Optical properties of the crystalline modifications of boron and boron-rich borides

prog. Crystal Growth and Charect. 1988, Vol. 16, pp. 179-223 Printed in Great Britain. All rights reserved

0146-3535/88 $0.OO + .50 Copyright © 1988 Pergamon Press plc

OPTICAL PROPERTIES OF THE CRYSTALLINE MODIFICATIONS OF BORON AND BORON-RICH BORIDES Helmut

Werheit

Universitat Gesamthochschuie Duisburg, Laboratorium f/Jr Festk6rperphysik, D-4100 Duisberg, FRG

1. INTRODUCTION T h e c r y s t a l l i n e m o d i f i c a t i o n s o f b o r o n a n d o f t h e b o r o n - r i c h b e r i d e s b e l o n g to t h e s o l i d s w i t h c o m p l e x c r y s t a l s t r u c t u r e s , w h o s e m a i n p h y s i c a l p r o p e r t i e s a r e to s o m e e x t e n d q u a l i t a t i v e l y d i f f e r e n t f r o m t h o s e o f s o l i d s w i t h s i m p l e p e r i o d i c s t r u c t u r e s . S i n c e s e m i c o n d u c t o r s a r e well k n o w n to r e s p o n d v e r y s e n s i t i v e l y to s t r u c t u r a l d e f e c t s o f a n y k i n d , it i s n o t s u r p r i s i n g t h a t especially the electronic and the transport properties of the semiconducting or iselatlng boron-type crystals differ even qualitatively from those of classical semiconductors. Obviously, f r o m t h e v i e w - p o i n t o f t h e s e c l a s s i c a l s e m i c o n d u c t o r s t h e a r r a n g e m e n t s of a t o m s in complex periodic structures imply such a strong disorder that its effect e.g. on the transport p r o p e r t i e s is c o m p a r a b l y s t r o n g a s t h e t r a n s i t i o n f r o m t h e c r y s t a l l i n e to t h e a m o r p h o u s p h m m destroying the crystalline long-range order completely. Such conclusion is especially suggested by the electronic transport properties of the modifications and compounds of boron a n d o f t h e b o r o n - r i c h b o r i d e s , be~.ause t h e p h y s i c a l t r a n s p o r t m e c h a n i s m s a r e a t l e a s t c l o s e l y r e l a t e d to t h o s e o f a m o r p h o u s s e m i c o n d u c t o r s [1-4]. The structural variety of the crystalline modifications of elementary boron and of the numerous beriden offer excellent prerequisites for systematic investigations of complex crystal structures and the influence of the degree of complexity on the physical properties. T h e s i m p l e s t s t r u c t u r e o f t h i s s e r i e s o f c o m p l e x c r y s t a l s is f o u n d in t h e a l p h a - r h o m b o h e d r a l m o d i f i c a t i o n o f e l e m e n t a r y b o r o n w i t h o n e Bit i c o s a h e d r o n p e r u n i t cell; t h e m o s t ,complex structure presently k n o w n belongs to borides of type Y B ~ with 104 Bts icoeahedra in the cubic unit cell. By this the structural relationship of the boron-related rxystais is already mentioned: Most of their structure families contain these nearly regular icosahedra, whose high symmetry is indicated b y 6 five-fold, 10 three-fold and 15 two-fold axis of rotation as symmetry elements. Besides fragments and condensed systems of the icomahedra are found in the structures. Periodical arrangements of these polyhedra in chains, layers or threedimensional structures form the main networks of the crystals. Moreover, these networks contaln a n u m b e r of voids which are large enough to accomodate additional boron or foreign atoms in periodical or statistical arrangement. M a n y borides and solid selutionn can thus be derived from the modifications of elementary boron, and 'consequently an especially narrow r e l a t i o n s h i p w i t h i n t h e s e s t r u c t u r e f a m i l i e s c a n be e x p e c t e d . T h e c i t e d i n t e r s t i t i a l a c c o m o d a t i o n o f f o r e i g n a t o m s o f f e r s a n o t h e r a s p e c t to i n v e s t i g a t e p r o p e r t i e s o f s u c h s t r u c t u r e s s y s t e m a t i c a l l y . On o n e h a n d t h e i n f l u e n c e o f d i f f e r e n t s i n g l e a t o m s on the properties of the boron structure can be compared; on the other hand the interaction o f t h e s i n g l e a t o m s , w h i c h a r e d e f i n i t e l y s e p a r a t e d in a h i g h l y i o o l a t i n g d i e l e c t r i c m e d i u m c a n be the object of investigation. T h e s y s t e m a t i c i n t e r r e l a t i o n b e t w e e n m a i n p h y s i c a l p r o p e r t i e s o f t h e s e s o l i d s p r o v e d to b e e x i s t e n t e . g . in t h e e l e c t r o n i c p r o p e r t i e s implied b y t h e s t a t i c p r o p e r t i e s o f t h e c r y s t a l s t r u c t u r e . I n s p i t e o f t h e u n p a i r e d e l e c t r o n in t h e o u t e r s h e l l o f t h e b o r o n a t o m , r e m a r k a b l y

179

180

H. Werheit

all m a t e r i a l s of i c o s a h e d r a l b o r o n s t r u c t u r e s hithertoo investigated are semiconductors, F u r t h e r m o r e t h e a l r e a d y m e n t i o n e d t r a n s p o r t p r o p e r t i e s seem to be s y s t e m a t i c a l l y i n t e r r e l a t e d , too, s i n c e in all s e m i c o n d u c t o r s of i c o s a h e d r a l b o r o n t y p e h o p p i n g p r o c e s s e s d o m i n a t e the electronic t r a n s p o r t (cp.[l-4]) I n spite, of t h e o b v i o u s l y s t r o n g b o n d i n g w i t h i n t h e i c o s a h e d r a , t h e b o r o n t y p e c r y s t a l s a r e not a t all molecular c r y s t a l s . 26 e l e c t r o n s o c c u p y t h e s t r o n g l y b o n d i n g o r b i t a l s w i t h i n t h e i c o s a h e d r o n a n d t h e r e m a i n i n g 10 e l e c t r o n s a r e a t t h e d i s p o s a l of t h e l i n k a g e to o t h e r i e o s a h e d r a o r s t r u c t u r e u n i t s [5]. T h e ionic c h a r a c t e r of t h e s e b o n d i n g s is not v e r y s t r o n g , w h i c h follows from t h e o s c i l l a t o r s t r e n g t h s of lattlee v i b r a t i o n s . N e v e r t h e l e s s t h i s i n t e r i c o s a h e d r a l b o n d s t r e n g t h is a s s u m e d to be e v e n s t r o n g e r t h a n t h e i n t r a i c o s a h e d r a l o n e [6], w h i c h a c c o u n t s f o r h i g h m e l t i n g I ~ i n t s , small e x t e n s i o n c o e f f i c i e n t s , g r e a t h u r d n v s s e s a n d r e l a t e d mechanical p r o p e r t i e s . T h e c o n t r a s t to molecular c r y s t a l s c a n c l e a r l y be s e e n in t h e p h o n o n s p e c t r a ( s e e s e c t i o n 4.4): I n t e r n a l a n d e x t e r n a l v i b r a t i o n s of t h e i c e s s h e d r a a r e s u p e r i m p o s e d a n d not s e p a r a t e d like in molecular c r y s t a l s . T h i s example a l r e a d y e m p h a s i z e s o p t i c a l p r o p e r t i e s to be a p o w e r f u l i n s t r u m e n t of a s y s t e m a tically c o m p a r i n g r e s e a r c h of t h e m o d i f i c a t i o n s a n d c o m p o u n d s of b o r o n b e e i n g o b j e c t of t h e p r e s e n t a r t i c l e . S u c h e x p e r i m e n t a l m e t h o d s ~tre e s p e c i a l l y n e c e s s a r y in c a s e s like b o r o n a n d its c o m p o u n d s , s i n c e t h e complex s t r u c t u r e s a n d g r e a t n u m b e r s of a t o m s p e r u n i t cell a g g r a v a t e t h e o r e t i c a l c a l c u l a t i o n s exceptionally~ a n d t h e r e f o r e a t p r e s e n t t h e y a r e a v a i l a b l e o n l y to a v e r y limited e x t e n t . By all m e a n s , a l s o t h e e x p e r i m e n t a l r e s u l t s a v a i l a b l e a r e f a r from s u p p l y i n g a c o m p r e h e n s i v e p i c t u r e of t h i s g r o u p of s o l i d s . N e v e r t h e l e s s t h e y a r e s u f f i c i e n t to s h o w a t l e a s t t r e n d s of s y s t e m a t i c i n t e r r e l a t i o n s . As will be s h o w n , e s p e c i a l l y t h e i n v e s t i g a t i o n of o p t i c a l p r o p e r t i e s y i e l d e d v a l u a b l e i n s i g h t i n t o t h e s e i n t e r r e l a t i o n s a n d allow a l s o to i n t e r p r e t o t h e r p h y s i c a l p r o p e r t i e s [7].

2. STRUCTURAL RELATIONSHIPS OF BORON TYPE CRYSTAI,S To g e t a s y s t e m a t i c a n d c o m p a r a b l e collection of t h e c r y s t a l p r o p e r t i e s j it is n e c e s s a r y to c l a s s i f y t h e m o d i f i c a t i o n s a n d c o m p o u n d s a c c o r d i n g to t h e i r s t r u c t u r a l r e l a t i o n s h i p s , e v e n t h o u g h t h e i n f o r m a t i o n o n t h e i r p r o p e r t i e s is n o t y e t complete.

I n t h e b a s i c s t r u c t u r e of a l p h a - r h o m b o h e d r a l b o r o n t h e c o r n e r s o f t h e u n i t cell a r e o c c u p i e d b y o n e BJa i c o s a h e d r o n , e a c h , b e e i n g i n s i g n i f i c a n t l y d i s t o r t e d in t h e d i r e c t i o n o f t h e t r i g o n a l axis. The a t o m s o n t h e e d g e s of t h e r h o m b o h e d r a l cell a r e covalently bonded, whereas the equatorial atoms of t h e i c o s a h e d r a f o r m t h r e e - c e n t e r b o n d s . The s t r u c t u r e may be c o n s i d e r e d a s a s l i g h t l y d e f o r m e d cubit: c l o s e p a c k i n g of i c o s a h e d r a . The s p a c e g r o u p is R 3 m (12 a t o m s p e r u n i t cell).

