit Dielektrikumsdicken d ä 1 (j.m

5. Vorstellungen wich-Kathoden

5.1 5.2 5.3 5.4 6.

und die feldabhängige

mit dünnen

über den Mechanismus

der Leitung

und der Emission

von

Sand-

Prozesse an der Barriere Metall 1 — Dielektrikum Prozesse im Dielektrikum Prozesse in der oberen Elektrode Emission ins Vakuum Schlußfolgerungen

Literatur

1. Einleitung I n den letzten Jahren wurde den sogenannten kalten Kathoden große Aufmerksamkeit gewidmet. U n t e r dieser Bezeichnung versteht man gewöhnlich die Kathoden, die bei Zimmertemperatur (oder sogar bei tieferer Temperatur) emittieren und deren Emission mit einem sehr starken elektrischen Feld verbunden ist, die also als Feldkathoden arbeiten. Feldstärken der notwendigen Größenordnung von 106 V/cm und höher können entweder durch eine starke Asymmetrie des Elektrodensystems oder durch Anlegen der Spannung an eine dünne Isolierschicht erreicht werden. Die erste Methode, bei der die Kathode durch eine feine Spitze mit einem Krümmungsradius Ä 1 [xm gebildet wird, wurde von Müller [1] entwickelt. Damit wurden bis jetzt viele interessante wissenschaftliche Ergebnisse erzielt, die zu wichtigen praktischen Anwendungen geführt haben. Nur wenige Arbeiten (z. B. [2]) behandeln die Emission aus einer Metallspitze, die mit einer dünnen Schicht eines Dielektrikums bedeckt ist. Die Arbeiten von Malter [3] können als Ausgangspunkt der zweiten Richtung, m i t der wir uns hier befassen wollen, betrachtet werden. Malter h a t nämlich an l'

4

L . ECKERTOVÄ

dem System Al-Al 2 0 3 -Cs 2 0, Cs beim Elektronenbeschuß eine anomale Sekundäremission mit einem Koeffizienten bis 1000 und einer sehr großen Trägheit beobachtet. E r h a t dies dadurch erklärt, d a ß sich die Oberfläche der Probe infolge der Sekundäremission bis auf ein von dem Kollektorpotential abhängiges Potential positiv auflädt, u n d d a ß das in der dünnen Isolierschicht entstandene Feld die Elektronen aus der Unterlage reißt. Die Potentialdifferenz u n d d a m i t auch die daraus resultierende Elektronenemission k a n n auch noch nach dem Ausschalten des Primärstrahles eine gewisse Zeit lang aufrechterhalten werden. I n dieser Richtung wurden weitere Arbeiten durchgeführt. Man h a t mit verschiedenen Isolierschichten gearbeitet u n d die Oberflächenladung in verschiedener Weise erzeugt. Die praktisch interessantesten Ergebnisse wurden m i t MgO-Kathoden erzielt, die nach einem kurzen Anfangsprozeß eine stabile selbständige Feldemission aufweisen. Eine andere Möglichkeit der Realisierung ebener F e l d k a t h o d e n bieten die sogenannten K o n d e n s a t o r k a t h o d e n oder Sandwich-Kathoden, bei denen m a n die Spannung direkt a n zwei Elektroden legt, die durch eine dünne Isolierschicht voneinander getrennt sind. Die untere Elektrode ist k o m p a k t , die obere ist entweder netzförmig oder wird durch eine aufgedampfte Metallschicht gebildet, die f ü r die Elektronen halbdurchlässig ist. Mahl [4] u n d später Yudynskii [5] haben zum erstenmal solche K a t h o d e n konstruiert. Physikalisch sind diese K a t h o d e n in verschiedener Hinsicht dem K o n t a k t zweier Metalle analog, von denen eines mit einer Oxidschicht bedeckt ist. Deshalb konnte m a n vor allem die theoretische Bearbeitung des Metallkontakts von Holm u n d Kirschstein [6] zur theoretischen Beschreibung der K a t h o d e n nutzen. Eine eingehende Übersicht der Arbeiten (bis 1958) auf dem Gebiet der Feldemission aus dünnen Isolierschichten, besonders den Malter-Effekt betreffend, enthält das von Elinson u n d Vasiliev [7] verfaßte Buch. I n der letzten Zeit erschien eine diesbezügliche Übersicht von Zernov u n d Yasnopolskii [8], Seit dem J a h r e 1960 sind besonders viele neue Arbeiten erschienen, denen wir hier vor allem Aufmerksamkeit widmen wollen. E s sei hier noch darauf hingewiesen, daß auch die Emission „heißer" Elektronen aus Halbleitern, besonders aus p-n-Übergängen, den Gegenstand der gegenwärtigen U n t e r s u c h u n g e n bildet (z. B. [9]). U n t e r s u c h t wird auch die Emission aus der Oberflächensperrschicht eines Halbleiters (GaP [10]) u n d aus e x t r e m d ü n n e n Metallschichten, in denen die Leitung durch den Tunneleffekt erfolgt [11]. Diese Probleme haben m i t der hier beschriebenen Emission viel Gemeinsames, können hier jedoch nicht näher erörtert werden. Wir f ü h r e n hier n u r eine kurze Übersicht der wichtigsten Ergebnisse der Malter-Emission, der Emission aus MgO- u n d MgO-ähnlichen K a t h o d e n u n d der Emission aus den mit einer Isolierschicht bedeckten Spitzen an. F e r n e r werden wir uns hier eingehender mit den Sandwich-Kathoden befassen. Abschließend legen wir die Vorstellungen über den Mechanismus der Emission dieser K a t h o d e n dar. 2. Der Malter-Effekt und die feldabhängige Sekundärelektronenemission Wie schon erwähnt, k o n n t e die Malter-Emission bei verschiedenen Materialien festgestellt werden. Die Dicken des Dielektrikums lagen der Größenordnung nach meist zwischen 1000 A u n d 1 [im. Die Abhängigkeit des Emissionsstromes

Geometry of the Formation of X - R a y Diffraction. Lines

43

where y[ is an ordinate of the point B , and y\' is that of the point D (see Fig. 1). Also, as follows from equation (5), y[ is direct related to y2. Hence, with all other parameters fixed, the least deflection on the recording line in the positive direction of the ordinate axis will be given by the ray from the point N2 and the largest one will be given by that from the point N r For the points N2 and Nj formula (5) takes the form ' "

y

and

z2 + (y + ») tg p

9

=

y

K

_l_xl(n - y + x2 tg P) + n)tgp •

'

ia\ (8)

Here the argument y corresponding to the minimum value of the function y[t —n(y) will be expressed as y = — n — x2 ctg /? + cosec ¡3 /a^ x2 .

(9)

Again, it is seen from (7) and (8) that the functions y{: = y[y _„(«/) and Vun = y'i,n (y) have points of discontinuity with the following values of y : y = y3 = - n -

and

y = yi

=

-X2

2

x

ct

g /S

(10)

ctg/?.

n

(11)

Hence, for the case of a real intersection of the ray with the recording line formula (9) becomes y = y5 = — n — x2 ctg (} + cosec /?

x2.

(12)

Similarly, for the minimum value of the function y'i,n(y) one obtains y = y6 = n — x2 ctg jS + cosec

x2 .

(13)

Using equation (8) with equality of the expressions m

_L ^(nm + x2 tg p) = _ x2 + (m — n) tg p

m

x1 (n + m+ x2 tg p) x2 — (m + n) tg p

one finds the focusing condition for the extreme rays : 1

3

z2(l+tg2/?)

*

'

With m > n in equation (15), there is a direct relation of x3 to x2 and 6. In the case where condition (15) holds, the formulae (12) and (13) can be expressed as follows : = ys

and ye-n

_ *2 - i K ^ t g P ) t gp ^ -

2

- m2 tg* p [

tg^

.

From (16) and (17) follows that ys and y6 are direct related to x2 and 6.

' (i/j

44

G. V . DAVYDOV

Again, Df,

with

—2n

x2

oo

and

and

ye -> 0

(18)

6 -> 90°,

which means, with condition (15) holding, that the quantities — 2 n and 0 are the largest limiting values of y& and ys, respectively. From equations (16) and (17) with n — 0 one obtains (19) Thus, under the condition expressed by (15) and with the point X - r a y source, the ray diffracted from the specimen at the point having coordinates (0, 0, 0) yields the smallest deflection on the recording line only with x2 oo or 6 90°. But in the real case, with m y5, the smallest deflection will be yielded by the ray diffracted from the point having coordinates (0, yh, 0), which is evident from the above considerations. As follows from (7), (15), and (16) this smallest ray deflection in the case of the plane problem will be given by the formula vi.M=

2 V(*2 - „wtif -

'" 2 tg2

(x, + xt)

"

( 2 o)

and the largest one, for the extreme ray according to equation (8), will be given by yi,.„ = m + *

( » - m + 3?too

or

e

90°.

(24)

Thus, using a monochromatic X - r a y source in our plane problem, the theoretically possible minimum instrumental broadening of the X - r a y diffraction line is equal to the linear dimension of the X - r a y source. This makes evident the significance of the fineness of the X - r a y tube focale spot or the reducing of its effective area. As it is seen from (15) and (22) one obtains with n = 0

and

• m © C © © IN © to i. (N Ml m I1Mi 00 00 I i-HI 1 © i-H< M N 1 M M ©© 1 in in in © © r- to in IN IN IN m lO © m in

©

£

S

1 1

1 1

1

1

© © © COCO lO lO m eo eo CO eo © © © © © © 1© © © 1© © © 1 TH j Tji M< © © © in in in IN IN CI to lO m i> i> r -

© © © ©

© ©

© © ©

© © © © © © © © © © © © © © © © ©

© © © © © ©

iM ')/o b im f ü r den Size-Effekt korrigierten Kohlerdiagramm bei 4,2 °K f ü r die verschiedenen Probendicken angegeben. Eine quadratische Abhängigkeit ist nicht zu finden. E s d ü r f t e an der offenen Fermifläche liegen, d a ß die Kanersche Theorie an Blei nicht bestätigt werden k a n n . So sind wir auf eine empirische K o r r e k t u r angewiesen. Zur K o r r e k t u r des Hallkoeffizienten wollen wir folgenden Ansatz machen: Rb

=

R

+

Rs.

(1)

Rh ist der bulk-Hallkoeffizient u n d der Definition n a c h von der Probendicke unabhängig, R ist der gemessene Hallkoeffizient u n d i? s der size-Hallkoeffizient. Glieder höherer Ordnung werden vernachlässigt. R s sei in Anlehnung a n die Theorie von Sondheimer [8, 11] eine F u n k t i o n von H d, zur K o r r e k t u r des ~ 1 ß a n ; u n d nach Sondheimer Widerstandseinflusses nehmen wir wiederum können wir fordern, daß das Korrekturglied eine F u n k t i o n von djl ist. So bekommen wir folgenden A n s a t z : Rs

=

f(QbdH).

(2)

10

§

w1 10''

W'3 QjBfr&cm/kG]

10'2

Flg. 7. Differenz der Widerstandsänderung in den beiden Stellungen bei Heliumtemperatur v o n Blei ( 9 9 , 9 9 9 + %) für Proben verschiedener Dicke in einem für den Size-Effekt korrigierten Kohlerdiagramm

QbdH(pttun2k0e)—

-

Fig. 8. Interpolierte Korrektur des Hallkoeffizienten für die Probendicke

64

R.

^ Blei, $9.3$}*% md'0.052 cm • d-0.012 cm 0W51 cm ^d'0.0022cm $S.9S% od-0.013 cm

LÜCK

F i g . 9. Feldabhängigkeit des für die endliche Probendicke korrigierten Hallkoeffizienten R\, bei Heliumtemperatur von verschiedenen Bleiproben in einem für den Size-Effekt des Widerstandes korrigierten Kohlerdiagramm

n o

Aus unseren Meßergebnissen der Fig. 6 wurde diese Funktion nach einem Interpolationsvero fahren ermittelt. Sie ist in Fig. 8 dargestellt. Hiermit war es möglich, unsere Meßkurven zu c? korrigieren. In Fig. 9 ist der für die endliche ® Probendicke korrigierte Hallkoeffizient i? b bei A 4,2 UK in einem für den SizeEffekt korrigierten T Kohlerdiagramm aufgetragen. Im Rahmen der 4 1 • Meßgenauigkeit ist jetzt abgesehen von den beiden >15 niedrigsten Feldstärken die Kohlerregel in bezug auf den Hallkoeffizienten bei veränderlicher Probendicke und bei veränderlichem Rest widerstand, aber konstant gehaltener Temperatur erfüllt. Wie 20 / sieht nun der Temperatureinfluß aus 1 Beide KorgJBIfjSicm/kß/ rekturen wurden auch für die Temperaturkurven durchgeführt, diese Korrekturen waren aber sehr gering. Die Temperatur kurven für gleich reine Proben unterschiedlicher Dicke sind im Rahmen der Meßgenauigkeit identisch. Als Beispiel ist in Fig. 10a die Temperaturabhängigkeit einer Probe für drei Feldstärken in einem Kohlerdiagramm dargestellt. Fig. 10 b zeigt das Entsprechende für eine weniger reine Probe. Aus diesen Abbildungen ist eindeutig der Schluß zu ziehen, daß bei veränderlicher Temperatur für den Hallkoeffizienten die Kohlerregel nicht mehr gilt. Bei der Widerstandsänderung können wir eine empirische Korrektur leichter durchführen. Da im nicht korrigierten Kohlerdiagramm (Fig. 3 und 4) die Kohlerregel erfüllt ist, braucht man nur eine Korrektur im Sinne der «^Korrektur anzubringen, etwa 0

(Ae)b =

Aef.

(3)

Öb

• •

J*

• •

•

5 1

•

o

• Erwärmung von fl. Helium aus. 2.0 k6 + . . . . ßOkO . . . . ¡2k6 13M

o

1 gJB(fiSlcmlkG)

O? 15 |

I a

o o°° c9

20

o°

d-0.005icm Blei.99.S9$ %. o Temperatur dl sfI.Heliums n f l . Helium aus, 6,0kC * Erwärmung vi • • • 92k6 • • • • ISMO 1 10'! QjBlijSicm/kSh

F i g . 10. Feld- und Temperaturabhängigkeit des für die endliche Probendicke korrigierten Hallkoeffizienten Rh von Blei in einem f ü r den Size-Effekt des Widerstandes korrigierten K o h l e r d i a g r a m m a) 99,909 + % , b) 9 9 , 9 9 % ,

d = 0 , 0 0 5 4 cm il = 0 , 0 1 3 cm

65

Galvanomagnetische Eigenschaften von Blei

5. Schlußfolgerungen Um diese eigentümlichen Ergebnisse verstehen zu können, muß man sie mit den Erscheinungen, wie sie bei der Messung galvanomagnetischer Eigenschaften an Blei-Einkristallen gefunden wurden [7], vergleichen. Die Widerstandsänderung zeigt am Einkristall unbegrenztes quadratisches Anwachsen mit B, lediglich in bestimmten Richtungen zeigt sie Sättigung. Voraussetzung hierfür ist, daß die Bedingung n+ = aufgehoben wird; dabei treten offene Bahnen mit unendlich kleiner Schichtdicke auf [13]. I s t dagegen die Schichtdicke endlich klein, kann es je nach Orientierung des Stromes zur Richtung der offenen Bahn wieder quadratisches Ansteigen geben. In der Richtung des quadratischen Ansteigens der Widerstandsänderung haben Alekseevskii und Gaidukov [7] einen linearen Zusammenhang der Hallspannung mit der magnetischen Feldstärke gefunden, wie es in der Theorie von Kaner [12] für den Hochfeldfall bei n+ = n_ gefordert wurde. In den übrigen Richtungen, in denen die Widerstandsänderung im Rotationsdiagramm ein Minimum zeigt, fanden Alekseevskii und Gaidukov [7] im Rotationsdiagramm der Hallspannung ein Maximum. Die Linearität ist in dieser Richtung aufgehoben. Eben diese Erscheinung haben wir auch am Vielkristall gefunden. Möglicherweise ist am Einkristall in diesen Richtungen wie bei uns ein Vorzeichen Wechsel bei geringen Feldstärken zu finden. Wenn beim Halleffekt der Einfluß der offenen Fermifläche so groß ist, muß geprüft werden, ob er nicht auch in der Widerstandsänderung zu finden ist. Für die Feldabhängigkeit folgt aus unseren Messungen Ao/p ~ B1,75 anstatt ~ 2 J 2 , wie es für die übrigen Richtungen am Einkristall durchaus gefunden wurde [7]. Die -B-Potenz der Feldabhängigkeit von (g(J5) — o { B ) 90 °)/o b steigt mit zunehmender Probendicke. Eindeutige Schlüsse können hieraus nicht gezogen werden, da verschiedene gegenläufige Einflüsse im Spiel sein müssen. Für die Abweichungen des Hallkoeffizienten von der Kohlerregel bei Temperaturänderung auch noch im Übergangsgebiet (a> r « 1) möchten wir ebenfalls die offene Fermifläche als Ursache ansehen, da der Einfluß der offenen Fermifläche auf den Halleffekt wesentlich größer ist als auf die Widerstandsänderung und diese ja auch bei Temperaturänderung der Kohlerregel genügt (Fig. 11). Ein zwingender Schluß ist aus dem Experiment jedoch nicht zu ziehen. Mit der gleichen Berechtigung könnten die Nichtexistenz der Relaxationszeit, die Anisotropie der Gitterschwingungen und deren Temperaturabhängigkeit oder der temperaturabhängige Übergang der Elektronen von einem Band ins andere als Erklärung herangezogen werden.

