Physica status solidi: Volume 25, Number 1 January 1 [Reprint 2021 ed.] 9783112496787, 9783112496770

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plrysica status solidi

V 0 L I M E 2:> - A t MIt I. il 1 - 1 9 6 i{

Classification Scherno 1. Structure of Solids 1.1 Alloys. Metallurgy 1.2 Solid-State Phase Transformations 1.3 Surfaces 1.4 Films 2. Non-Crystalline State 3. Crystallography 3.1 Crystal Growth 3.2 Interatomic Forces 4. Microstructure of Solids 5. Perfectly Periodic Structures 6. Lattice Mechanics. Phonons 6.1 Mossbauer Investigations 7. Acoustic Properties of Solids 8. Thermal Properties of Solids 9. Diffusion in Solids 10. Defect Properties of Solids (Irradiation Defects see 11) 10.1 Defect Properties of Metals 10.2 Photochemical Reactions. Colour Centres 11. Irradiation Effects in Solids 12. Mechanical Properties of Solids (Plastic Deformations see 10)see 10.1) 12.1 Mechanical Properties of Metals (Plastic Deformations 13. Electron States in Solids 13.1 Band Structure. Fermi Surfaces 13.2 Excitons 13.3 Surface States 13.4 Impurity and Defect States 14. Electrical Properties of Solids. Transport Phenomena 14.1 Metals. Conductors 14.2 Superconductivity. Superconducting Materials and Devices 14.3 Semiconductors 14.3.1 Semiconducting Films 14.3.2 Semiconducting Devices. Junctions (Contact Problems see 14.4.1) 14.4 Dielectrics 14.4.1 High Field Phenomena, Space Charge Effects, Inhomogeneities, Injected Carriers (Electroluminescence see 20.3; Junctions see 14.3.2) 14.4.2 Ferroelectric Materials and Phenomena 15. Thermoelectric and Thermomagnetic Properties of Solids 16. Photoconductivity. Photovoltaic Effects 17. Emission of Electrons and Ions from Solids 18. Magnetic Properties of Solids 18.1 Paramagnetic Properties 18.2 Ferromagnetic Properties 18.3 Ferrimagnetic Properties. Ferrites 18.4 Antiferromagnetic Properties (Continued on cover three)

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, Y. L. B O N C H - B R U E V I C H , Moskva, 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 , Warszawa, A. S E E G E R , Stuttgart, F. S E I T Z , Urbana, O. 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.DIETZE,Saarbrücken, J . D . E S H E L B Y , Cambridge,P.P. F E 0 F I L O V , Leningrad, J. H O P F I E L D , Princeton, 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, R. K U B O , Tokyo, M. M A T Y À S , 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. V. 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 25 • Number 1 • Pages 1 to 484, K1 to K66, and Al to A6 January 1, 1968

AKADEMIE-VERLAG•BERLIN

Subscriptions and orders for single copies should b e addressed t o A K A D E M I E - V E R L A G G m b H , 108 Berlin, Leipziger Straße 3 - 4 or t o B u c h h a n d l u n g K U N S T U N D W I S S E N , Erich Bieber, 7 S t u t t g a r t 1, Wilhelmstr. 4 - 6 or to Deutsche B u c h - E x p o r t u n d - I m p o r t G m b H , 701 Leipzig, Postschließfach 160

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Information I n response to the increasing number of submitted papers, in particular t h a t of Short Notes, the following changes will be introduced: 1. From Vol. 25, No. 1 (January 1, 1968) the offset part will no longer include the pre-printed abstracts of papers to be published in physica status solidi. (To give a survey of the papers to be published in the following issue we shall continue t o present the respective titles on the A-pages in the offset part.) 2. From Vol. 26, No. 1 (March 1, 1968) physica status solidi will no longer contain the titles and abstracts of papers which have appeared in ,,H3HKa TBepnoro Tejia" (Fizika Tverdogo Tela). I n Vol. 25, No. 2 the last of these titles and abstracts will be presented, thus completing the 1967 volume of Fizika Tverdogo Tela. With these changes we comply with the numerous requests for publication, thereby increasing the amount of information contained in this journal. We think t h a t , in view of the existence of international abstract journals with rather short publication times, the omission of the announcements in physica status solidi will not be a serious loss. We also point out t h a t from Vol. 25 E r r a t a published in physica s t a t u s solidi, will appear in the list of contents.

phys. stat. sol. 25 (1968)

Author Index G . B . ABDULLAEV M . K H . ALIEVA J . G . ALLPRESS E . E . ANDERSON G . ARLT H . L . ARORA

75 75 541 K31 323 223

P . BAILLY G . BALDINI T . 0 . BALDWIN R . W . BALLUFFI V . K . BASHENOV R . G . BESSENT M . J . BLACKBURN D . BOIS H . P . BONZEL J . G . BROERMAN H.A.BROWN

317 557 71 163 K51 K107 K1 691 . 209 757 KLL

G . CARTER W . J . CASPERS P . CHABSLEY T . CLAESON W . COCHRAN W . A . COGHLAN M . V . COLEMAN

:

449 721 531 K95 273 679 241

F. L. M. H. T.

DAVOINE DELAEY V . DMITRUK DUBLER R . DUNCAN

691 697 K75 109 K105

F. U. B. J. B.

M . EL-AKKAD ESSMANN L . EVANS H . EVANS L. EYRE

S. P. A. R. F.

P . FEDOTOV FELTHAM G . FITZGERALD FREUD FROHLICH

K51 K71, K107 263 K127 303

R. H. R. G. N. P. N. J.

GAUTHIER P . GESERICH GHOSH GIULIANI A . GJOSTEIN GORLICH A . GORYUNOVA GRABMAIER

691 741 K71 437 209 93, K 1 5 513 K7

K115 373, 395 417 K39 K39

F . C. GREISEN F . GUITR G . I . GUSEVA O. H. K. R. M. A. D.

HÄRLIN W . DEN HARTOG HERRMANN HERRMANN HÖHNE HOLZ M . HUGHES

G . JACOBS E . JÄGER V.JANKÜ A. JEAN C. J E C H G . JUNGK A . A . KAMINSKII H . KABRAS Y . KAZUMATA R . KELLY M . KESTIGIAN L. K . KEYS S.KIM K . KLEINSTÜCK N . P . KOLMAKOVA D . KOSKIMAKI G . KOTITZ R . KOWALCZYK G . KRENZKE P . KRISPIN D . KUHLMANN-WILSDORF U . KÜMMEL M . KURIYAMA

753 189, 2 0 3 775 359 KILL 655 427, 655, 661 K55 567 449 621 K43 K103 557 641 K59 K75 93, K 1 5 563 641 119 K5 223 607 787 K67 K15 233 K131 147 K105 711 667

T . J . LAW C. L E A A . D . W . LEAVER E . M . LEDOVSKAYA T . LEFFERS H.LEMK E I . LICEA Y U . A . LOGATCHOV R . K . LOMNES M . LUUKKALA

139 613 531 63 337 K131 461 763 583 K99

P . MACKUS E . F . MAKAROV B . MAKIEJ A . Z. MAMEDOVA M . J . MAKCINKOWSKI

331 607 K127 75 K67

804 F . W . MARSH

Author Index K91

V . B . SHIPILO

H J . MATZKE

641

G . P . SHPENKOV

G . M . MCMANUS

667

K . N . SHRIVASTAVA

H . MBCHBTTI

K65

C. H . B . M E E

613

H . MENNIGER

K59

N . N . SIROTA

L . MERTEN

125

Z . SMETANA

E . A . METZBOWER

403

A . SMILGA

M . A . MINDLINA A. I . M I T S E K J . T . MOORE G . O . MÜLLER L . E . MURR

K83 787 K105 K131 629

V . V . NEMOSHKALENKO H . NEUHÄUSER

K83 593

H . W . NEWKIRK W . D. NIX

K79 679

S. P. V. B. V.

OBERLÄNDER R . OKAMOTO M . ORLOV N . OSHCHERIN V . OSIKO

L. M. P. V. V.

PELSERS H . PILKUHN PINARD A . POVITSKII A. PRESNOV

345 81 513 K123 K75 621 9 691 607 K51

P . QUADFLIEG

323

R . T . RAMSEY R . RAUCH E . REGUZZONI M.F.ROSE J . RÖSELER

103 K15 437 103 311

J . SAK J . A . M . SALTER

155 531

J . V . SANDERS A . S . SASTRI W . R . SAVAGE P . SCHNUPP A . H . SCHOLZ M . SCHULZ D . SCHUMACHER

541 K67 131 455 285 521 359

W . SCHÜZ A . SCHWEIGHOFER A . SEEGER G. P . SEIDEL P . SEIFERT

253 K91 359 175 303

K23, K27 513 K47

A . SIKORA

K127

S . SIKORSKI

345 K23, K27 K87 331

D . K . SMITH

K79

L . SNIADOWER

233

V . I . SOKOLOVA

513

J . SÓLYOM

473

G . SPINOLO

557

M . STASIW

K55

M . J . STOWELL A . SUKIENNICKI V . A . SYCHUGOV

139 K19 K L 19

CH. SYMANOWSKI

93

T . A . TEVOSYAN D . J . D . THOMAS

K75 241

G . THOMAS H . TRÄUBLE E . V . TSVETKOVA

81, 263 373,395 513

P . ULLMANN G . VERTOGEN J . VISCAKAS T . E . VOLIN G. VÖLKEL Y U . K . VORONKO V . VYSÌN

93 721, 729 331 163 K35 K L 19 K103

A. D. WADSLEY F . F . Y . WANG

541 119

S . WANG C . WATSON F . L . WEICHMAN H. WEVER E . WIESER F . WILLIAMS J . C. WILLIAMS L . WOJTCZAK

223 K7 583 109 607 493 KL K19

R. H. T. YEH P . A . YOUNG

551,

K65 417

A . ZAWADOWSKI A . ZIA G . ZOLLFRANK

473 273 711

S . ZUKOTYÄSKI P . S . ZYRYANOV

583 775

Contents Pago

Beview Article M. H .

PILKUHN

The Injection Laser

9

Original Papers B . M . LEDOVSKAYA

On the Adiabatic Potential of a Complex System T.

O . BALDWIN

The Temperature Dependence of the Borrmann E f f e c t in Copper .

63 .

71

G . B . ABDULLAEV, M . K H . ALIEVA, n d a A . Z . MAMEDOVA

Thermal and Infrared Quenching of Photoconductivity in n-InSe Single Crystals

75

P . R . OKAMOTO a n d G . T H O M A S

On the Four-Axis Hexagonal Reciprocal Lattice and its Use in the Indexing of Transmission Electron Diffraction Patterns

81

P . GÖRLICH, H . KARRAS, C H . SYMANOWSKI, a n d P . ULLMANN

The Colour Centre Absorption of X-Ray Coloured Alkaline E a r t h Fluoride Crystals M. F . ROSE a n d R . T .

Higher Order Elastic Constants in H.C.P. Crystals H . DÜBLER u n d H .

93

RAMSEY

103

WEVER

Thermo- und Elektrotransport in ß-Titan und ß-Zirkonium . . . .

109

F . F . Y . WANG a n d M . KESTIGIAN

L.

MERTEN

Magnetic Properties of Al-Substituted Yttrium Orthoferrite. . . .