\

The e s s e n t i a l s t r u c t u r a l v a r i a t i o n s in t h e r e l a t e d b o r i d e s e x h i b i t a n a d d i t i o n a l c h a i n of a t o m s p o s i t i o n e d a l o n g t h e main d i a g o n a l o f t h e u n i t cell. I t c o n t a i n s one, two o r t h r e e a d d i t i o n a l atoms, w h i c h a r e s y m m e t r i c a l l y a r r a n g e d a n d seem to s a t u r a t e t h e t h r e e - c e n t e r b o n d s in t h e a l p h a - r h o m b o h e d r a l u n i t cell (fig. 1).

Fig. I. A r r a n g e m e n t of t h e a t o m s in t h e u n i t cell of t h e a l p h a - r h o m b o h e d r a ] b o r o n s t r u c t u r e family. ( ~ i c o s a h e d r a l a t o m s located on t h e e d g e s of t h e u n i t cell. •

equatorial atoms forming three-¢:entcr bonds.

•

c h a i n a t o m s located a l o n g t h e main d i a g o n a l of the u n i t cell.

Optical prope~ies chemical f o r m u l a . .

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181

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B (alpha-rhombohedral)

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references

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BL=

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[8, 9]

BL=X Ba=S .

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B12S .

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BI=X= Bs-~Si B6P B~J~s Bee

(B1=-aSinjSi= BI=P= B~=As=

B4C...BI0.6C

(BIt-nC,)C2B

BLtCaAi Bt2C=Si B4Si Bl~Pa BI~As=

BttCtAI ]hzC~Si (BuSi)SifB .BI~P*B Bt=As=B

B~=O=

[]I] [9, 12 - 15] [9, 12 - 18] [16 - 18]

BItXa see

([3] and ref. t h e r e i n , [ 14- 19] [14 - 19] [11, 14] [20, 21] [9, 12 - 15] [9, 12 - 15]

B o r o n c a r b i d e c a n be s y n t h e s i z e d w i t h i n t h e large, h o m o g e n e i t y r a n g e b,..ween B4C a n d a b o u t Bt0.6C [22 - 24]. Near t h e c a r b o n - r i c h limit o n e B atom s e a m s to be s u b s t i t u t e d b y c a r b o n [25], while t o w a r d s t h e b o r o n - r i c h limit a s t a t i s t i c a l s u b s t i t u L i o n of t h e C-B-C c h a i n s b y p l a n a r y B4 a r r a n g e m e n t s is a s s u m e d [26, 27]. The chemical c o m p o s i t i o n c a u s e s s t r o n g v a r i a t i o n s of t h e e l e c t r o n i c p r o p e r t i e s , w h i c h is i m p o r t a n t f o r a p p l i c a t i o n . [28 - 32]. B e c a u s e of t h e s t r u c t u r a l r e l a t i o n s h i p similar c o n d i t i o n s f o r i s o a t r u c t u r a l b o r i d e s c a n be e x p e c t e d , e v e n t h o u g h t h e y h~ve n o t y e t b e e n p r o v e d , p~.~rhaps a p a r t f r o m t h e b o r o n - s i l i t : o n c o m p o u n d s [20, 21, 33]. z_.z_._s t r ~ c t ~ r ~ l , fa_~dL~...o.f.~e .m-r b ~ ! b..p. he,d r a! .._b.p_ron In t h e u n i t cell of t h e b e t a - r h o m b o h e d r a l modification, t h e c o r n e r s a n d t h e medium p o i n t s of t h e e d g e s a r e o c c u p i e d b y Bl= Jcosah~;dra. On t h e m a y o r cell d i a g o n a l (c a x i s in Lhc h e x a g o n a l d e s c r i p t i o n ) two a g g r e g a t e s of t h r e e i c o s a h e d r a , w h i c h a r e g r o w n t o g e t h e r a t t h e i r s u r f a c e s , a r e a r r a n g e d symmetric:ally (~ a single, atom in t h e c e n t e r of t h e u n i t cell (fig. 2a).

t b

Fig. 2. C r y s t a l s t r l i c t u r e of t h e b e t a - r h o m b o h e d r a l b o r o n s t r u c t u r e

family.

a) A r r a n g e m e n t of Bl= i c o s a h e d r a a n d c o n d e n s e d s y s t e m s in t h e u n i t cell. b) Ba4 g i a n t i c o s a h e d r o n a s a s u b u n i t of t h e Unit cell. c) Bt0 s u b u n i t o f t h e u n i t u n i t cell.

182

H. Werheit

F o r g r o u p t h e o r e t i c a l c o n s i d e r a t i o n s a d i f f e r e n t d e s c r i p t i o n is more s u i t a b l e . A g i a n t B04 i c o s a h e d r o n c o n s i s t i n g o f a c e n t r a l B~t i c o e a h e d r o n b u s i n g linked with 12 h a l f - i c o s a h e d r a in t h e d i r e c t i o n of i t s f i v e - f o l d a x e s (fig. 2b) is a r r a n g e d a t t h e c o r n e r s of t h e u n i t cell in s u c h a way t h a t o n e of t h e t h r e e - f o l d a x e s c o i n c i d e s with t h e t r i g o n a l axis of t h e t h e s t r u c t u r e . T h e n t h e h a l f - i e c m a h e d r a on t h e e d g e s of t h e u n i t cell c o m p l e t e o n e a n o t h e r . The h a l f - i c o s a h e d r a d i r e c t i n g into t h e i n t e r i o r o f t h e u n i t cell a r e completed b y two t e n - a t o m i c s t r u c t u r e u n i t s c o n s i s t i n g of t h r e e c o n d e n s e d c a p s of i c o s a h e d r a (fig. 2c), which a r e s y m m e t r i c a l l y a r r a n g e d r e l a t i v e to a s i n g l e B atom in t h e c e n t e r of t h e u n i t cell. T h i s d e s c r i p t i o n of t h e b e t a - r h o m b o h e d r a l u n i t cell e x h i b i t s the s t r o n g s t r u c t u r a l r e l a t i o n s h i p to t h e B~tX~ t y p e o f t h e a l p h a - r h o m b o h e d r a l s t r u c t u r e from t h e v i e w p o i n t o f s y m m e t r y : The B~ i c o s a h e d r o n is r e p l a c e d b y t h e Bs~ i c o s a h e d r o n a n d t h e t h r e e - a t o n d c c h a i n b y a Bm-B-B~0 o r a c o r r e s p o n d i n g s u b s t r u c t u r e u n i t . T h e s p a c e g r o u p is R 3 m (105 a t o m s p e r u n i t cell). B o r i d e s of t h i s t y p e c a n be o b t a i n e d b y s y s t e m a t i c a l s u b s t i t u t i o n of b o r o n a t o m s in t h e s t r u c t u r e u n i t s w i t h i n t h e u n i t cell. Besides, it c a n be e x p e c t e d t h a t a s u b s t i t u t i o n a l o r i n t e r s t i t i a l d o p i n g w i t h c a r b o n m u s t be a s s u m e d in most of t h e c r y s t a l s a v a i l a b l e a t p r e s e n t , b e c a u s e r e m o v i n g of c a r b o n d u r i n g t h e u s u a l p u r i f i c a t i o n p r o c e s s e s is v e r y difficult. chemical f o r m u l a

structural formula

B (beta-rhombohedral) B~AIs B~.Si

Bs~(B~o)zB B~(BeAh)~AI Bs~(BvSis)~Si

references [9, 34, 35] [15, 36] [15, 36]

A n u m b e r of solid s o l u t i o n s in t h e b e t a - r h o m b o h e d r a l s t r u c t u r e was synthesized. They are formed b y t r a n s ition e l e m e n t s (So, Cr, Mn, Fe, Ni, Cu, Zr) a n d b y e l e m e n t s w i t h p a r t i a l l y filled p l e v e l s (Si, Ge, Se) a c c o m o d a t e d in t h e i n t e r s t i t i a l v o i d s o f t h e e l e m e n t a r y b e t a r h o m b o h e d r a l b o r o n s t r u c t u r e [37 - 43]. CusJM~.aB~0~ [38] is e s p e c i a l l y m e n t i o n e d , b e c a u s e it is t h e o n l y c o m p o u n d of t h i s t y p e , o n w h i c h oplAcal r e s u l t s will be reported.

I J

2.3. S t r u c t u r a l _f~m_i~ of alpha-tctr.~ggn!ll b~r~n_ [t~tr~_ ~

j e ~lb,OO~,~ ~oI Fig. 3. C r y s t a l s t r u c t u r e of a l p h a tetragonal boron ( p r o j e c t i o n on (001) [14, 44]. chemical f o r m u l a

T h e u n i t cell c o n t a i n s four Bit i c o s a h e d r a at 4(c) p o s i t i o n s (Wyckoff n o t a t i o n ) , w i t h o n e of t h e f i v e - f o l d a x e s o f t h e i c o e a h e d r a d i r e c t i n g p a r a l l e l to t h e c - a x i s . Additionally two s i n g l e B a t o m s o c c u p y 2(b) p o s i t i o n s (fig. 3). S p a c e g r o u p P 4 t / n n m o r P4 2m (50 a t o m s p e r u n i t cell). I n t h e s t r u c t u r e - r e l a t e d b o r i d e s t h e 2(b) p o s t i o n s a n d t h e a d d i t i o n a l i n t e r s t i t i a l void p o s i t i o n s a t 2(a) a r e completely or partly occupied by other, p r e f e r a b l y metal a t o m s . structural formula

B ( a l p h a - t e t r a g o n a l modifioation)(Bl2)4Bz

(Btz)4X2Yt BltBe B~4BeAI ~-AlB12 C2AIsBu 8} BsIC BtsC B2rJ~I BtBNi

references [14, 44]

(Btl)4Be4 (Btl)4BetAh (BI2)4Al. (Bll)4CtAh u) (BIs)4B3C (Blt)4BtC8

[14, 45] [14, 46] [54] [13, 15] [47] [ 47 ]

(B~)d~2N~ (BIz)4BINiz

[48]

[14, 49]