Fig. 11. 5

Feld- und Temperaturabhängigkeit der Widerstandsänderung von Blei (99,99 % ) in einem Kohlerdiagramm

physica

gjß(pttcm/k6}-

66

R . LÜCK: Galvanomagnetische Eigenschaften von Blei

Fest steht, daß im Niedrigfeldfall die Kohlerregel für den Hallkoeffizienten wieder gilt und dieser einem Wert zustrebt, der nicht einmal im Vorzeichen mit dem für n = 4 nach der Theorie quasifreier Elektronen geforderten übereinstimmt. Danach ist bei Blei die Abweichung der Fermiflache von der Kugel form ähnlich wie bei Indium und im Gegensatz zu Aluminium [1] groß. Eine bessere Erklärung bringt im Niedrigfeldfall qualitativ die Zweibandtheorie von Sondheimer und Wilson [14], nach der für n+ = n_ unter der Voraussetzung gleicher Leitfähigkeiten der beiden Bänder für den Hallkoeffizienten ein Wert nahe Null erwartet wird. Aus einer Abschätzung unserer Meßergebnisse des Hallkoeffizienten und der Widerstandsänderung im Magnetfeld mit dieser Theorie ergibt sich n+ = n_ = 0,3. Der wahre Wert dürfte zwischen diesem und dem von der Theorie quasifreier Elektronen geforderten (nach Fig. 1) liegen. Herrn Dr. Th. Ricker gilt mein Dank für zahlreiche anregende Diskussionen. Herrn Dipl.-Phys. K . E. Saeger und Fräulein I. Atmer sei für die tatkräftige Unterstützung bei der experimentellen Durchführung gedankt. Die Deutsche Forschungsgemeinschaft unterstützte dankenswerterweise diese Arbeit. Literatur [1] R. LÜCK, phys. stat. sol. 18, 49 (1966). Teil I der Dissertation, TH Stuttgart, 1966 [2] I. M. LIFSHITS, M. YA. AZBEL und M. I. KAGANOV, Soviet Phys. — J . exper. theor. Phys. 3, 143 (1956). [3] I. M. LIFSHITS, M. YA. AZBEL und M. I. KAGANOV, Soviet. Phys. — J . exper. theor. Phys. 4, 41 (1957). [4] A. V. GOLD, Phil. Trans. Roy. Soc. (London) A 251, 85 (1958). [ 5 ] L . MACKIKNON, M. J . TAYLOR u n d M. R . DANIEL, P h i l . M a g . 7, 5 2 7 ( 1 9 6 2 ) .

J . A. RAYNE, Proc. 8th Internat. Conf. Low Temperature Physics, London, 1963 (S. 204). A. R. MACKINTOSH, Phys. Rev. 131, 2420 (1963); Proc. Roy. Soc. A 271, 88 (1963). [6] R. C. YOUNG, Phil. Mag. 7, 2065 (1962). R. T. MINA und M. S. KHAIKIN, Soviet Phys. — J . exper. theor. Phys. 15, 24 (1962); 18, 896 (1964). [7] N. E . ALEKSEEVSKII und Y u . P. GAIDUKOV, Soviet Phys. — J . exper. theor. Phys. 9, 311 (1959); 10, 481 (1960); 14, 256 (1962). J . E . SCHIRBER, P h y s . R e v . 1 3 1 , 2 4 5 9 ( 1 9 6 3 ) . [ 8 ] E . H . SONDHEIMER, P h y s . R e v . 8 0 , 4 0 1 ( 1 9 5 0 ) . [ 9 ] D . K . C. MACDONALD u n d K . SARGINSON, P r o c . R o y . Soc. A 2 0 3 , 2 2 3 ( 1 9 5 0 ) .

[10] M. KOHLER, Ann. Phys. (Germany) 32, (5), 211 (1938). ] 1 1 ] E . H . SONDHEIMER, P h y s . R e v . 8 0 , 4 0 1 ( 1 9 5 0 ) .

]12] E. A. KANER, Soviet Phys. — J . exper. theor. Phys. 7, 454 (1958). [ 1 3 ] N . E . ALEKSEEVSKII,

Y u . P . GAIDUKOV,

I . M . LIFSHITS

und

V . G. PESHANSKII,

Soviet. Phys. - J . exper. theor. Phys. 12, 837 (1961). [14] E. H. SONDHEIMER, Proc. Roy. Soc. A 193, 484 (1948). E . H. SONDHEIMER und A. H. WILSON, Proc. Roy. Soc. A 190, 435 (1947). (Received May 11,

1966)

D.

BERGNER

67

und W. LANGE: Diffusion von Sn und Zn

phys. stat. sol. 18, 67 (1966)

und Institut

Forschungsinstitut für NE-Metalle Freiberg (a) für Metallkunde und Materialprüfung der Bergakademie

Freiberg

(b)

Diffusion von Sn und Zn in Sn und in einer homogenen Sn-Zn-Legierung Von D . BERGNEB (a) u n d W . LANGE (b)

Neben Ergebnissen methodischer Untersuchungen über die Anwendbarkeit der Fisherschen Methode zur Messung von Korngrenzendiffusionskoeffizienten werden die Ergebnisse von Messungen der Volumendiffusion von Sn und Zn und der Korngrenzendiffusion v o n Sn in Sn und in einer homogenen Sn-Zn-Legierung mit 0,5 Gew.% Zn mitgeteilt. An investigation is made of the applicability of the Fisher method for measuring grain boundary diffusion coefficients. Results are also given of volume diffusion measurements of Sn and Zn, and of the grain boundary diffusion of Sn in Sn and a homogeneous Sn-Zn alloy containing 0.5 w t % Zn.

Die Untersuchungen zur Selbstdiffusion in Metallen blieben bisher vorwiegend auf reine, d. h. unlegierte Metalle beschränkt. Dies gilt f ü r die Messung der Volumendiffusion wie auch insbesondere f ü r Untersuchungen zur Korngrenzendiffusion. F ü r das Verständnis des Verhaltens technischer Werkstoffe ist es jedoch nützlich, die Kenntnisse über das Diffusionsverhalten in Legierungen, und zwar sowohl f ü r homogene als auch heterogene Legierungen, zu erweitern. I n der vorliegenden Arbeit soll über die Ergebnisse von vorbereitenden methodischen Untersuchungen und von Messungen der Diffusion von Sn und Zn in Sn und einer homogenen Sn-Zn-Legierung mit 0,5 Gew% Zn berichtet werden. 1. Versuchsvorbereitung und Auswertungsverfahren Die Probenvorbereitung und die Versuchsdurchführung erfolgten in der bereits früher [1, 2] beschriebenen Weise. Die Auswertung der Volumendiffusionsmessungen wurde nach dem Schichtenteilungsverfahren durchgeführt [1], die der Korngrenzendiffusionsmessungen wie in [2] nach dem experimentell sehr einfachen Verfahren von Fisher [3].1) Lange u. a. [4] haben gezeigt, daß die mit der Fisherschen Methode gewonnenen Aktivierungsenergien etwa mit den Werten übereinstimmen, die an Bikristallen mit einem großen Orientierungsunterschied gemessen werden. E s schien daher möglich, auch die vorliegende Untersuchung unter Anwendung der Fisherschen Methode durchführen zu können. Hierfür war es notwendig, zu überprüfen, unter welchen Versuchsbedingungen man bei Anwendung der Fisherschen Methode eindeutige Ergebnisse erwarten kann. Die experimentellen Arbeiten wurden weitgehend im Institut für Anwendung radioaktiver Isotope der T U Dresden, zum Teil noch während der Zugehörigkeit der Autoren zu diesem Institut, durchgeführt. Für die Ermöglichung des Abschlusses der Arbeiten sei an dieser Stelle besonders gedankt. 5»

D.

BERGNER

67

und W. LANGE: Diffusion von Sn und Zn

phys. stat. sol. 18, 67 (1966)

und Institut

Forschungsinstitut für NE-Metalle Freiberg (a) für Metallkunde und Materialprüfung der Bergakademie

Freiberg

(b)

Diffusion von Sn und Zn in Sn und in einer homogenen Sn-Zn-Legierung Von D . BERGNEB (a) u n d W . LANGE (b)

Neben Ergebnissen methodischer Untersuchungen über die Anwendbarkeit der Fisherschen Methode zur Messung von Korngrenzendiffusionskoeffizienten werden die Ergebnisse von Messungen der Volumendiffusion von Sn und Zn und der Korngrenzendiffusion v o n Sn in Sn und in einer homogenen Sn-Zn-Legierung mit 0,5 Gew.% Zn mitgeteilt. An investigation is made of the applicability of the Fisher method for measuring grain boundary diffusion coefficients. Results are also given of volume diffusion measurements of Sn and Zn, and of the grain boundary diffusion of Sn in Sn and a homogeneous Sn-Zn alloy containing 0.5 w t % Zn.

Die Untersuchungen zur Selbstdiffusion in Metallen blieben bisher vorwiegend auf reine, d. h. unlegierte Metalle beschränkt. Dies gilt f ü r die Messung der Volumendiffusion wie auch insbesondere f ü r Untersuchungen zur Korngrenzendiffusion. F ü r das Verständnis des Verhaltens technischer Werkstoffe ist es jedoch nützlich, die Kenntnisse über das Diffusionsverhalten in Legierungen, und zwar sowohl f ü r homogene als auch heterogene Legierungen, zu erweitern. I n der vorliegenden Arbeit soll über die Ergebnisse von vorbereitenden methodischen Untersuchungen und von Messungen der Diffusion von Sn und Zn in Sn und einer homogenen Sn-Zn-Legierung mit 0,5 Gew% Zn berichtet werden. 1. Versuchsvorbereitung und Auswertungsverfahren Die Probenvorbereitung und die Versuchsdurchführung erfolgten in der bereits früher [1, 2] beschriebenen Weise. Die Auswertung der Volumendiffusionsmessungen wurde nach dem Schichtenteilungsverfahren durchgeführt [1], die der Korngrenzendiffusionsmessungen wie in [2] nach dem experimentell sehr einfachen Verfahren von Fisher [3].1) Lange u. a. [4] haben gezeigt, daß die mit der Fisherschen Methode gewonnenen Aktivierungsenergien etwa mit den Werten übereinstimmen, die an Bikristallen mit einem großen Orientierungsunterschied gemessen werden. E s schien daher möglich, auch die vorliegende Untersuchung unter Anwendung der Fisherschen Methode durchführen zu können. Hierfür war es notwendig, zu überprüfen, unter welchen Versuchsbedingungen man bei Anwendung der Fisherschen Methode eindeutige Ergebnisse erwarten kann. Die experimentellen Arbeiten wurden weitgehend im Institut für Anwendung radioaktiver Isotope der T U Dresden, zum Teil noch während der Zugehörigkeit der Autoren zu diesem Institut, durchgeführt. Für die Ermöglichung des Abschlusses der Arbeiten sei an dieser Stelle besonders gedankt. 5»

68

D . BERGNER u n d W . LANGE

2. Methodische Untersuchungen Nach Fisher [3] stellt sich unter der Bedingung, daß die Konzentration des diffundierenden Materials an der Probenoberfläche C 0 (0, t) — const

(1)

und der Korndurchmesser r groß gegen die mittlere Eindringtiefe durch Volumendiffusion ist, folgende Konzentrationsverteilung C(x, t) nach der Diffusionszeit t ein: C{x, i) = 4 C 0 ^ j 1 / 2 e x p {-y

x)

(2)

wobei (ö D t \-l/2

7

(3)

t die Diffusionszeit, 6 die Korngrenzenbreite und DK bzw. Dx den Korngrenzenbzw. Volumendiffusionskoeffizienten bedeuten. Die Größe y kann experimentell ermittelt werden, indem man den Logarithmus der spezifischen Aktivität a(x) gegen die Eindringtiefe x darstellt. Dann ergibt sich aus Gleichung (2) wegen a(x) ~ C(x) d In a(x) y = - - — . = tgc.

(4)

Bei den in dieser Arbeit beschriebenen Untersuchungen sollte das Auftragen dünner Schichten angewandt werden. E s mußte daher untersucht werden, inwieweit die Bedingung (1) C„ = const hierdurch nicht verletzt wird. Das Ergebnis war folgendes: 1. Fig. 1 zeigt, daß die bei 90 °C für die Diffusion von Sn in Sn für verschiedene Zeiten gefundenen Eindringkurven log a = f(x), wie in Gleichung (4) gefordert, linear verlaufen, wenn man von den oberflächennahen Bereichen absieht. Durch Quadrieren der Gleichung (3) ergibt sich für eine gegebene Temperatur ¿1/2

=

ffi*»*

=

const.

Trägt man y~2 gegen i 1 / 2 auf, so ist bei Gültigkeit der Fisherschen Theorie eine Gerade zu erwarten. Ist die auf der Probenoberfläche aufgetragene Menge durch

Jt f/umj-

Fig. 1. Änderung der spezifischen Aktivität a mit der Eindringtiefe x für verschiedene Zeiten i für die Selbstdiffusion von Sn bei 90 °C. Kurve 1: t — 14 h Kurve 4: t — 90 h Kurve 2: i = 25 h Kurve 5: t = 189 h Kurve 3: i = 46 h Kurve 6: l = 335 h

69

Diffusion von Sn und Zn in Sn und in einer homogenen S n - Z n - L e g i e r u n g Fig. 2. Abhängigkeit der Größen tg~2 « und aQ von t'i' für Sn-Selbstdiffusion bei 90 °C

y

H /

•

/

y•

2500

1//

/

/

200

400

600

Diffusion abgewandert, so wird damit die Randbedingung (1) verletzt, und man sollte Werte für y - 2 finden, die von der Geraden abweichen. In Fig. 2 sind für die Korngrenzendiffusion von Sn in Sn bei 90 °C für verschiedene Zeiten gemessene Werte von tg~ 2 0

^H X -N 0 + oh exp j ^ j .

From equations (10) and (14) exp

/ A 0 2 - AGA _ 7 w1 + 4 ^ ----j _ 7 + 4 ^

(14)

88

G. B. GIBBS : Solvent Diffusivity in a Dilute B.C.C. Binary Alloy

so that relation (9) may be written 3

3 exp

6 + 13 =

2 «>, + 3 ^ +

7

'2

•HT)

K

The first bracketed term on the right-hand side is approximately equal to m. Writing the second complete term as s, the impurity correlation factor is (15) Since jump frequencies are always positive s must be greater than zero. I t is largest for very weak perturbation and an upper limit s = 3 gives the correct self-diffusion correlation factor /„ = 0.727 when 6 = 0 and (DJDa) = 1 are substituted into equation (15). If b is several times 3, /, values consistent with 0 < s 3 occupy a narrow range. 5. Conclusions Lidiard's [4] analysis of solvent diffusivity in a dilute f.c.c. alloy may be applied to the b.c.c. lattice to yield equations of similar form. The solvent diffusivity in impure material is a linear function of solute concentration. The coefficient of proportionality b depends on altered vacancy jump frequencies and solute-vacancy binding energy at sites 2nd n.n. to the impurity. An approximate relationship exists between the solute correlation factor b and the diffusivity ratio (X)j/Z)s) in pure solvent. Acknowledgement

This paper is published by permission of the Central Electricity Generating Board. References [1 ] J. BARDEEN and C. HERRING, Imperfections in Nearly Perfect Crystals, Wiley, New York 1952 (p. 261). [ 2 ] A . D . LECLAIRE a n d A . B . LIDIARD, P h i l . M a g . 1 , 5 1 8 ( 1 9 5 6 ) .

[3] J. R. MANNING, Phys. Rev. 116, 819 (1959). [4] A. B. LIDIARD, Phil. Mag. 5, 1171 (1960). [ 5 ] R . E . HOWARD a n d A . B . LIDIARD, R e p . P r o g r . P h y s . 2 7 , 1 6 1 ( 1 9 6 4 ) .

[6] N. L. PETERSON and S. J. ROTHMAN, U.S.A.E.C. Rep. ANL-6568 (1965). (Received June 20, 1966)

S. B. MCRICKARD: Twinning Deformation in Iron at 4 °K

89

phys. stat. sol. 18, 89 (1966) Brookhaven National

Laboratory,

Upton, New

York

Twinning Deformation in Iron at 4 "K1) By S. B . MCRICKABD

Fine grained, Perrovac iron has been found to deform in a discontinuous manner when tested in tension at 4.2 °K. The deformation has been found to be a unique continuous twinning process which is initiated by a critical stress of 137000 psi and from which a uniform strain energy release of 0.01 eV/mol for each twinning event has been calculated. The characteristics of this twinning deformation are discussed and comparisons with other reported low temperature phenomena are made. Fein gekörntes Ferrovac-Eisen wird durch eine Zugspannung bei 4,2 °K in diskontinuierlicher Weise deformiert. Es wird gefunden, daß die Deformation ein einmaliger kontinuierlicher Zwillingsbildungsprozeß ist, der durch eine kritische Spannung von 137000 psi eingeleitet wird, woraus ein gleichbleibender Dehnungsenergiegewinn von 0,01 eV/mol für jeden Zwillingsbildungsvorgang berechnet wird. Die Charakteristiken dieser Zwillingsbildungsdeformation werden diskutiert und mit anderen veröffentlichten Tieftemperaturphänomenen verglichen.

1. Introduction I n the course of low temperature deformation studies of Ferrovac iron at Brookhaven the occurrence of serrated stress-strain curves at 4.2 °K was observed. The relatively large deformation produced and unique characteristics of the deformation process prompted a more extensive experimental examination of the phenomenon. While sporadic twinning had been observed in fine grained Ferrovac iron when tested below 130 to 140 °K only a t 4.2 °K was continual twinning deformation obtained. Extensive twinning deformation has been reported in b.c.c. Mo-Re alloy [1], and sporadic twinning in Fe and Fe-Si has been studied by other investigators [2, 3, 4], Smith and Rutherford [5] have observed discontinuous deformation in zone refined iron of medium grain size (ASTM 3 to 6 or 0,1 m m dia. avg.). They obtained as much as 10% strain at 4.2 °K principally as t h e result of mechanical twinning although they did present some metallographic evidence of slip. Prestrain at higher temperatures did not inhibit twinning and they presented arguments against the hypothesis t h a t the serrated curves are the result of thermal instability from adiabatic heating. This latter view has been proposed by Basinski [6] and Wessel [7]. The decrease in specific heat a t low temperatures suggests t h a t the energy released during deformation can produce a significant local rise in temperature. Basinski observed heat pulses with each load drop in aluminium and an Al-Mg alloy tested at 4.2 °K. The load drops occurred after considerable plastic deformation and no mechanical twins were found in the specimens. On t h e other hand the d a t a of Smith and Rutherford show t h a t deformation in iron at 4.2 and even at 77 °K is initiated by twinning. Haasen [8] has stated t h a t ') This work was performed under the auspices of the U.S. Atomic Energy Commission.