119

Zur Richtungsdispersion der optischen Gitterschwingungen des tetragonalen BaTi0 3 bei Zimmertemperatur

125

W I L L I A M R . SAVAGE

Field Electron Microscope Study of Niobium Surfaces M . J . STOWELL a n d T . J .

P.

KRISPIN

J . SAK

LAW

The Growth and Defect Structure of Gold Films Formed on Molybdenite in Ultra-High Vacuum

139

Zur Photoimpedanz von ZnS-Phosphoren

147

Temperature Dependence of the Band Structure

155

T . E . VOLIN a n d R . W .

BALLUPFI

Annealing Kinetics of Voids and the Self-Diffusion Coefficient in Aluminum G.

P.

SEIDEL

131

163

Einfluß von Neutronenbestrahlung auf die plastische Verformung von Eisen-Einkristallen mit unteschiedlichem Kohlenstoffgehalt (I)

175

F. Guiu

Deformation of Molybdenum Single Crystals a t Slow Rates of Strain

189

F. Guiu

The Stress Dependence of the Dislocation Velocity in Molybdenum

203

H . P . BONZEL a n d N . A . GJOSTEIN

Surface Self-Diffusion Measurements on Copper

209

H . L . ARORA, S . K I M , a n d S . W A N G

Optical Absorption of the Impurity Ag and Models for the E-Band in an Alkali Halide

223

4

Contents Page

L . ¡SNIADOWER a n d R .

KOWALCZYK

M a g n e t o p l a s m a Reflection in Mercury Telluride

233

M . V . COLEMAN a n d D . J . D . THOMAS

W . SCHÜZ

T h e S t r u c t u r e of A m o r p h o u s Silicon N i t r i d e Films

241

Ü b e r v e r b o t e n e Reflexe bei E l e k t r o n e n b e u g u n g a n v a k u u m a u f g e d a m p f t e n Tellurschichten

253

A . G . FITZGERALD a n d G . THOMAS

E l e c t r o n Microscope Observations of Twinning a n d P h a s e T r a n s f o r m a t i o n s in I n d i u m Sulfide Crystals

263

W . COCHRAN a n d A . Z I A

A. H .

SCHOLZ

S t r u c t u r e a n d D y n a m i c s of P e r o v s k i t e - T y p e Crystals

273

C o m p u t e r Calculations for Vacancies in Alkali Halides w i t h NaCl Structure

285

F . FRÖHLICH a n d P .

SEIFERT

P o t e n t i a l Differences in NaCl Crystals due t o Plastic D e f o r m a t i o n b y I n d e n t a t i o n E x p e r i m e n t s between R T a n d 20 °K

303

J.

RÖSELER

A N e w Variational Ansatz in t h e P o l a r o n T h e o r y

311

F.

BAILLY

Energies of F o r m a t i o n of Metal Vacancies in I I - V I Semiconducting Tellurides (HgTe, CdTe, ZnTe)

317

G . ARLT a n d P . QUADFLIEG

Piezoelectricity in I I I - V Compounds w i t h a Phenomenological Analysis of t h e Piezoelectric E f f e c t

323

J . VISCAKAS, P . MACKUS, a n d A . SMILGA

T . LEFFERS

Space-Charge-Limited Currents a n d H i g h Electric Field E f f e c t in Vitreous Selenium Films

331

C o m p u t e r Simulation of t h e Plastic D e f o r m a t i o n in F a c e - C e n t r e d Cubic Polycrystals a n d t h e Rolling T e x t u r e Derived

337

S . OBERLÄNDER a n d S . SIKORSKI

On t h e M a t h e m a t i c a l Description of Electrical Processes in Semic o n d u c t o r s (I)

345

D . SCHUMACHER, A . SEEGER, a n d O . H Ä R L I N

Vacancies, Divacancies, a n d Self-Diffusion in P l a t i n u m H . TRÄUBLE u n d U .

ESSMANN

Fehler im Flußliniengitter v o n Supraleitern zweiter A r t H . TRÄTJBLE u n d U .

359 373

ESSMANN

Der d i r e k t e Nachweis v o n Flußlinienbewegungen in s t r o m d u r c h flossenen Supraleitern

395

E . A . METZBOWER

N o n c e n t r a l Force Model for H e x a g o n a l Close-Packed Crystal L a t tices (II)

403

B . L . EVANS a n d P . A . YOUNG

R.

HERRMANN

Delocalized E x c i t o n s in T h i n Anisotropic Crystals

417

U n t e r s u c h u n g der Fermifläche des Molybdän d u r c h Zyklotronresonanzmessungen

427

G . GIULIANI a n d E .

REGUZZONI

X - R a y P r o d u c t i o n a n d T h e r m a l Annealing of Frenkel P a i r s i n KCl

437

Contents

5 Page

D . M . H U G H E S a n d G . CARTER

T h e E f f e c t s of Oxygen Adsorption a n d Low E n e r g y I o n B o m b a r d m e n t on t h e Electrical P r o p e r t i e s of C a d m i u m Sulphide T h i n Films P.

SCHNUPP

I . LICEA

A

449

Model for Tunneling t h r o u g h a Non-Crystalline T h i n Dielectric

Film

455

E l e c t r o n T e m p e r a t u r e in Polar Semiconductors

461

J . SÖLYOM a n d A . ZAWADOWSKI

On t h e Specific H e a t of Dilute Magnetic Alloys

473

Short Notes J . C. WILLIAMS a n d M . J .

L. K . KEYS

BLACKBURN

T h e I d e n t i f i c a t i o n of a Non-Basal Slip Vector in T i t a n i u m a n d T i t a n i u m - A l u m i n u m Alloys

K1

E S R of T i 2 0 3

K5

J . GRABMAIER a n d C. WATSON

H . A.

BROWN

P l a s t i c D e f o r m a t i o n in Single Crystals of Mg-Al-Spinel: Slip P a r allel t o t h e {111} P l a n e

K7

T h e Critical a n d H i g h - T e m p e r a t u r e Behavior of a Heisenberg Ferromagnet

Kll

P . GÖRLICH, H . KARRAS, G . KOTITZ, a n d R .

RAUCH

Phonon-Assisted Colour Centre Fluorescence of Additively Coloured Alkali E a r t h Fluoride Crystals

K15

A . SUKIENNICKI a n d L . WOJTCZAK

Magnetic D o m a i n S t r u c t u r e in U n i a x i a l Thin F i l m s

K19

N . N . SIROTA a n d V . B . SHIPILO

H i g h - P r e s s u r e a n d T e m p e r a t u r e E f f e c t on t h e Electrical P r o p e r t i e s of p - T y p e Gallium Antimonide N . N . SIROTA a n d V . B .

K23

SHIPILO

High-Pressure E f f e c t on t h e Electrical Conductivity a n d H a l l Cons t a n t of I n d i u m Antimonide

K27

Concerning t h e Uniqueness of a Molecular Field Solution for Two Magnetic Sublattices

K31

Die Beeinflussung der E P R - S p e k t r e n von Cu 2 + -dotierten Seignettesalz-Einkristallen d u r c h ein äußeres elektrisches F e l d

K35

E . E . ANDERSON

G . VÖLKEL

J . H . EVANS a n d B . L .

E.

JÄGER

K. N.

EYRE

T h e E f f e c t of T u n g s t e n Content on t h e Residual R e s i s t i v i t y of Zone R e f i n e d Molybdenum

K39

J a h n - T e l l e r E f f e c t a t L a t t i c e of Sites of S y m m e t r y D 3 d

K43

SHRIVASTAVA

T e m p e r a t u r e Dependence of P a r a m a g n e t i c Resonance T r a n s i t i o n P r o b a b i l i t i e s : Mn 2 + in MgO

K47

V . K . BASHENOV, V . A . PRESNOV, a n d S. P . FEDOTOV

Some R e m a r k s on t h e Antishielding E f f e c t in I I I - V Compounds . .

K51

M . H Ö H N E a n d M . STASIW

Detection of Ag 2 + in Doped AgCl Crystals b y E S R

K55

6

Contents Page

G. JUNGK u n d H .

MENNIGER

Zum Einfluß der Energiebandstruktur auf Photoeffekte in Silizium K59 R . H . T . Y E H a n d H . MECHETTI

Superconductive Current in a Superconductor-Normal Metal Superconductor Junction K65

Pre-printed Titles of papers to be published in this or in the Soviet journal ,,®H3HKa TBepgoro T e j i a " (Fizika Tverdogo Tela) A1

Contents

7

Systematic List Subject classification: 1. 2

Corresponding papers begin on the following pages (pages given in italics refer to the principal subject classification): 263

1. 3

131

1.4

139, 241,253

2

241

3

81

3. 1

139, K39

3.2

103, 403

4

163,263

5

71, 81, 253, K43

6

71, 125, 273, 311, 403, K15

8

155,403, 473

9

109, 163, 209, 359

10

285, 303, 317, K47

10.1

109, 139,175, 189, 203, 337, 359, 373, 395, Kl, K7

10.2

93, 223, 437, K15, K55

11

175,437

12.1

103, 403

13

63, 461, K43

13.1

155, 233, 311, 427, 455, K59

13.2

417

13.3 13.4

449 9, 75, 147, 223, 331

14.1

K39

14.2

373,395,

14.3

323, 345, 461, K23, K27

14.3.1

K65

449,455

14.3. 2

9, 345

14.4.1

9, 147, 331

14.4.2

125, 273

16

75, 147, K59

17

131

18

233, 473, K31

18.2

119, Kll,

18.3

119, K43

K19

18.4

119

19

427, K5, K35, K47, K51, K55

8

20.1

Contents

233, 417

20.2

9

20.3

9, K15

21

103, 109, 131, 163, 189, 203, 373, 395, 427, K l , K39

21.1

71, 209, 337, 403

21.1. 1

175, K19

21.6

139, 359

22

75, 241, 263, 417

22.1. 2

K59

22.1.3

253, 331

22.2

461, K51

22.2.1

9,323

22.2. 2

9

22.2.3

9, 323, K23, K27

22.4

461

22.4.1

147,449

22.4.3

233, 317

22.5. 1

K55

22.5.2

223, 285, 303, 437

22.5.3

93, K15

22.6

K5, K7, K47

Contents of Volume 25 Continued on Page 487

Review

Artide

phys. stat. sol. 25, 9 (1968) Subject classification: 14.4.1 and 20.2; 13.4; 14.3.2; 20.3; 22.2.1; 22.2.2; 22.2.3 Institut fur Elektrophysik der Technischen Hochschvle Braunschweig

The Injection Laser By M. H . PlXKTJHN1) Contents 1.

Introduction

2. General

considerations

2.1 Radiative recombination in semiconductors 2.2 Laser conditions 3. Spontaneous

light emission

and p-n junction

properties

of GaAs

3.1 Photo- and cathodoluminescence near the bandgap 3.2 p-n junction properties and electroluminescence 4. Different

injection

5. Theoretical

geometry

Length dependence of threshold current density. Optical gain and losses Emission characteristics at different laser length Quantum efficiency Influence of laser width Optimum laser geometry

7. Influence of impurities

7.1 7.2 7.3 7.4 8.

structures

results

6. Influence of laser

6.1 6.2 6.3 6.4 6.5

laser

and junction

structure

Laser fabrication procedures Recombination mechanism at the junction Influence of impurities on quantum efficiency Threshold current density

Conclusion

References Present address: Physikalisches Institut der Universität Frankfurt a. M..

