Physica status solidi: Volume 5, Number 3 1964 [Reprint 2021 ed.] 9783112499962, 9783112499955

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pliysica status solidi

V O L U M E S • N U M B E R 3 • 1964

Contents Review Article P . GÖRLICH, H . KARRAS, G . KOTITZ, a n d R . LEHMANN

Spectroscopic Properties of Activated Laser Crystals (I)

Original Papers G. R I C H T E B Theorie des stationären photomagnetoelektrischen Effektes in InSb . H. GÖTZ Die Anregung von Scherschwingungen im NaC10 s -Einkristall mit Hilfe eines magnetischen Hochfrequenzfeldes . E. PBAVECZKI Definition and Use of Ferromagnon Annihilation Operators in the BLOCH-DYSON Theory of Ferromagnons

Page 437 463 477 481

D . MATZ a n d F . GAECIA-MOLINEB

Non-Ohmic Transport in Semiconductors in a Magnetic Field . . . .

495

B . DORNER a n d H . STILLER

Frequency Spectrum in Solid Methane at 6.6 °K

511

E . FORTIN a n d F . L . WEICHMAN

Photoconductivity in Ag 2 0

515

N . SEXER e t J . TAVERNIER

Phénomènes de transport dans les matériaux à bande non parabolique mais à symétrie sphérique

521

H . M . O T T E , J . D A S H , a n d H . F . SCHAAKE

Electron Microscopy and Diffraction of Thin Films: Interpretation and Correlation of Images and Diffraction Patterns J. LITWIN X-Ray Examination of the Binary System CdTe- CdSe J . LAGOWSKI Local Changes of the Work Function of Germanium and Silicon due to Dislocations

527 551 555

R . PERTHEL u n d H . J A H N P . PETRESCU

Über das paramagnetische Verhalten von Mn 3 0 4 und Co 3 0 4 Sur quelques phénomènes transitoires présentés par l'émission exoélectronique photostimulée des couches minces de Â1203

563 569

P . KBISPIN a n d W . LUDWIG

M. RÖSLER

Photodielectric Investigations on ZnS Phosphors Zur Theorie des elektrischen Widerstandes ferromagnetischer Metalle bei tiefen Temperaturen

573 583

R . GEVERS, P . DELAVIGNETTE, H . BLANK, J . V A N LANDUYT, a n d S . AMELINCKX

Electron Microscope Transmission Images of Coherent Domain Boundaries (II)

595

M . BALKANSKI e t M . NUSIMOVICI

B.

SELLE

Interaction du champ de rayonnement avec les vibrations de réseau aux points critiques de la zone de Brillouin du silicium Das Abklingen der Lumineszenz von ZnS :Mn bei Anregung im Gebiet der Mn-Eigenabsorption Photospannungen in Photoleitern ( I ) Photospannungen in Photoleitern ( I I )

635 649

W . RUPPEL W . RUPPEL G . LÜCK a n d R . SIZMANN

657 667

The Radiation Annealing of Frenkel Defects B. G I B B S The Thermodynamics of Creep Deformation G. ScHOTTKY Zur Anwendung des PEiERLSschen Modells auf schwach gekrümmte Versetzungen

683 693

G.

697

Short Notes (listed on the last page of the issue) Pre-printed Titles and Abstracts of Papers to be published in this or in the Soviet journal ,,H3HKa TBepnoro T e j i a " (Fizika Tverdogo Tela).

physica status solidi B o a r d of E d i t o r s P. A I G R A I N , Paris, S. A M E L I N C K X , Mol-Donk, W. D E K E Y S E R , Gent, W. F R A N Z , Münster, P. G Ö R L I C H , Jena, E. G R I L L O T , Paris, R. K A I S C H E W , Sofia, P. T. L A N D S B E R G , Cardiff, L. N É E L , Grenoble, A. P I E K A R A , Poznan, A. S E E G E R , Stuttgart, 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 , Cambridge, R. C O E L H O , Fontenay-aux-Roses, H.-D. D I E T Z E , Aachen, J . D. E S H E L B Y , Birmingham, G. J A C O B S , Gent, J . J A U M A N N , Köln, E. K L I E R , Praha, E. K R O E N E R , Clausthal-Zellerfeld, M. M A T Y A S , Praha, H. D. M E G A W , Cambridge, T. S. M O S S , 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 5 • Number 3 • Pages 435 to 712 and K 105 to K 160 1964