S. B. MCRICKARD: Twinning Deformation in Iron at 4 °K

89

phys. stat. sol. 18, 89 (1966) Brookhaven National

Laboratory,

Upton, New

York

Twinning Deformation in Iron at 4 "K1) By S. B . MCRICKABD

Fine grained, Perrovac iron has been found to deform in a discontinuous manner when tested in tension at 4.2 °K. The deformation has been found to be a unique continuous twinning process which is initiated by a critical stress of 137000 psi and from which a uniform strain energy release of 0.01 eV/mol for each twinning event has been calculated. The characteristics of this twinning deformation are discussed and comparisons with other reported low temperature phenomena are made. Fein gekörntes Ferrovac-Eisen wird durch eine Zugspannung bei 4,2 °K in diskontinuierlicher Weise deformiert. Es wird gefunden, daß die Deformation ein einmaliger kontinuierlicher Zwillingsbildungsprozeß ist, der durch eine kritische Spannung von 137000 psi eingeleitet wird, woraus ein gleichbleibender Dehnungsenergiegewinn von 0,01 eV/mol für jeden Zwillingsbildungsvorgang berechnet wird. Die Charakteristiken dieser Zwillingsbildungsdeformation werden diskutiert und mit anderen veröffentlichten Tieftemperaturphänomenen verglichen.

1. Introduction I n the course of low temperature deformation studies of Ferrovac iron at Brookhaven the occurrence of serrated stress-strain curves at 4.2 °K was observed. The relatively large deformation produced and unique characteristics of the deformation process prompted a more extensive experimental examination of the phenomenon. While sporadic twinning had been observed in fine grained Ferrovac iron when tested below 130 to 140 °K only a t 4.2 °K was continual twinning deformation obtained. Extensive twinning deformation has been reported in b.c.c. Mo-Re alloy [1], and sporadic twinning in Fe and Fe-Si has been studied by other investigators [2, 3, 4], Smith and Rutherford [5] have observed discontinuous deformation in zone refined iron of medium grain size (ASTM 3 to 6 or 0,1 m m dia. avg.). They obtained as much as 10% strain at 4.2 °K principally as t h e result of mechanical twinning although they did present some metallographic evidence of slip. Prestrain at higher temperatures did not inhibit twinning and they presented arguments against the hypothesis t h a t the serrated curves are the result of thermal instability from adiabatic heating. This latter view has been proposed by Basinski [6] and Wessel [7]. The decrease in specific heat a t low temperatures suggests t h a t the energy released during deformation can produce a significant local rise in temperature. Basinski observed heat pulses with each load drop in aluminium and an Al-Mg alloy tested at 4.2 °K. The load drops occurred after considerable plastic deformation and no mechanical twins were found in the specimens. On t h e other hand the d a t a of Smith and Rutherford show t h a t deformation in iron at 4.2 and even at 77 °K is initiated by twinning. Haasen [8] has stated t h a t ') This work was performed under the auspices of the U.S. Atomic Energy Commission.

90

S. B . MCRICKARD

considerable localized plastic flow must occur before the t e m p e r a t u r e will rise. Basinski also argued against t h e proposals of Seeger [9] a n d Haasen [8] t h a t this plastic instability is t h e result of a catastrophic breakdown of dislocation barriers. F u r t h e r m o r e he suggested t h a t mechanical twinning might serve as a nucleating agent for subsequent adiabatic deformation. I n a recent paper [10] Boiling and R i c h m a n have reported a unique deformation process which t h e y have defined as continual mechanical twinning (CMT). Their experimental results were based predominantly on their observations of continuing twinning deformation in single crystals of a quenched supersaturated F e - B e alloy. T h e y also reported continual twinning deformation in single crystals of Au, Ag, and Cu after extensive slip deformation a t low temperatures. Some of the more pertinent characteristics of CMT listed b y Boiling a n d Richman a r e : deformation governed b y a resolved shear stress for twinning; slip while not excluded plays a minor role ( < 1% of deformation); negative strain r a t e sensitivity. These features were looked for in t h e deformation behavior of Ferrovac iron a t 4.2 °K. 2. Testing Procedure The Ferrovac iron used in this s t u d y was 99.94% pure a n d contained 30 p p m C , 5 p p m N, 55 p p m 0 , a n d 70 p p m Al. I t is now generally realized t h a t low t e m p e r a t u r e ductility depends on the oxygen content being below t h a t of the total deoxidizers present. The iron was cold swaged to a p p r o x i m a t e size and machined into subsize tensile specimens 3 inches long with a uniform gage section 1 inch long and 0.120 inches in diameter. The specimens were t h e n annealed a n d f u r n a c e cooled which produced a final grain diameter of approximately 0.025 m m . The twinning phenomenon could be produced in coarser grained specimens b u t p r e m a t u r e f r a c t u r e was frequently encountered. The tests were conducted on an I n s t r o n tensile machine with a double walled helium dewar enclosing the specimen and grip assembly. The specimens were immersed in liquid helium and the t e m p e r a t u r e was recorded b y means of a germanium resistor thermometer. To obtain test t e m p e r a t u r e s above 4.2 °K a controlled flow of s a t u r a t e d helium vapor was p u m p e d into the dewar until equilibrium conditions were indicated. After completion of the test the specimens were measured in an optical comp a r a t o r to t h e nearest 0.0001 inches a n d t h e n sectioned for metallographie examination. Twin density measurements were made using q u a n t i t a t i v e metallography techniques. 3. Results The shape of the load-elongation graphs obtained for specimens tested below 30 °K is shown in Fig. 1. Observe t h a t while deformation above 4.2 °K m a y include some sporadic twinning only a t 4.2 °K is deformation restricted to continual plastic instability as revealed b y the serrated test curve. I t is also interesting t o note t h a t if t h e peaks of each load drop a t 4.2 °K are connected by a continuous curve one would obtain a deformation curve containing an a p p a r e n t Luders b a n d extension and work hardening curve typical of slip deformation in iron near room temperature.

91

Twinning Deformation in Iron at 4 ° K

From each graph in Fig. 1 the stress required for the first twin and the first observable slip has been calculated and listed in Table 1. These were determined from the first twin load drop and the first departure from the elastic slope respectively. Table 1 Summary of low temperature twinning and yield stresses Temperature (°K) 26 24 17 12 4

First twin psi

First slip psi

124900 130200 129600 134000 132000-139000

127500 129600 117400 133900 ...

After a specimen is thus tested at 4.2 °K the gage length is found to contain a series of constrictions along its length. The specimen diameter adjacent to these contrictions reduces very little from the original dimensions while the minimum diameter within the constrictions is fairly uniform from one to the next with the exception of the large reduced section which contains the eventual fracture. To study the deformation process in more detail four specimens were tested and examined: two after one load drop, one after three load drops, and one after four load drops respectively. The results from these tests and others carried to fracture at 4.2 °K produced the following observations: (i) After some initial sporadic twinning the deformation process proceeds by means of continuous plastic instability. (ii) The deformation has the form of a "Luders band" extension followed by a strain hardening curve. (iii) The maximum stress during the "Luders" extension and the stress drop (orelastic energy release) are remarkably uniform. iv) The "Luders" extension is physically characterized in the specimen gage section by the formation of distinct constrictions which progress down its length. Until the strain hardening portion of the curve has begun the minimum diameters of the constrictions are approximately equal. Each load drop results in the formation of one constriction. (v) The volume element defined by the constriction is marked by a high Fig. 1. Typical load-extension curves for Ferrovac iron tested below 30 °K

92

S . B . MCRICKARD

Fig. 2. Top: Typical Microstructurc before testing. Middle and bottom: Microstructures a f t e r testing adjacent to and inside a constriction, respectively (600 x )

X

/

0.02 e

0.002è è - injmin

0.02c

Time Fig. 3. The effect of strain rate on twinning behavior of iron a t 4.2 °K

93

Twinning Deformation in Iron at 4 °K

density of twins most of which are quite thick as shown in Fig. 2. Outside the constricted regions t h e twin density is virtually zero. W h e n allowed t o continue into t h e strain hardening range the deformed regions begin t o overlap and twins are produced t h r o u g h o u t the gage length. (vi) A small stress concentration a t one end of the specimen gage length probably initiates the first twinning burst just as yielding is initiated a t higher temperatures. After the first constriction is formed, the size of which is governed b y the energy required for this process, the stress climbs t o t h e critical value again. P r o b a b l y affected somewhat b y the preceding constriction the a d j a c e n t undeformed volume repeats the event and so on. There is little metallographic evidence of grain t o grain twin propagation. Instead, grain orientation is most likely t h e deciding factor in twin propagation t h r o u g h t h e localized deformed volume. (vii) An order of magnitude increase in strain r a t e 0.002 t o 0.02 produces a small b u t significant decrease in peak stress which suggests a negative strain r a t e sensitivity as defined by Boiling a n d Richman [10]. This can be seen in Fig. 3. The deformation d a t a for several specimens has been listed in Table 2 along Table 2 Summary of tensile test data at 4.2 °K Spec. No.

A (in.2)

Drop No (in./min)

0.02 1 2 0.02 3 0.02 0.02 4 5 0.02 0.02 6 0.02 7 0.02 8 0.02 9 1 0.02 176 0.0115 0.02 2 0.02 3 4 0.002 5 0.002 6 0.002 7 0.002 0.02 8 177 0.0115 1 0.02 1 0.02 178 0.0115 175 0.0115 0.02 1 0.02 2 3 0.02 0.02 4 181 0.0116 1 0.02 2 0.02 0.02 3 Average at 0.02"/min 171

0.0115

*) Average

ff

max (psi)

°min (psi)

137300 135000 138000 137700 137400 137800 141000 137400 141000 138500 138500 138100 139400 139200 139000 140700 138300 131900 130300 138500 139200 139000 155600 135200 134100 134600 137100

121300 126400 122500 122500 126200 124200 121700 124500 124600 115900 118200 117900 113800 116400 118500 117600 120500 117500 115300 123400 124500 117700 115600 119600 118300 117800

A • -^min (in-2)

E Length E (erg/unit vol.) (in.) (cal/mol)

0.0102*) 6 . 5 9 x 1 0 ' 0.0102 3.54 x 10' 0.0102 6.40 x 10' 0.0102 6.27x10' 0.0102 4.62x10' 0.0102 5.60x10' 0.0102 7.95x10' 0.0102 5.32x10' 0.0102 6.75x10' 0.0107*) 5.52 x 10' 4.96x10' 0.0107 0.0107 4.93x10' 0.0107 6.25 x 10' 0.0107 5.58x10' 0.0107 5.01x10' 0.0107 5.64 x 10' 0.0107 4.35x10' 4.05x10' 0.0106 0.0105 4.70 X10' 0.0104 5.21 x 10' 0.0102 6.05 x 10' 4.60x10' 0.0108 11.2 x l O ' 0.0106 0.0107 4.34 x 10' 0.0109 3.38x10' 3.59x10' 0.0109

0.217 0.256 0.158 0.160 0.170 0.160 0.252 0.205 0.221

182 181 329 384 259 680 166 157 154

cal/mol cal/mol cal/mol cal/mol cal/mol cal/mol cal/mol cal/mol cal/mol

94

S. B.

MCRICKARD

with calculated energy releases produced by the twinning loads drops. Only load drops during the " L u d e r s " extension are listed. In specimens tested to fracture a reduction in area as high as 32% was measured. 4. Discussion The data obtained in this investigation permits a more detailed study of low temperature discontinuous deformation in iron than possible from previous studies. It is fairly certain that at least for iron this type of deformation is not the result of thermal instability but is the result of localized twinning bursts. The characteristics of this unique deformation such as the stress requirements, energy released, and manner of propagation have therefore been analyzed. 4.1 Characteristics

of

deformation

The data in Table 1 indicate that there is a constant stress level below at least 30 ° K in this fine grained iron at which macroscopic deformation is initiated. The deformation may propagate either as mechanical twinning or slip. This behavior has been discussed in an earlier paper [11]. In quenched iron it was shown that slip deformation was apparently initiated by twinning below 120 ° K without affecting the yield stress-temperature curve. Macroscopic slip does not appear possible in Ferrovac iron at 4.2 ° K . However, it is probable that both twinning and macroslip are nucleated by dislocations moving to the first energy barrier. Cross slip is believed to be inhibited in b.c.c. metals at low temperatures [12] and it has been proposed that twinning and cross slip be considered as alternate low temperature stress relief mechanisms in f.c.c. metals [13]. If this hypothesis can also be applied to iron, then at 4.2 ° K twinning becomes the controlling mechanism independent of strain or strain rate. However, this does not exclude any small relaxation of stress along the twin-matrix interface by means of slip dislocations as suggested by Sleeswyk and co-workers for example [14]. The irregular shape of the twins shown in Fig. 2 suggest this is the case. The relative contribution of this slip to the total observed strain could not be estimated. The average peak stress which sets off the load drops listed in Table 1 at 0.02 in./min is 137100psi with a standard deviation of ± 2600 psi ( + 2%). The high reproducibility of this stress level from specimen to specimen indicates that this represents the critical stress for continuous twinning deformation. The effect of strain rate on this critical stress is small but from the typical deformation curve shown in Fig. 3 there is a definite increase in critical stress with decreasing strain rate. This behavior indicates a negative strain rate dependence of approximately the same magnitude as that of Fe 3 Be as shown by Boiling and Richman. The effect of temperature on this phenomenon could not be studied because it is observed only at 4 ° K in iron. Since the load drops produced in the " L u d e r s " extension are extremely uniform and the strained volume of specimen can be determined with reasonable accuracy the strain energy released per twinning burst can be calculated. The

Twinning Deformation in Iron at 4

strain energy is given by

and in terms of experimentally determined parameters 'max

where for each load drop 2/max is the peak load, LmSn is the load after the drop, A0 the original cross sectional area, and Amin the minimum area in the constriction. This energy is the maximum per unit volume since the minimum cross section in the constriction is used. The energy thus calculated is listed in Table 2. In addition to the minimum diameter the length of each constriction was measured in several specimens. This was accomplished by sectioning the specimens longitudinally and counting the twin density along the gage length using quantitative metallography. A statistical distribution was thus obtained bewteen the maximum density in the center of the constrictions and the background density (usually nil) in the unaffected regions of the gage length. The half-peak height location was chosen as the boundary of the twinned region. A typical constriction in a specimen after one drop in load is shown in Fig. 4. In specimen 175 the four constrictions were relatively distinct from one another and the twin zone length determined metallographically was in good agreement with profile measurements taken before sectioning. On the other hand the three constrictions observed in specimen 181 indicated considerable overlap. The twinned zones in the outer constrictions were larger than the interior zones and if the constrictions are assumed to be symmetrical then the maximum reduction in diameter in the second and third constrictions must have occurred right at the boundary of the preceeding constriction. Taking into account these considerations the approximate volumes of the twin zones of several

Fig. 4. Composite of specimen gage l e n g t h showing one constriction

96

S . B . MCRICKARD

specimens were determined and the energies previously calculated were converted into calories per mol as listed in Table 2. The average energy (excluding the fourth drop of specimen 175) is 227 cal/mol or approximately 0.01 eV/mol. This energy is much lower t h a n had been anticipated but it is interesting to note t h a t almost identical values have been obtained for the activation energy for deformation in iron from an internal friction study [15] and from an empirical expression for the thermal dependence of the yield strength of iron b y this writer [16]. A possible explanation for a relationship between t h e energy required for yielding and for continuous twinning at 4 °K may be as follows. At 4 °K the thermal assistance to the stress in surmounting the energy barrier for slip is quite small and if the barrier energy for continuous twinning is slightly less t h a n the energy for slip at 0 to 4 °K then the observed approximate energy equivalence of the two processes is possible. A dissociated or twin dislocation may be expected to require somewhat less energy to move away from an impurity atmosphere t h a n a full slip dislocation. 4.2

Mechanism

We know t h a t the deformation process is carried out almost completely by localized twinning which progresses discontinuously down the length of the specimen. Some strain hardening then appears t o occur due to the same twinning process. Therefore this is definitely not a catastrophic slip process. Furthermore the d a t a in Table 1 show t h a t the same approximate stress level is attained at higher temperatures where sporadic twinning followed b y slip occurs but continuous twinning deformation does not. One or two twins usually occur before slip begins at a lower stress level. This is important because it rules out an avalanche of thermally activated slip at a critical stress level. This level is attained well above 4 °K. Also the argument of Basinski for adiabatic bursts or thermal instability causing the discontinuous deformation may not be applicable. A significant amount of deformation must occur before the temperature will rise locally and while twinning could nucleate such a process, there is no sign of twinning or slip occuring in the gage length before the advancing twin zones or constrictions. Furthermore the subsequent appearance of a strain hardening process is h a r d to explain in terms of thermal instability. Instead, the m a n y common features of this deformation with t h a t of Boiling and Richman's CMT suggests this m a y be a form of their twinning deformation mechanism. First, there is a critical resolved stress for this twinning process and, in fact, by extrapolating data obtained b y Richman and Boiling [17] on polycrystalline F e - 2 5 Be a t 77 °K to the same grain size approximate agreement in this stress level is obtained ( « 100 kg/mm 2 ). I t is believed t h a t t h e thermal dependence is quite small between 4 and 77 °K. Second, a small b u t negative strain rate dependence is also observed. Finally, CMT has a positive temperature dependence and since this twinning phenomenon can only be observed in iron at 4 °K the implication of the d a t a is t h a t the temperature dependence is not negative or continuous twinning might be expected to occur at some higher temperatures instead of slip. 5. Conclusions 1. The yielding and subsequent straining of fine grained Ferrovac iron at 4 °K is characterized by a discontinuous deformation process instead of the

Twinning Deformation in Iron at 4 °K

97

sporadic twinning followed by slip observed a t slightly higher temperatures. This deformation consists of continual mechanical twinning with no observable macroslip and it progresses through the specimen in a manner similar t o t h a t of slip at room temperature. 2. The twinning deformation exhibits a critical stress of approximately 137000 psi, produces localized strains of approximately 0.10, has a small negative strain r a t e sensitivity and can produce work hardening before fracture. These characteristics show a similarity with continual mechanical twinning process described by Boiling and Richman. 3. The energy release per load drop is approximately 0.01 eV/mol and this m a y represent the situation where the athermal stress component is almost equal to the total barrier energy for dislocation breakaway at 4 °K. Acknowledgements

The author is indebted to R. Richman and G. Boiling for very helpful discussions and to D. Beshers and R. Rosenberg for their advice. References [1] A. LAWLEY and R. MADDIN, Trans. AIME 224, 573 (1962). [ 2 ] H . W . PAXTON a n d A . T . CHURCHMAN, A c t a m e t a l l . 1, 4 7 3 (1953).

[3] D. HULL, Proc. Roy. Soc. A 274, 5 (1963). [4] A . W . SLEESWYK, J . N . HELLE, a n d A . DEGEUS, J . I r o n S t e e l I n s t . 2 0 2 , 3 3 0 (1964). [ 5 ] R . L . SMITH a n d J . L . RUTHERFORD, T r a n s . A I M E 2 1 1 , 8 5 7 (1957).