M. H. PlLKUHN

10

1. Introduction Although the phenomenon of light emission from semiconductors had been known for a large number of years and the possibility of stimulated emission had been repeatedly discussed, it was not before 1962 t h a t the first semiconductor lasers were actually built [1, 2, 3]. These first lasers were p - n junction or "injection" lasers. GaAs was the first material where laser action was achieved, and it was followed soon b y other direct gap I I I - V compounds. Stimulated emission from semiconductors has been obtained in later years also by optical [4] and electron beam excitation [5]. The injection laser, however, still receives its special attention, because it represents a simple and small device for converting electrical energy efficiently into coherent light energy. The discovery of the injection laser provided the impact for intensive investigations of radiative recombination and laser properties not only of I I I - V compounds, b u t also of other semiconductors. Radiative recombination in I I I - V compounds has been reviewed recently by Gershenzon [6, 7], and stimulated emission from semiconductors by Burns and N a t h a n [8], Unger [9], Stern [10], Dumke [11], H a k e n [12], and N a t h a n [13]. A detailed understanding of injection lasers and p - n junction luminescence is often difficult to achieve, because one is dealing with a situation which is more complicated t h a n in homogeneous solids: The radiative recombination takes place in or near a region where the semiconductor varies from n - t o p-type, which causes changes in the optical and electrical properties of the material. Furthermore, the emission process is intimately connected with the charge carrier transport mechanism across the p-n junction. This article is mainly concerned with those structural parameters which influence p - n junction luminescence and laser performance. The number of these parameters is large and comprises, for instance, doping levels, impurity gradients, depletion layer widths, influence of deep levels, and absorption coefficients. Beside this, the role of geometrical parameters like laser length and width will be studied. Different types of injection lasers and materials will be discussed, b u t the major emphasis will lie on Fabry-Perot lasers and on GaAs, on which most of the work has been done. Special attention is given to the question of a) internal laser losses, b) the optical gain, and c) the q u a n t u m efficiency. Practical performance aspects, like problems of high temperature operation or maximum coherent light power, are discussed. I t was tried to cover most of the more recent references u p to about March 1967. 2. General Considerations 2.1 Radiative

recombination

in

semiconductors

The energy which is released in a semiconductor, when minority carriers recombine with majority carriers, can be emitted as light (photon assisted recombination), it can be given as heat to the lattice (phonon assisted recombination), or as kinetic energy to another free carrier (Auger recombination). I n order to get light emission, one obviously first has to create minority carriers. This can be done by external short wavelength light (photoluminescence), electron beams (cathodoluminescence) or electrical fields (electroluminescence). This article is predominantly concerned with the latter method. The most efficient way to generate minority carriers electrically, is b y means of a p - n junction. Other methods, for instance, a) Schottky contact injection, b)

The Injection Laser

11

Fig. 1. Schematic classification of radiative recombinations in semiconductors. (1) Bandto-band. (2) Band to shallow or deep impurity level. (3) Donor-acceptor pair recombination

thin film tunneling, and c) impact ionization are also possible and have been employed or proposed for electroluminescent devices or injection lasers [14], but no improvements over the p-n junctions have been obtained. These methods gain importance in materials where no p-n junctions can be made [14]. We shall confine our discussion to p-n junction luminescence and p-n junction lasers. Fig. 1 shows the pertinent energy bands and levels of a semiconductor and the possible radiative recombinations. They can be classified as follows: 1. band-to-band recombination (or free excitons), 2. band to shallow or deep impurity level recombination (or bound excitons), and 3. donor-acceptor pair recombination. We shall first discuss the band-to-band transitions. The quantities to be considered are the rate of spontaneous recombination rapon(E) and the rate of stimulated recombination rsi;m(E), together with its inverse process rate, the rate of induced absorption rabs(E). The transition probabilities determining these rates can be reduced to essentially the squared matrix elements of the momentum operator [15], For band-to-band transitions they are l P l ' = (vv| | v | v c ) 2 ,

(1)

where y>c and yiv are the Bloch functions of the conduction and valence band, respectively. In semiconductors there is a continuous range of allowed states with wave vector k and energy E' in the conduction or valence band between which recombination giving a particular photon energy hv — E can take place. This is an essential difference to the conventional solid state laser with sharp energy levels. In order to determine transition rates, one has to know, besides |P|2, the density of states Dc, Dv and the occupation probabilities fc, fv of conduction and valence band. In general, they are all functions of k and E'. The spontaneous recombination rate is proportional to the product of the probability / c , that Dc conduction band states are occupied, and the probability (1 — / v ), that Dy valence band states are empty: r 8 P o n ~ |P| 2 -D C 2> V / C (1 - / v ) .

(2)

In order to obtain the net stimulated emission rate, one has to subtract the induced absorption rate (proportional to fv (1 — / c )) from the induced emission rate (proportional to fc (1 — / v )) which yields r.am~

\P\2DCDV

(/C-/V).

(3)

M. H. PlLKUHN

12

The complete expressions for r s p 0 n and r 8t j m are obtained after integration of (2) or (3) over all k- (or energy) values. The occupation probabilities / c and / v are temperature dependent. In semiconductors they can be described by Fermi-Dirac statistics, and quasi-Fermi levels Fa, Fp have to be used for electrons and holes, if we are dealing with the non-equilibrium case. Dc and Dv depend on the effective masses of electrons and holes, and in highly doped material also on the impurity bands which can give rise to "tailing effects". Significant differences in the transition probabilities arise from the band structure of the material. If the conduction band minimum and valence band maximum lie at the same k-value in ¿-space, k is essentially conserved for the band-to-band recombination, if we neglect the spread of ¿-values around the band extrema. Such transitions are called "direct", and the corresponding materials "direct gap materials" (examples are GaAs, InAs, I n P etc., see also Fig. 2 b). If conduction band minimum and valence band maximum lie a t different places in ¿-space ("indirect gap materials"), large changes in k can occur during band-to-band recombination (examples are Ge, Si, or GaP). Since the emitted photon has a negligibly small momentum, momentum conservation can only be obtained by phonon emission (or absorption). This phonon co-operation is apparent, for instance, in the emission spectra: For indirect gap materials, zero phonon lines are forbidden, i.e. only phonon satellites are observed. On the other hand, direct gap materials have a strong zero phonon line and weak phonon satellites, if any. Phonon co-operation is also the reason t h a t the probability for radiative indirect transitions is smaller than for direct transitions [16]. The probability for indirect transitions contains an additional matrix element describing the electron phonon interaction during recombination [17, 18]. The variation in band structure leads to the following significant differences in absorption (and consequently emission): I n direct gap materials, the absorption coefficient a is proportional to (E — -Eg)1'2, i-e. it rises steeply with increasing energy E near the bandgap Eg, and reaches typical values of 106 cm - 1 . I n indirect gap materials, a is proportional to {E — Eg)2, i.e. it rises rather slowly near the bandgap and has values between 102 and 103 cm - 1 . The absorption coefficient a is related to the stimulated emission rate by the relation [15] (n index of refraction). Furthermore, the following relation exists between stimulated and spontaneous emission rate [19, 20]: r,tim(E) = ravon(E) (1 - e/«•), (5) where AF = Fn — Fp is the difference between electron and hole quasi-Fermi levels. Dumke [16] has first pointed out that, because of the high band-to-band absorption coefficient, direct gap materials should have the highest stimulated emission rates (see equation (4)) and the highest optical gain. They are therefore more suited for injection lasers than indirect gap materials. Similar statements can be made about the spontaneous emission (see (4) and (5)). In fact, this can already be seen from the detailed balance considerations of van Roosbroeck and Shockley [21] which connect absorption with the total spontaneous emission rate f rspon(E) d E . Low absorption (as in indirect gap E

The Injection Laser

13

materials) means low radiative recombination rate or large radiative lifetime. Competitive non-radiative processes gain importance if the radiative lifetime is large which leads to low quantum efficiencies. For band to impurity level recombinations, the second process of Fig. 1, the differences between direct and indirect gap materials should be less pronounced than in the case of band-to-band transitions. This is, because impurity levels represent more or less localized states through which momentum can be conveyed to the lattice. Zero phonon emission lines, for instance, are now allowed for indirect transitions. Theories have been worked out only for shallow levels [22]. Shallow acceptors, for instance, can be represented by the Bloch functions of the close-by valence band states. Since their ¿-values have only a limited spread around the ¿-value of the nearest band extremum, shallow states are actually not sufficiently localized to take up the total fc-change occuring in indirect transitions. I n other words, indirect band to shallow level transitions are still less favoured t h a n direct transitions. The theory of conduction band-acceptor transitions has been worked out by Dumke [23] and Eagles [24], As in the case of band-to-band transitions, the transition probability contains the interband matrix elements of the momentum operator and has the form [23] |P| 2 /(1 + m n E J m p i? a ) 4 . E& is the acceptor binding energy, Ea = h2 k2j2 ma the kinetic energy of an electron in the conduction band, and m n , m p the effective electron and hole masses. Deep levels are more localized, and therefore should be more favourable for radiative indirect transitions, but little is known quantitatively. Still more favourable are donor-acceptor fair transitions, which is the third process of Fig. 1. Now both, electrons and, holes are trapped at impurity levels and recombine subsequently, ¿-conservation should be still easier than in the previous case, where one carrier remained free. In fact, in indirect gap materials like GaP, the donor-acceptor pair transition has been found to be the most efficient radiative transition [25]. Again, deep level recombination should lead to the highest efficiency, which indeed has been observed [26, 27] experimentally. I t has not been definitely resolved so far, whether even in direct gap materials donor-acceptor transitions are more efficient than the band-to-band recombination. This is quite conceivable, because free carriers can have a larger kvalue spread than trapped ones. The effect, however, if it exists, will not be large, and generally no momentum selection rules are assumed for direct transitions. 2.2 Laser

conditions

The following two laser conditions will be discussed and specifically applied to the case of semiconductor junction lasers: 1. The "first laser condition", which gives net light amplification by stimulated photon emission. 2. The "second laser condition", which gives actual light oscillation. To illustrate the first laser condition for solid state lasers, we assume two (sharp) energy levels 1 und 2. Transitions between level 1 and 2 can occur under emission or absorption of a photon h v (Fig. 2a). I n order to populate the upper level 2 heavily, we pump electrons from level 1 first to a (broad) level 3, from where they drop to 2 ("three-level laser"). I n order to depopulate level 1, often a "ground level" is introduced below 1 ("four-level laser"). Let level 1 be occupied by nx and level 2 by n2 electrons; spontaneous transitions are neglected. Due to absorption (transitions from 1 to 2) the intensity I of light with frequency v is

M. H. PlLKUHN

14

a

b

c

Fig. 2. Comparison between a four-level laser and a semiconductor laser, a) Four-level laser, b) Energy bands of a direct gap semiconductor (E energy, k quasi-momentum). c) Degenerately doped p — n junction under high forward bias (I" applied external voltage)

decreased by the amount I —S(v) rij B12.

(6a)

Stimulated emission (transitions from 2 to 1 ) will increase the intensity by Ih^-S{v)n2B21.