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

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UND

W I S S E N , E r i c h B i e b e r , 7 S t u t t g a r t S . W i l h e l m s t r . 4 - 6 or t o D e u t s c h e B u c h - E x p o r t u n d - I m p o r t G m b H , L e i p z i g C 1, P o s t s c h l i e ß f a c h 2 7 6

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Schriftleiter und verantwortlich f ü r den I n h a l t : Professor Dr. Dr. h. c. P. G ö r l i c h , Berlin C 2, Neue Schönhauser Str. 20 bzw. J e n a , Humboldtstr. 26. Redaktionskollegium: Dr. S. O b e r l ä n d e r , Dr. E. G u t s c h e , Dr. W. B o r c h a r d t . Anschrift der Schriftleitung: Berlin C 2, Neue Schönhauser Str. 20, F e r n r u f : 42 6788. Verlag: Akademie*Verlag G m b H , Berlin W 8, Leipziger Str. 3 - 4 , F e r n r u f : 220441, Telex-Nr. 011773, Postscheckkonto: Berlin 35021. Die Zeitschrift „physica status solidi" erscheint jeweils am 1. des Monats. Bezugspreis eines Bandes DM 60, — . Bestellnummer dieses Bandes: 1068/5. Für 1964 sind 4 Bände vorgesehen. Gesamtherstellung: V E B Druckerei „ T h o m a s Müntzer" B a d Langensalza. — Veröffentlicht unter der Lizenznummer 1310 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik.

Review

Article

phys. stat. sol. 5, 437 (1964) Jena, Carl-Zeiß-Straße

1

Spectroscopic Properties of Activated Laser Crystals (I)

Contents 1.

Introduction

2. General crystals 3.

remarks

on the geometry

and excitation

of laser resonators

of

activated

Theory

3.1 3.2 3.3 3.4 3.5

Spontaneous transitions Induced transitions Optical excitation Laser condition Spectroscopic requirements

4. Spectroscopic emission

data

on absorption,

4.1 Group of Lanthanides 4.1.1 General considerations 4.1.2 Experimental data 4.1.2.1 Cerium, 4.1.2.2 Praseodymium, 4.1.2.3 Neodymium, 4.1.2.4 Promethium, 4.1.2.5 Samarium, 4.1.2.6 Europium, 4.1.2.7 Gadolinium, 4.1.2.8 Terbium, 4.1.2.9 Dysprosium, 4.1.2.10 Holmium, 4.1.2.11 Erbium, 4.1.2.12 Thulium, 4.1.2.13 Ytterbium, 4.1.2.14 Lutetium, 4.1.3 Glasses 4.2 Actinide group 4.3 Transition metals 4.4 Semiconductors 29*