[6] Z. S. BASINSKI, Proc. Roy. Soc. A 240, 229 (1957). [7] E . T . WESSEL, T r a n s . A S M 4 9 , 149 (1957). [8] P . HAASEN, T r a n s . A I M E 2 1 2 , 4 2 (1958).

[9] A. SEEGER, Dislocations and Mechanical Properties of Crystals, Wiley, New York 1957 (p. 206). [ 1 0 ] G . F . BOLLING a n d R . H . RICHMAN, A c t a m e t a l l . 1 3 , 709 (1965). [ 1 1 ] S. B . MCRICKARD a n d J . G. Y . CHOW, T r a n s . A I M E 2 3 3 , 147 (1965).

[12] N. BROWN and R. A. EKVALL, Acta metall. 10, 1101 (1962). [ 1 3 ] B . RAMASWAMI, J . a p p l . P h y s . 3 6 , 2 5 6 9 (1965). [14] A . W . SLEESWYK, A . DEGEUS, a n d J . N . HELLE, A c t a m e t a l l . 1 1 , 3 3 7 (1963).

[15] H. D. GUBERMAN, Thesis, Columbia University, New York, 1963. [16] S. B. MCRICKARD, Phil. Mag. 13, 433 (1966). [17] R . H . RICHMAN a n d G. F . BOLLING, private communication, 1966. (Received

7 physlca

June 22,

1966)

R.

CHANG

and L. J.

GRAHAM:

Edge Dislocation Core Structure in

B.C.C.

Fe

99

phys. stat. sol. 18, 99 (1966) North American Aviation Science Center, Thousand Oaks,

California

Edge Dislocation Core Structure and the Peierls Barrier in Body-Centered Cubic Iron By R . CHANG a n d

L. J .

GBAHAM

The core structure and Peierls barrier for an edge dislocation lying in the {110} plane with Burgers vector along in body-centered cubic iron were investigated numerically with the aid of a high speed computer using an anharmonic potential. The core radius is about 5 A and the corresponding core energy is 2.7 eV per identity distance along the dislocation line (six atom planes). The Peierls barrier is about 0.03 eV and the Peierls stress for dislocation motion at absolute zero is computed to be 5.36 X10 9 dyn/cm 2 or 0.0066 of the shear modulus. Die Kernstruktur und Peierls-Barriere für eine Stufenversetzung in der {110} -Ebene mit dem Burgersvektor in -Richtung in kubisch raumzentriertem Eisen wird mit einem elektronischen Rechenautomaten untersucht, wobei ein anharmonisches Potential benutzt wird. Der Kernradius beträgt ungefähr 5 Ä und die entsprechende Kernenergie 2,7 eV pro Identitätsabstand längs der Versetzungslinie (sechs Atomebenen). Die Peierls-Barriere ist ungefähr 0,03 eV und die Peierls-Spannung für Versetzungsbewegung beim absoluten Nullpunkt wird zu 5,36 X10 9 dyn/cm 2 oder 0,0066 des Schubmoduls berechnet.

1. Introduction Although the t r e a t m e n t of dislocations within the framework of isotropic elasticity theory has proven very fruitful in the past, there has been an increasing need to improve the t r e a t m e n t by taking into consideration the intrinsic anisotropy of crystal elasticity [1 to 3] and the nonelastic distortions in the dislocation core region. There have been several a t t e m p t s to d^al with t h e nonlinear core region of an edge dislocation. The method of Gallina, et al. [4] based on a perturbation of the phonon field of a perfect crystal b y t h e lattice distortion near the core region of an edge dislocation appears t o be interesting b u t is of limited applicability. The first a t t e m p t to obtain the configurations and energies near the core region for both screw and edge dislocations in sodium chloride based on interactions among the various ions was made by H u n t i n g t o n and Dickey [5]. Cotterill and Doyama obtained the core energies and atomic configurations for both screw and edge dislocations in face-centered cubic copper numerically using a truncated Morse potential [6, 7]. Very recently t h e core structures and configurations of an edge dislocation in body-centered cubic metals have been probed numerically with the aid of several interatomic potentials by Chang [8, 9] and by Cotterill and Doyama [10]. 2. The Anharmonic Potential and the Edge Dislocation Core Structure in Body-Centered Cubic Iron A prerequisite of the application of numerical methods to study defect configurations and energies in solids with the aid of interatomic potentials is the 7*

R.

CHANG

and L. J.

GRAHAM:

Edge Dislocation Core Structure in

B.C.C.

Fe

99

phys. stat. sol. 18, 99 (1966) North American Aviation Science Center, Thousand Oaks,

California

Edge Dislocation Core Structure and the Peierls Barrier in Body-Centered Cubic Iron By R . CHANG a n d

L. J .

GBAHAM

The core structure and Peierls barrier for an edge dislocation lying in the {110} plane with Burgers vector along in body-centered cubic iron were investigated numerically with the aid of a high speed computer using an anharmonic potential. The core radius is about 5 A and the corresponding core energy is 2.7 eV per identity distance along the dislocation line (six atom planes). The Peierls barrier is about 0.03 eV and the Peierls stress for dislocation motion at absolute zero is computed to be 5.36 X10 9 dyn/cm 2 or 0.0066 of the shear modulus. Die Kernstruktur und Peierls-Barriere für eine Stufenversetzung in der {110} -Ebene mit dem Burgersvektor in -Richtung in kubisch raumzentriertem Eisen wird mit einem elektronischen Rechenautomaten untersucht, wobei ein anharmonisches Potential benutzt wird. Der Kernradius beträgt ungefähr 5 Ä und die entsprechende Kernenergie 2,7 eV pro Identitätsabstand längs der Versetzungslinie (sechs Atomebenen). Die Peierls-Barriere ist ungefähr 0,03 eV und die Peierls-Spannung für Versetzungsbewegung beim absoluten Nullpunkt wird zu 5,36 X10 9 dyn/cm 2 oder 0,0066 des Schubmoduls berechnet.

1. Introduction Although the t r e a t m e n t of dislocations within the framework of isotropic elasticity theory has proven very fruitful in the past, there has been an increasing need to improve the t r e a t m e n t by taking into consideration the intrinsic anisotropy of crystal elasticity [1 to 3] and the nonelastic distortions in the dislocation core region. There have been several a t t e m p t s to d^al with t h e nonlinear core region of an edge dislocation. The method of Gallina, et al. [4] based on a perturbation of the phonon field of a perfect crystal b y t h e lattice distortion near the core region of an edge dislocation appears t o be interesting b u t is of limited applicability. The first a t t e m p t to obtain the configurations and energies near the core region for both screw and edge dislocations in sodium chloride based on interactions among the various ions was made by H u n t i n g t o n and Dickey [5]. Cotterill and Doyama obtained the core energies and atomic configurations for both screw and edge dislocations in face-centered cubic copper numerically using a truncated Morse potential [6, 7]. Very recently t h e core structures and configurations of an edge dislocation in body-centered cubic metals have been probed numerically with the aid of several interatomic potentials by Chang [8, 9] and by Cotterill and Doyama [10]. 2. The Anharmonic Potential and the Edge Dislocation Core Structure in Body-Centered Cubic Iron A prerequisite of the application of numerical methods to study defect configurations and energies in solids with the aid of interatomic potentials is the 7*

100

R . CHANG a n d L . J . GRAHAM

availability of a reliable potential between the atoms. I t is the first intent of the present paper to demonstrate the importance of this point and to introduce an empirical interatomic potential taking into consideration the anharmonic properties of the solid. The details of obtaining the anharmonic potential for bodycentered cubic iron have been reported by the same authors previously [11]. The potential is given by the following equation: 0 j ( r ) = -0.15614 r4 + 0.815729 r3 + 1.24594 r2 (2.40 A < r ^ 3.3894 A ) ,

12.2404 r + 16.0183 (1)

where 0 is in electron volts and r in Angstroms. The potential is graphically shown in Fig. 1. A cut-off distance at r = 3.3894 A, nearly midway between the second and third nearest neighbors where the potential and its first derivative are both zero, was adopted. The potential is not very accurate, however, at interatomic distances less than about 2.40 A. A second potential 02 which matches the value, slope and curvature with the potential 0X at r = 2.40 A and with the Johnson potential [12] at r = 2.20 A was included. The potential 02 has the form 02(r)

= 34.0878 r4 -

327.9263 r3 -f 1184.7172 r2 -

(r ^ 2.40 A)

1905.7946 r + 1152.0160 (2)

and is also shown graphically in Fig. 1. Since the distance of closest approach between the atoms in the fully relaxed position was never less than about 2.30 A , the inclusion of the second potential 0 2 will have little effect on the final results and is for the convenience of machine computation only. The I B M 7094 computer with digital display was used. The edge dislocation lies in the slip plane (110) with Burgers vector along [111], the dislocation line being parallel to [112]. A crystallite containing 648 atoms per identity distance along the dislocation line (six atom planes) and made infinitely long along the line by the use of periodic boundary conditions was embedded in an elastic continuum with fixed atomic coordinates determined by continuum elasticity theory. Of the 648 atoms, the inner 414 atoms were

cut-off Fig. 1. The anharmonic interatomic potentials for body-centered cubic iron

Edge Dislocation Core Structure and the Peierls Barrier in B.C.C. Fe

101

permitted to relax according to the potentials shown in Fig. 1, while the remaining 234 boundary atoms were kept fixed but were included in the energy and force computations. Since a previous calculation indicated that the core radius is of the order of one to two Burgers vectors, this size of crystallite was felt to be sufficiently large to include an intermediate region where the theory of anisotropic elasticity applies. Table 1 Procedures to relax the atoms according to prescribed interatomic potentials 1. Obtain coordinates of atom ijk

and its N neighbors.

2. Compute forces acting on atom ijk

by the N neighbors ( F

3. Obtain new coordinates of atom i j k such that the sum total of forces from step 2 is

4. Go to next atom and repeat steps 1 to 3. 5. Repeat step 4 until whole array is covered. 6. Repeat steps 1 to 5 until the changes in coordinates of all atoms in array between successive iterations are within preset limits (0.000001 A).

The procedures used in evaluating the positions of the atoms within the crystallite are summarized in Table 1. The fully relaxed atomic configurations will not be reproduced here. Figure 2, taken directly from the digital display of the computer, shows the energy of the dislocation within a radius r as a function of the radial distance r from the dislocation line. Superimposed onto this plot is a straight line described by the following equation E = 3.236 x 10" K b3 log10 (r/r0) = 6.40 log10 (r/r0)

eV

per identity distance along the dislocation line (K = 1.293 X 1012 dyn/cm2 , r0 = 1.94 x 10"8 cm),

F i g . 2. D i s l o c a t i o n e n e r g y versus r a d i a l distance f r o m d i s l o c a t i o n line, d a t a t a k e n d i r e c t l y f r o m 7094 d i g i t a l d i s p l a y , b o d y - c e n t e r e d cubic i r o n

r(A)~

(3)

102

R . CHANG a n d L . J . GRAHAM

where K is a constant dependent solely on the dislocation configuration and the elastic properties of the material [3], b is the Burgers vector of the dislocation and r0 is a parameter given by the intercept of the straight line with the abscissa. The data points in the region beyond a distance of about 5.0 A from the center of the dislocation line thus agree remarkably well with those predicted from continuum anisotropic elasticity. If a small variation of about 0.2 eV is permitted, the straight line region can be extended to a radial distance of about 3.0 A. The core radius therefore lies between 3.0 to 5.0 A (roughly one to two Burgers vectors) with a corresponding core energy of 1.6 to 2.7 eV according to the results shown in Fig. 2. Similar calculations using other potentials such as the Johnson [12], Morse [13] and Lennard-Jones [14] potentials showed that these potentials do not yield the right value for K in Equation (3) and that both K and r0 are very sensitive to the potential used. The dislocation core energy relaxed according to the Johnson potential (graphically shown in Fig. 3), for example, is shown in Figure 4.

Fig. 3. The Johnson potential for body-centered cubic iron (see lleference [121). 1.90 £ r £ 2.40: S>,(r) = -2.195976 (r - 3.097910)' + + 2.704060 r - 7.436448, 2.40 £ f £ 3.00:

F2

,

/JTi(y)dy + / J M ( y ) d y . 0

(12)

(13)

F,

We are, of course, interested in the case in which tunnelling is the predominant mechanism, that is to say the case in which the second integral is negligible compared to the first and we require only to integrate J j i ( y ) over values of y corresponding to the conduction band spike. A good approximation to the integration is available when we note (Fig. 5) that the function J x i ( y ) is dominated by a sharp peak so that we can take the height of this peak which is given by J-n(y), maximized by choosing a value of the variable y = ?/max say, as being linearly related to the area under it, and hence the integral. I.e.

/ ~

J"Tl(«/max) ~

and i/max is given by

[Py

&y

Q4)

Bylymzx

= 0, [es

— \—IP] / Jymax ß y l which can be expressed analytically as

so that

2/max

kT

/'I IS . 2/max

2 2/max

+

s =

öS / „ 1

— ay

t' 2

B

.

=

0

8 m*

(15)

(16)

On the right hand side at temperatures we might reasonably expect to encounter, ¡îî> ~ ,

which is tantamount to saying that the effect of density of states

variation with energy is not a significant contribution, we then have (17)

2/ma

Fig. 5. Tunnelling current distrif bution. A Boltzman tail of occupation probability. B Parabolic density of states. C Tunnelling probability. D Tunnelling current distribution as function of energy

Conduction

Energy distribution of

T. L. Tansley

110 and equation (14) becomes I ~ T

VI"

exp [({»

= T Vi' 2 exp

^

(k

-

T)

j exp

exp [ ( ? | (F 2 T ) -

[(p

F> k T

i

_

7>

(18)

F2)]

but F 2 = Fd2 — Fa2 and the fractions of the applied voltages falling on either side of the junction can be defined in terms of a constant such that Fd2 — Fa2 = = A ( F d — F a ) . As the exponential terms dominate, equation (18) can then be written exp [ - ( § + £ F D A)] exp [ F a

( M - L

T)] exp

[L

FD

T]

with L = — ? k X i k

which in the special case of —4 equation / =

and

(19)

M = £X

1

— can be written in the form of the Newman T

/ooexp(^)exp(g

(20)

and X h

4 ¿Y2 !

T „

=

4A h2 2 k 32 i' m*

V

D

The constant I 0 0 has turned out to be slightly temperature dependent but otherwise the qualitative form is that observed experimentally in many cases. 4. Experimental A series of junctions have been grown by the epitaxial deposition of Galliumarseno-phosphide from the vapour phase as a high resistivity (IVD small) n-type layer on a substrate of p-type Gallium-arsenide with an acceptor impurity concentration of about 10 16 c m - 3 (zinc doped). Of this series five good layers were selected with phosphorus contents ranging from approximately 10 to 25 a t %

200

400

woo Forward bias (mV)

Tig. 6. Forward characteristics at six temperatures

m -

Temperature f°KI Fig. 7. Variation of current with temperature at fixed forward bias

Forward Bias Current-Voltage Characteristics for a Heterojunction

111

representing lattice mismatch of less t h a n 1% in all cases, thereby avoiding a significantly high density of interface mismatch states [11]. Measurements were carried out in absolute darkness in an evacuated temperature controlled enclosure b y vibrating reed electrometer and picoammeter. Fig. 6 shows detailed current-voltage characteristics for one of these units plotted semi-logarithmically a t six different temperatures. The curves are divisible into three distinct regions. At low bias of the order of a few tens of millivolts, a deviation from linearity occurs. Above these low values a set of parallel straight lines is observed with a corresponding V0 of about 45 mV and this region stretches to the point where the series resistance of the bulk material begins to dominate the characteristics. The series resistance ranges from 120 £2 at 360 °K to 80 k Q at 100 °K corresponding to an activation energy of about 0.1 eV in the linear region of a semilogarithmic plot of series resistance versus reciprocal temperature. Fig. 7 shows the semilogarithmic plot of forward current against temperature at a fixed bias (300 mV). The curve exhibits no significant deviation f r o m linearity over seven orders of magnitude of current flow. The slope of the line corresponds to a T0 of approximately 10 °K. The relationship between current, voltage, and temperature can therefore be written as in equation (20) F II =T J 0 0 e x p - -A— e x Pp

0.045 V

10 °K '

I t is of interest to note t h a t the experimental ratio TJ F 0 is observed to be about 200 °K e V - 1 , while the theoretical derivation p u t s in a value of 4/f k F D (equation (19)). Substituting for f we have f 1/ h and the appropriate • 1 i tC t r tj 1/ O Tib numerical values are "r 4 - = 4.8 X10- 1 1 deg s, k

m

and

'— = 0.072 for electrons in GaAs, TO m c2 = 5 X 105 eV for an electron.

If we now substitute the experimentally observed value of T0j V0 = 200 °K e V - 1 we have a barrier width constant of 3 x l 0 ~ 6 cm e V - 1 ' 2 with a corresponding zero-bias width of 30 nm. This value is well within the range within which one would expect the treatment to be valid, so t h a t there seems to be some quantitative agreement between the calculated result and experiment. A minor difficulty does arise, however, in t h a t the condition for equation (20) to be a good approximation to equation (19) is not as effectively surpassed as one would hope. Empirical equations such as equation (20) are themselves open to question although in this case the fit to a variety of results [3, 4, 8, 12] is good. The similarity of the present results t o those of Shibata [13] for GaAs tunnel diodes is notable, as are the results of Padovani and Sumner [14] on Schottky emission at a GaAs-Au interface, which, although apparently comparable to those in Fig. 7, are better fitted by an empirical equation of the form I ~

above the low voltage tail with

T2 exp

Fb) k (T •

T0 ~ 50 °K .