Z)The a t t r i b u t i o n of t h e C2AiaBu s a m p l e s i n v e s t i g a t e d to t h e a l p h a - r h o m b o h e d r a l b o r o n g r o u p may be i n c o r r e c t , s i n c e a c c o r d i n g to [50] o n l y t h e h i g h t e m p e r a t u r e phase belongs to this structure-group, while a t low t e m p e r a t u r e s two c r y s t a l l o g r a p h i c a l l y r e l a t e d t w i n n e d o r t h o r h o m b i c p h a s e s exist. ~)3 of 4 AI s i t e s a r e s t a t i s t i c a l l y c~:cupied

Optical properties With s o m e r e s t r i c t i o n s ,

the following crystals

183

can be attributed

Ah~0Cu0.~B:s (B~)~AI~Cu: ~) (distorted icosahedra)

to t h i s s t r u c t u r e

family:

[51, 52]

~) metal s i t e s o n l y s t a t i s t i c a l l y o c c u p i e d BeC ((Ba:)aCs)a o r ~ (by the additional C atoms the structure is no more tetragonal but orthorhombic) 2.4. S t r u c t u r a l

f a m i l y of b e t a - t e t r a g o n a l

The 3-dimensional framework of the B~ i c o s a h e d r a a l t e r n a t e l y a l l i g n e d d o u b l e - i c o s a h e d r a B:: l i n k e d to t e n neighboured double icosahedra. The g r o u p P4~ o r P 4 : ( 1 9 0 a t o m s p e r u n i t

[47, 53, 55] becomes disproportionately

distorted

and

b o r o n ( t e t r a g o n a l II o r I l l )

b e t a - t e t r a g o n a i b o r o n m o d i f i c a t i o n c o n s i s t s o f c h a i n s of in t h e a a n d b a x i s d i r e c t i o n s , a n d o f t w i n n e d a d j a c e n t B~: i c o s a h e d r a in t h e s e c h a i n s a n d to f o u r r e m a i n i n g b o n d s lead to s i n g l e B a t o m s (fig. 4). S p a c e cell).

. . . . .

0

~

I I

b 0 B in B~t (z,O.lO) o B ;n B~ (z-O.BS}

e B in B~ (z.0.35) • interslitiol B,~

Fig. 4. C r y s t a l s t r u c t u r e

• B in 8~ (z-0.60)

of beta-tetragonal

b o r o n [56].

a) C h a i n s o f Bta i c o s a h e d r a . b) t w i n n e d B:~ d o u b l e i c o s a h e d r a . I n t h e r e l a t e d b o r i d e s c e r t a i n a t o m i c p o s i t i o n s in t h e d o u b l e i c o s a h e d r a r e m a i n u n o c c u p i e d ; t h e m e t a l a t o m s a r e s t a t i s t i c a l l y d i s t r i b u t e d in i n t e r s t i c e s o r r e p l a c e p a r t l y t h e s i n g l e B

atoms. chemical formula

structural

formula

B ( b e t a - t e t r a g o n a l m o d i f i c a t i o n ) (B:I*2BI:)4(B:.s)4 a l p h a - A I B t : (AIs~B~ (BIs$2BII),AIt: Ah.IBeo.TBI: (Bm*2Btz)4BtA1RBem

references [56]

[57, 58] [59, 60]

f o r a r e c e n t s o m e w h a t d i f f e r e n t d e s c r i p t i o n , s e e [51] possibly

gamma-AIB1: is a l s o to b e a t t r i b u t e d

PCG~-M

( (B,:)4B~*B:oA16.~s to t h i s g r o u p .

[51 ]

184

H. Werheit

2.5. S t r u c t u r a l family of o r t h o r h o m b i c b o r i d e s The b o r o n f r a m e w o r k c o n s i s t s o f Bu i c o e a h e d r a a r r a n g e d in a d i s t o r t e d simple h e x a g o n a l p a c k i n g with isolated b o r o n a t o m s a t i n t e r s t i t i a l s i t e s . The metal a t o m s o c c u p y f u r t h e r i n t e r s t i t i a l v o i d s (fig. 5). The s p a c e g r o u p is Imam (64 a t o m s p e r u n i t cell] chemical f o r m u l a

structural formula

MM~Bt~ .

.

.

.

.

.

.

references

(MM~*B=ZBt~)4 .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

LiAIB~, NaBis Mg~B~ MgAIBt~

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

(LiAI*B=*Bt=),

(NaB*B=XB~=)~ (MgtZB~=Bt~.)4

(MEAI=B==B~)~

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

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.

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.

.

.

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.

.

[61] [62 - 64] [65] [14, 15, 66]

SiBe z~

NOB~5 YBB~ 15B Boron atoms

1Ytlrlum olom

~B 8oron otoms

2~

........... M a

Fig. 5. C r y s t a l s t r u c t u r e of t h e o r t h o r h o m b i c b o r i d e s (NaBts [62 - 64] a s a r e p r e s e n t a t i v e , projected on the xy plane).

Fig. 6. Simplified schematic r e p r e s e n t a t i o n o f the YB~ t y p e c r y s t a l s t r u c t u r e w h e n looking a l o n g t h e [1001 axis [67].

2.6. S t r u c t u r a l family of YBu t y p e . P e r i o d i c a l l y a r r a n g e d Blm a n d BN u n i t s a n d s t a t i s t i c a l l y d i s t r i b u t e d metal a t o m s on c e r t a i n p o s i t i o n s ( s e e fig. 6) f o r m a u n i t cell with 104 Bl= I c o s a h e d r a o r 1248 B atoms. B e s i d e s Y most o f t h e l a n t h a n i d e a n d some a c t i n i d e Atoms a r e k n o w n to f o r m t h i s s t r u c t u r e [4, 67, 68]. S p a c e g r o u p : Fm3c (1248 B a t o m s p e r u n i t cell). A d d i t i o n a l l y to t h e i c o s a h e d r a l s t r u c t u r e families, t h e r e a l s o s t r u c t u r e s a r r a n g e m e n t s of B a t o m s exist: 2.7. Metal h e x a b o r i d e s t r u c t u r e

with o t h e r polyhedral

family.

T h e u n i t cell is c u b i c a n d c o n t a i n s o n e f o r m u l a w e i g h t of MBe. The b o r o n a t o m s f o r m r e g u l a r o c t a h e d r a o c c u p y i n g t h e c o r n e r s o f t h e u n i t cell, while t h e metal a t o m s a r e p o s i t i o n e d in t h e c e n t e r (fig. 7). All t h e r a r e - e a r t h m e t a l s a n d m o r e o v e r Ca, S r , Ba, T] a n d Pu f o r m t h e s e l e o s t r u c t u r a l h e x a b o r i d e s ( s e e [69, 70, 72]). H e x a b o r i d e e with d i v a l e n t metals h a v e b e e n p r e d i c t e d to be s e m i c o n d u c t o r s o r i s o l a t o r s , a n d h e x a b o r i d e s w i t h t r i v a l e n t m e t a l s to be metallic (see [72]). 2.8. Metal d.o_decaboride s t r u c t u r e

family.

T h e s t r u c t u r e may be d e s c r i b e d in t e r m s of a modified fcc u n i t cell w i t h t h e metal a t o m s in t h e c e n t e r s of r e g u l a r c u b o - o c t a h e d r a h a v i n g B a t o m s a t e a c h of t h e i r 24 v e r t i c e s , o r a l t e r -

Optical properties

Fig. 7. Metal [69, 70, 72].

hexaboride

crystal

185

Fig. 8. Metal d o d e c a b o r i d e ( : r y s t a l s t r u c t u r e [71, 72].

structure

natively, b y a modified NaCl-type structure a r r a n g e m e n t s o f 12 B a t o m s , e a c h o c c u p y i n g t h e A c t i n i d e s a s well a s a lot of o t h e r metal a t o m s D o d e c a b o r i d e s with d i v a l e n t m e t a l s a r e e x p e c t e d w i t h t r i v a l e n t metals to be metallic (see [72]), h u t

w i t h metal a t o m s a n d cubo-octahedral s t r u c t u r e p o s i t i o n s (fig. 8). L a n t h a n o i d e s , a r e a b l e to f o r m t h i s s t r u c t u r e [71, 72]. to be a e m i c o n d u c t i n g o r i n s u l a t i n g , t h o s e c o m p a r e s e c t i o n 4.3.

3. INTERACTION BETWEEN LIGHT AND CRYSTALS The collective d e s i g n a t i o n " o p t i c a l p r o p e r t i e s of s o l i d s " c o m p r i s e s all t h e p h y s i c a l p h e n o m e n a , in w h i c h , in a n y w a y p h o t o n s r e s p e c t i v e l y w a v e s a r e i n v o l v e d , w h i c h a r e a t t r i b u t e d to t h e o p t i c a l r a n g e of t h e e l e c t r o m a g n e t i c s p e c t r u m . T h i s m e a n s t h a t i n v e s t i g a t i o n s w i t h t h e e x p e r i m e n t a l m e t h o d s of o p t i c s c a n be p e r f o r m e d . Since t h e s e m e t h o d s a r e s t e a d i l y d e v e l o p e d w i t h r e s p e c t to t h e g e n e r a t i o n of r a d i a t i o n a s well a s to t h e s e n s i t i v i t y of d e t e c t i o n , a t p r e s e n t , b e c a u s e of t h e u s e of s y n c h r o t o n r a d i a t i o n , t h e s h o r t w a v e limit of o p t i c s r e a c h e s t h e r a n g e o f s o f t X - r a y s , while i t s l o n g - w a v e limit o v e r l a p s w i t h t h e m i c r o w a v e r a n g e of m m - w a v e s , b e c a u s e of r e s c e n t d e v e l o p e m e n t of F o u r i e r s p e c t r o s c o p y i n s t r u m e n t s . T h e s i g n i f i c a n c e of o p t i c a l i n v e s t i g a t i o n s f o r c l a r i f y i n g p h y s i c a l p r o p e r t i e s of s o l i d s is b a s e d o n t h e f a c t t h a t in wide r a n g e s o f t h e o p t i c a l s p e c t r u m t h e l i g h t beam is a p r o b e w h i c h is o n l y w e a k l y c o u p l e d w i t h t h e p h y s i c a l p r o c e s s e s , a n d h e n c e it c l o s e l y a p p r o a c h e s t h e ideal of u n d i a t u r b i n g m e a s u r e m e n t . N e v e r t h e l e s s e.g. t h e a b s o r p t i o n m e a s u r e m e n t s b e l o n g to t h e most s e n s i t i v e d e t e c t i o n m e t h o d s , s i n c e a l r e a d y in r a t h e r t h i n s a m p l e s t h e i n t e r a c t i o n b e c o m e s multiplied b e c a u s e o f t h e s h o r t w a v e l e n g t h s . Of c o u r s e t h i s a d v a n t a g e i n c l u d e s t h e p r o b l e m t h a t a l r e a d y small p e r t u r b a t i o n s o r d e v i a t i o n s i n f l u e n c e t h e m e a s u r e m e n t s a n d may a g g r a v a t e t h e i n t e r p r e t a t i o n of e x p e r i m e n t a l r e s u l t s . T h e r e f o r e , b e c a u s e of t h e u n c e r t a i n t i e s in m a t e r i a l purity and preparation methods at present, simultaneous investigations with different methods or use of really comparable samples are necessary. Hitherto the subsequent optical phenomena modifications and compounds of boron.