spontaneous

Z = 58 Z = 59 Z = 60 Z = 61 Z = 62 Z = 63 Z = 64 Z = 65 Z = 66 Z = 67 Z = 68 Z = 69 Z = 70 Z = 71

fluorescence,

and

stimulated

438

P . GÓRLICH, H . KARRAS, G. KOTITZ, and R . LEHMANN

5. Instrumental

set-up

6. General optical and thermal 7. Problems of crystal growth 8.

properties

Conclusion

1. Introduction In 1 9 5 8 SCHAWLOW and TOWNES [1] suggested the extension of the maser concept to the optical spectral region. The basic foundation of the phenomenological theory had already been outlined by them and their work includes several concrete suggestions as to the substance, the geometry of the resonators, and the excitation. In 1 9 6 0 , MAIMAN [2] used cylindrical ruby crystal resonators and achieved a distinct reduction of the half width for the R 1 emission line (6943 Á, 300 °K) by producing the necessary inverted population by means of high-pressure flash lamps. In rapid succession papers were published on the laser action of gas mixtures (HeNe) and gases, of activated crystals, glasses, and liquids and, in an ever increasing degree, of semiconductor crystals. The rapid development of experimental research led to the desire for accounts reviewing the state of knowledge in short time intervals. Survey articles on the theoretical foundation, the experimental results in the field of gaseous and solid state lasers and the possibilities of their technical utilization were published by KAISER [3, 4 ] and Y A R I V and GORDON [5]. A survey of the laser properties of rare-earth activated crystals was given by JOHNSON [6], while the basic spectroscopic characteristics of solid state lasers were discussed by SUGANO [7]. Because of the large number of papers published in this field we think it useful to restrict our report on the spectroscopic properties of laser substances to the activated crystals and to give only a brief account of glasses and semiconductors. For the work on non-linear optics which will likewise only be dealt with in passing, reference should be made to the publication of BLOEMBERGEN [8]. With regard to the nomenclature of the formulas used by us we are leaning on YARIV and GORDON [ 6 ] and KAISER [4], Furthermore, we confine ourselves in this contribution to papers published after 1960; the earlier literature on absorption, coloration, and luminescence of the alkaline earth halides has been referenced in the review article by GÓRLICH, KARRAS, and LEIIMANN [9]. 2. General Remarks on the Geometry and Excitation of Laser Resonators of Activated Crystals Laser resonators of activated crystals essentially consist of three components : 1. the atomic system, in which the emission processes take place, 2. the crystalline host, in which the atomic systems (activators) are embedded in atomic distribution, 3. a suitable resonance structure for sustaining an electromagnetic field configuration. As activators may be used ions of the transition metals, of the rare earths and the actinide group which are incorporated in the crystalline host during crystal synthesis. In most cases the activator ion is incorporated by isomorphous substitution of an intrinsic ion. The activated crystal is then optically processed to obtain the required geometry of the resonator which in many technical applications consists of a polished cylinder with silvered plans or curved end faces.

Spectroscopic Properties of Activated Laser Crystals (I)

439

The host crystals must have a large spectral range of transmission. Therefore the bonds of the crystal lattice usually are of the heteropolar or homopolar type or a combination of both types (sapphire, A1203). In each case crystals are involved, in which the energy gap between the conduction and the valence bands is several eV. The pure host crystals are therefore completely colourless optical media. The activator ions give rise to energy states, which come to lie in the forbidden gap between the energy bands of the crystal. The structure of the energy states is primarily determined by the activator. Interaction with the lattice leads to multiplet splitting; it is strongest for ions of the transition metals. As a direct consequence of the incorporation of activators the activated crystals exhibit both selective absorption regions, which frequently fall into the visible spectral region and then give rise to the deep colours of the laser crystals, and spontaneous fluorescences. The absorption and fluorescence spectra as well as the pertinent energy diagrams are complex structures, which are discussed in Section 4. In order to assist in understanding the laser emission we may for the present content ourselves with a schematic simplification of a four and three-level system, (Fig. 1) on the basis of which we shall briefly describe the excitation mechanism. The activators are excited by optical frequencies (optical pumping), which gives rise to the transition from the ground state 0 to the excited state 3. For simplicity the latter has been shown as a discrete state. Naturally the transition may also lead to the conduction band of the crystalline host. The population of the metastable state 2 is accomplished by nonradiative transitions from state 3. A high quantum efficiency for the non-radiative transition is an essential requirement for achieving high population densities in metastable states. Maser action leads to a high transition probability to the terminal state 1, which without emitting radiation is depopulated by transitions into the ground state 0. For threelevel lasers the terminal state of emission will under ideal conditions coincide with the ground state 0. Practically one refers to a four-level laser, if E1 8 kT and to a three-level laser, if E1 kT.

3 PUMP LEVEL E3 1 METASTABLE LEVEL Ez

C: § hvL*E2-E,

Fig. 1. Four-level laser diagram. 3—>2 and 1 —• 0 non-radiative transitions

440

P . GOKLICH, H . KAKRAS, G . KOTITZ, a n d R . LEHMANN

B y heavy cooling a three-level laser will under favourable conditions be converted into a four-level laser. These rather general statements already allow some important conclusions to be drawn: Relative to host crystals:

a) The unactivated host crystals should have neither intrinsic nor extrinsic absorptions for the excitation regions (transition 0 — 3 ) and the emission regions (transition 2 1). To avoid unnecessary heating by the absorption of the exciting light usually irradiated in a broad band the host crystals should also outside the excitation and emission regions be free from selective absorption. b) The host crystals should have a high thermal conductivity so that the energy conducted to the lattice without emitting radiation (transitions 3 - ^ - 2 and 1 - > 0) and the energy absorbed by irradiation may rapidly be dissipated. c) The host crystals are required to have good optical as well as mechanical properties. Schlieren, strains, and bubbles increase the threshold, at which maser action starts. Chemical impurities may give rise to parasitic absorption bands and an unwanted interaction with the activator. Unsufficient hardness and poor chemical stability of the host crystals enhance the difficulty of producing the required resonant structure, which calls for close geometrical (planeness, parallelism) and optical (index of refraction) tolerances. d) The lattice structure of the host crystals should allow the incorporation of activators. The more favourable the geometrical conditions (ion radii) are, the greater the activator concentration may be without giving rise to noticeable optical disturbances (strain, inhomogeneous distribution of activators, colloidal precipitation of activators). e) Different valencies of the activator ions and the substituted host ions should allow of being compensated by the incorporation of additional foreign ions or by non-stoichiometric intrinsic ions (in interstitial positions). f) Host crystals should permit of being obtained from synthesis in large specimens, which must have an excellent optical quality. Relative to activators:

g) The pumping light (transition 0 - 5 - 3 ) should strongly be absorbed (great oscillator strengths). h) The spectral region of excitation is required to coincide with the peak of the spectral energy distribution of the pumping light source. i) The quantum efficiency of the non-radiative transition 3—^2 should be as great as possible ( « 1 ) , so that a high population density of the metastable state is achieved. k) The average lifetime of the metastable state 3 should not be too small in the interest of a high population density. 1) The terminal state 1 should be depopulated as quickly as possible into the ground state 0. m) To avoid unnecessary heating by radiation the activator should not exhibit any selective absorption bands apart from those necessary for excitation. n) The wavelength of the laser light should be appropriate to the respective application, e. g. infrared: for telecommunications (little scattering in atmosphere) and as light sources in place of energetically unfavourable thermal radiators;

Spectroscopic Properties of Activated Laser Crystals (I)

441

visible region: for technical and scientific investigations in conjunction with visual-optical measuring instruments (microscopes, precision measuring instruments) ; ultraviolet : for photochemical and biological processes. Relative to the instrumental

set-up :

o) Greatest accuracy and closest tolerances in the production of laser resonators ; p) Largest possible resonators for the respective resonance geometry ; q) Optimal coupling, e. g. reflectors of dielectric layers instead of metal layers (absorption of the laser light in the metal layer) ; r) Pumping light sources of high energy and durability with a spectral energy distribution favourable for the respective activator ; s) Optimum optical transmission of the pumping light into the resonator by suitable construction of the reflectors and by using highly reflecting and stable coatings or walls, respectively; t) Use of absorption filters for eliminating the light not required for excitation to avoid unnecessary heating; u) Suitable constructive elements for dissipating the thermal energy from the resonator rod ; v) Development of cryostats for the laser action at low temperatures. A number of these statements may qualitatively and quantitatively be derived from theory. From the spectroscopic properties of activated crystals information may be obtained relative to their suitability as laser medium. So as to exactly define the spectroscopical requirements the theoretical fundamentals have to be dealt with in some more detail. 3. Theory 3.1 Spontaneous

transitions

Expressions for the transition probability A of a spontaneous emission process may be derived from quantum theory [10]. For the simplest case of dipole oscillations first the quantity of the electric dipole moment SR has to be determined, which is the product of oppositely equal charges and the distance of the centres of gravity of electric charges : 2R = e r ,

(1)

e electron charge. The mean distance r is determined as _ _ / V* x V dr 1 ~ f '/'* '¡',lr with the wave function ¥ = y)

e~2nivt

^ ^

and the conjugate complex function

obeying the Schrôdinger equation d2W a r = - 4

(3)

Solutions for the stationary states of the atomic systems can be obtained from the Schrôdinger equation merely for definite eigenfrequencies V{ and for definite eigenvalues of the energy E{ — h i>£. To these eigenfrequencies and eigenvalues

442

P . GORLICH, H . KARRAS, G . KOTITZ, a n d R . LEHMANN

of the eigenfunctions

belong the normalization and orthogonality conditions f Vi V>t* dr = 1; iy>mW*dt =

0.

The dipole moment corresponding to the transition from En to Em is obtained from the relations (1) and (2) as follows: 2Jt = e2ni(-'m~ "»>' e J y>* r y>m dr;

(4)

9!)t = e » ' « " - - T l n m (Stat) .