+

(6b)

(v = cjn = velocity of light in a medium with refractive index n, S(v) — normalized spectrum of the spontaneaus emission.) The Einstein coefficients B12 and B21 for induced absorption or emission are identical except for the ratio of degeneracies of levels 1 and 2 B1S =

B21G2jG1.

(7)

Thus the net variation of intensity becomes dI =

S(v)B21

(n2 - n, GJGJ

dx.

(8)

Light amplification (dI > 0) is achieved when the induced emission is larger than the absorption; the first laser condition in its simplest form is therefore Wj G2 < n2 Gi,

(9)

i.e., the well known postulate of "population inversion". The following Boltzmann distribution demonstrates t h a t population inversion is impossible in thermodynamic equilibrium: , GJn, G2 = e

or a Gaussian shape [63, 115] approximated by D(E)

= D0

e

(18b)

.

In many cases, the latter - expression yields better agreement with the experiments. I n the peak shift of the emission lines (Fig. 3), the gap shrinkage effect competes with the Burstein-Moss shift. Morgan [64] showed that, because of the large effective hole mass, p-GaAs should exhibit a larger gap shrinkage than n-GaAs. The Burstein-Moss shift, on the other hand, has the opposite trend, i.e. it is small for large effective masses. This explains why we see predominantly a Burstein-Moss shift for n-material in Fig. 3, and a gap shrinkage effect for the p-material. Still, a Burstein-Moss shift may be noticed in p-material in absorption [46, 59] and in cathodoluminescence [47, 50]. At high acceptor concentrations, the cathodoluminescent line has a high energy shoulder — line (4) in Fig. 3 — shifting to higher energies with increasing acceptor density. Line (4) is interpreted as a donor to valence band transition. 3.2 p-n junction

properties

and

electroluminescence

The most important methods for p - n junction fabrication are : diffusion, epitaxy, and alloying. Diffusion of Zn into n-material at temperatures near 850 °C [65, 66] is probably the most widely used procedure for GaAs. Epitaxial layers may be grown on n- or p-type substrates from the liquid phase (Ga solutions) [67] or from the vapour phase. Alloyed junctions can be produced by alloying Sn, Sn-Zn, Au-Zn, for instance, into n- or p-GaAs at temperatures between 300° and 500 °C [68, 69]. By varying the fabrication procedure, the doping levels near the junction, the degree of compensation, and the impurity gradient at the junction can be altered. Abrupt or nearly abrupt junctions may 2*

20

M. H.

PlLKUHN

be easier to obtain by the latter two techniques, epitaxy and alloying. Diffusion in many cases gives graded junctions. Experimentally, the impurity gradient may be determined by capacitancevoltage measurements using the relation 1 OQ « 2

(19a)

for linearly graded junctions (a = gradient), and for abrupt junctions weget 1/Ci» =

W

qt

a

)(F

d

-F).

(19b)

These relations and also the built-in-voltage Vd (tv EJq) may be changed in the presence of deep levels in GaAs [71, 72], A t high forward bias and high temperatures, the current-voltage characteristics of GaAs p-n junctions generally have the form J = J0 (etrinkT _

(20)

For a diffusion current, i.e. when the injected minority carriers recombine outside the depletion region, n is one [73]. A t high forward bias, this can be found in some lightly doped GaAs diodes [68, 69], particularly, if surface leakage is avoided by special guard-ring structures [74]. However, most diodes with intermediate to high doping levels have n-values around 2 [75]. n = 2 is interpreted as a recombination in the depletion layer of the junction [76, 77], A t low forward bias, low temperatures, and especially in heavily doped junctions, the current may be due to tunneling [75, 78, 79, 80], In this case too, an exponential current-voltage relationship exists: J = J0eiri*,

(21)

but 0 is independent of temperature which is significant for tunneling. Fig. 4 shows a plot of In J vs. V for epitaxial GaAs diodes at various temperatures [80]. I t may be noticed that at high forward biases, the slope of the curves changes with temperature, whereas it becomes temperature independent at low bias. A combination of all mechanisms of current flow, often makes a definite interpretation of the J-V characteristics impossible. We now discuss the near edge emission for high forward bias and high temperature. The emission line near the band gap is the principal emission for most laser type diodes. The intensity of this line, i.e. the radiative current component, and the total diode current mostly do not show the same voltage dependence. This indicates that there are different mechanisms for radiative and non-radiative recombinations [81, 82]. In the high bias region it is quite

Voltage (V)

Fig. 4.

Current—voltage relationship at different temperatures for a solution grown epitaxial GaAs diode. A f t e r [80]

The Injection Laser

21

c o m m o n , t h a t the intensity changes as e'' vlkT with the voltage [74, 81 to 84, 89], which means t h a t the radiative current is a diffusion current. If, for the total current, n of equation (20) is equal to 2, a quadratic intensity-current dependence results ( I ~ J 2 ) [93], which is very frequently observed. The diffusion current in G a A s diodes merits some further comments. I t consists of contributions from an electron current jn and a hole current jv, and for the "injection r a t i o " the following relation holds: (22)

(np , pn = minority carrier concentrations on the p- and n-side, r p , r n = lifetimes). Since the electron mobility in GaAs is about 20 times larger than the hole mobility, _Dn, the electron diffusion coefficient is much larger than Z>p, and hence electron injection into the p-side is favoured. Furthermore, the different magnitudes of a) Burstein-Moss shift and b) g a p shrinkage effect in n- and pmaterial also favour electron injection a t high doping level [64]: a) The Fermi-level for electrons is shifted higher into the b a n d than it is in the case of holes, thus i educing the effective barrier heigh for electron injection. b) T h e effective energy g a p is smaller on the p-side, because of the larger g a p shrinkage [64], The influence of these two effects can be accounted for b y multiplying Sj with exp(— A E J I c T ) , where AEg is the effective b a n d g a p difference between p- and n-side. The injection ratio now has the form which applies to heterojunctions [235]. Experimentally one finds in f a c t t h a t electron injection prevails in highly doped diodes, which means t h a t the p-side of the junction is the spatial origin of the luminescence [9, 91, 95]. The width of the light emitting region is given b y the electron diffusion length \/~Dn r n . Since minority carrier lifetimes are short in highly doped G a A s [23, 85, 86] (in the order of 10~ 9 s for 10 18 carriers/ cm 3 ) the diffusion length is small. Experimental values lie between 0.5 and 3.0 ¡xm [75, 87, 88], If electron injection dominates, the electroluminescent line resembles the photo(cathodo)luminescent line of the p-material [48, 49], a s demonstrated in F i g . 3. I t s peak lies a t 1.483 eV (77 ° K ) a t low doping levels 10 16 to 10 17 c m - 3 ) , and moves to a lower energy a s the doping level is increased [48, 49, 78, 112, 113]. (jVa = Nt j a t the junction for diffused diodes). T h e majority of laser diodes do show this junction luminescence due to electron injection into the p-side. Only for very low donor concentrations m a y hole injection into the n-side be observed [83, 89 to 91, 110, 111], preferably a t high temperatures. The electroluminescent spectrum in that case shows an emission line typical of the photo(cathodo)luminescence of n-material, i.e. a t 1.51 eV (< 10 16 c m - 3 and 77 ° K ) . This is illustrated in F i g . 5 which depicts the emission from a diode doped such t h a t both, hole injection into the n-side (line A) and electron injection into the p-side (line C), m a y be observed. Care must generally be taken in the identification of electroluminescent lines, since re-absorption in the crystal m a y alter the shape of the emission lines or even create " g h o s t p e a k s " [83, 91, 92]. Besides the near edge emission, several broad lines can be often found a t low energy due to deep level recombination. They have exponential intensity-voltage characteristics in particular voltage regions, but n of equation (20) usually changes from low to high values with increasing voltage [72, 94, 96]. The theory

22

M. H .

PlLKUHN

GoAsa85P015 i2°K Q+

400mA^ Fig. 5. Spontaneous emission spectrum from a diffused GaAs diode ( p + - n - n + structure) at 4.2 ° K . Donor density on the n-side: iVd = 2.8 x 1 0 " c m " ' . After [90] Fig. 6.

Emission spectra of a Fabry-Perot type injection laser (GaAszPi — i , Zn-diffused, Nd = 1 0 " c m - 8 ) at 4 . 2 ° K showing peak shift and line narrowing. After [189]. Different arbitrary intensity units

7200

7300

7400 m

7500 —

of radiative deep level recombination in p - n junctions has been worked out recently by Nelson [97] and by Morgan [98]. According to Morgan and for recombination in the depletion region, n should change from 1 to 2 at a characteristic kink voltage Vk which depends on the distance Et of the deep level from the valence band edge, and on the capture cross sections of electrons and holes

vk = EJq

-

2 EJq

- ^

In (CJCJ

.

At a higher characteristic voltage, F^, ti becomes still larger than. 2. Morgan gives the expression Vd = 2 In (NJn,) ± In (CJCJ

(23) For (24)

(no degeneracy assumed, plus sign for deep donors, minus sign for deep acceptors). For low temperatures and low forward bias, phenomena which are mostly associated with tunneling are observed. With increasing diode current (or voltage), the peak of the near edge emission shifts to higher energy, particularly in heavily doped diodes. I n Fig. 6, this peak shift may be seen for a GaAs^Px _ x laser diode (x = 0.85) at 77 ° K . The following two models which explain the peak shift shall be discussed: a) The "band-filling model" [78, 79, 99, 102, 115] for high forward bias, and b) "photon assisted tunneling" [78, 100 to 103] which becomes important at low forward bias. a) The band-filling model may be easily understood recalling the BursteinMoss shift of n-type material (Fig. 3). I n the electroluminescence of heavily doped diodes, injected electrons fill up the empty conduction band on the p-side or in the depletion layer [105] of the junction, and hence the electron quasi-Fermi level moves to higher energy as the current is increased. The peak shift

23

The Injection Laser

d(hv)IQj will be large, if tail states are present at the conduction band edge, i.e. if the density of states Dc is small. The following relation holds [104]: (25) (25> q d Dc which in turn can be used to determine Dc from the experimental peak shift data, if the electron lifetime r n and the width of the active region d are known. Determined in this manner, Dc is in fact much smaller than expected for parabolic bands and the known effective electron mass, which implies that tail states are indeed involved [104]. I f the conduction band tail states extend deep into the forbidden gap, these deep states are filled with electrons through tunneling [78, 102] rather than by thermal injection. Tail states h a v j been seen as much as 0.1 eV below th . conduction band [79, 112], The process itself has been refered to as "horizontal tunneling" [78, 102], but it has not been resolved whether it is genuine tunneling, impurity conduction or a similar process. Here, as well as for photon assisted tunneling the peak energy changes as the applied voltage:

_

8j

hv

=

V±qV„.