have

become

important

in

the

case

of

the

3.1. Optical a b s o r p t i o n a n d r e f l e c t i v i t ~ T h e i n t e r a c t i o n of t h e e l e c t r o m a g n e t i c w a v e s w i t h a solid is c o n v e n i e n t l y d e s c r i b e d in t h e c a s e o f a p u r e d i e l e c t r i c o n t h e b a s i s of t h e c l a s s i c a l t r e a t m e n t of L o r e n t z , w h o c o n s i d e r e d t h e solid a s a n a s s e m b l y of o s c i l l a t o r s , w h i c h a r e i n d u c e d to f o r c e d v i b r a t i o n s . T h i s l e a d s to t h e w e l l - k n o w n a b s o r p t i o n a n d d i s p e r s i o n c u r v e s in t h e o p t i c a l s p e c t r a . [f f r e e c a r r i e r s a r e p r e s e n t , t h e D r u d e - L o r e n t z t h e o r y d e s c r i b e s t h e f r e q u e n c y d e p e n d e n c e of t h e d y n a m i c a l c o n d u c t i v i t y (cp. e.g. [73]). F o r a d e t a i l e d c o m p a r i n g d i s c u s s i o n of t h e d i f f e r e n t t h e o r i e s o n f r e e - c a r r i e r a b s o r p t i o n , s e e [74]. The i n f l u e n c e of t h e collision f r e q u e n c y of f r e e c a r r i e r s o n t h e r e f l e c t i v i t y s p e c t r u m is s h o w n in fig. 9. If t h e collision f r e q u e n c y is v e r y small c o m p a r e d w i t h t h e p l a s m a r e s o n a n c e f r e q u e n c y o f t h e f r e e c a r r i e r s , a s t e e p " p l a s m a e d g e " o c c u r s . But it s m e a r e s o u t , w h e n t h e collision f r e q u e n c y i n c r e a s e s . T h e s e i n t e r a c t i o n s of p h o t o n s w i t h b o u n d complex d i e l e c t r i c f u n c t i o n

electrons,

phonons and free

c a r r i e r s lead to t h e

H. Werheit

186 lOO

E(w)

R % 8o

--~' + i ~ "

from w h i c h via t h e complex f r e q u e n c y - d e p e n d e n t refractive index the a b s o r p t i o n coefficient K and the reflectivity R are derived.

10

I n t h e o p p o s i t e way, e x p e r i m e n t a l i s t s t r y to d e r i v e t h e p a r a m e t e r s and mechanisms of elementary i n t e r action processes from experimentally obtained absorption and reflectivity spectra.

,I.........L.........1.........1 200

3.2. P h o t o e l e c t r i c e f f e c t s 000 ¢m -I wave n u m b e r

~00

1000

Fig. 9. I n f l u e n c e of t h e f r e e c a r r i e r c.o]]iaion f r e quency w v on the plasma edge in the r e f l e c t i v i t y s p e c t r u m . P a r a m e t e r s : ~ = 8.8; b~ p = 10 L4 s-l; b~r : 1010 (1), l0 IL (2), 10 Ls (3), 5,101~ (4), 10 la (5), 5,1013 (6), l 0 1 4 (7), 5,1014 (8), 10 l~ (9) s-L

In the external photoelectric effect e l e c t r o n s o f a solid a r e excited b y l i g h t to l e a v e t h e surface, and h e n c e i n f o r m a t i o n on t h e r e l a t i v e p o s i t i o n of t h e e n e r g y l e v e l s of electrons in t h e c r y s t a l to t h e v a c u u m level of e l e c t r o n s c a n be obtained.

In t h e i n n e r p h o t o e l e c t r i c a f f e c t , w h i c h is i m p o r t a n t in c o n n e x i o n with t h e i n t e r p r e t a t i o n o f e n e r g y b a n d s t r u c t u r e a n d t r a n s p o r t p r o p e r t i e s , f r e e c a r r i e r s a r e g e n e r a t e d b y optic:ally i n d u c e d t r a n s i t i o n s a c r o s s t h e e n e r g y g a p r e s p e c t i v e l y from e n e r g y l e v e l s in t h e b a n d g a p . The e x c i t a t i o n s p e c t r a c o n t a i n v a l u a b l e i n f o r m a t i o n o n t h e e l e c t r o n i c s t r u c t u r e a n d on r e l a x a tion p r o c e s s e s .

By t h e l i g h t - i n d u c e d a b s o r p t i o n ( p h o t o a b ~ o r p t i o n ) a t h e r m a l l y n o n - e q u i l i b r i u m s t a t e is e s t a b l i s h e d , w h o s e a b s o r p t i o n s p e c t r u m is o b t a i n e d . D e p e n d i n g on t h e r e l a x a t i o n time c o n s t a n t it is m e a s u r e d b y p e r i o d i c a l excitation a n d h e n c e l i g h t - i n d u c e d modulation of t h e t r a n s m i t t e d l i g h t beam ( l o c k - i n t e c h n i c ) o r , in t h e c a s e of g r e a t r e l a x a t i o n time c o n s t a n t s , a s e.g. in b e t a - r h o m b o h e d r a l b o r o n a t low t e m p e r a t u r e s , b y c a l c u l a t i n g t h e d i f f e r e n c e of t h e s p e c t r a of t h e excited a n d t h e u n e x c i t e d s t a t e .

I f t h e c a r r i e r c o n c e n t r a t i o n in a n t h e r m a l n o n - e q u i l i b r i u m s t a t e is h i g h e n o u g h , in t h e c a s e of a s u f f i c i e n t a m o u n t o f r a d i a t i n g r e c o m b i n a t i o n o r l u m i n e s c e n c e , t h e s p e c t r a l d i s t r i b u t i o n of t h e e m i t t e d H g h t c a n be m e a s u r e d y i e l d i n g i n f o r m a t i o n on b a n d s t r u c t u r e a n d s t a t e s in t h e band gap. 3~5_. ! n . ~ ] ~ t i c . . ]igh£. sc_atterinng I n f o r m a t i o n on o p t i c a l a n d a c o u s t i c p h o n o n s c~tn be o b t a i n e d b y i n e l a s t i c l i g h t s c a t t e r i n g . I f t h e p o l a r i z a b i l i t y of t h e s t r u c t u r e c a n be c h a n g e d b y light, S t o k e s a n d a n t i - S t o k e s l i n e s of s c a t t e r e d l i g h t a r e o b s e r v e d , w h i c h a r e s y m m e t r i c a l l y d i s p l a c e d r e l a t i v e to t h e f r e q u e n c y of t h e i n c i d e n t l i g h t b y a m o u n t s e q u a l to t h e f r e q u e n c i e s of t h e p h o n o n s i n v o l v e d . If s u c h p h o n o n s a r ~ a c o u s t i c , t h e p r o c e s s is called Brillouin s c a t t e r i n g ; if t h e p h o n o n s a r e optical, t h e p r o c e s s is called Raman s c a t t e r i n g . B a c a u s e o f kphot0m ~ Ir/a { k - v e c t o r a t t h e b o u n d a r y of t h e l e t Brillouin zone) i n f o r m a t i o n is mainly r e s t r i c t e d to t h e c e n t e r of t h e Briliouin zone. 4. R E S U L T S

OF OPTIOAL MEASUREMENTS

AND THEIR INTERPRETATION

4.1 I n t e r a c t i o n with i n n e r e l e c t r o n s I n t h e s p e c t r a l r a n g e of h i g h p h o t o n e n e r g y o n l y t h e i n t e r a c t i o n of t h e e l e c t r o m a g n e t i c r a d i a t i o n with i n n e r e l e c t r o n s is i m p o r t a n t . T h e r e f o r e t h e i n f l u e n c e of c r y s t a l l i n e s t r u c t u r e a n d chemical b o n d i n g o n t h e o p t i c a l p r o p e r t i e s is e x p e c t e d to be i n s i g n i f i c a n t in c o m p a r i s o n with t h e i n f l u e n c e o f t h e i n d i v i d u a l a t o m s , a n d in t h e modifications o f b o r o n a s well a s in t h e b o r o n - r i c h b o r i d e s t h e B a t o m s will d o m i n a t e t h e o p t i c a l p r o p e r t i e s .

Optical properties

187

S y s t e m a t i c i n v e s t i g a t i o n s o n t h e i n f l u e n c e of n o n - b o r o n a t o m s o n t h e o p t i c a l p r o p e r t i e s a r e n o t a v a i l a b l e . T h e t r a n s m i s s i o n o f a 1000 A t h i c k b o r o n film [75] i s s h o w n in fig. 10 a, b. T h e h a t c h e d l i n e in f i g . 10 s w a s c a l c u l a t e d f r o m t h e p h o t o a b s o r p t i o n c r o s s s e c t i o n [76]. T h e t r a n s m i s s i o n m i m i n u m b e t w e e n 50 a n d 100 A w a v e l e n g t h s e e m s to c o r r e s p o n d to t h e 183 oV (68 A) l s - 2 p t r a n s i t i o n d e r i v e d f r o m X - r a y R a m a n s c a t t e r i n g in a m o r p h o u s b o r o n [77].