(5)

2

In the transition of the atom from state En to state Em an oscillation of the charge density is produced having the frequency vnm jj* m » Vnm = Vn — Vm = (6) h Planck's constant, which according to the classical electrodynamics is associated with the radiation of an electromagnetic wave. 2JiMm (stat) stands for the temporally constant (stationary) electric dipole moment. The energy loss per second of the radiation source consisting of N dipoles is

c velocity of light, or upon differentiation of (4) and taking the mean value cos2 (2 tt vnm — d) = 1/2 (d phase constant) we obtain — 64 7il Vnm &= (stat)] 2 , (8) = \efw* *Vmdz\ = e t „ .

(9)

According to the transition probability Anm which is defined as the probability of a spontaneous radiative transition of an atom per unit time from the state En to the state Em the following formula is obtained for the energy radiated per second (J/s): (10)

S = AnmNhvnm and by using the relations (8) and (9) nm

_

64 ji4 3 ^

3 nm

2

2nm

'

y*1)

The components (matrix element) of the integral term xnm Xnm = iv*

xfmdz,

Ynm = / rp* y Wm dr ,

Znm

= J rp* z y>m dt,

allow the calculation both of the selection rules and the polarization properties of the spontaneous emission. For the forbidden dipole transitions all matrix elements tend to zero. The decay time or mean spontaneous lifetime of the excited state En is obtained from the transition probability: fnm Zn — a '

r nm lnm — y a '

M

The oscillator strength fnm is unity, if only one transition from En to Em is possible.

Spectroscopic Properties of Activated Laser Crystals (I)

443

From (11) and (12) we obtain r

_

4 3 2 2 64 Ut jiJl vv nm ee rl n m for the mean spontaneous lifetime of the excited state.

"

3.2 Induced

transitions

The electron oscillating with the frequency vnm (according to Equation (4)) during its transition from En to Em may interact with an external electric field 6 = ®0 e2"iH

(14)

and may absorb or lose energy depending upon the phase position 0 between electron and field. The time constant of this "induced" process is _ Tind

~

2 7i mv0rnm e|®0| cos 0 '

(1,J)

m electron mass. Due to its dependence upon the external field this process may be designated as an induced process. While the spontaneous process takes place only in one direction, in emission (En -> Em), the induced process may take place in two directions, namely in emission (En —> Em) and in absorption (Em —> En). At thermal excitation and thermodynamic equilibrium the number Nn of the excited atoms is determined by the Boltzmann statistics: _ En W Nn = N0e , (16) k Boltzmann constant, T absolute temperature. In the case of degeneracy the energy state of the quantum number J consists of (2 J + 1) coinciding states, which under the influence of magnetic (Zeeman effect) and electric (Stark effect) fields are split up into term components. Taking into consideration the statistic weights of the ground state g0 and the excited state 9n (9j = 2 J + 1) the population numbers then are determined as Nn=N0^e

_ kEil T

9o

.

(16a)

In addition to the spontaneous emission transitions caused by the influence of the thermal radiation field, both induced emission and induced absorption transitions are likewise partaking in the establishment of the equilibrium condition. On the assumption that the induced transition probabilities are proportional to the blackbody radiation density per unit frequency interval q(v) (J/m 3 Hz) the following relation [11] can be set up for the emission and absorption processes: (induced) absorption = spontaneous emission + induced emission; Q(v) BmnNm

= AnmNn

+ q(V) BnmNn.

(17)

Hence it follows by using the relations (16) and (17) Q(Vnm) = T>

„

kl1

(18) i?

444

P . GÔRLICH, H . KAEEAS, G . KÔTITZ, a n d R . LEHMANN

and from a comparison of the coefficients with the Planck radiation formula (unpolarized blackbody radiation density per unit frequency interval (J/m 3 Hz)) Q(Vnm)

8 nhv* m

=

1

ci

e

'

kT

- 1

for the coefficients in relation (17): B m n = Bnm

and

(20)

3

c

Bnm ~ ¿j

i 3 h vfim

on

Anm .

(21)

Thus Equation (17) can be rewritten as follows: Wind

=

Anm

N

n

+ W

l

i n i

N

n

(22)

,

where W^d is the transition probability for the induced processes. By rearranging and substituting the spontaneous transition probability Anm (11) we obtain t Wind

=

N

-^n m nm

^ m

1

=

A

nm

IVT

~hVn

e

64 n* ( Wind = g ^ y vnm

1

!

m

kT

m ~hr~

(23)

-1 1

>

fàty

8 ji3 e2 t 2 WU=H7l^h"nmp(vnm).