112

T. L. TANSLEY: Forward Bias Current-Voltage Characteristics

5. Conclusions The problem of accounting for the forward characteristics of heterojunctions has been approached by the formulation of a mathematical model from which simple tunnelling theory derives an analytical relationship relating voltage, current, and temperature. The model requires the postulation of an exponential band profile which m a y arise from the diffusion of impurities across the interface or possibly from minority carrier effects at the interface. The result shows qualitative and quantitative success when confronted with detailed experimental results. Another idealized case, for which N D N A , results in a space charge distribution which can be approximated by a step function and this leads to a parabolic band profile. B o t h cases result in tunnelling currents in which the impressed variables, voltages and temperature, are separable. The former yields t h e result we have examined in detail and fits experiment. The end point of the latter is a tunnelling current of the form

We must conclude t h a t the dominant current transport mechanism in heterojunctions is one of tunnelling through the conduction band spike and t h a t the band profiles at the interface are best represented by an exponential function, probably due to exchange of impurities between the materials. Acknowledgements

I would like for the growth P. C. Newman, support of this edged.

to record m y t h a n k s to Miss C. A. Fisher and Mr. R. A. Ford of diodes and Ir. Ch. v a n Opdorp and Messrs. H. C. Wright, and J . R. A. Beale for assistance in interpretation. The partial work by the Ministry of Defence (Naval) is gratefully acknowlReferences

[1] R. L. ANDERSON, IBM J.I Res. Developm. 4, 283 (1960) ; Solid State Electronics 6, 341 (1962). [2] CH. VAN OPDORP, private communication. [ 3 ] T . L . TANSLEY a n d P . C. NEWMAN, i n p r e s s .

[4] L. J. VAN RUYVEN, Thesis, T. H. Eindhoven 1964. [5] P. C. NEWMAN, Electronics Letters 1, 265 (1965). [6] P. J. PRICE, Proc. Internat. Conf. Physics of Semiconductors, Exeter, 1962 (p. 99). [ 7 ] H . WANNIER, P h y s . R e v . 5 2 , 1 9 1 ( 1 9 3 7 ) . [ 8 ] R . H . REDIKER, S . STOPEK, a n d J . H . R . W A R D , S o l i d S t a t e E l e c t r o n i c s 7 , 6 2 1 ( 1 9 6 4 ) .

[9] E. SPENKE, Electronic Semiconductors, McGraw-Hill, 1958 (p. 87). [10] E. SPENKE, Z. Phys. 126, 67 (1949). [ 1 1 ] W . G . OLDHAM a n d A . G. MILNES, S o l i d S t a t e E l e c t r o n i c s 7 , 1 5 3 ( 1 9 6 4 ) . [ 1 2 ] Z . I . ALFEROV, P i z . t v e r d . T e l a 6 , 2 3 5 3 ( 1 9 6 4 ) .

[13] A. SHIBATA, Japan. J. appi. Phys. 3, 711 (1964). [14] P. A. PADOVANI and C. G. SUMNER, J. appi. Phys. 36, 3744 (1965). (Received June 30,

1966)

V. RADHAKRISHNAN : Tunnelling in a Two-Band Model Superconductor

113

phys. stat. sol. 18, 113 (1966) Tata Institute

of Fundamental

Research,

Bombay

Tunnelling in a Two-Band Model Superconductor By V.

RADHAKRISHNAN

The Josephson and ordinary tunnelling currents are evaluated for two-band model superconductors. Numerical calculations of these currents are made for a niobium-tin junction, for which this model is applicable, and compared with experiments. Der Josephson- und normale Tunnelstrom werden für Zweibandmodell-Supraleiter berechnet. Numerische Berechnungen dieser Ströme für eine Niobium-Zinn-Verbindung, für die dieses Modell anwendbar ist, werden mit Experimenten verglichen.

1. Introduction Transition element superconductors differ from the other superconductors in many properties, even though broadly they share the main features with them. To study this group of superconductors, Suhl, Matthias, and Walker [1] have suggested a two-band model extension of the BCS theory. Recently, Sung and Yun Lung Shen [2] have made specific heat measurements on superconducting niobium, and have fitted their data in a two-band model, by adjusting the two gap parameters. Since their result seems to favour the two-band model, it is interesting to examine whether this two-band model can also explain some peculiarities in the tunnelling characteristics of junctions involving one transition element superconductor and a non-transition element superconductor. The special feature is t h a t the negative resistance region is almost flat, and the cusp is less pronounced [3, 4], I n this paper we present a calculation of the tunnelling current for a junction made of a transition element superconductor and a non-transition element superconductor. We apply the two-band model to the transition element. As an example, we have made numerical computations for a niobium-tin junction. 2. Tunnelling Formalism The theory of tunnelling for superconductors is based on the Hamiltonian H

— H0 + Ht

H0 = Hl

,

(1)

+ Hn ,

(2)

where H L , H t i are the full Hamiltonians for the left-side and right-side superconductors respectively, and H T is the tunnelling Hamiltonian. We take the transition element to be on the left side and assume t h a t the left side is at 8

phyaica

V. RADHAKRISHNAN : Tunnelling in a Two-Band Model Superconductor

113

phys. stat. sol. 18, 113 (1966) Tata Institute

of Fundamental

Research,

Bombay

Tunnelling in a Two-Band Model Superconductor By V.

RADHAKRISHNAN

The Josephson and ordinary tunnelling currents are evaluated for two-band model superconductors. Numerical calculations of these currents are made for a niobium-tin junction, for which this model is applicable, and compared with experiments. Der Josephson- und normale Tunnelstrom werden für Zweibandmodell-Supraleiter berechnet. Numerische Berechnungen dieser Ströme für eine Niobium-Zinn-Verbindung, für die dieses Modell anwendbar ist, werden mit Experimenten verglichen.

1. Introduction Transition element superconductors differ from the other superconductors in many properties, even though broadly they share the main features with them. To study this group of superconductors, Suhl, Matthias, and Walker [1] have suggested a two-band model extension of the BCS theory. Recently, Sung and Yun Lung Shen [2] have made specific heat measurements on superconducting niobium, and have fitted their data in a two-band model, by adjusting the two gap parameters. Since their result seems to favour the two-band model, it is interesting to examine whether this two-band model can also explain some peculiarities in the tunnelling characteristics of junctions involving one transition element superconductor and a non-transition element superconductor. The special feature is t h a t the negative resistance region is almost flat, and the cusp is less pronounced [3, 4], I n this paper we present a calculation of the tunnelling current for a junction made of a transition element superconductor and a non-transition element superconductor. We apply the two-band model to the transition element. As an example, we have made numerical computations for a niobium-tin junction. 2. Tunnelling Formalism The theory of tunnelling for superconductors is based on the Hamiltonian H

— H0 + Ht

H0 = Hl

,

(1)

+ Hn ,

(2)

where H L , H t i are the full Hamiltonians for the left-side and right-side superconductors respectively, and H T is the tunnelling Hamiltonian. We take the transition element to be on the left side and assume t h a t the left side is at 8

phyaica

114

V . RADHAKBISHNAN

a potential difference V above the right side, then we have 4 cka ck a — F SS £ ckf cLk[ A, k. ct+ ak.„ + H.C.] + 2 [Tik- &L ak.a + H.C.] ,

ky k', a

k, k', a

Nl = 2 dko dfco + 2 Cka Cka , kf a k, a NR = 2 ato ako , k, a

where cka, dk„ are the creation operators for the states of the s- and the d-band of the left-side superconductor, and ak„ the creation operator for the states of the right-side superconductor. T\t K is the tunnelling matrix element for the i-th band. The other quantities have the same meaning as in reference [1]. The tunnelling current is given by [5] i

I = e ; but at low temperatures yq (5, - 4 ) and (4, - 4 ) ^ (5, - 3 ) coincide 9

physica

H(k6)—F i g . 3. Breit-Babi diagram for the divalent T l " 5 ion, I = 1/2, in cubic ZnS, showing the two allowed transitions observed a t 9 GHz

130

A . R Ä U B E R a n d J . SCHNEIDER

2.4 Optical excitation

and thermal

decay of the

centres

The paramagnetic centres observed can be produced by 365 nm illumination. In the case of the gallium centre, radiation peaked at 450 nm proved to be more effective in some samples, as reported by Fair et al. [7]. After turning off the exciting radiation at 77 °K, the intensity of the ESR-spectrum of the thallium centre was found to be somewhat reduced, but then remained almost constant. A slow continuous decay of the indium centre was observed under these conditions, and a much faster decay for the gallium centre. The fast primary decay of the signals may possibly be attributed to recombination of photoexcited free carriers with the paramagnetic centre, whereas the slow decrease of the ESR-signal may represent the thermal decay of the centre. At liquid helium temperatures, part of the ESR-intensity of the spectrum of the gallium centre remains frozen-in. 2 ) 2.5 Hexagonal

ZnS

The Hamiltonian (1) is not appropriate for those a S 1 / 2 -state ions which reside in the non-cubic domains of the ZnS crystal. Here, the ion experiences an additional axial crystalline electric field. The isotropic parameters g and A in the Hamiltonian (1) should therefore be replaced by axially symmetric tensors which reflect the non-cubic symmetry of the crystalline environment. In addition, nuclear electric quadrupole interaction has now to be considered. The nuclear Zeeman interaction may be assumed to remain isotropic. If the magnetic field is oriented parallel to the symmetry axis, H || c, the eigenvalues of this generalized Hamiltonian can be again expressed in closed form. However, we note that the system can no longer be described in terms of a single zero-field splitting parameter, as in the cubic case. In zero magnetic field, each state F is now further split into F + 1 sublevels, for even F, and into F + 1/2, sublevels, for odd F. No closed solutions of the eigenvalue equation can be obtained for H c, with the exception of the special case I = 1/2. The angular dependence of the E S R spectrum of the T l 2 + ion in hexagonal ZnS has been studied in detail. We find g-ll = 2.0093 + 0.0005 , gL -g]] = 0.0010 + 0.0002 , A\\ (Tl 203 ) = 71.30 ± 0.03 GHz , An (Tl 205 ) = 71.98 + 0.03 GHz , A± - An = 0.016 ± 0.002 GHz . A similar analysis for the Ga 2 + and I n 2 + centre in hexagonal ZnS was not attempted, since the evaluation of the E S R spectra is somewhat more complicated, as a result of the nuclear electric quadrupole interaction, present if I > 1/2. Exploratory measurements have shown that deviations from cubic symmetry must be again small. In the case of the Ga 2 + centre, the E S R lines were found to shift by not more than 6 G when the crystal was rotated from H || c to H _J_ c. However, the mean value for the hyperfine interaction, A (Ga 71 ) = = 7.87 GHz, was found to be definitelv larger than that reported for cubic ZnS, A (Ga 71 ) = 7.716 GHz. 2

) W. C. Holton, private communication.

Localized 2S1/2-State Centres in ZnS

131

A similarly small angular dependence was observed for the ESR spectrum of the In2+ centre in hexagonal ZnS, where JT(In 115 ) = 9.72 GHz. The fact that the hyperfine interaction is systematically enhanced in hexagonal ZnS may be correlated with the somewhat greater covalent character of the bonds in cubic ZnS. Co valency implies that the unpaired electron has a finite probability at the ligands and, in this way, the spin density at the central nucleus will be decreased. A similar characteristic difference has been found to occur between the hyperfine interaction of the Mn2+ ion in cubic and hexagonal ZnS [9]. 3. Discussion

The gr-factors of the centres observed in ZnS exhibit only small deviations from the free spin value, 2.0023, as expected for a 2 S m -ion. Watanabe [10] has recently presented theoretical arguments to explain the small ¿/-shifts of the isomorphous Si3+ andGe 3+ centre, recently observed in ZnS by Sugibuchi and Mita [11]. Here, it is assumed that the unpaired electron is localized in an antibonding molecular orbital formed by interaction with the four nearest sulfur ligands. The spin density of the unpaired electron at the site of the nucleus, |yn(0)l2> as defined by (2), should increase with the nuclear charge, for a given n, since the wavefunction will then be more concentrated towards the nucleus. This is found to be the case for the isoelectronic n = 3, 4, 5 atoms and ions listed in Table 1. For comparison, we have also included the values for free neutral sodium [5], copper [12], silver [5] and gold [5] atoms, determined by atomic beam measurements. However, this increase is seen to be much less than one might expect on the basis of the Goudsmit relation [5], |y(0)|2oc Zt Z\, where Zj represents the effective nuclear charge close to the nucleus and Za that outside the closed shell. In addition, for n = 6, the spin density of the neutral gold atom is seen to exceed that of the divalent thallium ion. This shows that the effects of the crystalline environment must be considered. In the ionic picture, the cubic crystalline electric field will admix some g-character to the Table 1 unpaired electrons wavefunction, ESR parameters of 2S1/2-state ions in cubic thus reducing its s-character. A ZnS. The values for the neutral atoms have further reduction of |^(0)!2 has to been determined by atomic beam be expected when the unpaired elecmeasurements tron is admitted to form partially covalent bonds with the sulfur lig|v(0)2| 9 ands [10]. (1024 cm-3) The isotropy of the ESR spectra Ga2+ 4 s1 38.3 1.9974 in cubic ZnS makes it appear rather 2+ In 5 s1 64.8 1.9930 likely that the divalent paramagTl2+ 6 s1 186.5 2.0095 netic ions occupy substitutional zinc sites of purely cubic symmetry. Si3+ 3 s1 14.9 2.0047 An interstitial site for the divalent Ge3+ 4 s1 39.4 2.0086 impurity ion is not chargecompensated and appears to be energeticNa° 3 s1 5.06 2.00228 ally less favourable. However, there Cu° 33.5 4 s1 is some ambiguity with respect to 63.9 Ag° 5 s1 2.00224 the question of the charge state of 268.2 Au° 6 s1 2.00412 9'

132

A. RÄUBER and J . SCHNEIDER: Localized 2 S!/ 2 -State Centres in ZnS

these centres before optical excitation. I t should be kept in mind that stable compounds of gallium, indium and thallium exist for the monovalent as well as for the trivalent charge state of the metal ion. As an example, we mention the stable monosulfides GaS, I n S and possibily, TIS, which are diamagnetic, containing an equal number of both monovalent and trivalent metal ions. In contrast, stable aluminium compounds always contain the metal ion in its trivalent charge state. This difference in the chemical behaviour might possibly also account for the fact that strong hyperfine interaction, indicating considerable localisation of the unpaired electron at the nucleus, is observed for the divalent gallium, indium and thallium center in ZnS, but none for the shallow aluminum donor [3], However, it is surprising that gallium in CdS [13], [3] as well as indium in ZnO [3] form shallow donor states of which only the single E S R line, characteristic for mobile electrons, has been detected. The optical excitation spectrum of the paramagnetic centres reported should be different, depending whether the original charge state of the metal ion was monovalent or trivalent. I n some of the gallium-doped samples investigated, the paramagnetic divalent metal ions were preferentially created under 450 nm illumination, as already reported by Fair et al. [7]. However, there were also samples in which 365 nm excitation proved to be more effective. We have not attempted a systematic investigation of the optical excitation spectrum of the paramagnetic centres. ZnS, activated with gallium, indium or thallium can exhibit a luminescence which preferentially occurs in the orange-red region of the optical spectrum. When the exciting light is turned off, the luminescence is found to decay much faster than the paramagnetic centres. Acknowledgements

We would like to thank F . Friedrich for the preparation of the samples and O. Schirmer for comments on the manuscript and for active help in the numerical calculations. References [1] A. RÄUBER and J . SCHNEIDER, Phys. Letters (Netherlands) 3, 230 (1963). [2] J . SCHNEIDER,

A. RÄUBER,

B . DISCHLER,

T . L . E S T L E , a n d W . C. HOLTON,

J.

chem.

[6]

Phys. 42, 1839 (1965). K. A. MÜLLER, and J . SCHNEIDER, Phys. Letters (Netherlands) 4, 288 (1963). P. H. KASAI, J . chem. Phys. 43, 4143 (1965). N. P. RAMSEY, Molecular Beams, Clarendon Press, Oxford 1956. H. KOPFERMANN, Kernmomente, Akademisch® Verlagsgesellschaft, Frankfurt/Main 1956. R . A. ZHITNIKOV and N. V. KOLESNIKOV, Soviet Physics — Solid State Phys. 6, 2645

[7]

H . D . FAIR, JR.,

[3] [4] [5]

(1965). R . D . EWING, a n d F . E . WILLIAMS, P h y s . R e v . L e t t e r s 1 5 , 3 5 5 (1965).

[8] A. RÄUBER and J . SCHNEIDER, Phys. Rev. Letters 16, 1075 (1966). [9] J . SCHNEIDER,

S . R . SIRCAR, a n d A . R Ä U B E R , Z . N a t u r i . 1 8 a , 9 8 0

(1963).

[10] H. WATANABE, Phys. Rev., in press. [ 1 1 ] K . SUGIBUCHI a n d Y . MITA, P h y s . R e v . 1 4 7 , 3 5 5 ( 1 9 6 6 ) . [12]

Y . TING a n d H . L E W , P h y s . R e v . 1 0 5 , 5 8 1

(1957).

[13] J . DIELEMAN, Proc. l l t h Colloque Ampère 1962, Eindhoven, North-Holland Pubi. Comp., Amsterdam 1963. (Received, August 1, 1966)

W. DREYBKODT and K. FUSSGAENGER: Ag + Dipole Strengths in Alkali Halides

133

phys. stat. sol. 18, 133 (1966) Physikalisches

Institut der Universität

Frankfurt)Main

Interpretation of the Temperature Dependence of Ag + Dipole Strengths in Alkali Halides By W . DREYBBODT a n d K .

FUSSGAENGEB

The absorption spectrum of Ag + doped RbCl is measured a t various low temperatures in t h e spectral range from 6.4 to 5.0 eV. The temperature dependence of t h e Ag + dipole strength in RbCl is different from t h a t found in NaCl, KCl, and K B r . The different temper a t u r e behaviours are interpreted as being due to the different lattice potentials seen b y t h e Ag + in t h e various host crystals. The static potentials for Ag + in the crystals are calculated b y a semiphenomenological method. For NaCl, KCl, and K B r a potential minimum is found a t t h e symmetry centre. For RbCl, however, a potential barrier exists a t the symm e t r y centre and potential mimima occur a t 0.25 A away from t h e centre in the [111] direction. Das Absorptionsspektrum von Ag + -dotiertem RbCl wird bei verschiedenen tiefen Temperaturen im Spektralbereich von 6,4 bis 5,0 eV gemessen. Die Temperaturabhängigkeit der Dipolstärke zeigt einen anderen Verlauf als er bisher von Ag+ in NaCl, KCl und K B r bekannt ist. Die verschiedenen Temperaturabhängigkeiten der Dipolstärke werden mit verschiedenen Gitter-Potentialen erklärt, die auf das Ag+ Ion in den verschiedenen Wirtskristallen wirken. Mit einer halbklassischen Methode werden die statischen Potentiale f ü r das Ag+ Ion in den Kristallen berechnet. Die Berechnungen ergeben f ü r NaCl, KCl und K B r ein Potentialminimum im Symmetriezentrum, f ü r RbCl dagegen einen Potentialberg im Symmetriezentrum und Potentialminima 0,25 Ä außerhalb in [ l l l ] - R i c h t u n g .