,

~

Boron

!

a,

l

or0n

,am1

1

! Io0o

IAVlV.ZX~ I ~ w°

Fig. 10a, b: O p t i c a l t r a n s m i s s i o n o f a 1000 A t h i c k b o r o n film [75]. 4.2 P u n d a m e n t a l a b s o r p t i o n a n d a b s o r p t i o n

edge

Many of the electrical and optical properties of pure semiconductors and isolators can be d e s c r i b e d in t e r m s o f t h e i r b a n d s t r u c t u r e E(k), w h i c h e x h i b i t s t h e r e l a t i o n b e t w e e n e n e r g y and momentum of electrons and holes in the different possible states of the conduction and v a l e n c e b a n d s f o r t h e v a r i o u s s y m m e t r y p o i n t s in t h e f i r s t B r i l l o u i n z o n e o f t h e r e c i p r o c a l l a t t i c e ( s e e f i g . 12). T h e e n e r g e t i c a ] d i f f e r e n c e b e t w e e n t h e a b s o l u t e e x t r e m e o f c o n d u c t i o n a n d v a l e n c e b a n d i s t h e b a n d g a p , w h i c h is i m p o r t a n t in c o n n e c t i o n w i t h o p t i c a l a n d transport properties. The curvatures of the bands yield according to l/m* = 1/h s *

~ l E / ~k s

the effective mass of the carriers. T h e p a r t o f t h e e l e c t r o m a g n e t i c s p e c t r u m i m m e d i a t e l y r e l a t e d to t h e b a n d s t r u c t u r e i s c a l l e d t h e r a n g e o f f u n d a m e n t a l a b s o r p t i o n , w h i c h c o m p r i s e s all t h e p h o t o n - i n d u c e d t r a n s i t i o n s o f electrons from occupied states in the valence band inte empty states in the conduction band. In t h e c a s e o f c r y s t a l l i n e b o r o n a n d b o r o n - r i c h b o r i d e s t h i s s p e c t r a l r a n g e e x t e n d s f r o m a b o u t ] to a b o u t 30 eV. Since the valence electrons are relevant for the chemical bonding of the crystals, the fundamental absorption yields information on the energetical distribution of bonding states, i n d e e d in c o m b i n a t i o n w i t h t h e a c t u a l d i s t r i b u t i o n of s t a t e s i n t h e c o n d u c t i o n b a n d . The absorption edge reflects the distribution of states near the the band edges, which are r e l e v a n t f o r t h e e l e c t r o n i c t r a n s p o r t p r o p e r t i e s . By e x t r a p o l a t i o n to z e r o a b s o r p t i o n t h e b a n d g a p c a n be o b t a i n e d . D e p e n d i n g o n t h e k i n d a n d p o s i t i o n o f t h e b a n d e x t r e m e d i f f e r e n t k i n d s of t r a n s i t i o n s m u s t b e d i s t i n g u i s h e d : In the transition processt the total energy and the momentum of the whole system must be c o n s e r v e d . If t h e m o m e n t a k , a n d k¢ o f t h e e l e c t r o n i n t h e i n i t i a l s t a t e E(kv) o f t h e v a l e n c e b a n d a n d in t h e f i n a l s t a t e E(ke) in t h e c o n d u c t i o n b a n d a r e t h e s a m e , o n l y t h e s u p p l y of e n e r g y is n e c e s s a r y f o r t h i s " d i r e c t " t r a n s i t i o n , w h i c h c a n b e s i m p l y r e a l i z e d b y a b s o r p t i o n o f a p h o t o n (kptotn : 0). B u t if kv a n d kc a r e d i f f e r e n t , f o r m o m e n t u m c o n s e r v a t i o n t h e t r a n s i t i o n r e q u i r e s t h e a d d i t i o n a l i n t e r a c t i o n w i t h a p h o n o n , w h i c h k i n d o f t r a n s i t i o n is c a l l e d "indirect". Moreover quantum mechanical selection rules must be considered, which means that t h e t r a n s i t i o n p r o b a b i l i t y d e p e n d s o n t h e k i n d o f t h e s t a t e s i n v o l v e d (e.g. s o r p). T h e d i f f e r e n t k i n d s of t r a n s i t i o n s c a n b e d i s t i n g u i s h e d b y t h e f r e q u e n c y d e p e n d e n c e of t h e e d g e a b s o r p t i o n (cf. e . g . [78]). F o r t h e s u b s e q u e n t l y d i s c u s s e d e x p e r i m e n t a l r e s u l t s only t h e f o l l o w i n g t y p e s ttre i n t p o r t a n t :

188

H. Werheit

d i r e c t allowed t r a n s i t i o n s : K

(h~o-

~

,0 I~)u2

i n d i r e c t allowed t r a n s i t i o n s :

1 K

~

.

.

ZI E) 2

(~."- k e .

.

.

BW

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-~ (exp (@/T) - I )

(positive sign for phonon a b s o r p t i o n and negative sign for phonon emission) Hence f r o m t h e f r e q u e n c y a n d t e m p e r a t u r e d e p e n d e n c e of t h e a b s o r p t i o n c o n s t a n t K w i t h i n t h e a b s o r p t i o n e d g e o n e c a n o b t a i n t h e b a n d g a p , t h e e n e r g y of t h e p h o n o n s i n v o l v e d a n d t h e k i n d of t r a n s i t i o n . S t r u c t u r a l family of a l p h a - r h o m b o h e d r a l b o r o n Here we h a v e t h e s i t u a t i o n , w h i c h is u n u s u a l f o r i c o s a h e d r a l b o r o n - t y p e c r y s t a l s , t h a t t h e o r e t i c a l c a l c u l a t i o n s a r e more c o m p r e h e n s i v e t h a n t h e e x p e r i m e n t a l w o r k h i t h e r t o o p e r f o r m e d , T h e e l e c t r o n i c b a n d s t r u c t u r e s a n d t h e d e n s i t i e s of s t a t e s f o r a l p h a - r h o m b o h e d r a l b o r o n , BIzP2 B18As~ a n d Bl~ CBC w e r e c a l c u l a t e d b y A r m s t r o n g e t al. [79, 80] a n d g i v e a v a l u a b l e b a s i s f o r t h e i n t e r p r e t a t i o n of e x p e r i m e n t a l w o r k . As a n example, fig. 12 s h o w s t h e c a l c u l a t e d b a n d s t r u c t u r e of a l p h a - r h o m b o h s d r a l b o r o n f o r t h e p o i n t s of h i g h s y m m e t r y a n d i n t e r m e d i a t e p o s i t i o n s in t h e f i r s t BriUouin z o n e (fig. 11). The total d e n s i t i e s of t h e e l e c t r o n i c s t a t e s o f t h e s e c r y s t a l s d e p e n d i n g on e n e r g y a r e c o m p a r e d in fig. 16. Peak p o s i t i o n s in t h e d e n s i t y of s t a t e s c u r v e s (figs. ]3) a r e r o u g h l y c o n f i r m e d b y X - r a y e m i s s i o n p e a k s [81].

--

~

=_~

=-

--~

~

•

~W

•

N

L

~

!

(Y

•

$

u

Fig. 11. Brillouin z o n e of r h o m b o hedral lattices. J

Fig. 12. Calculated b a n d s t r u c t u r e of a l p h e - r h o m b o h e d r a l b o r o n for symmetry points of the Brillouin z o n e in fig. 11 a n d c o n nectirig l i n e s [75].

~

v

] ,J

¢

T

^

r

=

v '

G e n e r a l l y w a s f o u n d t h a t mainly p s t a t e s form t h e t h e b a n d e d g e s o f t h e conduction and v a l e n c e b a n d s of t h e s e c r y s t a l s . Hence t h e a b s o r p t i o n is e x p e c t e d to b e h a v e a c c o r d i n g to indirect forbidden transitions. Possibly this explains that the a b s o r p t i o n values within the e d g e of a l p h a - r h o m b o h e d r a l b o r o n (fig. 14) a r e r a t h e r low. Optical e d g e a b s o r p t i o n s p e c t r e a r e a v a i l a b l e o f a l p h a - r h o m b o h e d r a l b o r o n a n d b o r o n c a r b i d e o n l y (figs. 14, 15)

Optical

189

properties

Band gaps theoretical

experimental

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

alpha-rh.

B

2.4 eV ( i n d i r e c t ) 2.88 eV ( i n d i r e c t )

[79] [82]

1.97 eV (E II c) 1.9 eV (E J c) 2.0 eV

[3, 83] [3, 83] [4, 84]

B~=P=

2.5

eV ( i n d i r e c t )

[79]

BttAs~

2.3

eV ( i n d i r e c t )

[79]

3.3 eV (E _~ c) 3.35 oV 3.47 eV

[87] [207] [207]

boron carbide

4

eV ( i n d i r e c t )

[80]

0.48 eV ( i n d i r e c t )

[4, 85, 86]

Apart from the good agreement between calculated a n d o p t i c a l l y m e a s u r e d g a p s , in t h e b a n d s t r u c t u r e of alpha-rhombohedral boron a rather unusual situa t i o n in c o m p a r i s o n w i t h c l a s s i c a l s e m i c o n d u c t o r s can be seen: The absolute valence band maximum shows a rathar strong curvature, while the conduction band at its minimum is very narrow. From this w o u l d r e s u l t t h a t maI << m0s, a n d h e n c e t h e hole 2.10 3

, ,~--/eY

-25-

_20

-~.~

-'/~

.Y

0

Fig. 13. Calculated densities of s t a t e s [79, 80]. m o b i l i t y s h o u l d be d i s t i n c t l y h i g h e r t h a n t h e e l e c t r o n m o b i l i t y . I n d e e d , G o l i k e v a e t at. [88] m e a s u r e d a Hall m o b i l i t y o f 120 cmZV-ts-1, w h i c h is m u c h h i g h e r t h a n in any other icosahedral boron-type crystal hithertoo investigated, and the reported sign of the thermoelectric power indicates p-type conduction. But unfortunately~ the s i g n o f t h e Hall e f f e c t in t h i s p u b l i c a t i o n is n o t c l e a r l y indicated, so that there remains some uncertainty. Nevertheless the calculated band structure of alpha-rhombohedral boron shows a fairly good agreement not only with the optical results but also with the transport properties. T h e o p t i c a l l y m e a s u r e d b a n d g a p o f BziP2, too, a g r e e s matisfactorily with that obtained from the band structure calculation. B u t in t h e c a s e o f b o r o n c a r b i d e t h e c a l c u l a t e d a n d t h e optically measured band gaps d i f f e r . T h e low v a l u e d e r i v e d f r o m t h e optic'~fl a b s o r p t i o n (fig. 15) w a s a d d i t i o nally confirmed by modulated reflection measurements [89]. By t h i s m e t h o d , w h i c h c a n b e u t i l i z e d to o b t a i n c r i t i c a l p o i n t s in t h e c o m b i n e d d e n s i t y o f s t a t e s , w a s demonstrated t h a t t h e c h a n g e of c h e m i c a l compcmition causes a certain change of the density of states. Besides t h e g a p , a n a d d i t i o n a l t r a n s i t i o n a t a b o u t 1.5 eV i s d i s t i n c t l y i n d i c a t e d (fig. 16). M o r e o v e r t h e g a p is a l s o m a r k e d in t h e d i f f e r e n c i a ] r e f l e c t i v i t y s p e c t r u m o b t a i n e d

/'xc 2

i0zl 1.9

2.0

}

2.1

Z2

Z3

2.4

Z.5 eV L6

hm-,

Fig. 14. A l p h a - r h o m b . s o r p t i o n e d g e [83]. 1000 cm~ g00 800

° o

boron;

ab-

Boron c o r b i d e A e A A t,

7#

0 0

%,

~ ~C !