(25)

The ratio n of the induced and spontaneous transition probabilities is ~ " - A

Wind n

m

1 =

~

c

kT

Q{V)

&nT*

•

( 2 6 )

- 1

In the thermal radiation field the fraction of the induced processes for the optical frequencies is known to be very small due to v~3. At a temperature of T = 300 °K and a wavelength a = 1 [i.m n « 10~78. I t is only at sufficiently large wavelengths that the ratio becomes h > 1 (n = 1 for A = 69.3 ¡¿mat T = 300 °K). In deriving the formulas (23) through (25) for the induced transition probability PPind the line profile and the difference in the spectral width of the laser emission as compared with the atomic absorption region was not taken into consideration. Our further calculations are closely following those of Y A B I V and G O E D O N [5]. The function of the line profile be g{v). Defining g(v) dv as the probability of absorbing or emitting a photon with an energy between h v and h (v dv) for any given transition, we obtain oo / g(v) dv = 1 . (27) 0 Rearrangement of (26) and insertion of g(v) dv leads to the following expression: Wind g(v) dv = q(v) g ^ r ^ Anm g(v) dv .

(28)

Spectroscopic Properties of Activated Laser Crystals (I)

445

Anm g(v) dv is the spontaneous transition probability for the emission of a photon having an energy between h v and h (v + dv). W\ai g(v) dv is the induced transition probability caused by the spectral energy density Q(V) in the frequency range between v and v + dv. The energy flux (W/m2) between the frequencies v and v + dv is J(v) dv = c Q(V) dv . (29) If the transition is induced by a monochromatic radiation /„ (W/m2), one may also write I(v) = /„

30

1 10(77°K) physica

[6]

—2000

[6] [16] [6]

~2000

3

0.002 15

~2000

1.5 0.6

0.5

3/2

12

2

11/2

11/2

~2000 — 2000 -2000 -2000 263 270 90

[22] [6] [23] [6] [6] [5, 12, 13] [24] } [25] } [26] } [27]

35

[5]

452

P . GÖRLICH, H . K A K B A S , G . K O T I T Z , a n d R . L E H M A N N

Table 1 Activator

Host substance

Ho 3 +

2.046 2.059 2.047 2.092

77

1.612

CaW0 4

4.2 20 77 4.2 27 77

Ca(Nb0 3 ) 2 SrF2

77 77

1.1160 1.1153 1.1153 1.116 1.116 1.911 1.916 1.91 1.972 1.918

CaW0 4

77

Ca(Nb0 3 ) 2 CaF2

77 77

Er 3 +

CaW0 4

Tm 2 +

CaF,2

CaF2 Tm 3 +

Nd 3 +

glass

300

Gd 3+

glass

78

Ho 3 +

glass

77 77

Yb3+

glass

U U

CaF2

U3+ XJ3 +

CaF2 CaF2 CaF2

78 78 300

U3+

CaF2

77

u

u XJ3 +

U

!

o

3

i i Si Si

I I I I I

Ni 2 +

l 1 !

M g F

2

20

pulsed

800

J

0.28—0.63 0.28-0.63

pulsed pulsed pulsed cont. cont. pulsed pulsed pulsed pulsed pulsed

50 450 800 600 1000 60 73 125 1600

J J J W W J J J J

pulsed

50

0.274; 0.277

pulsed

4300

J

0.44-0.46

pulsed

3600

J

pulsed

1300

J

pulsed pulsed

2.51 2.61 2.57

pulsed pulsed pulsed 0.88-0.92

2.613

(20 ° K ) (77 ° K ) (90 ° K ) (20 ° K )

1.1; 1.5

i 1 ! i i I

°K) °K) °K) °K)

8 32 38

0.44-0.64

1.622

2 (20 3.78 (77 4.35 (90 «¿1200 (300

0.4-0.6; 1 — 1.3 pulsed

0.6934(jBj) 0.6929(-R2)' 0.6934(Ä1)' 0.701 1 0.704 1

J

cont.

i 1

2.556

0.2542 0.3856 0.4128 0.4131

Si0 2 (glass)

0.38—0.52

0.91; 0.95; 0.98

1.015

2.407

2

J J J J

2.5 2.6

SrF2

a i

0.44—0.46

80-300 250 90 260

0.42-0.7 0.42—0.7 0.46-0.48 1.7 - 1 . 8

>1.95

Threshold (J or W )

pulsed pulsed pulsed pulsed

0.44 •••0.46

1.06

2.5 2.24

300 290 77 77

Pumping region Operating mode (|xm)