1. Introduction The areas below the uv absorption bands of the Ag + doped alkali halides NaCl, KCl, and KBr show a strong temperature dependence [1 to 4], which is interpreted by the temperature dependence of the dipole strength of the electronically forbidden, vibronically allowed Ag + transitions 4d 10 to 4d 9 5s [5 to 12]. Assuming linear electron-phonon interaction the dipole strength follows a coth (h a>_/2 k T) law [7 to 11]. Nonlinear electron-phonon interaction caused by two-phonon processes adds a quadratic term in T to the dipole strength; this becomes noticeable at T jg: 0 D /3 [4]. a)_ is the effective frequency of the odd phonon corresponding to an Ag+ resonance mode or an asymmetric lattice mode. The far infrared resonant modes of Ag+ have been measured in NaCl by Weber [13] and in KCl, KBr, and K I by Sievers [14], For all Ag+ doped systems investigated so far the ionic radius of the impurity is comparable with that of the cations. So the vibrating Ag+ ion occupies an equilibrium position at a lattice site. If the ionic radius of the impurity is much smaller than that of the replaced ion, the impurity ion may occupy a new equilibrium position away from the lattice site. For a-Agl Hoshino [15] has shown from the temperature dependence of the intensities of X-ray reflections that 42 possible off centre positions exist for the two Ag + ions in the elementary cell, favouring the disordered structure model first proposed by Strock [16]. For Li+ in KCl electrocaloric measurements

W. DREYBKODT and K. FUSSGAENGER: Ag + Dipole Strengths in Alkali Halides

133

phys. stat. sol. 18, 133 (1966) Physikalisches

Institut der Universität

Frankfurt)Main

Interpretation of the Temperature Dependence of Ag + Dipole Strengths in Alkali Halides By W . DREYBBODT a n d K .

FUSSGAENGEB

The absorption spectrum of Ag + doped RbCl is measured a t various low temperatures in t h e spectral range from 6.4 to 5.0 eV. The temperature dependence of t h e Ag + dipole strength in RbCl is different from t h a t found in NaCl, KCl, and K B r . The different temper a t u r e behaviours are interpreted as being due to the different lattice potentials seen b y t h e Ag + in t h e various host crystals. The static potentials for Ag + in the crystals are calculated b y a semiphenomenological method. For NaCl, KCl, and K B r a potential minimum is found a t t h e symmetry centre. For RbCl, however, a potential barrier exists a t the symm e t r y centre and potential mimima occur a t 0.25 A away from t h e centre in the [111] direction. Das Absorptionsspektrum von Ag + -dotiertem RbCl wird bei verschiedenen tiefen Temperaturen im Spektralbereich von 6,4 bis 5,0 eV gemessen. Die Temperaturabhängigkeit der Dipolstärke zeigt einen anderen Verlauf als er bisher von Ag+ in NaCl, KCl und K B r bekannt ist. Die verschiedenen Temperaturabhängigkeiten der Dipolstärke werden mit verschiedenen Gitter-Potentialen erklärt, die auf das Ag+ Ion in den verschiedenen Wirtskristallen wirken. Mit einer halbklassischen Methode werden die statischen Potentiale f ü r das Ag+ Ion in den Kristallen berechnet. Die Berechnungen ergeben f ü r NaCl, KCl und K B r ein Potentialminimum im Symmetriezentrum, f ü r RbCl dagegen einen Potentialberg im Symmetriezentrum und Potentialminima 0,25 Ä außerhalb in [ l l l ] - R i c h t u n g .

1. Introduction The areas below the uv absorption bands of the Ag + doped alkali halides NaCl, KCl, and KBr show a strong temperature dependence [1 to 4], which is interpreted by the temperature dependence of the dipole strength of the electronically forbidden, vibronically allowed Ag + transitions 4d 10 to 4d 9 5s [5 to 12]. Assuming linear electron-phonon interaction the dipole strength follows a coth (h a>_/2 k T) law [7 to 11]. Nonlinear electron-phonon interaction caused by two-phonon processes adds a quadratic term in T to the dipole strength; this becomes noticeable at T jg: 0 D /3 [4]. a)_ is the effective frequency of the odd phonon corresponding to an Ag+ resonance mode or an asymmetric lattice mode. The far infrared resonant modes of Ag+ have been measured in NaCl by Weber [13] and in KCl, KBr, and K I by Sievers [14], For all Ag+ doped systems investigated so far the ionic radius of the impurity is comparable with that of the cations. So the vibrating Ag+ ion occupies an equilibrium position at a lattice site. If the ionic radius of the impurity is much smaller than that of the replaced ion, the impurity ion may occupy a new equilibrium position away from the lattice site. For a-Agl Hoshino [15] has shown from the temperature dependence of the intensities of X-ray reflections that 42 possible off centre positions exist for the two Ag + ions in the elementary cell, favouring the disordered structure model first proposed by Strock [16]. For Li+ in KCl electrocaloric measurements

134

W . DKEYBRODT a n d K . FUSSGAENGER

by Lombardo and Pohl [17] and measurements of the dielectric constant by Sack and Moriarty [18] show t h a t the Li + ion goes to an off centre position in [100] or [111] direction [19]. Matthew [20] has calculated the potential for the Li + ion in KC1 by expanding the energy after displacement of the impurity ion from the lattice site u p to sixth order and has found t h a t off centre positions are possible. Dienes, Hatcher, Smoluchowski, and Wilson [21] have improved this calculation by taking into account additional displacements of the nearest neighbors. They find an off centre position of Li + at about 12% from the origin along the [100] axis. Orgel [22] has pointed out t h a t in AB 6 complexes interaction with symmetric lattice vibrations may mix s and d orbitals, thus changing the coordination number and causing off center positions. uv absorption spectra of Cu + in KC1 and K B r at varios temperatures measured recently by Kraetzig, Timusk, and Martienssen [23] and discussed by Dultz [24] also require an off centre behaviour for the temperature dependence of the Cu + dipole strength. I n this paper we report results on the temperature dependence of the Ag + dipole strength in RbCl which are different from those obtained b y Fussgaenger, Martienssen, and Bilz [4] and by Kraetzig, Timusk, and Martienssen [23]. Our results can be explained by assuming the Ag + ion in RbCl to be in an off centre position. I n the case of Ag + doped alkali halides we assume t h a t the different temperature behaviours arise from different potentials seen by the Ag + ion in the different host crystals. Therefore we t r y to calculate these potentials b y a semiphenomenological method used by Das, J e t t e , and Knox [25]. From these potentials the temperature dependence of the dipole strength is calculated, showing t h a t anharmonicity of the potential has the same influence on the dipole strength as non-linear electron-phonon interaction. The results of the calculation are compared with the experimental d a t a of Ag+ in NaCl, KC1, and K B r [4, 13, 14] und in RbCl. 2. Experimental Techniques +

The Ag doped RbCl crystal was grown in air from Merck reagent grade material by the Kyropoulos technique. 0.2Mol-% AgCl were added to t h e melt. The density Nv of Ag + centres per cm 3 in the crystal was determined b y colorimetric chemical analysis [4], The crystal was heated for 48 hours at 600 °C in hydrogen chloride ambient. Before the measurements the crystal was heated again to 600 °C and quenched between copper plates to destroy colloidal centres. The absorption measurements in the spectral range 6.4 to 5.0 eV were carried out with a Perkin Elmer Spectrophotometer 350. 3. Experimental Results The u v absorption spectrum of RbCl:Ag + at four temperatures is shown in Fig. 1. At 5.5 °K a set of four bands is observed. The bands at 5.34 eV, 5.73 eV, and 5.88 eV have nearly the same height. The band at 6.25 eV is about 5 times greater in its absorption constant and 2 times greater in its area t h a n t h e other bands. W i t h increasing temperature the maximum of the band at 5.34 eV does not shift in energy, the maximum of the band at 6.25 eV shifts to lower photon energies. The areas below the bands are temperature dependent. I n Fig. 2 the total dipole strength of the observed Ag + bands in RbCl is plotted versus tem-

Temperature Dependence of A g + Dipole Strengths in Alkali Halides

200

2.5

210

Wavelength Inm I— 240 220 230

A /\ /1

2.0

Rbü:Ag'

' A.1

Nr-0.7'10"c/n3 b T'5.5°K

'ML % 15

I § 10

-

/

f

1 T-57°K 3- T=K3°K 4- T=253"K

.1 2 J 4

k

1\

\

A / 1

\ 3.5

6.4 -

135

6.2 6.0 5.8 5.6 Photon energy (eVi

VV 5.4

5.2

200

100

5.0

Temperature !°K) ~

Fig. 2. Temperature dependence of the dipole strength in R b C l : A g +

Fig. 1. Absorption spectrum of R b C l : A g + at four temperatures. ATv = 0.7 x 1 0 " c m - 3

perature. The total dipole strength D [5] is computed by the equation D

=

27Vc

n( e) 2

(i)

+ 2]2

hf

K(e)

N,

27 e„

de

h c

n(X)

[n2(A) + 2]2 Nv i

«2

4

(i)

For the index of refraction of RbCl we use the temperature independent value n = 1.6734 at the average wavelength X = 214.4 nm (average photon energy e = 5.78 eV), and at 48 °C [26]. K is the absorption constant, 2Vv is the density of Ag+ centres per cm 3 listed in Fig. 1. The total dipole strength D is related to the total oscillator strength / by ,

2

m*

-

/ = - - — 2- e ' 3 Ä

D

.

(2)

m* is the effective electronic mass, s is the centre of gravity energy of the bands. Following (2) the temperature dependence of the oscillator strength is equal to that of the dipole strength, if the temperature dependence of i is negligible. For Ag + in NaCl, KC1, and K B r this is the case, since Ag + bands of nearly equal height and opposite energy shifts of the band maxima have been found [4], In R b C l : A g + , however, the bands are of different height and asymmetric energy shift. Therefore we prefer to plot the dipole strength as a function of temperature rather than the oscillator strength. The extrapolated dipole strength at T = 0 ° K is D(0) = 10.5 x 10~ 2 A 2 corresponding to an oscillator strength /(0) = 5.3 X 10" 1 after (2). With increasing temperature the dipole strength of the Ag+ bands in RbCl first decreases, passes a minimum at about 60 ° K , and then increases linearly. 4. Calculation of Static Potentials The different temperature behaviours of the oscillator strengths of the Ag + doped systems NaCl, KC1, and K B r on the one hand, and that of the system RbCl: Ag + on the other hand can be interpreted as arising from different types

136

W . DREYBRODT a n d K . FUSSGAENGER Fig. 3. Schematic representation of the displacements in the A g + Halg" complex, x is the displacement of the A g + ion in [111] direction, u and v are the resulting displacements of the nearest neighbours in [100] directions

of potentials seen by the A g + ion in the different sytems. Therefore we try to calculate these potentials with a semiclassical method reported by Das, Jette, and Knox [25], For this purpose we calculate the change of crystal energy for a given displacement of the Ag + ion from a lattice site, allowing its nearest neighbours to relax and to determine new equilibrium positions. The change of the total crystal energy is given by A E m = A # e + AE t + A£\.dw + \H P .

(3)

A E c is the electrostatic Coulomb energy, A ET the Born-Mayer repulsive energy. AEyc are compared with the resonant frequencies cores measured by Weber [13] and Sievers [14] and with the effective frequencies wq derived from the temperature dependence of the oscillator strength [4]. The calculated frequencies are in good agreement with the experimental ones. For NaCl and KC1 coq, however, they are higher than the resonant frequencies. This seems to indicate that odd lattice phonons of higher frequency than the Ag + resonant frequency contribute to the dipole strength resulting in a higher effective frequency a>q. The ratios (D!2jD1)c due to the anharmonicity of the potentials calculated by (18) with the constants b of Table 3 are compared in Table 4 with the experimental ratios (/2//i)i [4]- For Ag + in NaCl good agreement is obtained. The deviations for Ag+ in KC1 and K B r may be understood by contributions from nonlinear electron-phonon interaction [4] which adds a term

to the dipole strength in (17). D'/ contains electronic transition moments, energies and the coupling constant of the nonlinear electron-phonon interaction. a>+ is the frequency of the single representative symmetric lattice mode. For simplicity, a linear model with one symmetric and one asymmetric lattice mode is assumed. In order to test the influence of anharmonicity and nonlinear electron-phonon interaction, we have tried to refit the temperature dependence of the Ag + dipole strength in KC1 [4] in three ways: (i)

if only anharmonicity contributes, (17) with co_ = cores = 0.73 xlO 1 3 s _ 1 measured by Sievers [14] should fit the experimental data. No agreement, however, to the low temperature data is achieved, (ii) if only nonlinear electron-phonon interaction contributes, the dipole strength is given by the sum of (15) and (20) with a)_ = OJ . The fit yields OJ+ = 7.3 X 10 13 s _ 1 , which is an unreasonable result, (iii) if both mechanisms contribute, the data can be fitted by the sum of (17) ICS

142

W . DREYBRODT a n d K . FUSSGAENGER

Table 5 Calculated and experimental values for Ag+ in RbCl Quantity

Calculated 0.25 X 10-10 m 1.9 x IO -21 Ws 7.9 X 1010 m" 1 2.4 x IO1 N/m 1.1 x IO13 s - 1 96 ° K 7.9 X 1 0 - 2 A 2

xe

V0 a kG

COc

• T min D( 0)

and (20) with

Experimental

: CO.

— —

0.3 x IO13 s- 1 60 ° K 10.5 x io- 2 A2

= o)g and D'2 -f- D'2' = Z)2 resulting in = Dx coth

D(T)

—

+ D2 coth2 (

«

(21)

2 kT

(21) fits the experimental data [4] with wq = 1.0 X 1013 s" 1 , D t = 3.63 X 10~3 A 2 and D2 = 0.22 x 10"3 A 2 . The temperature behaviour of the dipole strength for the "off centre" potential of case 2 (Ag + in RbCl) is quite different. A t zero temperature the A g + vibrates about the off centre position xe with the frequency o>e determined by (14). A t temperatures where the average thermal energy k T is smaller than the barrier energy V0 of the double well potential the behaviour of the A g + is mainly determined by only one well of the potential. The potential for x 2; 0 may be crudely approximated by a Morse potential V(x) = F 0 (1 - e- 8 «*»-*)) 2 -

F„ .

(22)

The constant a is determined by the barrier height V0 and the curvature ke of the potential (14) at xe. The calculated values of x„, F„, k., co., and a are listed in Table 5. The dipole strength is given by D(T)

=

DA(V

=

De]

|Z2| » > =

DE\

(0) of Ag+ in RbCl can be estimated roughly with the calculated values and a, if we assume that the observed A g + absorption bands in RbCl and KC1 arise from equivalent A g + transitons and by inserting the value Z)ei = 1-25 of A g + in KC1. The calculated and experimentally observed (Fig. 2) zero temperature dipole strengths of A g + in RbCl listed in Table 5 are in the right order.

Temperature Dependence of Ag + Dipole Strengths in Alkali Halides

143

I n the temperature range where the average thermal energy of the vibrating Ag + exceeds the barrier energy F 0 , the motion of the Ag + is determined b y t h e double well potential (14) and the probability for finding the Ag + at lattice site grows rapidly, diminishing the "off centre" effect. At higher temperatures the mean square deviation from lattice site increases, resulting in an increase of t h e dipole strength. The minimum of the dipole strength (Fig. 2) is given b y the condition lcT^=V

0

-~hw

e

.

(24)

The calculated and experimental values of f m are listed in Table 5. The linear increase of the experimental curve (Fig. 2) at high temperatures t h u s far can not be understood simply in our approximation. 6. Conclusion The results seem t o indicate, t h a t the observed temperature dependences of Ag + dipole strengths in different host crystals m a y be understood by calculations of static potentials. This includes the case of Ag + in RbCl where the temperature dependence, b u t also the static potential, is even qualitatively different from NaCl, KC1, and K B r . Acknowledgements

We are very indebted to Professor Dr. W. Martienssen, Professor Dr. H . Bilz and Priv.-Doz. Dr. W. Gebhardt for m a n y stimulating discussions and suggestions and t o Professor Dr. H . A. Müser for reading the manuscript. Many t h a n k s are due to W. Dultz, V. NüBlein, and E. Zybell. We t h a n k t h e Deutsches Rechenzentrum, Darmstadt, for computing the low temperature characteristics of the used carbon-composition thermometer. The crystals have been grown b y the F r a n k f u r t Unit for Crystal Growing. We t h a n k the Deutsche Forschungsgemeinschaft for the financial support to this group. References [ 1 ] M . FORRÓ, Z . P h y s . 5 6 , 5 3 4 ( 1 9 2 9 ) ; Z. P h y s . 5 8 , 6 1 3 ( 1 9 2 9 ) .

[2] W. MARTIENSSEN, Proc. Int. Conf. Semicond. Phys., Czech. Acad. Sci., Prague 1960

(p.

760).

[ 3 ] R . ONAKA,

A . FUKUDA,

K . INOHARA,

T . MABUCHI,

and

Y . FUJIOKA,

Japan.

J.

appl. Phys. 4, Suppl. 1, 631 (1965). [ 4 ] K . FUSSGAENGER,

W . MARTIENSSEN, a n d H . BILZ, p h y s . s t a t . s o l . 1 2 , 3 8 3 ( 1 9 6 5 ) .

[5] R. S. MULLIKEN, J. chem. Phys. 7, 14 (1939). R . S . MULLIKEN a n d C. A . R I E K E , R e p . P r o g r . P h y s . 8 , 2 3 1 ( 1 9 4 1 ) .

[6] F. SEITZ, Rev. Mod. Phys. 23, 328 (1951). [7] R. KUBO and Y. TOYOZAWA, Progr. theor. Phys. 13, 160 (1955). [ 8 ] A . D . LIEHR a n d C. J . BALLHAUSEN, P h y s . R e v . 1 0 6 , 1 1 6 1 ( 1 9 5 7 ) .

C. J. BALLHAUSEN, Introduction to Ligand Field Theory, McGraw-Hill Book Company, Inc., New York 1962 (p. 186). [9] R. S. KNOX, J. Phys. Soc. Japan 18, Suppl. II, 268 (1963). [ 1 0 ] J . M . CONWAY, D . A . GREENWOOD, J . A . KRUMHANSL, a n d W . MARTIENSSEN, J . P h y s .