(26)

q

F 0 is a small voltage (in the order of millivolts) accounting for the fact that recombinations do not all take place exactly between the quasi-Fermi-levels. At T = 0, F 0 should be negative but, at high temperature, F 0 may be positive if there is a sufficient number of states occupied above the quasi-Fermi-level [19]. I f recombination includes an isolated acceptor level, then q F 0 can be always negative and close to the acceptor binding energy (30meV) [106]. The intensityvoltage relationship is given by (21). 0 is temperature independent and lies between 8 and 20 meV [78, 79], Whether the total peak shift is attributable to the filling of the conduction band only, has not been completely decided [47, 114]. The valence band edge can also have large tails which may be filled with holes at high injection levels, if the radiative recombination extends into the depletion region [47, 114]. Tailing is strongest in heavily compensated materials, e.g. in diodes with Sidoped [36] or Sn-Zn-doped [69] p-regions, where the electroluminescent emission peak lies at unusually low energies (1.27 eV at 300 ° K is reported in [69]). b) I n 'photon assisted tunneling, also refered to as "diagonal tunneling", electron-hole tunneling between the conduction band states on the n-side and the valence band states on the p-side of the junction takes place [78,100 to 102, 110]. Peak shifts with current are generally larger as in the case of horizontal tunneling, and 0 of (21) lies between 20 and 100 meV [78, 102], As before, the peak energy shifts with the applied voltage according to relation (26), and peak shifts up to 0.3 eV have been observed [78]. Theoretical treatments of diagonal tunneling [102, 103] give as important features the exponential intensityvoltage relationship (21) and expressions for 0 which agree with experimental results. 0 is inversely proportional to wlt a characteristic width parameter of the depletion layer [102], 4 tft h 1 0 = - ' (abrupt junctions) yqmnwi _ 9 1 6 and 0 = — ( F 0 — F) ' —'— w (linearly graded junctions) vqmn i (m tunneling mass).

I I

-

(27

»

'

24

M. H. PlLKTJHN

For linearly graded junctions 0 is slightly voltage dependent. The actual depletion layer width w changes with voltage and is related to wx in the following way: w = w1 (V0 — V)n, where n = \ or for abrupt and linearly graded junctions, respectively. In the tunneling regime, GaAs diodes may show an additional emission line with peak energies h v larger than the applied voltage, and with h v — q V kT [103, 108], This high energy line shows no peak shift and merges with the "regular" emission line at high bias. I t has been explained by an Auger recombination process which takes place at defects in the junction [103, 108]. High energy lines have also been interpreted by electroluminescent refrigeration [109]. 4. Different Injection Laser Structures The Fabry-Perot laser with two parallel reflecting sides perpendicular to the p-n junction is the most important laser structure. The reflecting sides can be made by proper polishing or cleaving of the crystal. GaAs cleaves easily along the (110) plane, and for a junction perpendicular to the cleaved faces, e.g. a (100) plane should be chosen. Because of the high refractive index of most semiconductors (n = 3.6 at 8400 A for GaAs [116]) the semiconductor-air interface has a sufficiently large reflection coefficient, which may be still improved further by silvering. The remaining two laser sides are roughened (e.g. sawed), in order to discourage ordered reflections from them. In Fig. 6, typical line narrowing at the laser threshold can be seen in a low resolution experiment. I n high resolution experiments and for good optical quality of the reflecting sides, mode structure due to interference in the FabryPerot cavity can be seen in the spontaneous emission near threshold [117, 119]. The mode spacing A/ is found from the interference condition m A/2 n = L (m integer): A2 AA = (28) 2L(n-kdn/M) " Contrary to other solid state lasers, dnjdX is not small for semiconductor lasers and must be included particularly in the neighbourhood of the band gap [118]. At threshold, the modes near the intensity maximum increase superlinearly; laser emission eventually occurs in one or several modes at or near the peak of the spontaneous emission line [119]. The width of the laser line is extremely small and, according to interferometric measurements, less than 1.2 x 1 0 - 3 A at 4.2 ° K [120]. The laser line shows very little peak shift with increasing current. Instead, new modes appear at higher energy, or, if heating effects dominate, at lower energy. Above threshold all additional energy is pumped into the laser line and the spontaneous line saturates [9, 121], Lasher and Stern [19] have theoretically determined the lineshape of stimulated and spontaneous emission and shown that their peaks coincide for high gain (77 °K). For low gain, the stimulated emission line occurs at the low energy side of the spontaneous line. This may indeed be observed experimentally at 77 ° K . Low gain at threshold is achieved by silvering the laser ends [9], or making the lasers very long (see Section 6.1). For very high gain at threshold, as for instance in lasers with high internal losses (GaAsj.Pi _ x ) , the stimulated line may even be shifted to the high energy side of the spontaneous emission [122], However, this may also be caused by a Burstein-Moss shift, if the line narrows slowly.

The Injection Laser

25

Besides the lineshape, stimulated and spontaneous emission from Fabry-Perot lasers differ in coherence and angular distribution. Coherence of the stimulated emission has been demonstrated nicely by the interference experiments of Michel and Walker [123]: Two light beams leaving the laser diode at opposite ends were superimposed and showed interference when the diode current exceeded the threshold value. The angular distribution of the stimulated emission is often quite complex and depends on the structure of the light emitting region. I t has been thoroughly studied both experimentally [124, 126, 128, 134] and theoretically [11, 126]. Beam angles 6 are generally large for p-n junction lasers, because the width of the active region, d, is very small, and 6 is of the order of Ajd. The larger beamspreads have therefore been found perpendicular to the junction plane ("vertical" beamspread) rather than in the junction plane ("horizontal" beamspread). For conventional Zn-diffused GaAs lasers vertical beamspreads are 15° to 25° at 77 °K and horizontal beamspreads about 3° [124]. The horizontal far field pattern is usually complex and shows symmetries only in special cases [125, 134], Extremely narrow lasers (prepared by a stripe geometry contact), show a regular symmetry (Hermite-Gaussian) 2 ) [125] indicating either slightly curved mirrors or refractive index variations giving a "lens-effect". In p-n junction structures of p + -j>-n + type (where v signifies a 5 to 10 ¡xm thick layer of low n- or p-doping) higher order transverse modes may be established [90, 126, 127] (cf. also Sections 5 and 7). Laser emission from these structures has smaller vertical beamspreads [90, 126] (4° to 6°), and the vertical far field pattern may show a small subsidiary beam [126], Additional fine structure in the vertical far field pattern of GaAs lasers was observed by Ludman and Hergenrother [128], and was interpreted through variations of the light path within the junction cavity. Application of Fermat's principle yields a series of closely spaced emission angles with an angle spacing proportional to X - 1 ' 4 . Interference effects at the diode mount may give a similar structure [1, 125, 107]. Spontaneous light is emitted isotropically from the junction and usually exhibits no particular angular distribution outside the diode. In large diodes the strong re-absorption in the p-layer and the refractive index variations at the junction may produce a certain far field pattern [129]. The high absorption loss for the spontaneous emission accounts also for the fact that its quantum efficiency is lower than for stimulated emission (cf. Section 6.3). I n Fabry-Perot lasers, the integrated light intensity therefore increases sharply at the threshold current density jt. This is one of the easiest experimental methods to determine j t . Near field investigations of Fabry-Perot lasers show that the spontaneous emission is distributed unifo mly across the p-n junction, whereas the laser emission occurs in "spots" or "filaments" [124, 130, 131]. Usually one such spot appears at threshold and several more above threshold. The spots have diameters of 3 to 10 ¡i.m, and are most likely caused by optical inhomogeneities or local current distortions. Relations between diffusion faults (dislocations) or crystal mosaic structure and laser spots have been demonstrated [132, 133, 258]. I t has been suggested that mutual coherence between the different laser filaments exists [134]. Despite the confinement of the laser emission to filaments, the horizontal beamspread is smaller than the vertical. In very narrow junctions and for good junction perfection, single filamentary lasing can be achieved [135]. 2

) I am indebted to Dr. D'Asaro for a preprint.

26

M. H.

PlLKUHN F i g . 7. E m i s s i o n s p e c t r a of an all-side reflecting l a s e r ( G a A s , Zn-diffused, Nd = 10 1 8 c m " ' ) a t 77 ° K . T h e laser mode of t o t a l i n t e r n a l reflection is seen on t h e long w a v e length side of t h e s p o n t a n e o u s line. A f t e r [ 1 4 0 ] . A r b i t r a r y intensity units

Laser action perpendicular to the p-n junction plane, i.e. in the direction of the current flow, has been obtained in InSb by Melngailis [136]. With the aid of a longitudinal magnetic field a long injection plasma providing sufficient optical gain perpendicular to the junction could be produced. Large volume injection lasers of this type are most promising for high power generation at small beam angles. In GaAs injection lasers the transverse gain has been measured by Burrell et al. [70] but no structures for transverse laser oscillations have been built. A different type of injection laser is the quadratic all-side reflecting laser (see Fig. 7), which may for instance be fabricated by proper cleavage of all four crystal faces. In this structure modes due to total internal reflection with very small end losses In (1 /i?j R 2 ) exist. The light impinges at an angle on the crystal surface which is larger than the angle for total internal reflection, i.e. larger than 16° (measured against normal incidence) for the GaAs-air interface. Threshold current densities of these structures are smaller than those of FabryPerot lasers prepared from the same material. In Fig. 7, the emission spectra below and above jt can be seen at 77 °K for GaAs. The laser line emerges on the low energy side of the spontaneous line because the gain at threshold is low. This is in agreement with Lasher and Stern's [19] theoretical results. At low temperatures (4.2 °K and below) both lines have the same peak positions. The stimulated emission of the all-side cleaved laser has no particular angular distribution, and filamentary laser action, as for Fabry-Perot lasers, is not observed. Furthermore, the integrated light output does not increase noticeably at threshold which makes this laser type unsuitable for high power generation. B y changing its geometry from quadratic to bar shaped and increasing its length, one again approaches the Fabry-Perot laser type. Fabry-Perot modes can be observed especially, if lossy irregularities lie in the path of the modes of total internal reflection. The temperature dependence of jt is qualitatively similar in all injection laser types: At low temperatures (T < 20 to 50 °K), jt remains constant, and at high temperatures (T > 30 to 100 °K) jt rises approximately as Ts [137 to 139]. Continuous operation therefore becomes difficult at high temperatures. I n GaAs it has been obtained without too much trouble up to 77 °K [139 to 141] and, with special stripe geometry, up to 200 °K [125]. In triangular lasers, Fabry-Perot modes and modes due to total internal reflection can both occur simultaneously. Such lasers have been fabricated from

The Injection Laser

27

GaAs as equilateral triangles [142, 143] and as isosceles triangles with one right angle [144], A closed light path, being totally internally reflected a t each side, may be drawn into these triangles. The corresponding modes of total reflection are observed at low current density [143, 144], i.e. the stimulated emission has no special angular distribution and occurs on the low energy side of the spontaneous line (77 °K). More often, however, one encounters Fabry-Perot modes, i.e. the laser beam is normal to one or two of the triangle sides. Isosceles right-triangular lasers can emit directional laser beams from one side, two sides (angle between beams = 90°), and also from all three sides [144], Equilateral triangular lasers emit directional beams from two sides (angle between beams = 120°). Triangular lasers offer the possibility of controlling the direction of the emitted laser beam by proper variation of the current distribution across the p-n junction plane. Cylindrical injection lasers have been developed using GaAs [145]. I n this case, stimulated emission is of course independent of angle in the plane of the p-n junction. "Radial modes" whose spacing is determined by the radius of the cylindrical cavity can be observed in the emission spectrum [145], The quantum efficiency of the stimulated emission is presumably better than t h a t of comparable Fabry-Perot structures, because the laser light can leave the cavity in all directions (in the p - n junction plane) [145, 146]. 5. Theoretical Results Amongst the important theoretical problems relating to injection lasers are: a) the nature of the electronic transitions, b) the optical gain, and c) the laser losses. We shall concentrate on the latter two problems which are essential for the understanding of the threshold current density and its dependence on temperature and structural parameters. The major difficulties confronting theoretical work and the consequent simplifying assumptions are: 1. Insufficient knowledge of the band structure at the band edges of highly doped materials. Parabolic bands, exponential tails, and Gaussian tails have been assumed. 2. More must be known about the transition probabilities. Neglect of ¿-selection rules and constant matrix elements have turned out to be successful approximations. 3. I t is difficult to treat non-uniform systems, such as p - n junctions, where quasi-Fermi levels as well as band tails and optical constants vary with distance. With the exception of the "sandwich" models used in the theory of laser losses, recombination in uniform systems is mostly considered. The major features of injection lasers are understood theoretically, but a quantitative understanding of many of the details has not yet been reached. A low temperature approximation for the optical gain g and the threshold current density jt may be derived easily from equation (8), if the Einstein coefficients are expressed in the following way [147, 148]:

O 71 V*

fl V T 2 1

r 21 is the electron lifetime in the upper state, and for injection lasers with predominant electron injection into the p-side it corresponds to the radiative electron

M. H. PlLKUHN

28

lifetime r n . For g we obtain from (8) O 71 V

(30)

r2l

We now assume nx G2IG1 n2 which implies that we need fairly high population inversion, i.e. gain, at threshold in order to overcome the laser losses. The total current density flowing through the junctions is =




In this formula th? internal quantum efficiency rj takes non-radiative transitions into account. Assuming a general lineshape 8(v) = 1/Av for the spontaneous emission (Av = linewidth), we get the following expression for the low temperature gain: *

8jiqn2v2

Avd,

(T

0).

(32)

We see that the optical gain increases linearly with diode current density at T = 0. Inserting (32) into the second laser condition (17a), one obtains a simple relation for the threshold current density at T = 0 [149]: 2 2 / i

F

w

=

f i * -

y)112

e

~y°

d

y •

(38)

— oo

E c is t h e energy of t h e n o m i n a l conduction b a n d edge, a n d r/a a tail s p r e a d i n g energy r e l a t e d t o t h e screening length r0 b y r]c — (q2le) (4 n Nd r^)1!2. If r]c a p p r o a c h e s zero, we r e t u r n t o case (37) of u n p e r t u r b e d p a r a b o l i c bands. " E x p o n e n t i a l t a i l s " of t h e t y p e (18a) h a v e b e e n also used, for instance, f o r d e t e r m i n i n g t h e lineshape of t h e s p o n t a n e o u s emission [19], a n d for considering t h e effect of tail s t a t e s on t h e t e m p e r a t u r e d e p e n d e n c e of jt [151]. H e r e is a brief a c c o u n t of t h e results o b t a i n e d a f t e r e v a l u a t i o n of t h e integrals of e q u a t i o n s (34) a n d (35) using t h e d e n s i t y of s t a t e s f u n c t i o n s (37) or (38) [19, 150, 170] (all e x a m p l e s given refer to G a A s ) : a) E n e r g y d e p e n d e n c e of t h e emission r a t e s . A t low t e m p e r a t u r e s , t h e m a x i m u m of t h e s p o n t a n e o u s a n d of t h e s t i m u l a t e d emission lie a p p r o x i m a t e l y a t t h e same energy, i n d e p e n d e n t of gain. A t higher t e m p e r a t u r e s , however, t h i s is only t r u e for high gain (large separation of quasi-Fermi-levels). F o r low gain, t h e m a x i m u m of t h e s t i m u l a t e d emission is s h i f t e d f r o m t h a t of t h e s p o n t a n e o u s emission t o lower energy, in agreement w i t h e x p e r i m e n t a l results. F o r calculations w i t h ¿-selection rules applied, q u a l i t a t i v e l y similar v a r i a t i o n s of emission r a t e s w i t h energy are o b t a i n e d [130], T h e a g r e e m e n t b e t w e e n calculated lineshapes (no ¿-selection) a n d experiment a l ones (in heavily d o p e d laser diodes) is b e t t e r in t h e case of b a n d tails (exponential [19] a n d Gaussian [150]) t h a n in t h e case of p a r a b o l i c b a n d s [19]. Various theoretical t r e a t m e n t s of t h e f o r m a t i o n of axial laser modes [153 t o 155] a n d of laser line w i d t h s [155, 156] have been carried o u t , p a r t l y b a s e d on t h e m e t h o d s j u s t m e n t i o n e d , b u t t h e r e are h a r d l y a n y possibilities for a c o m p a r i s o n with experiments. b) T h e dependence of t h e m a x i m u m optical g a i n on t h e n o m i n a l c u r r e n t dens i t y j' = j rj r/d is d e p i c t e d in Fig. 8 for d i f f e r e n t t e m p e r a t u r e s , r is a q u a n t i t y (discussed later in t h i s section) characterizing t h e light c o n f i n e m e n t in t h e active layer. T h e low t e m p e r a t u r e experimental s i t u a t i o n m a y be o f t e n a p p r o x i m a t e d b y rj = 1, d = 1 ¡xm, a n d T fa 1, i.e., f = j. W e see t h a t a t low t e m p e r a t u r e s a n d in a g r e e m e n t w i t h (32), g is proportional t o f (independent of t h e d e n s i t y of s t a t e s f u n c t i o n assumed). A t high t e m p e r a t u r e s , parabolic b a n d s a n d Gaus300

Fig. 8. Theoretical dependence of the maximum optical gain on the nominal current density j' =j -q rid at various temperatures. The dashed lines refer to the assumption of parabolic bands, the full lines to the assumption of ''Gaussian tails". After Stern [150]

j'(A/cm2 ¡im}-

30

M. H .

PlLKUHN

sian tails lead to different results (represented by full and dashed lines in Fig. 8). In the first case, g increases more than linearly with f , e.g., at 80 ° K as f 2 and at 300 ° K even as j ' 5 (in the region around 100 c m - 1 ) , and we shall write this as g •—> j'p. I n the second case, p is not too far from unity up to about 160 ° K , and reaches 2.8 at 300 ° K (in the case of Fig. 8). This is in better agreement with experiments (see Section 6.1). Higher doping levels bring p even closer to one at high temperatures (e.g. for N& = 3.3 x l O 1 9 and Nd = 3 x l 0 1 9 c m - 3 , p is 1.04 at 300 °K). At very high current densities, "saturation" of the gain (absorption) should be taken into account, and this has been treated theoretically by several authors [i >7 to 159], c) The theoretical temperature dependence (for Gaussian tails) of the nominal current density j't (giving a gain of 100 c m - 1 ) has been plotted in Fig. 9 for different doping levels [150]. As a common feature, the curves of Fig. 9 (as well as those corresponding to parabolic bands [19]) have a temperature independent region (at low temperatures) and a region where j't increases strongly, which may or an increase of the be approximated by an exponential increase (j't ~ type j't ~ Tm (m « 3). With increasing number of tail states (higher compensation), the low temperature threshold current density becomes larger, and the one at high temperatures becomes smaller, which leads to the "crossover" of the curves in Fig. 9. Apparently, the technically very desirable weak temperature dependence of jt is achieved by a narrow funnel of tail states. This was also deduced in a more qualitative way and for exponential tails by Dousmanis et al. [151]. For parabolic bands, basically the same temperature dependence of j't results, but a crossover as in Fig. 9 is not seen [19]. The exponent m of the region where j't ~ Tm, decreases with increasing gain (for example from 2.6 to 1.8 as g changes from 30 to 10 3 cm" 1 ) [19]. These theoretical results, based on many simplifying assumptions, basically describe the experimentally observed temperature dependence of jt. F o r a more complete theory, the following refinements should be introduced: 1. The rate constant B decreases with temperature [23], 2. the laser losses a increase with temperature (see Section 6.1), 3. active layer width d and mode confinement constant r depend on temperature, although this has been estimated to be a weak effect [150], 4. the internal quantum efficiency rj decreases towards high temperatures (see Section 6.2), and 5. the actual situation at the p - n junction should 1

1

1

Na-Nd'3*101scm-3

A'

103

g-100cm~1

i

' / /

/ / / ! / i < / / / / /_

j'-pl 1 ' / / / / / / / > I l l J / /// A

fo'PlO19, 1019\ 3*10\

i

3

" i

10

i

30

100 T(%)

30l

Fig. 9. Theoretical temperature dependence of the nominal current density j' which is necessary to reach a gain of 100 c m - 1 . After Stern [150]

31

The Injection Laser

be considered, instead of a uniform system. Here the problem will be rather complex, depending on details of the junction structure and the recombination mechanism. The latter often changes with temperature (e.g. from tunneling to thermal injection, or from predominant electron to hole injection, etc.) which should be taken into account. Attempts have been made to account for details of the junction structure. Pikus [160], for instance, has calculated the temperature dependence of j t for linearly graded junctions, assuming that all recombination takes place in the junction region itself, and t h a t quasi-Fermi-levels remain constant in that region. Although no band tailing was assumed, the results resemble those of Stern [150], depicted in Fig. 9. An increase of the junction gradient leads to an increase of j t a t low temperatures, but to a decrease at high temperatures, i.e., to a similar crossover as in Fig. 9. Mayburg [161] also considers recombination in the junction region, but reaches a relation jt.—• T1-5. I t is interesting to note t h a t calculations based on the validity of ¿-selection rules [130] lead to an unsatisfying temperature dependence of jv namely jt ~ T, which definitely does not agree with experimental results. Theoretical models concerning the optical losses of injection lasers have to assume well defined junction structures. I n these models, the p - n junction is substituted by a multi-layer structure containing a central active layer with gain (negative absorption, — a a c t ), and adjacent passive layers which are lossy (positive absorption, a n , a p ). The refractive index may also vary from one layer to another. The laser losses are predominantly "penetration" (or "diffraction") losses (cip) whose magnitude is determined by the degree of electromagnetic wave penetration into the lossy p- and n-regions. A small amount of laser losses may be traced to free carrier absorption in the active layer (a fc ). I t turns out t h a t the most simple model, namely t h a t of three layers which only differ in a but not in refractive index, is apparently insufficient, because the calculated penetration losses are nearly one order of magnitude higher than the experimental values [149, 162], Smaller penetration losses are obtained if differences in refractive index are assumed between the various layers. This may be understood as follows: If the active layer has the largest index, a waveguide effect leads to a confinement of the electromagnetic modes. Under these conditions, the waves do not penetrate as much as in the case of constant index into the lossy regions. Calculations for thin active layers have been performed by McWhorter et al. [163], and Yariv and Leite [164], for active layers of variable thickness by Stern [165], Antonoff [166], and Anderson [167]. Anderson shows that the net gain of a three layer structure may be expressed in the following way: G = - F1 (a n + F2 a a c t + 1), t h a t no mode guiding takes place below a certain critical width of the active layer, which is about 3.3 jim for a GaAs laser at room temperature (y = 34). The strongest mode confinement is reached for symmetric structures. As expected, the net gain G increases with the width of the active layer if all other parameters are kept constant.