~

o° ° 0t

'~oc

• r-l13K

30~ 2~

0

~.5

!. " ""

ZD

o

295 K

=

~50 K

,

25

I

3.0

3.5 ~.,

~(450K) -# (295 K ) -

"l"

z.O

4.5

um

5.5

Fig. 15. B o r o n c a r b i d e ( c h e m i c a l c o m p o s i t i o n a p p r o x i n 0 a t e l y Bl=Oa). Absorption coefficient at different temperatures in the. spec:tral r a n g e o f t h e a b s o r p t i o n e d g e [4, 85, 8 6 ] .

190

H. W e r h e i t

b y o p t i c a l m o d u l a t i o n (fig. 20). A p a r t f r o m t h e s e q u a n t i t a t i v e r e s u l t s , t h e h i g h r v f l e c t i v i t y t h r o u g h o u t t h e v i s i b l e s p e c t r a l r a n g e , w h i c h c a n immediately b e s e e n b y e y e a t a p o l i s h e d s u r f a c e , c o n t r a s t s with t h e c a l c u l a t e d b a n d g a p of 4 eV.

15tJ

l°Z

i

,/ -

l°Z Boroncorb]de

lO

---

2

I

lO

t~0-'

- - - - 1

~X Boroncorbide

/I ~2 0.5

0

e/

-3

-lO 05

l.O

I~5

28

Z5

3.0

hta

15

eV 4.0

1.0

1.5

eV

~.0

Fig. 17. B o r o n c a r b i d e . L i g h t - i n d u c e d c h a n g e of t h e d i f f e r e n t i a l r e f l e c t i v i t y of t h e b o r o n c a r b i d e s a m p l e s 1 a n d 4 (see fig. 16)[89].

Fig. 16. B o r o n c a r b i d e . Relative d i f f e r e n c e of t h e r e f l e c t i v i t y s p e c t r a of b o r o n c a r b i d e of various chemical composition; zero of the ordinate arbitrarily chosen. Nominal compositions: 1: Bt~,94Cz.¢~, 2: Bi~.~sOt.6s, 3: BI~.~aC~.I~, 4: B1s.1~C2.s~, 5: Blt.oiC~.~, 6: Blo.MO4.4t [89]. N e v e r t h e l e s s some q u a l i t a t i v e f e a t u r e s of t h e c a l c u l a t e d b a n d s t r u c t u r e c a n c o n t r i b u t e to t h e i n t e r p r e t a t i o n of e x p e r i m e n t a l r e s u l t s . The c a | c u l a t e d Fermi level lies below t h e v a l e n c e b a n d e d g e , i n d i c a t i n g t h a t b o r o n c a r b i d e s h o u l d be a d e g e n e r a t e d s e m i c o n d u c t o r . T h i s r e s u l t a g r e e s w i t h t h e a b n o r m a l t h e r m o e l e c t r i c p o w e r , w h i c h is v e r y h i g h a n d i n c r e a s e s n e a r l y p r o portional to T e v e n to very high temperatures. T h e very n a r r o w b a n d s agree with the low carrier mobility [28 - 32]. Structural family of beta-rhombohedral boron Beta-rhombohedral boron is the only representative of the icosahedral boron-type crystals, which has been available already for a n u m b e r of years in form of single crystals of appreciable size to perform all kinds of solid state investigations. Nevertheless, w h e n such results are interpreted, it must be taken in mind that these single crystals are not of such high q u a l i t y a s e.g. t h e comntercially a v a i l a b l e Si s i n g l e c r y s t a l s . T h i s h o l d s f o r t h e d e g r e e of p u r i t y a n d f o r s t r u c t u r a l d e f e c t s a s well. As a n i m p u r i t y , e s p e c i a l l y c a r b o n m u s t be m e n t i o n e d , w h o s e c o n c e n t r a t i o n in t h e p u r e s t c r y s t a l s a v a i l a b l e is a b o u t 50 to 100 p p m , a n d a s s t r u c t u r a l d e f e c t s t w i n s seem too be s c a r c e l y a v o i d a b l e .

N u m e r o u s e x p e r i m e n t s w e r e p e r f o r m e d p b u t b e c a u s e of t h e complex s t r u c t u r e , s t r u c t u r e c a l c u l a t i o n s a r e n o t a s y e t available. S i d o r i n e t el. [90] m e a sured the reflectivity s p e c t r u m in t h e r a n g e of f u n d a m e n t a l a b s o r p t i e s u p to 40 eV, f r o m which by Kramers-Kronig relation the absorption spectrum a n d the dielectric functic,n w e r e d e r i v e d (fig. 18 a, b). The a b s o r p t i o n maxim a could be attributed to trsnsitions b e t w e e n d e n s i t y of s t a t e s maxima of v a l e n c e band and conduction band (fig. 19), w h i c h w e r e k n o w n from X-ray investigat i o n s like XPS ( X - r a y photoelectron spectrum) and SXS ( b o r o n Ke m i s s i o n spectrum)

theoretical band

I

/

% , l ~ 40

I

/ ~ - -x 0 ./ \\< ........ \,~-~ n 0

2o

J

0 o

_

~ '~'~-~-~.-~

/

10

20 ho~

30 eV

40

5

lO

15

20

25

30

35 ev ~0

Fig. 18. B e t a - r h . b o r o n , a) A b s o r p t i o n c o e f f i c i e n t K, r e f l e c t i v i t y R a n d r e f r a c t i v e i n d e x n v s . p h o t o n e n e r g y (K c a l c u l a t e d from R b y g r a m e r s - K r o n i g r e l a t i o n J90]. b) Real a n d i m a g i n a r y p a r t s of t h e d i e l e c t r i c f u n c t i o n ( d e r i v e d from r e f l e c t i v i t y in fig. 18 a.

191

Optical properties

showing preferably s-type respectively p-type valence band states and Bremsstrahlung isoc h r o m a t e a n d p h o t o e l e c t r i c yield i n d i c a t i n g p r e ferably the unoccupied p-type respectively s - t y p e s t a t e s of t h e c o n d u c t i o n b a n d (see [91] and references therein).

8

XP5

5X5

'~sochromote 6/ O

! ^

I

photoeleclric

y~,l~

Transitions: a' - - > a' - - >

A' B'

b' - - > b' - - >

A' C'

3.4 e V

4.6 eV 6.0 eV 9.0 eV

P o s s i b l y b e c a u s e of t h e p r o b l e m s to p r e p a r e 35 ev 30 ideally r e f l e c t i n g s u r f a c e s w i t h o u t c o n t a m i n a lion, t h e r e i s s o m e inaccuracy in t h e absolute v a l u e s of t h e d e r i v e d s p e c t r a of r e f r a c t i v e index, absorption coefficient and dielectric Fig. 19. E v a p o r a t e d boron. Qualitative d e n f u n c t i o n in fig. 18, which follows from c o m p a r i s i t y of states in t h e s u r r o u n d i n g of the. son with the s u b s e q u e n t r e s u l t s obtained by b a n d g a p o b t a i n e d b y d i f f e r e n t e x p e r i m e n t a l o t h e r e x p e r i m e n t a l m e t h o d s , which a r e only little sensitive against surface conditions. m e t h o d s [91]. ,

-v~

-~

-5'

,

,

5 I0 15 £-E, =

0

~

In f i g s . 23 a, b t h e r e f l e c t i v i t y a n d t h e r e f r a c t i v e i n d e x r e s u l t s o b t a i n e d b y s e v e r a l a u t h o r s i n t h e low e n e r g y r a n g e of t h e f u n d a m e n t a l a b s o r p t i o n a n d i n t h e a d j a c e n t t r a n s p a r e n t r e g i o n a r e collected [92 - lOlL In t h e l a t t e r r a n g e t h e most p r e c i s e method to g e t t h e r e f r a c t i v e index, n a m e l y b y o b t a i n i n g i n t e r f e r e n c e s f r o m multiple r e f l e c t i o n s in p l a n e - p a r a l l e l s a m p l e s , was u s e d (fig. 21) [102]

Fig. 22 s h o w s t h e a b s o r p t i o n e d g e of b e t e - r h o m b o h e d r a l b o r o n a s m e a s u r e d b y s e v e r a l a u t h o r s [92 - 95, 98, 99, 101, 103, 104]. Some a b s o r p t i o n i s o t h e r m s of t h e e d g e a n d t h e a d j a c e n t s p e c t r a l r a n g e a r e plotted in fig. 23 [96, 97]. T h e a n i s o t r o p y of t h e a b s o r p t i o n e d g e is r a t h e r low (fig. 24), i n d e e d it is d i s t i n c t in t h e e d g e tail, which s e e m s to be i n f l u e n c e d b y impurities. T a k i n g t h e t h e o r e t i c a l l y d e r i v e d f r e q u e n c y d e p e n d e n c e s i n t o a c c o u n t (see s e c t i o n 4.2) t h e abBorption e d g e c a n be a n a l y s e d b y d e c o m p o s i n g t h e m e a s u r e d a b s o r p t i o n s p e c t r a a s is s h o w n in fig. 25 [94]. As u s u a l , w i t h i n c r e a s i n g t e m t z s r a t u r e s t h e a b s o r p t i o n e d g e a n d h e n c e t h e t r a n s i t i o n e n e r g i e s , too, s h i f t to lower e n e r g i e s ; h u t a s a p e c u l a r i t y of b e t a - r h o m b o h e d r a l b o r o n a n e d g e tail w a s a n a l y s e d , w h i c h b e h a v e s like a d i r e c t t r a n s i l J o n , b u t w h o s e a b s o r p t i o n i n c r e a s e s s t r o n g l y with t e m p e r a t u r e . T h e r e s u l t of t h e e d g e a n a l y s i s l e a d s to t h e s u b s e q u e n t e q u a t i o n d e s c r i b i n g t h e e d g e a b s o r p tion p r e s i s e l y a t l e a s t in t h e t e m p e r a t u r e r a n g e from 70 to 850 K [1, 3, 92, 93]. (he;- k S , - ~ Ez(T))*

I h~

I

(bw-

k g , - ZI E,(T))'

+ C, ---

K (w,T) : C, ---

I - exp(-9 ~IT)

l - e x p ( - g z/T)

hw

(h~¢- Z} E'(T)) wa + C~ ..................... e x p (~ Eo/kT)

Parameters" Index

ZI E(T=O)

2 3

1.562 ' 0.004 eV 1.33 : 0.02 eV 0.37 ± 0.02 eV

0

0.94

1

O~

79400 t 200 4270 ~" 100 232000 1 1~000 -

g D

860 K 374 K

ke

74 meV 32 meV

-

-

-

192

H. Werheit

Fig.