Chem. Solids 24, 239 (1963). [11] A. GOLD and J. P. HERNANDEZ, Phys. Rev. 139, A 2002 (1965). [ 1 2 ] W . GEBHARDT a n d E . MOHLER, p h y s . s t a t . s o l . 1 5 , 2 5 5 ( 1 9 6 6 ) .

144

W. DREYBRODT and K. FUSSGAENGER: Ag + Dipole Strengths in Alkali Halides

13] R. WEBER, Phys. Letters (Netherlands) 12, 311 (1964). 14] A. J . SIEVERS, Phys. Rev. Letters 13, 310 (1964). S. TAKENO and A. J . SIEVERS, Phys. Rev. Letters 15, 1020 (1965). 15] S. HOSHINO, J . P h y s . S o c . J a p a n 1 2 , 3 1 5 (1957).

16] L. W. STROCK, Z. phys. Chem. (Leipzig) B 25, 441 (1934); B 31, 132 (1935). 17] G. LOMBARDO a n d R . 0 . POHL, P h y s . R e v . L e t t e r s 1 5 , 2 9 1 (1965). 18] H . S. SACK a n d M. C. MORIARTY, S o l i d S t a t e C o m m . 3 , 9 3 (1965).

19] N. E. BYER, P. S. WELSH, and H. S. SACK, Bull. Am. Phys. Soc. 11, 229 (1966). 20] J . MATTHEW, Cornell University (Ithaca), Materials Science Center Report No. 373, 1965 (unpublished). 2 1 ] G. J . DIENES, R . D . HATCHER, L e t t e r s 1 6 , 2 5 (1966).

R . SMOLUCHOWSKI, a n d

W . WILSON, P h y s .

Rev.

22] L. E. ORGEL, J . chem. Soc. 4186 (1958); J . Phys. Chem. Solids 7, 276 (1958). 2 3 ] E . KRAETZIG, T . TIMUSK, a n d W . MARTIENSSEN, p h y s . s t a t . sol. 1 0 , 7 0 9 (1965) a n d

private communication. 24] W. DULTZ, Diplomarbeit, Physikalisches Institut der Univerisät Frankfurt/Main 1966. 25] T. P. DAS, A. N. JETTE, and R. S. KNOX, Phys. Rev. 134, A1079 (1964). B. G. DICK and T. P. DAS, Phys. Rev. 127, 1053 (1962). 26] Z. GYULAI, Z. Phys. 46, 80 (1927). 2 7 ] N . F . MOTT a n d M . J . LITTLETON, T r a n s . F a r a d a y S o c . 3 4 , 4 8 5 (1938).

28] M. SACHS, Solid State Theory, McGraw-Hill Book Company, Inc., New York 1963 (p. 60). 29] M. BORN and K. HUANG, Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford 1956 (p. 16). 3 0 ] J . E . MAYER, J . c h e m . P h y s . 1, 3 2 7 (1933).

31] A. G. MAKHANEK, Opt. and Spectr. 11, 6 (1961). 32] W. SHOCKLEY, Phys. Rev. 70, 105 (1946). (Received

August

4,

1966)

145

V. D. EGOROV: Anklingen der Photoleitfähigkeit in Halbleitern phys. stat. sol. 18, 145 (1966) Physikalisch-Technisches

Institut der Deutschen Akademie

der Wissenschaften

zu

Berlin

Zum linearen Anklingen der Photoleitfähigkeit in Halbleitern Von V . D . EGOBOV 1 ) Es wird der Einfluß der Oberflächenrekombination und einer parasitären Kapazität auf den Anfangsteil des Anklingens des Photosignals betrachtet. Die Ergebnisse ermöglichen Aussagen über die Verwendbarkeit eines Germanium-Photowiderstandes zur Messung der Intensität von Lichtimpulsen. PaccMaTpHBacrcH

BJIHHHHC

n0BepxH0CTH0i!

peKOMÖwiaijHH

H

napa3HTHOH

EMKOCTH Ha Ha^ajibHbiö yqacTOK carHajia HECTAUHOHAPHOII «JOTOIIPOBOUHMOCTH. CooömaeTCH o npHMeHeHHH repMamieBoro oToconpoTHBJieHHH

HJJH

H3MepeHHH

H H T e H C H B H O C T e Ö H M n y j l b C H b l X CBCTOBblX n O T O K O B .

1. Einführung und Fragestellung Bekanntlich (siehe z. B. [1]) wächst die Konzentration 8 p der zusätzlichen Ladungsträger in Abwesenheit eines Hafteffektes bei homogener Anregung eines homogenen Halbleiters durch einen Rechtecklichtimpuls linear mit der Zeit t, solange t klein ist im Vergleich zur Lebensdauer r (solange die Rekombinationsgeschwindigkeit 8plr klein ist im Vergleich zur Paarerzeugungsrate). I m Falle einer merklichen Absorption des Lichtes erfolgt die Anregung des Halbleiters ungleichmäßig in der Tiefe, und infolge der Diffusionsbewegung der Ladungsträger bleibt die Proportionalität zwischen 8p und t im allgemeinen nicht erhalten, auch bei t r. Erhalten bleibt jedoch die lineare Abhängigkeit der Gesamtzahl der Elektron-Loch-Paare w SP = / 8p{x, o

t) äx ,

die in dem Halbleiter pro Einheit der beleuchteten Oberfläche erzeugt werden, von der Zeit (Fig. 1): SP = g0 • t ,

wobei g0 das Integral der Paarerzeugungsrate über die Probendicke ist. Wenn das Licht vollständig in der Probe absorbiert wird, so gilt 9o =

(! -

Ä) y ?Ph •

Hier ist j Ph die Dichte des auffallenden Photonenstromes, R der Reflexionskoeffizient, y die Quantenausbeute. Die zusätzliche elektrische Leitfähigkeit der Probe bei t r ist =

-

ü)yj

F h

t,

(1)

mit der Länge l und der Breite b der Probe, den Beweglichkeiten der Elektronen fj,n und Löcher und der Elementarladung e. Die Spannungsänderung 1

) Gast von der Lomonossov-Universität Moskau.

10 pliysica

145

V. D. EGOROV: Anklingen der Photoleitfähigkeit in Halbleitern phys. stat. sol. 18, 145 (1966) Physikalisch-Technisches

Institut der Deutschen Akademie

der Wissenschaften

zu

Berlin

Zum linearen Anklingen der Photoleitfähigkeit in Halbleitern Von V . D . EGOBOV 1 ) Es wird der Einfluß der Oberflächenrekombination und einer parasitären Kapazität auf den Anfangsteil des Anklingens des Photosignals betrachtet. Die Ergebnisse ermöglichen Aussagen über die Verwendbarkeit eines Germanium-Photowiderstandes zur Messung der Intensität von Lichtimpulsen. PaccMaTpHBacrcH

BJIHHHHC

n0BepxH0CTH0i!

peKOMÖwiaijHH

H

napa3HTHOH

EMKOCTH Ha Ha^ajibHbiö yqacTOK carHajia HECTAUHOHAPHOII «JOTOIIPOBOUHMOCTH. CooömaeTCH o npHMeHeHHH repMamieBoro oToconpoTHBJieHHH

HJJH

H3MepeHHH

H H T e H C H B H O C T e Ö H M n y j l b C H b l X CBCTOBblX n O T O K O B .

1. Einführung und Fragestellung Bekanntlich (siehe z. B. [1]) wächst die Konzentration 8 p der zusätzlichen Ladungsträger in Abwesenheit eines Hafteffektes bei homogener Anregung eines homogenen Halbleiters durch einen Rechtecklichtimpuls linear mit der Zeit t, solange t klein ist im Vergleich zur Lebensdauer r (solange die Rekombinationsgeschwindigkeit 8plr klein ist im Vergleich zur Paarerzeugungsrate). I m Falle einer merklichen Absorption des Lichtes erfolgt die Anregung des Halbleiters ungleichmäßig in der Tiefe, und infolge der Diffusionsbewegung der Ladungsträger bleibt die Proportionalität zwischen 8p und t im allgemeinen nicht erhalten, auch bei t r. Erhalten bleibt jedoch die lineare Abhängigkeit der Gesamtzahl der Elektron-Loch-Paare w SP = / 8p{x, o

t) äx ,

die in dem Halbleiter pro Einheit der beleuchteten Oberfläche erzeugt werden, von der Zeit (Fig. 1): SP = g0 • t ,

wobei g0 das Integral der Paarerzeugungsrate über die Probendicke ist. Wenn das Licht vollständig in der Probe absorbiert wird, so gilt 9o =

(! -

Ä) y ?Ph •

Hier ist j Ph die Dichte des auffallenden Photonenstromes, R der Reflexionskoeffizient, y die Quantenausbeute. Die zusätzliche elektrische Leitfähigkeit der Probe bei t r ist =

-

ü)yj

F h

t,

(1)

mit der Länge l und der Breite b der Probe, den Beweglichkeiten der Elektronen fj,n und Löcher und der Elementarladung e. Die Spannungsänderung 1

) Gast von der Lomonossov-Universität Moskau.

10 pliysica

146

V. D. EGOROV Fig. 1. Schaltung des Photowiderstandes

Licht

bei Belichtung ist proportional der zusätzlichen elektrischen Leitfähigkeit W

8C7=

1/r0 *6G(tj

solange

C.j

V

"

W


Die Bedingung t r ist hinreichend für das lineare Anklingen der Photoleitung im Falle der ausschließlichen Volumenrekombination. Das Ziel der vorliegenden Arbeit ist die Berücksichtigung des Einflusses der Oberflächenrekombination der Nichtgleichgewichtsladungsträger sowie einer Parallelkapazität auf die Zeitabhängigkeit des Photosignals im Anfangsstadium des Anklingens. Die Bedingung t r wird dabei zunächst als erfüllt betrachtet. r

2. Einfluß der Oberflächenrekombination Wir betrachten die Funktion g(t) = dSP/di, die proportional der Anklinggeschwindigkeit der Photoleitfähigkeit der Probe ist. In Abwesenheit einer Rekombination gilt g(t) = g0. Unter Berücksichtigung der Oberflächenrekombination ist g(t) = g

0

-

S08p(0,

t) -

Sw8p(w,

t) .

Dabei wird angenommen, daß die Rekombination lediglich an der belichteten Seite sowie der gegenüberliegenden Seite stattfindet. S0 und Sw sind die entsprechenden Oberflächenrekombinationsgeschwindigkeiten. Uns interessiert, bei welcher Dauer des Lichtimpulses t man die Größe g(t) mit dem gegebenen Genauigkeitsgrad konstant und gleich g0 annehmen kann (was dem linearen Anwachsen der Photoleitfähigkeit entsprechen würde). Die Oberflächenkonzentrationen 8p(0, t) und Sp(w, t) könnten aus der Kontinuitätsgleichung für Elektron-Loch-Paare erhalten werden: 68p — St

r,

82Sp

— D—~ dx2

= g0 « e " '

„

;

1

5,0

l >

0,

0

< x < w , '

mit der Anfangsbedingung 8p(x, 0) = 0 und den Randbedingungen z AOX ' - D

ex

x=

0 =s =

0

z

P

(o,t),

Sw 8p(iv,

t) .

Hier ist D der ambipolare Diffusionskoeffizient der Ladungsträger und a. der Absorptionskoeffizient des Lichtes. Die Lösung dieser Gleichung würde jedoch zu unübersichtlichen Ausdrücken führen. Deshalb seien einige verhältnismäßig

Zum linearen Anklingen der Photoleitfähigkeit in Halbleitern

147

einfache Abhängigkeiten aufgezeigt, die die Funktion 8g(t) = g0-

g(t) = £ 0 $î>(0, 0 +

Sp(w, t)

von oben und unten beschränken. Wenn die injizierten Ladungsträger sich gleichmäßig über die Probe verteilten und keine Oberflächenrekombination stattfände, wäre die Konzentration der Nichtgleichgewichtspaare , t) < ^

.

Man kann zeigen, daß die Konzentration an der beleuchteten Oberfläche « m

W

1u

\

D

ist. Hier ist der zweite Summand die Lösung der Kontinuitätsgleichung bei x — 0 für einen unendlichen Halbraum bei Oberflächenanregung des Halbleiters (a oo) und in Abwesenheit einer Rekombination an der Oberfläche. Wir erhalten also ^


So + ^

w

) =

28,

S2

]/n

are the fluxes of neutrons of wavelengths A and A/2, respectively, Fhki and _F2a, 2k, 21 are the structure factors for the two reflections (hkl) and

(2 h.2k,2 Dhk, =

I), ' e~ 2Mhki.

The suffix A and A/2 refer to the two radiations under consideration. By solving (CI)

t>i __ "

Phkl 2

COS 6X

3

r t i i cos e,!2

8

The measurements, done for both the (111) and (200) reflections, gave equal to 6.94 and 7.07, respectively. The integrated intensity powers of the reflections (111) and (222) of the Si single crystal are expressed by 1

Pni =

/

Y \

l d

C0S

1 /

+

TZ d 2 ~2

\ Sin

N

7t d 2

0 >

\

X2 sin 2 0 n l A2

P222 = Y U d cos 0222 + — sin 02221 N 0 A s i n 2 ^ + \ (

l d

C0S

+ ^

Sin

N

F l n

°

m

'

^222 A22 +

T sin

2

^

'

where I and d are the length and the diameter of the Si crystal, N is the number of the elementary cells per unit volume, 0 n l and 0222 are the Bragg angles of the (111) and (222) reflections for the wavelength A and the other symbols have the meaning as specified above. We used the formulae of the intensities diffracted by a perfect non absorbing crystal and we took into account the geometrical factors [26]. The contribution of the intensities by the (444) reflection of the radiation of A/2 wavelength is

166

B . COLELLA a n d A. MERLINI : (222) " F o r b i d d e n " B e f l e c t i o n in Ge a n d Si

included in P2i2. F222 ^ldc __222 __ 9 Fni ,,

Since the ratio / J 2 2 2 / P n l was equal to 3.3 x 10~ 2 , it follows

o s e

7i d2 m + ~2~ sin e m

„

cog

, "d« — sm 0222

'222p P222 D ,t> fhnlllijv I Z5 111 ,1 tf\. TP M cos 0 m P m 4

^ 0 .

If the experimental error is taken to be equal to + 1 0 % , which is in effect greater than the effective error., the value of F222!Fm is /

M

=

250 '

This upper limit is slightly smaller if one uses for the (222) reflection of the A radiation the formula of the mosaic crystal instead of the formula of the perfect crystal. References [1] W. H. BRAGG, Proc. Phys. Soc. 33, 304 (1921). [2] M. BENNINGER, Z. Phys. 106, 141 (1937). [3] P . P . EWALD a n d H . HÖNL, A n n . P h y s . 2 5 , 281 (1936).

[4] M. BENNINGER, A c t a cryst. 8, 606 (1955). [5] B . BRILL, Acta cryst. 13, 275 (1960).

[6] Described by T. H. CRAWFORD, JR. and M. C. WITTELS, Proc. Internat. Conf. on Peaceful Uses of Atomic Energy, United Nations, 7, 659 (1956). [7] M. C. WITTELS, Solid State Division Annual Progress Beport, OBNL-3017, 1960 (p. 7 7 ) . [8] M. C. WITTELS, X-Bay] Diffraction Studies of Beactor Irradiated Germanium and Silicon. Abstracts of the Communications to the Seventh International Congress of the International Union of Crystallography, Moscow, U.S.S.B., July 1966; to be published in Acta cryst.. [9] H. C. SCHWEINLER, private communication. [10] H . COLE, P . W . CHAMBERS, a n d H . M. DUNN, A c t a c r y s t . 1 5 , 138 (1962). [11] P . B . HIRSCH a n d G. N . BAMACHANDRAN, A c t a c r y s t . 3 , 187 (1950). [ 1 2 ] B . COLELLA a n d A . MERLINI, p h y s . s t a t . sol. 14, 8 1 (1966).

[13] M. BENNINGER, Z. Krist. 113, 99 (1960). [14] A. A. MARADUDIN, E . W . MONTROLL, a n d G. H . WEISS, Theory of Lattice D y n a m i c s in

the Harmonic Approximation, Solid State Phys., Suppl. 3, Academic Press 1963 (p. 249). [15] J . J . DE MARCO and B. J. WEISS, Phys. Bev. 137, A1869 (1965). [16] S. GÖTTLICHER, B . KUPHAL, G. NAGORSEN, a n d E . WÖLFEL, Z . p h y s . C h e m . 2 1 , 133

(1959). [17] H . J . GRENVILLE a n d K . LONSDALE, N a t u r e 1 8 1 , 7 5 8 (1958). [18] S. T . KONOBEEVSKII a n d F . P . BUTRA, J . n u c l e a r E n e r g y 1 1 , 4 8 ( 1 9 5 9 / 6 0 ) . [19] H . C. BOLTON a n d J . W . HEATON, P r o c . P h y s . Soc. 78, 2 3 9 (1961).

[20] B . J . WEISS, P h y s . L e t t e r s 12, 293 (1964).

[21] G. LÉMAN, Thèse, Université de Paris, Masson & C. Editors 1963. [22] J . P. LELIEUR, Thèse, Faculté des Sciences d'Orsay, France 1965. [23] L. KLEINMANN a n d J . C. PHILLIPS, P h y s . B e v . 125, 819 (1962). [24] K . H . BENNEMANN, P h y s . B e v . 1 3 3 , A 1 0 4 5 1964.

[25] A. MERLINI and E. VAN DER VOORT, in Study of Lattice and Electronic Defects in Crystalline Solids, Euratom Beport 1643.e 1964. [26] W. H. ZACHABIASEN, Theory of X-Bay Diffraction in Crystals, John Wiley & Sons, Inc., N e w York 1945 (formulae (3.152) a n d (3.168)).

(Received June S, 1966)

J . V A N L A N D U Y T et al.: Determination of the Nature of Stacking Faults

167

phys. stat. sol. 18, 167 (1966) S.C.K.

- C.E.N.,

Mol

On the Determination of the Nature of Stacking Faults in F.C.C. Metals from the Bright Field Image By J . V A N L A N D U Y T 1 ) , R . GEVEES, a n d S. AMELINCKX1)

A method is described whereby the nature of stacking faults in f.c.c. structures can be determined from the bright field image and the fine structure of the diffraction spots. The method is applied to stainless steel where the faults are found to be intrinsic. The procedure can easily be extended to similar problems where the slope of a planar interface has to be determined. Eine Methode wird beschrieben, durch die die Art der Stapelfehler in k.f.z.-Strukturen aus dem Hellfeldbild und der Feinstruktur der Beugungspunkte bestimmt werden kann. Die Methode wird für rostfreien Stahl benutzt, wobei die Fehler als gittereigen gefunden werden. Das Verfahren kann leicht auf ähnliche Probleme ausgedehnt werden, bei denen die Neigung einer ebenen Zwischenfläche bestimmt werden muß.