32

M. H. PlLKTJHN

Stern [165] has treated the problem in a general way, assuming that the refractive index and the absorption coefficient are arbitrary functions of the position coordinate x, perpendicular to the p-n junction. If the electromagnetic wave lies in the plane of the junction and propagates in the z-direction, its 2J-vector may be expressed by Ey(x, y, z, t) = u(x) eikz + 1l2Gi ~imt (40) (k = propagation vector). For the simplest solution of the wave equation, the intensity |u(x)\2 has its maximum at the centre of the active layer (first order transverse modes). For second order transverse modes, two intensity maxima at the sides of the active layer a -e found. Second and higher order transverse modes will be more probable for wide active regions. The net gain G may be written, according to Stern [165], in the following way: + °o e?=—i- J 0i(x) \u(x)\2 dx , J \u(x)\2dx=l. (41) — 0o If the refractive index is independent of x, we get k = co n/c, and G then becomes the average value of the negative absorption. Stern [165] showed that G may be approximated by a linear dependence on a a c t and the penetration losses G = - /Xct

-

= - r [ - g(j) + a f c ] -

•

(42)

r is a dimensionless function depending on the widths and optical constants of the layers. Comparing (42) and (41), one can see that _T is the relative fraction of light intensity contained in the active region. In many cases, r is close to unity, e.g., for Zn-diffused GaAs lasers r = 0.9 [165], The penetration losses are lowest for dielectric symmetry, and they decrease with increasing width of the active layer. In Table 1, the calculated values for the first and second order transverse modes (a'p, a.'p') are given for GaAs, assuming Am = 0.02, and experimental values for the optical constants. The agreement with experimental values for the internal laser losses is good. Quantitatively, the comparatively large variations in refractive index, which are necessary for a successful theory of the penetration losses, have not yet been fully explained. If changes in the free carrier concentration are taken as a source of the dielectric discontinuity the resulting variations turn out to be too small [164]. Larger variations result from the general dependence of refractive index on the shape and position of the absorption edge [168], The latter depends on Table 1 Theoretical dependence of laser penetration losses on width of the active region, after Stern [165], The following parameters were assumed (GaAs and 77 °K): X = 8400 A, Jiact = 3.61, An = 0.02, a. This is in fact the case, as we seen from Table 3. The experimental temperature dependence of 1//3 and jt gives good qualitative agreement with the theoretical results outlined in the previous section (cf. Fig. 9). The weak increase of the laser losses with temperature is in line with the experimentally observed changes in absorption coefficients a n , a p [46, 52], For a more refined discussion, the variation of mode confinement with temperature should be taken into account, although this is probably a minor effect. A different point of view, namely that absorption losses determine predominantly the temperature dependence of jt, has also been published [185]. The actual magnitude of the gain factor at low temperature may be estimated from (32) ^ [cm/A] = 1.59 X

1

0

-

(

4

4

)

If we assume?? = l,n = 3.6, E = 1.46eV,AjE = 1.98 x 10" 2 eVandd = 10" 4 cm, we obtain fi = 2.9 X 1 0 1 cm/A for GaAs. This value corresponds to the " b e s t " experimental values at 4.2 °K (not those of Tab. 2), which may be obtained for doping levels < 10 18 c m - 3 . Lower values for the gain factor have been reported for GaAs x Pi _ ,e (fj = 0.9 to 1.0 X 10" 2 cm/A) [188, 189] and for I n P injection lasers (j8 = 3.4 X 10" 3 cm/A) [186] at 77 °K. Of course, an experimental comparison of gain factors for different semiconductors is difficult to make, because one should refer to the same junctions structure. According to (44), the laser energy E (which is close to the bandgap) should be an important materials parameter. Assuming that all other parameters in (44) do not change, this means that small gap semiconductors, like InSb or InAs, should have high gain factors and low threshold current densities. So far, this has not been confirmed [190, 191], possibly, because junction planarity or quantum efficiency were too poor in these cases. The values of the internal losses gained from the length experiments agree well with those obtained by other methods. Variation of the reflection coefficients, either by silvering of the laser ends [9], or by evaporation of antireflective films [177 to 179], yields losses between 12 and 40 cm" 1 for GaAs and 77 °K. A value of 15 c m - 1 was obtained from the intensity ratios of the oscillation of the spontaneous emission [177], For comparison with theoretical results, we shall choose one of the smaller "typical" values like 13 c m - 1 , because higher losses may be caused by irregularities. If we set r pa 1 in (42), the internal losses become the sum of free carrier absorption in the active region and penetration losses. For standard Zn-diffused GaAs lasers, free carrier losses may be estimated in the following way: As before, we assume d = 10~4 cm and electron injection into the p-side. From the known Zn-diffusion profile [61] one gets an average acceptor concentration of 2 x 10 18 c m - 3 in the active region. A similar concentration is obtained if the measured lifetime r n = 1 ns [86] is introduced in to Dumke's concentration-lifetime relation [23]. Free carrier losses of 4 c m - 1 result, which leaves 9 c m - 1 for the penetration losses if we take our "typical" case of 13 c m - 1 . Comparing this with the theoretical values of Stern [165] listed in Table 1, we see agreement with the

37

The Injection Laser

penetration losses of the first order transverse modes for d = 10~ 4 cm. Lasers with wider and lower doped active regions should have smaller internal losses. One should note that a width near 10~ 4 cm for the active layer of diffused GaAs lasers, gives satisfactory theoretical values for otp and also fo- /i, and that it agrees well with the experimental values for the electron diffusion length [75, 87, 88]. Not many investigations of injection laser losses have been carried out for materials other than GaAs. From the length dependence of jt, losses of 29 c m - 1 or higher were reported for GaAs^Pj x [188, 197], and 15 c m - 1 for epitaxial InAs lasers [192] (in contrast to 150 c m - 1 for diffused InAs lasers [190]). For I n P a value of 8 c m - 1 was found by the silvering technique [186]. These values, which all refer to 77 °K, are similar to those of GaAs. Higher losses should be probably attributed to junction imperfections and not to the material itself. 6.2 Emission

characteristics

at different

laser

length

The wavelength of the stimulated emission at threshold changes with the length of the laser, or, more generally speaking, with the Q of the cavity [139, 193, 194], Fig. 12 depicts the wavelength change with L (at 77 °K), and one sees that the wavelength decreases as the laser becomes smaller. This offers a possibility to tune an injection laser to a particular wavelength by choice of a suitable L. Similar wavelength shifts can be produced by variation of the reflection coefficients [193] or the internal losses [194], The data of Fig. 12 can be interpreted with the band filling model: Short lasers have high threshold current densities, and the conduction band tail will be filled up to high energies. If we plot A of Fig. 12 as a function of jt we get the regular peak shift curve of the spontaneous emission (as measured for diodes without laser geometry). The wavelength shift with laser length or reflection coefficient is larger at high temperatures [139, 193]. This has been explained through the larger current density changes, and through the variation of the electron distribution function in the band tails [193]. The relative positions of spontaneous and stimulated emission peaks also depend on the laser length [184], Short Fabry-Perot lasers have their stimulated emission line near the peak of the spontaneous emission at 77 °K (cf. Fig. 6). For long lasers (L > 2 mm), the total gain per unit length at threshold becomes

Fig. 12. Wavelength of the stimulated emission at threshold as a function of laser length at 77 °K (same laser data as in Fig. 10)

'

. L lmml

""

38

M. H. PlLKTTHN

small, and the laser line can be observed on the low energy side of the spontaneous line. This agrees, of course, with the results for all side cleaved lasers (cf. Fig. 7) and with the theory of Lasher and Stern [19]. Silvering of the reflecting sides gives a similar effect [9], Finally, the axial mode spacing, A2/2 L (n — X dw/dx), is a function of the laser length. For long lasers it becomes quite small, for example, 0.34 Â at 8400 Â for a 2 mm long GaAs laser. For very short lasers the mode spacing may be wide enough to assure single mode operation [196], The adjacent modes will then fall outside the frequency range having sufficient gain. Single mode operation can also be observed in long lasers over a certain current range. 6.3 Quantum

efficiency

We shall define as quantum efficiency the number of photons generated in one recombination step, i.e. its maximum value is one. In electroluminescent devices it becomes equivalent to the power conversion efficiency if the photon energy corresponds to the applied bias. Generally, we shall only count photons of the principal emission line near the band gap and neglect low energy lines. The external quantum efficiency ry,,xt, measured outside the diode, is always smaller than the internal quantum efficiency rj-v because a fraction of the light generated is again re-absorbed. Quantitative relations between ^ e x t and r/j, taking diode geometries and absorption losses into account, have been deduced [198 to 200]. Absorption losses are particularly high for the spontaneous emission, besause it is emitted isotropically and traverses highly doped regions of the crystal. Furthermore, unless special diode geometries are used, the average transmission coefficient i' a v is quite small, and only a small fraction of the light will escape from the crystal. In the case of the GaAs-air interface, I' a v = 0.1 is obtained by averaging over all angles of incidence [201]. Under the conditions of uniform photon density within the crystal and a small average absorption coefficient a, the following expression has been derived for the spontaneous emission [198, 201]: j

(»i - n2)2 (»1 + n2)2

(45)

( F crystal volume, A crystal surface area, w, refractive index of the semiconductor, n 2 of the ambient). V¡A has the dimension of a length, and jyext(spon) should become larger for small VjA, i.e. for small diodes. A size effect of this nature has been noticed, for instance, for the red emission of GaP diodes [202], Quantitative determination of ^¡(spon) or a on the basis of equation (45) is difficult. The most promising method of determining rjl is to change the refractive index n% of the ambi nt [203]. The external quantum efficiency of the stimulated emission is larger than that of the spontaneous emission for the following reasons: 1. The transmission coefficient is larger, because the laser beam is perpendicular to the reflecting surface of the Fabry-Perot laser. 2. The light is confined to the active region and therefore only attenuated by the laser losses a. 3. The internal quantum efficiency of the stimulated emission may be larger than that of the spontaneous emission. Under the assumption of uniform light generation and absorption over the laser length the following formula has been derived by

The Injection Laser

39

Stern [198] : W

—

)

e

i ^ r « £ _ 1 + Tn

««)

Often one can use an approximation for small total losses ( « L ^ l ) where upon (46) becomes I 4.- \ ijj(stim) ^(stim) = . (47) The following relation has been derived by Biard et al. [199] : »¡¡(stim) -

jy ext (stim) = 1 -

,

(X I j

In (1 -

(48)

Tu)

and it is identical to the approximation (47) for a small transmission coefficient Experimentally, the external quantum efficiency is usually determined by an integrating sphere where diffuse reflection at the walls produces a uniform photon density which is measured by a calibrated detector [204, 205], For Fabry-Perot lasers a considerable increase of the external quantum efficiency is found at the threshold [198, 204], I f t] are incremental values, the total light power of a laser diode above threshold, QJ; (j > jt), and the total power conversion efficiency, r/p, are (h v « q F) Qi = »Jext(spon) jt LWV + )? cx t(stim) (j — jt) % = QiH LWV = »; cxt (stim) -

LWV,

[^ e x t (stim) — r/ ext (spon)]

.