0.3125

CaF2 CaF2

j BaF 2

3 +

Cr3+

1

4.2

C a F ,

XJ3 +

Laser wavelength (fim)

Temperature (°K)

0.1—0.2

pulsed

12

i 1 j j |

pulsed pulsed cont. pulsed pulsed

800

i j ! 1

pulsed pulsed pulsed pulsed

j pulsed

J

Spectroscopic Properties of Activated Laser Crystals (I)

453

continued Lifetime (spont.) (ms)

Linewidth (spont.) (cm" 1 )

Position of terminal state (cm- 1 )

Laser transition

References

230

[28] [5] [16] [6] [29] [30] [6] [6]

2.2 375 556 0

4 4

} [31] 325

[32] [5] [16] [6] [5] [33] [35] [36] [37] [13]

~2000 4 0.7 1.5 136 (4 ° K )

4

2-^7/2 -^5/2 f 11/2 ^ 4 JT9/2

l

T

2

T

4

11/2 ^ 130 (4.2 130 (20 130 (77 95 (90 j Lorentzkräfte mit ö

i = \ g r a d fc 6» •

Ein dem Rekombinationsglied

entsprechendes Generationsglied ist hier

nicht angeführt, d a wir die Generationsprozesse in F o r m von Randbedingungen berücksichtigen werden. 1 ) Ein Zweiensemblemodell ist das einfachste, welches einen nichtverschwindenden P M E - E f f e k t liefert. U m konkret zu sein, wollen wir die Funktionen 6J(!) mit dem von KANE [5] gegebenen Modell identifizieren u n d mit drei Valenzbändern rechnen. Den Index i = 1 ordnen wir dem Leitungsband z u : 6j = € n ; die Indizes i = 2, 3, 4 beziehen sich auf die Valenzbänder. Entsprechend nennen wir e 2 = e p l , e 3 = e 4 = e p 3 . Wir nennen die Verteilungsfunktion f ü r Elektronen f1 u n d f ü r Defektelektronen [11] /J (i = 2, 3, 4). Wir machen die Voraussetzung A : Stoßprozesse sind sehr viel häufiger als Rekombinationsprozesse. Die unter Stößen sich einstellende Verteilung betrachten wir als einen Quasigleichgewichtszustand, der durch Rekombinationsprozesse in das eigentliche Gleichgewicht übergeht. Die Elektronen bzw. Defektelektronenverteilungsfunktionen, die den Zustand des Quasigleichgewichtes beschreiben, seien

/? = —

1

l + expe»-CW kT

ij+1 —

£(R\ _ E . ' 1 + exp

/ — 1, 2, 3 .

kT

Wir wählen die Wellenlänge des Lichtes zur Erzeugung der Elektron-Loch-Paare so, daß vollständige Absorption in einer sehr dünnen Oberflächenschicht des Kristalles erfolgt. Verallgemeinerung auf volumenmäßige Generation bereitet keine prinzipiellen Schwierigkeiten (GÄRTNER [12]).

Theorie des stationären photomagnetoelektrischen Effektes in InSb

465

Sie unterscheiden sich von der Fermiverteilung durch die Ortsabhängigkeit der Potentiale £(t) und |(r). E s sei betont, daß auf Grund dieses Ansatzes die Quasigleichgewichtsverteilungen der drei Valenzbänder durch ein gemeinsames Potential |(r) beschrieben werden. Nach S H O C K L E Y [ 1 3 ] nennen wir F und £ Quasifermipotentiale. Die Gleichungen (1) schreiben wir abkürzend in der Form i = 1, 2, 3, 4 .