1. Introduction Since electron microscopy has become the most extensively used method in t h e s t u d y of lattice defects, much effort has been made to derive as much information as possible from the electron micrographs. A striking example is t h e development of methods for determining t h e character of stacking f a u l t s (intrinsic or extrinsic) in f.c.c. structures. Howie [1] suggested the principle of a m e t h o d for determining this character b y contrast experiments. Art et al. [2| developed a practical procedure whereby information is used f r o m the bright field image, the dark field image, a n d f r o m t h e diffraction p a t t e r n . I n a later paper the same authors showed t h a t the dark field image a n d the diffraction p a t t e r n provided in fact sufficient information t o determine the n a t u r e [3j. I n the present paper we describe a method whereby only the bright field image and the associated diffraction p a t t e r n are required for t h e determination. The additional information is obtained from the fine structure of the diffraction spots resulting f r o m the presence of faults. A detailed s t u d y of this fine struct u r e has been submitted for publication [4j.

2. Principle of the Method and Application We shall first summarize the method developed in [3|. The method reduces to the following procedure: Orient the diffraction p a t t e r n correctly with respect to the image. Determine the indices of the diffracting planes a n d deduce t h e J

) Also Laboratorium voor Algemene Natuurkunde en Vaste Stoffen Fysika Rijksuniversitair Centrum — Antwerp (Belgium), Middelheimlaan 1.

J . V A N L A N D U Y T et al.: Determination of the Nature of Stacking Faults

167

phys. stat. sol. 18, 167 (1966) S.C.K.

- C.E.N.,

Mol

On the Determination of the Nature of Stacking Faults in F.C.C. Metals from the Bright Field Image By J . V A N L A N D U Y T 1 ) , R . GEVEES, a n d S. AMELINCKX1)

A method is described whereby the nature of stacking faults in f.c.c. structures can be determined from the bright field image and the fine structure of the diffraction spots. The method is applied to stainless steel where the faults are found to be intrinsic. The procedure can easily be extended to similar problems where the slope of a planar interface has to be determined. Eine Methode wird beschrieben, durch die die Art der Stapelfehler in k.f.z.-Strukturen aus dem Hellfeldbild und der Feinstruktur der Beugungspunkte bestimmt werden kann. Die Methode wird für rostfreien Stahl benutzt, wobei die Fehler als gittereigen gefunden werden. Das Verfahren kann leicht auf ähnliche Probleme ausgedehnt werden, bei denen die Neigung einer ebenen Zwischenfläche bestimmt werden muß.

1. Introduction Since electron microscopy has become the most extensively used method in t h e s t u d y of lattice defects, much effort has been made to derive as much information as possible from the electron micrographs. A striking example is t h e development of methods for determining t h e character of stacking f a u l t s (intrinsic or extrinsic) in f.c.c. structures. Howie [1] suggested the principle of a m e t h o d for determining this character b y contrast experiments. Art et al. [2| developed a practical procedure whereby information is used f r o m the bright field image, the dark field image, a n d f r o m t h e diffraction p a t t e r n . I n a later paper the same authors showed t h a t the dark field image a n d the diffraction p a t t e r n provided in fact sufficient information t o determine the n a t u r e [3j. I n the present paper we describe a method whereby only the bright field image and the associated diffraction p a t t e r n are required for t h e determination. The additional information is obtained from the fine structure of the diffraction spots resulting f r o m the presence of faults. A detailed s t u d y of this fine struct u r e has been submitted for publication [4j.

2. Principle of the Method and Application We shall first summarize the method developed in [3|. The method reduces to the following procedure: Orient the diffraction p a t t e r n correctly with respect to the image. Determine the indices of the diffracting planes a n d deduce t h e J

) Also Laboratorium voor Algemene Natuurkunde en Vaste Stoffen Fysika Rijksuniversitair Centrum — Antwerp (Belgium), Middelheimlaan 1.

168

J. Van Landtjyt, R. Geyers, and S. Ameltnckx

class (A or B) f r o m T a b l e 1. Table 1 Diffraction vector

Class

[200] [222] [440] [400] [111] [220]

A A A B B B

The n a t u r e of t h e f a u l t s can t h e n be deduced u n a m b i g u o u s l y f r o m t h e last column of T a b l e 2, using t h e aspect of t h e d a r k field image. I n T a b l e 2 t h e diff r a c t i o n vector is a s s u m e d t o p o i n t t o t h e r i g h t of t h e intersection line of t h e foil plane a n d t h e f a u l t plane. This a s s u m p t i o n does n o t i n t r o d u c e a n y loss in generality. T h e rule can be f o r m u l a t e d as follows: If t h e d i f f r a c t i o n vector points t o w a r d s a b r i g h t fringe in t h e d a r k field i m a g e a n d if t h e reflection belongs t h e class A t h e f a u l t is intrinsic. Changing either t h e n a t u r e of t h e fringe or of t h e class of reflection leads t o t h e opposite conclusion. Table 2 Bright field

Dark field

B

Extrinsic ^

D

D

B

B

B

D

D

B

B

B

D

D

B

D

D

B

Bright field

Dark field A

Intrinsic

B

B

B

D

D

D i i 1

B 1 i |

B i 1 1

D

D

D

B

B

D i 1

B j

B

D

:

1

fringe corresponding to top surface, fringe corresponding to bottom surface. B and D mean bright and dark, respectively, i.e. the nature of the extreme fringes.

169

Determination of the Nature of Stacking Faults in F.C.C. Metals

An inspection of this table reveals immediately that after determining the sense of the slope of the fault plane by an independent method, the nature of the outer fringes in the bright field would allow to determine the character of the stacking fault. Such a method can be based on the fine structure of the diffraction pattern. As suggested first by Hirsch et al. [5] and developed in more detail by the present authors [4], even a single fault produces satellite spots associated with the main spots. The lines connecting the satellites with the main spots, have directions perpendicular to the intersection line of the stacking fault plane with the foil plane. Their distance to the main spot depends upon the inclination of the plane and on the orientation of the foil. According to the two-beam dynamical theory and in the symmetrical Laue case the distance in reciprocal space between the main spot and its satellite is given by [4] u = + fftg

0

'

Gz

C

o-

down

r.

°

\

s d0. Then by using equation (1) and integrating over all primary energies we get for the total cross section for cluster production Tm

W(E,da)

(2)

E2 I t is convenient to define a reduced cross section , _

(M,

+

M2f

4 Z\ Zi M\

(3)

'

Equation (2) in terms of the reduced cross section becomes rr'tT

f

W(E,d0

dE .

(4)

0 We see that a's(d0) particle.

does not depend on the charge and mass of the irradiating rV-

W

vS» ° " o F i g . J. T h e r e d u c e d c r o s s s e c t i o n f o r b l a c k s p o t f o r m a t i o n in gold as a f u n c t i o n o f t h e m a x i m u m p r i m a r y k n o c k - o n energy

0.01

Protons Deuterons Alpha's Fission fragments 0.1

1.0

10 TJMeVi-

100

176

K . L . MERKLE Fig. 2. The probability for spot f o r m a t i o n in gold W{E) as a f u n c t i o n of t h e p r i m a r y recoil energy

E(keV)

•0.7MeV

protons

•2.5 MeV protons

4.0 MeV cc Fost

particles

neutrons

Fig. 3. Size spectra of defect clusters in Gold for charged particle a n d neutron irradiations. The curves for t h e 0.7 MeVp ( T m - 14 keV), the 2.5 MeV p ( jTm = 50 keV), a n d the 4 MeV a-irradiation ( T m = 312 keV) show the increase in spot size with increasing 7 ' m

120 KO 160 Cluster diameter (Âl~

3.3 Black

spots

in gel i

From the experiments we find the cross section a s for the formation of spots with d Si 20 A a n d according to equation (3) we f o r m a's . Fig. 1 shows the results of irradiations of gold with protons, deuterons, a-particles, a n d fission f r a g m e n t s at a n u m b e r of different energies. We have plotted the reduced cross section for black spot production in gold as a function of Tm . As indicated b y equation (4) the d a t a f r o m irradiations with different particles and energies fall onto one curve. Fig. 2 shows W(E) as determined from a best fit to the d a t a [3]. For t h e time being a n d for comparison with the d a t a on copper we will a p p r o x i m a t e W(E) b y a step function. This means t h a t we assume a sharp threshold energy ET {d0 ) with W{E) = 0 for E < E T a n d W(E) = 1 for E ^ E T . Then equation (4) becomes 4;iaS Ei j 1 1 \ The threshold energy for gold a t d0 & 20 A (see Fig. 3) is ET = 2.7 X 104 eV. Because of the increase in the p r i m a r y recoil cross sections toward low energies, this sharp threshold energy is somewhat lower t h a n the energy where W(E) = 0 . 5 . The latter energy is 3.4 x 104 eV.

Radiation-Induced Point Defect Clusters in Cu and Au (I)

177

Fig. 4. Recoil spectra in gold for the charged particle and neutron irradiations of Fig. 3. The scale on the ordinate is different for all 4 spectra

The maximum in Fig. 1 clearly shows that the black spots are directly produced I in energetic displacement cascades. Processes I involving the long-range migration of de1 fects can be excluded because the observed number of spots would be a monotonic funcI§ tion of the total number of point defects if the size distribution stayed the same and 3 the clusters were due to the migration and I clustering of point defects. The maximum observed in this case would be close to the threshold energy for displacement. The size distribution of the spots (see Fig. 3) as a Ì function of the recoil spectra, shows that the defect clusters are smallest in bombardments with low Tm. The irradiations with low Tm 20 50 100 200 Recoil energy (keV) are just the ones with the highest concentration of single defects, but smallest cascades. The size of the clusters therefore clearly reflects the size of the cascades. Therefore the other alternative that submicroscopic nuclei are produced in cascades and grow later on by the accumulation of point defects can also be excluded. In this case the defects would be larger in the 0.7 MeV proton irradiation than in the 4 MeV a-irradiation because the number of Frenkel pairs produced in the former is about a factor of 10 higher than in the latter. Fig. 5, 6, and 7 clearly demonstrate that the reverse is true: The average spot size increases

Fig. 5. Gold film of (001) orientation irradiated with 0.7 MeV protons to a dose of 1.41 xlO 1 / protons/cm ! 12

physiea

TOO À

178

K . L . MERKLE Fig. 6. Gold film of (001) orientation irradiated with 2.5 MeV protons to a dose of 1.41 x 101® protons/cm 2

c w

P

k

•

€ m

m Fig. 7. Single crystalline gold film of 4 X 10~2 MeV to any ET < 20 keV if we adjust the probability for the formation of a spot accordingly. Nevertheless it is significant to note t h a t if the spots in copper are produced in cascades the data indicate a threshold energy ET iS 20 keV and connected with such a cascade a probability p sg 0.2 for the formation of a cluster t h a t is large enough to be visible by transmission electron microscopy. We find E T l p ** 100 keV. 4. Discussion 4.1 Differences

between

black spots

in copper

and

gold

The reason for the differences in cluster size in copper and gold for cascades of similar energy may be found in the difference of the size of a cascade in copper and gold. The range of the primary atom gives a lower limit for t h e diameter of the region t h a t is being affected by the collision cascade. If one examines the ranges of a 50 keV gold and copper atom in their respective lattices, one finds t h a t the range of the gold atom is about 60 A while the range of the copper atom is about 250 A [8, 9]. Therefore the defect density in a gold cascade is expected to be considerably higher t h a n in copper, v. J a n [10] has treated the problem of the defect distribution in a displacement cascade theoretically on the basis of a statistical model. He finds t h a t subcascades appear in copper already a t 2 keV and in gold a t 10 keV. We interpret the appearance of spots in copper as due to interference of subcascades: the vacancy cluster formed in a subcascade is too small to be observed. However, when two or more subcascades overlap or the vacancy clusters are close enough to collapse into one, t h e resulting defect cluster becomes large enough to be resolved by the transmission electron microscope technique (see Fig. 13). If the subcascades are sufficiently separated this will occur with a probability smaller than 1. I n very energetic cascades secondary cascades might also overlap and thereby provide the possibility for the formation of rather large clusters ( « 50 A) t h a t are sometimes observed (see Fig. 13). This will also occur when a cascade is initiated at the site of an already existing one. Then as seen in Fig. 14 the observed average cluster size increases just due to this overlapping effect. I n gold, as can be seen from Fig. 3, the cluster size covers a wide range up to about 150 A. The size of the cluster increases here with increasing primary knock-on energy. Evidence for several clusters being formed in one cascade has been found at knock-on energies above 100 keV. However, even at higher energies (up to » 200 keV) one generally finds one large cluster associated with

183

Radiation-Induced Point Defect Clusters in Cu and Au (I)

several small ones. This indicates considerable overlapping of subcascades in gold up to rather high energies. In view of this v. J a n ' s estimate [10] of lOkeV for the formation of subcascades in gold might be somewhat low. However, there is the possibility that neighboring clusters interact to form one cluster upon warming to room temperature. Unfortunately the low temperature experiment of Howe and McGurn [11] did not introduce energetic enough cascades to decide this point. Further experiments of this type would therefore be of interest. >'ig. 13. A schematic representation of the defect arrangement in collision cascades of copper of different primary energy Ep. The vacancy clusters produced in the subcascades are thought to be too small to be visible in the electron microscope. Two conditions have to be fulfilled for the production of visible clusters: 1. The total number of vacancies produced in a cascade has to be large enough to produce a visible spot when brought together. 2. Vacancy clusters from subcascades have to be close enough to collapse into a single cluster. Relatively large clusters can also be formed by overlapping of secondary cascades

Ep-10+eV

Ep

m

10 eV

Vacancy dusters K y f formed in subcascades

Secondary cascades

I'ig. 14. Copper film irradiated with 4 x 101! cm-' Xe ions of 200 keV energy (290000 x)

184

K . L . MERKLE

4.2 Comparison

with theoretical

cascade

models

Brinkman [12] first pointed out t h a t energetic knock-ons can produce a rather concentrated region of severe damage in the crystalline lattice. He also noted t h a t dislocation loops are likely to be formed in a displacement spike. Seeger [13], in taking into account the effect of dynamic crowdions, predicted that a so-called depleted zone should be produced in the core of a displacement cascade, v. J a n ' s [10] statistical model and the computer simulations of cascades of Beeler and Besco [14] also indicate vacancy rich regions. This is also indicated by the experiment as we will see in the next section. To get an estimate for the number of vacancies remaining at room temperature in the depleted zone we take the total number of Frenkel defects produced in the cascade and multiply this by the fraction of the electrical resistivity increase remaining after a room temperature anneal [15]. We will now estimate in this way the number of vacancies contained in a cascade of energy ET. This should then correspond to the number of vacancies present in a spot of about 20 A in diameter. Using a Kinchin and Pease model [16] and displacement threshold energies of 25 and 35 eV, we get JVCu =

E 2 Ed

E

x 0.07 = 14 and iVAu = — x 0.1 = 2 Ea

= 49. The value for gold is in good agreement with a cluster size of 20 A provided the defects are arranged in a planar configuration. That the small defects are dislocation loops is also indicated by their diffraction contrast behavior. In the case of copper we get d = 10 A for a circular loop on a (111) plane or 12 A for a triangular loop. The values for copper are somewhat low. The reason for this might be a threshold energy for the production of spots ET > 10 4 eV or more than 7% of the defects might be left in a cascade at room temperature. In any case this comparison seems to indicate t h a t at a cascade energy of 104 eV in copper practically all of the subcascades have to be close enough to contribute to the visible spot. Some time ago Brinkman [12] already pointed out t h a t when the energy of a primary knock-on falls below a certain value Es it collides with practically every atom in its path., producing a displacement spike. The collision cascade would in this picture not be split up into subcascades for E < Es. Brinkman's result of Es = 8 x 104 eV for gold is in agreement with our preliminary observations t h a t bunches of spots do not appear at E < 10® eV. The value of 2.3 X 104 eV for copper is clearly too high; however, Brinkman's later values [17] of 4 x 103 and 103 eV are in agreement with the observations, v. J a n [10] estimates t h a t several vacancy regions are formed in copper above 2 keV and in gold above 10 keV. v. J a n ' s model generally agrees with our observations except that the energy for subcascade formation in gold is probably greater than 10 keV, possibly even higher than Brinkman's value of 80 keV. The theories do not give any information as to the interference between subcascades. For example it would be interesting to know the probability for the formation of larger vacancy clusters due to the collapse of closely spaced subcascade clusters. 4.3 The nature

of

spots

Several of the models for displacement cascades indicate t h a t a vacancy cluster should be formed at the site of a cascade. I t is therefore of importance to determine the nature of the defect clusters that are observed by transmission

Radiation-Induced Point Defect Clusters in Cu and Au (I)

185

electron microscopy. According to Ashby and Brown [18] one can determine the sign of the strain around a defect from dark-field observations. Defects that are within 1/4 of an extinction distance (f) of either surface of the foil show anomalous dark-white contrast and the orientation of this contrast with respect to the reciprocal lattice vector determines whether a cluster is of vacancy or interstitial type. Application of this method to black spot defects in copper [19, 20] and gold [5] indicated that most of the defects are of vacancy type. However, some defects of opposite contrast behavior were also observed [5, 19]. In gold this minority of reversed images was always present, regardless of the imaging conditions, whenever Tm was large compared to the threshold energy for the formation of spots. However, in irradiations where we observe only very small clusters we have found imaging conditions where all of the spots show vacancy character. Fig. 15 shows an example of this. We presume that all of the clusters with dark-white contrast are within 1/4 f of the surface in this case. The question arises now whether the reversed images are truly anomalous contrast effects from defects near the surface and therefore indicative of interstitial clusters or whether they are vacancy clusters at certain distances from the surface where a reversal of the image occurs [18, 21]. Recently Essmann and Wilkens [19] made a detailed study of the diffraction behavior of the spots in copper. They found that the images could be attributed to dislocation loops on (111) planes with a Burgers vector [111]- Dislocation loops on (111) planes had also been found in gold [8], Some of these loops in gold were identified as being triangular in shape with