(49)

For standard Zn-diffused GaAs lasers which are heavily doped, r; ex t(spon) is usually quite low. Experimental values range between 10~ 2 and 6 X 10~ 2 at 77 ° K [80, 92, 172, 206], and they are still lower by a facto • of 10 to 20 at 'oom temperature. They are influenced by the impurity density as discussed in Section 7.3. If one wants to improve ?7ext(spon), without being able to change ^¡(spon), two possibilities suggest themselves from equation (45): 1. The average absorption should be as low as possible. One way to do this is to keep the emission energy lower than the bandgap of the diode regions in which re-absorption takes place. For instance, different materials (GaAs, GaAszPj _ x , Ga^Ini _ ^As) can be used in such a way that the junction region has the lowest bandgap. Probably the same effect is reached when the junction region is heavily compensated, i.e. large band tails exist [36, 69]. I n highly compensated Si-doped GaAs diodes, for example, room temperature efficiencies of up to 0.1 have been reached [36]. Possibly, ^¡(spon) incr.ases simultaneously in Si-compensated GaAs [247a]. 2. The average transmission coefficient can be increased by use of antireflective films [207, 208] or by special diode geometries [209]. Diodes with hemispherically shaped n-regions were particularly successful [209], In these structures the isotropically emitted spontaneous light falls nearly perpendicular on the surface, i.e. T a v approaches T n . For laser diodes the transition from ^ e x t (spon) to ^ c x t (stim) is sometimes abrupt but more often gradual and associated with an increase in the number and

M. H. PlLKUHN

40

60 As

300 °K

020~

\ + \

"^ext(slim)

0.1 -

_ 0.10-

\ -J

I 200

I I 300 400 L(jm)-

I

Fig. 13. Dependence of the differential q u a n t u m efficiency of the stimulated emission on the laser length. Solution grown epitaxial GaAs lasers (jVj = 6 x 1 0 1 8 c m - 3 ) a t room temperature. After [ 8 0 ]

i

\^ext(spcn)

\ V

Fig. 14. T e m p e r a t u r e dependence of iji(stim), i? e x ^(stim), a n d i? e x ^(spon) for an epitaxial GaAs laser of the series oi Fig. 13. ( £ = 1 . 0 5 x 1 0 " ! c m ) . After [80]

size of laser filaments above threshold. Inhomogeneous light distribution over the laser width is probably responsible for the large scatter of efficiency data which one usually finds. However, the increase of r/cxt(stim) with laser length, as postulated in (46) and (47), can be observed experimentally [80, 198], I n Fig. 13, for instance, the differential quantum efficiency ^ e x t (stim) has been plotted as a function of laser length for a series of epitaxial GaAs lasers (300 °K). The shortest lasers have the highest efficiencies. From this length dependence the internal losses a and the internal efficiency may be estimated using equation (47). For example, from the best efficiency values of Fig. 13 (upper dashed curve) one obtains ^¡(stim) = 0.4 and a = 100 c m - 1 for room temperature. The losses determined from the length dependence of jt are in most cases lower. This may be expected, because threshold is first reached in those regions of the p-n junction where the losses are lowest. Nevertheless, similar conclusions about the general magnitude and the temperature dep:ndence of the losses can be reached from both efficiency and threshold data at variable laser length. The temperature dependence of the internal quantum efficiency »^(stim), together with that of rjext(stim) and ^ cxt (spon), is depicted in Fig. 14 for the laser series of Fig. 13. The external efficiencies refer to one of the shorter lasers of that series. One sees that at low temperatures ^¡(stim) is practically unity and that it drops slightly to about 0.4 at room temperature. The latter value is not too accurate because of the large correction for re-absorption, and most likely, even higher values than 0.4 may be expected. The internal quantum efficiency of the spontaneous emission rj (spon), has a similar magnitude and temperature dependence as ^¡(stim): For low temperatures ^¡(spon) has been estimated to be one [210, 213, 214], and (with the exception of [211]) no strong temperature dependence has been reported [92, 206, 210]. Hill [210] gives room temperature values of 0.37 to 0.63. This result, that the internal quantum efficiency is one or close to one under favourable conditions (e.g. optimum doping level, see Section 7.3) and over a large temperature range, can be considered as quite significant for radiative recombination between tailed conduction and valence bands in direct gap semi-

The Injection Laser

41

conductors. In indirect gap materials the efficiency of the near edge emission due to shallow donor-acceptor pair emission drops very rapidly towards high temperatures [26, 27, 212], As demonstrated in Fig. 14, the external efficiencies of GaAs laser diodes are smaller and show a stronger decrease with increasing temperature than ^¡(stim). Besides the small temperature dependence of the internal efficiency they contain the temperature dependence of the laser losses in the case of ^ ext (stim), or the averaged diode losses a in the case of r/ext(spon). The number of non-radiative transitions in direct gap materials will change, of course, with the crystal properties (e.g. defect concentration or surface condition) and with the recombination mechanism. For example, recombination via tunneling or recombination in "pure" materials involving isolated impurity levels have smaller internal efficiencies than the ones reported here for laser type materials (see Section 7.3). 6.4 Influence

of laser

width

Theoretically, threshold current density and quantum efficiency of FabryPerot lasers with uniform junction properties should not depend on the laser "width" W, i.e. the lateral extension of the p n junction plane. Experimentally it was found, however, that both, jt and r/c.xt(stim), drop when W is increased [215]. In Fig. 15, jt and ^ ext (stim) at 77 °K have been plotted as a function of the width to length ratio WjL for a GaAs laser series of different width but of constant length. For WjL > 1 both jt and ^ ( . xt (stim) begin to decrease, and for very wide lasers ^ ext (stim) approaches ?^xt(spon). In that case, the differential quantum efficiency does not increase at all at the laser threshold. In order to understand this width effect one has to look at the spectral and spatial distribution of the stimulated emission from lasers with very large ratios WjL. The spatial distribution is not that of a usual Fabry-Perot laser; most of the light leaves the laser under a very wide angle through the sawed instead of the cleaved sides. The spectral distribution shows that the stimulated emission occurs on the low energy side of the spontaneous line, similarly as is seen for modes of total internal reflection. Apparently, light is sufficiently amplified on the path parallel to the reflecting sides, or under an angle leading to total reflection. This results in stimulated emission with low quantum efficiency at current densities lower than the threshold current density for Fabry-Perot modes. These off-axis non Fabry-Perot modes can be suppressed by interrupting the optical path which is parallel to the reflecting sides. The simplest approach is to cut grooves through the p-n junction, or to introduce absorbing layers perFig. 15. Threshold current density and external quantum efficie ncy of the stimulated emission as a function of the width to length ratio WjL at 77 °K. Zn-diffused GaAs laser Series with constant length L = 2.5 x 1 0 " ! cm. See [215]

M. H. PlLKUHN

42

pendicular to the cleaved faces. Such laser structures do not exhibit the described width effect [215]. I n " b a d " lasers containing junction irregularities the optical path parallel to the cleaved sides may already be interrupted, and these can therefore have better quantum efficiencies than planar structures if WjL ^>1. A special structure, in which the non Fabry-Perot modes are suppressed through curvature of the p-n junction, is the "cylindrical" Fabry-Perot laser [215], In this case, the laser diode is fabricated by epitaxial growth of a p-layer on the outside of a cylinder of n-material. These, and similar designs, gain technological importance if large size injection lasers with high total light power are built, because the junction width may be increased without deterioration of the quantum efficiency. 6.5 Optimum

laser

geometry

The question of optimum laser geometries is raised when the limits of power conversion efficiency, ryp, or the highest possible light powers obtainable from injection lasers, Qlt are considered. According to (49), the parameters determining Qi and ryp are the threshold current density and the differential efficiencies of spontaneous and stimulated emission. These, in turn, are functions of laser length and reflection coefficient if we neglect the width effect described in the previous section. In order to find the length dependence of QY or 7jp the following three effects have to be considered: 1. At a given current density and laser width the total power input (j L W V) may be increased with L. 2. The external quantum efficiency r/ ex t(stim) decreases with increasing length. 3. The threshold current density decreases with increasing length. The situation becomes more complicated if one includes thermal limitations on optimum laser performance. Thermal limitations are obvious, because heat is generated during laser operation which leads to an increase of jt and, to a minor extent, a dsc - a s s of the quantum efficiencies. These thermal limitations, which include the influence of heat flow and series resistance, have been treated in a number of papers [161, 216 to 220], and optimum conditions for all laser geometries, not only the length, are the result. The influence of the laser length on the power conversion efficiency rjv was considered by Akselrad [221]. By assuming the expressions (43) and (48) for jt and ^ cxt (stim), respectively, and taking the current density of operation, j, as a constant parameter an optimum length for maximum rjv was derived Lopt =

yj a P - a

,

(^(stim) = 1).

(50)

For a typical set of experimental data (a = 20 cm - 1 , fi = 2.5 x 10~2 cm/A, R = 0.25, and j = 105 A/cm 2 ), refering to 77 °K, one obtains Z op t = 0.68 X X lO - 2 cm. The maximum power efficiency a t L = L o p t is [221] (VvU,=

{l - i W l f -

(51)

The influence of the laser length on the total stimulated light power was considered by Pilkuhn and Rupprecht [215], As a constant parameter the dissipated heat power per unit area qh = (j L W V — Qi)jL W was used. If Joule heating due to series resistance is neglected, this can be written as Qh =

v

h [1 - W s p o n ) ] + V {j - jt) [1 - Jjext(stim)].

(52)

The Injection Laser

43

As long as threshold current density and external quantum efficiency are independent of W, it is useful to consider the total stimulated light output per unit width QJW = L V (j — jt) ?7ext(stim) and, taking the relation (46) for

^ext(stim), one o b t a i n s

9jw

-

-

< 5 3 )

(n ex t(spon) 1). In Pig. 16, the length dependence of QJW has been plotted taking the usual relation (43) for the threshold current density and the same set of experimental data as just above. The light output per unit width goes through a maximum and then reaches a constant value when the diode length is increased. If the internal efficiency is unity the maximum is very flat. The optimum length ¿opt depends on the parameter gh/ V, which is equal to j for long units, and, taking qJV = 105 A/cm 2 , we get Lapt = 2 x 10~ 2 cm. This value is slightly larger than the one derived for optimum power efficiency from equation (50) for the same data. An example for the case ^¡(stim) < 1 is included as a dashed curve in Fig. 16. The maximum of QJW is less flat and L0pt increases as the internal efficiency becomes smaller, for instance, L o p t = 8 x 10~ 2 cm, for ?^(stim) = 0.9 and all other parameters remaining unchanged. I t should be noted that the values derived for L o p t all lie within the usual experimental range of injection laser lengths. For infinitely long units QJW approaches the limiting value QJW (L^oo)

= ^

(j - ,

(2)

where H ( P is the operator of the electronic energy of the subsystem I: H ? w i y i * Relation (A 10) also gives the angle between the reciprocal lattice planes (Mx Vx t1 WJ)* and (U2 V2 t2 W2)*. Similarly, the angle between two planes corresponding to the reciprocal lattice vectors gt = [hx kx it Zj]* and g2 = \h2 Jc2 i2 ¡21 is given by 1

cos In

a \ -

^

-

W

+ h^

+ ^iz+X^lJt

|0xl ' 10.1 W + * ! + -

250

F i g . 6. Bleaching curve of S r F , coloured at 37 ° K with 1 x 10 s r, heated to TH, measured a t 37 ° K ,