(2)

Die Lösung der Gleichungen (2) soll in zwei Schritten durchgeführt werden: In einem ersten Schritt wird die f-Abhängigkeit der Verteilungsfunktionen bestimmt. Aus der Voraussetzung A schließen wir, daß Rekombinationsprozesse keinen wesentlichen Einfluß auf die f-Abhängigkeit der Verteilungsfunktionen ausüben. Man erhält gute Näherungen durch Lösen der Gleichungen

Die Gleichungen (3) werden für eine beliebig vorgegebene Ortsabhängigkeit der Quasifermipotentiale gelöst. Nach der Berechnung der f-Abhängigkeit der Verteilungsfunktionen fi wird in einem zweiten Schritt die Ortsabhängigkeit der Quasifermipotentiale aus den Gleichungen

«'•MIHI),-

bestimmt. Die Gleichungen (4) entstehen aus den Gleichungen (2), wenn man annimmt, daß die Rekombinationsprozesse nicht von Feldern beeinflußt werden. 3. Lösung der Transportgleichungen Über die Stoßterme machen wir Voraussetzungen derart, daß man die Gleichungen (3) mit dem Ansatz

unabhängig voneinander nach bekannten Verfahren lösen kann. Vorausgesetzt wird die Existenz einer nur energieabhängigen Relaxationszeit [19, 20] für jedes Ladungsträgerensemble und isotrope, aber nicht unbedingt quadratische f-Abhängigkeit der Energie. Während wir die elektrische Feldstärke auf den Bereich linearer Effekte abgrenzen, darf das magnetische Feld beliebig große Werte annehmen, vorausgesetzt nur, daß Quanteneffekte vernachlässigt werden können. Die in den Boltzmanngleichungen auftretenden Ortsgradienten der Verteilungsfunktionen fi ersetzen wir näherungsweise durch die Ortsgradienten der QuasiGleichgewichtsverteilungsfunktionen dadurch können wir die Gleichungen (3) in der Form i(3f,gradf/i.)-(f)s( = 0

(5)

schreiben. Die %* entstehen aus den % it indem wir in ^ bzw. (j = 1, 2, 3) das elektrostatische Potential e cp durch die ortsabhängigen Quasi-Fermipotentiale C(r) bzw. £(r) ergänzen (Feldstärke g = — Vr ?)• Lösungen der Gleichungen

466

G. RICHTER

(5) erhalten wir nach üblichen Verfahren. Nach B A R R I E [14] läßt sich das Lösungsverfahren von WILSON [15] auf nichtparabolische Bandstruktur verallgemeinern. Die Ergebnisse dieser Methode sind bei HARMAN und HONIG [16] ausführlich dargestellt. Zu gleichen Ergebnissen führt das allgemeinere Lösungsverfahren von LIFSHITZ, A Z B E L und KAGANOW [17]. Die Ergebnisse dieser Methode für den Fall e(f) = e(|f|) sind ausführlich bei KOLODZIEJCZAK [18] dargestellt. 4. Bestimmen der Quasi-Fermipotentialc 4.1

Kontinuitätsgleichung

In diesem Abschnitt werden Gleichungen abgeleitet, die wir zur Bestimmung der Ortsabhängigkeit der Verteilungsfunktionen verwenden. Mit bekannter f-Abhängigkeit der Verteilungsfunktionen /,(!,£)

und

/i+1(U)

werden wir jetzt jede der Gleichungen (4) über f integrieren. Hierbei machen wir von der Tatsache Gebrauch, daß Stöße die gesamte Teilchenzahl innerhalb des Elektronenensembles und innerhalb des Defektelektronenensembles nicht ändern, d.h. JiZ,

ffl^l"-"BZi

«"»

Die Integration führt zu den Gleichungen -|div|gradte„/1^ = / ( f ) / !

)

(7 a)

-jidivfgradf ^fi+i dt=+i / Pin*dt •

(7b)

Die Summation über i in der Gleichung (7 b) ermöglicht es, die Bedingung (6 b) auszunutzen. Nach Einsetzen der Lösungen der Gleichungen (5) erhalten wir - - i d i v a » ( ( £ - y V r i ) = Un , y d i v o ^ f ö - y V t f ) = Up.

(8a) (8b)

Abkürzend steht für die rechten Seiten der Gleichungen (7 a) bzw. (7 b) Un 3 bzw. Uv. Wenn das Magnetfeld in z-Richtung liegt, sind an und av = av i = 1

Tensoren der Form Oxx

0X y

a

Oxx

x y

N

0

1

«9

0

0 0

Theorie des stationären photomagnetoelektrischen Effektes in InSb

467

mit entsprechenden Indizes n, pv p2, p3, wo oo

r(») B dk , 1 + (TW a>nY

a'ir. = c J 8en\dk)

' m lde«Y J 8en\dk) 1 ,

Oü J

CWtteX Sen\dk)

' ndk ,

k 1

+

(T W N Y

T