20.

Beta-rhomb,

vs,

photon

energy;

"/' -

long-dashed

~97~ . b) r e f r a c t i v e i n d e x of p o l y c r y s t a l s ; [98), {lool,

.. " ~ . ' ~

,

-

-

line: [96J,

(3)

196), (S) { ~ o l ] .

(4)

~

z

1

E ~

j

3

~

htu

I i

1 1

~ ~v

Fig. 21, B e t a - r h o m b . b o r o n . R e f r a c t i v e i n d e x o f s i n g l e c r y s t a l s o b t a i n e d f r o m o p t i c a l i n t e r f e r e n c e s [102],

1

2

I

I

3 hto

i

4

5

eV

10~

I lo~, 3.0

, ,~o-~ ~ .

~ _ ~ _

2.8 2.~ 0

•

I I 1

Z

I 4

3

5 k

6

I0;

j

=

E i!c EJ_c ] 8 ~lrn

7

g

., 1

Fig. 22. B e t a - r h n m b . b o r o n . A b s o r p t i o n

edge;

absorption

2

3 hoe

~.

c(mffici(.'nt o f p o l y c r y s t a l s

5

eV

6

vs. photon

energy; (I) [99], (2) [I03], (3) [I04], (4) {96J, (5) [92, 93, 94, 9S], (6) {1011. lO"

?~ "T. . . . . . . . . .

N~

. . . . . .

,,.';'"::',~:~4~ ( Fig. P.3. Beta-rhombohedral boron. Isotherms of the optics] absorption of polycrys t a l s [92, 93}. .

50r |

i

O~

1'

'C

p";

um

20

Fig. 24. Beta-rhomb. boron. Anisotropy within the absorption edge |94].

of the optical

%,~i

,~

absorption

,

'~

,!0

!~

-//~,,oa

20

0

Fig. 25. B e t a - r h o m b . b o r o n . 1)e(;ompositiotl of t b c a b s o r p t : i ( m e d g e (F, [ c a s e x a m p l e ) ; (1)(2) i n d i r ( : c t t r a n s i t i o n s , (3)(3') m e a s u r e d v a l u e s , (4) e x t r a p o l a t e d l o n g - w a v e tail [94], ~0

15

zO

ZY

30

193

Optical properties According investigations

on single crystals

(fig.

transition

E II c

1

1.635 z 0.015 e V

1.615 z

2

1.42

1.37

24, 25) y i e l d e d t h e a n i s o t r e p y

E/. c

t

0.03

eV

anisotropy 0.015

0.20 ± 0.005

z 0.03

0.05

........................................................................

Like in o t h e r s e m i c o n d u c t o r s

~

t

0.02

t h e g a p e n e r g i e s v a r y a s T z [1, 2, 92, 93]: LIR(T)

with

[94]:

=

/JE(T=0)-

~ * T'

= 3.24 Z I0 -7

T h e c o e f f i c i e n t B i s s m a l l e r t h a n in Ge b y a f a c t o r 3, w h i c h r e s u l t s s t r e n g t h in B.

from the higher

bonding

Since the surface recombination strongly affects the spectral dependence of photoeonductivity, only a qualitative verification of the position of the absorption edge can be realized by this experimental method. As expected, the photoconductivity peaks are near the steepest g r a d i e n t o f t h e a b s o r p t i o n e d g e (fig. 26) [1, 103, 105 - 113], B u t a d d i t i o n a l maxima w e r e

units

, 1\- o 1+1

4/

s

.

8

~X~

rise

:

12.22]

~ [aot, l o ~f,~o.f]

I

103

decoy :

'?)\

,o_+,

)a

• [.;'z

31

~/

Io"

I

O

Fig.

26.

0.3

0.6

0.9

Beta-rhomb.

1.2

15

)

boron.

5 pm

7

Photo¢on-

d u c t i o n a t 300 K p l o t t e d v s . w a v e l e n g t h . The different curves are shifted relativ e to e a c h o t h e r . (a) [1, 105], (b)

[1o6], (c) [IO7], (d) [1o3]

0

I00

200

300 T - -

~00

K

500

Fig. 27. B e t a - r h o m b . b o r o n . Photc~:ond u c t i v i t y r i s e a n d d e c a y time c o n s t a n t s v s . t e m p e r a t u r e [ l , 103, 105, 106, 222, 223]

found at higher and lower photon energies, which latter are to be considered in connection w i t h t h e g a p s t a t e s ( s e e below). A p e c u l a r i t y o f b e t e - r h o m b o h e d r a l b o r o n a r e t h e g r e a t r i s e a n d d e c a y t i m e c o n s t a n t s o f p h o t o c o n d u c t i o n [1, 103, 105, 106, 11], 113 - 116, 222, 223]. T h e y i n c r e a s e to v e r y h i g h v a l u e s t o w a r d s l o w e r t e m p e r a t u r e s (fig. 27). T h i s s h o w s t h a t o p t i c a l l y i n d u c e d n o n - e q u i l i b r i u m s t a t e s p e r s i s t a t low t e m p e r a t u r e s f o r v e r y l o n g t i m e s . A further confirmation of the edge position is given by the spectral distribution of the e m i t t e d r a d i a t i o n a s s o c i a t e d w i t h s w i t c h i n g [117, 118]. E m i s s i o n maxima w e r e o b s e r v e d a t 1.25, 1.30 a n d /.55 eV, c o n f i r m i n g s o m e o f t h e t r a n s i t i o n s o b t a i n e d b y o t h e r m e t h o d s , b u t it m u s t be considered that switching conditions resemble a non-equilibrium state at high temperatures. Taking the analysis of the absorption edge and the results on the gap states (see subsequent s e c t i o n ) a s well a s t h e t r a n s p o r t p r o p e r t i e s i n t o a c c o u n t , t h e b a n d model in fig. 28 c a n be

194 derived [1, 3], w h i c h allows a w i d e l y c o n s i s t e n t d e s c r i p t i o n of t h e most e l e c t r o n i c p r o p e r t i e s of b e t a rhombohedral boron:

H. Werheit

conduction bond

The two i n d i r e c t t r a n s i t i o n s w i t h i n t h e a b s o r p t i o n e d g e lead f r o m t h e valence band and from the partly occupied hole trapping level in f r o n t o f it with a h i g h d e n s i t y of o.9~eV~ ~ 7,~~.~~ ~ ~ ~_ states into the conduction band. ~ ~ ~'~'Z ~ ' ~ ~ ~ ~.~ hoLe Optical t r a n s i t i o n s from t h e v a l e n c e band and from the neighbouring hole t r a p p i n g level i n t o t h e e l e c t r o n trapping level a r e s t r o n g l y f o r b i d d e n . But a t h i g h e r t e m p e r a t u r e s t h e e l e c t r o n t r a p p i n g level b e c o m e s vatence bQnd more a n d more o c c u p i e d b y t h e r m a l excitations, and then the additional Fig. 28. B e t a - r h o m b . b o r o n . Widely a c c e p t e d b a n d d i r e c t t r a n s i t i o n l e a d i n g from t h i s model. The e x p e r i m e n t a l l y obtained transitions, e l e c t r o n t r a p p i n g level i n t o t h e c o n w h i c h a r e c o n s i s t e n t with t h e d i f f e r e n t l e v e l s a n d d u c t i o n b a n d a p p e a r s a t lower e n e r their relative energetical positions, are indicated g i e s . T h e s t r e n g t h of t h e r e l a t e d (for r e f . s e e [ l , 3]. absorption depends on the thermally i n d u c e d o c c u p a t i o n of t h i s level. At low t e m p e r a t u r e s an o c c u p a t i o n of t h e e l e c t r o n t r a p p i n g level c a n be a c h i e v e d b y o p t i c a l p r e e x e i t a t i o n of e l e c t r o n s into t h e c o n d u c t i o n b a n d a n d s u b s e q u e n t t r a p p i n g . S i n c e no d i r e c t r e c o m b i n a t i o n of t r a p p e d e l e c t r o n s w i t h h o l e s is p o s s i b l e ( s e e a b o v e ) a n d h e n c e t h e r m a l e x c i t a t i o n i n t o t h e c o n d u c t i o n b a n d is r e q u i r e d , w h i c h e x c i t a t i o n is s t r o n g l y r e d u c e d a t low t e m p e r a t u r e s , t h i s n o n - e q u i l i b r i u m s t a t e h a s a v e r y l o n g r e l a x a t i o n time e n a b l i n g to m e a s u r e t h e o p t i c a l e x c i t a t i o n s p e c t r u m (see s e c t i o n 4.3). The t r a n s p o r t p r o p e r t i e s of b e t e - r h o m b o h e d r a l b o r o n h a v e n o t be~,n clarified d e f i n i t e l y . T h e y a r e mainly a f f e c t e d b y t h e t r a p p i n g l e v e l s h a v i n g h i g h d e n s i t i e s of s t a t e s , it is a s s u m e d t h a t t h e e l e c t r o n i c t r a n s p o r t a t low t e m p e r a t u r e s t a k e s place mainly w i t h i n t h e p a r t l y o c c u p i e d level n e a r t h e v a l e n c e b a n d b y v a r i a b l e r a n g e h o p p i n g [1, I l l , 119 - 124, 126, 127]. A p a r t f r o m t h i s v a r i a b l e - r a n g e h o p p i n g model, a s c r e e n e d P o o l e - F r e n k e ] model [125, 126] a n d m a c r o s c o p i c p o t e n t i a l b a r r i e r s [128] h a v e b e e n d i s c u s s e d to d e s c r i b e t h e e l e c t r o n i c t r a n s p o r t a t low t e m p e r a t u r e s . At h i g h e r t e m p e r a t u r e s hole c o n d u c t i o n d o m i n a t e s , w h i c h is c a u s e d b y t h e t h e r m a l e x c i t a t i o n of e l e c t r o n s f i r s t i n t o t h e level n e a r t h e v a l e n c e b a n d a n d l a t e r i n t o the level n e a r t h e c~>nduction b a n d e d g e . I n t h i s r a n g e of t e m p e r a t u r e s t h o s e l e v e l s a c t a s t r a p p i n g l e v e l s of h i g h d e n s i t y of s t a t e s a n d lead to v e r y small mobilities of f r e e c a r r i e r s . Real i n t r i n s i c c o n d u c t i v i t y w i t h n = p is n o t r e a l i z e d u p to t e m p e r a t u r e s of 2000 K. A p a r t from some e a r l y i n v e s t i g a t i o n s on b e t a - r h o m b o h e d r a l b o r o n of l a r g e l y u n d e f i n e d i m p u r i t y [129, 130] only p - t y p e m a t e r i a l h a s boon o b t a i n e d . But a r e c e n t i n v e s t i g a t i o n on Fe]3~,.5 (solid s o l u t i o n of Fe in b o r o n with i n t e r s t i t i a l Fe a t o m s ) s h o w s t h a t n - t y p e material c a n b e o b t a i n e d b y s u i t a b l e d o p i n g [131, 132]. O b v i o u s l y an o v e r c o m p e n s a t i o n of t h e u n o c c u p i e d s t a t e s in t h e hole t r a p p i n g level is n e c e s s a r y . I t is n o t c l e a r a t p r e s e n t , w h e t h e r t h i s t r a n s p o r t b y f r e e e l e c t r o n s t a k e s p l a c e w i t h i n t h e electrer, t r a p p i n g level, w i t h i n new l e v e l s i n d u c e d b y t h e Fe a t o m s o r w i t h i n t h e c o n d u c t i o n b a n d . I n s p i t e of t h e s u c c e s s of t h i s p h e n o m e n o l o g i c a l d e s c r i p t i o n b y a s s u m i n g t r a p p i n g l e v e l s in t h e b a n d g a p , t h e p r o b l e m r e m a i n s to e x p l a i n t h a t o p t i c a l t r a n s i t i o n s i n t o t h e e l e c t r o n t r a p p i n g l e v e l s a r e so s t r o n g l y f o r b i d d e n , t h a t n o t t h e l o w e s t a c c o r d i n g a b s o r p t i o n could h i t h e r t o o be d e t e c t e d . I t s e e m s t h a t n o t e v e n q u a n t u m - m e c h a n i c a l s e l e c t i o n r u l e s a r e s u f f i c i e n t to explain a s u c h s t r o n g l y f o r b i d d e n t r a n s i t i o n . T h e r e f o r e a c c o r d i n g to r e c e n t i n v e s t i g a t i o n s we t a k e t h e h y p o t h e s i s i n t o c o n s i d e r a t i o n , t h a t t h e e l e c t r o n t r a p p i n g level d o e s n o t r e a l l y e x i s t in t h e norm~ll b a n d s t r u c t l i r e , b u t it b e c o m e s o n l y d e v e l o p e d w h e n f r e e e l e c t r o n s a r e excited, w h i c h , b e c a u s e o f t h e i r g r e a t e f f e c t i v e m a s s e s , s u b s e q u e n t l y f o r m small p o l a r o n s . T h e s e p o l a r o n l e v e l s would h e p o s i t i o n e d below t h e c o n d u c t i o n b a n d b y t h e p o l a r o n b i n d i n g e n e r g y a n d b e h a v e a t h i g h t e m p e r a t u r e s similarly to t r a p p i n g s t a t e s . T a k i n g i n t o a c c o u n t t h a t t h e u n i t cell of b e t a - r h o m b o h e d r a l b o r o n c~tn be a p p r o x i m a t e l y a s s u m e d to c o n s i s t of e i g h t u n i t cells of a l p h a - r h o m b o h e d r a l b o r o n (cp. s e c t i o n 4.4), p e c u l a r i t i e s of t h e e l e c t r o n i c b a n d s t r u c t u r e of a l p h a - r h o m b o h e d r a l b o r o n (fig. 12) may be a l s o valid f o r b e t a - r h o m b o h e d r a l b o r o n , a n d t h e u n u s a l r a t i o of t h e e f f e c t i v e m a s s e s

Optical properties

195

m,' >> rap"

w o u l d s u p p o r t s u c h a n a s s u m p t i o n , b y l e e d i n g to t h e s m a l l e r b o n d i n g e n e r g y o f a c c o r d i n g l y a s s u m e d hole p o l a r o n s i n s t e a d o f t h e h o l e t r e p p l n g level mentioned above. But the i n t e r p r e t a t i o n o f t h i s level is c o m p l i c a t e d b e c a u s e 0 a t o m s o b v i o u s l y i n t r o d u c e o c c u p i e d e l e c t r o n i c s t a t e s in e n e r g e t i c p o s i t i o n s a d j a c e n t to t h e hole p o l a r o n r e s p e c t i v e l y h o l e t r a p p i n g level within the gap. F o r t h e b o r i d e s , w h i c h a r e i t ~ s t r u c t u r a l to b e t a - r h o m b o h e d r a l b o r o n , n o o p t i c a l i n v e s t i g a t i o n s in t h e r a n g e o f f u n d a m e n t a l a b s o r p t i o n a n d a b s o r p t i o n e d g e a r e a v a i l a b l e . B a n d g a p e , m o r e o r less roughly estimated from the temperature dependence of the electrical conductivity at high t e m p e r a t u r e s a r e l i s t e d in t h e f o l l o w i n g t a b l e : Boride

band gap (estimated)

SiB14

references

F~ ~e~r. = 0.26 eV

[133]

Ei e l ~ , = 0.54 eV B g . ~ t r . = 0.79 eV Et .iectr. = 0.8 eV

[2, 134] [135] [131]

solid s o l u t i o n s : CUB=, ZnBu FeBn.s

Structure family of alpha-terra,oriel boron. O n l y a f e w r e s u l t s a r e a~Lilable, w h i c h a r e n o t s u f f i c i e n t f o r a c o m p a r i n g d i s c u s s i o n . T h e r e f l e c t i v i t y s p e c t r u m a n d t h e d e r i v e d d i e l e c t r i c f u n c t i o n o f O=AI~B~ a r e s h o w n in f i g s . 29a, b [136]. M e a s u r e d , r e s p e c t i v e l y e s t i m a t e d b a n d g a p s a r e l i s t e d i n t h e f o l l o w i n g t a b l e :

35 30 25

/

2o

~5 -I

Io

Io ~

i

~o

i

i

2o

i

3o

~

E,

eV

4o

5 i

5

i

i

15

|

L

25

35 E,

eV

Boride

band gap

S.-AIBle

Eg opt. = 2 , 5 eV Rt .~-tr. = 2.5 eV Eg opt > 3 eV

CtA|sBu*)

Fig. 29. C~,lsBu. a) r e f l e c t i v i t y , b) r e a l ( E l ) a n d i m a g i n a r y ( E = ) p a r t o f t h e d i e l e c t r i c f u n c t i o n in t h e s p e c t r a l range of fundamental absorption.

references [137] [127, 138] (estimated from yellowish transparency)

=)cf. r e m a r k in s e c t i o n 2,3 S t r u c t u r e .fa .a~_.'ly.~t" b~t~ .~t,et,~a~ona] b o r o n O p t i c a l b a n d g a p s w e r e m e e s u r e d o n two r e p r e s e n t a t i v e s BemAlsBIa (fig. 30) [137, 139]. Here, too, t h e r e s u l t s a r e discussions.

o f t h i s g r o u p , alphe-AIBlffi a n d not sufficient for comparing

196

H. Werheit

b a n d gap

Boride Be=AIsBt=

E~, o~t.

= 2.12 eV

~ g electr,

~¢-AIBi=

references

:

2.1

eV

BI 0~¢. = 1.9 eV b~,~opt. = 1.96 eV El e l e c t r . --- 2.2 e V

4/ ~ 1

[139] [139] [137] [139] [140] 2.i~

S t r u c t u r e famil~..o~f ort.h.orhp, m.bic bpxides Only r e s u l t s of p r e l i m i n a r y o p t i c a l i n v e s t i g a t i o n s on LiAIBa4 a r e a v a i l a b l e (fig. 31) [141]. C o m p a r i s o n of t h e a b s o r p t i o n m e a s u r e m e n t with p h o t o c o n d u c t i v i t y s h o w s t h a t t h e o p t i c a l l y o b t a i n e d t r a n s i t i o n e n e r g y of 1.82 eV is not t h e b a n d g a p . T h i s is c o n f i r m e d b y e l e c t r ( m b s o r p t i o n m e a s u r e m e n t s in fig. 32 []42], w h i c h m e t h o d w a s u s e d f o r t h e f i r s t time in c a s e on a boron-type crystal. Obviously, at least one additional level in t h e b a n d g a p is e x i s t e n t . Without t r a n s p o r t i n v e s t i g a t i o n s it c a n n o t be d e c i d e d , w h e t h e r it is positionend near the valence or near the conduction b a n d . F u r t h e r maxima in t h e e l e c t r o a b s o r p t i o n spa(:t r u m p o s s i b l y i n d i c a t e o t h e r l e v e l s in t h e b a n d g a p .

% 1.1o61 . . . .

! 6

iiJ I .!

2.0 hp leV]

2.2

fig. 30. alpha-AIB12 (I) a n d (Be, AI)BI~ c o m p o u n d (2). A b s o r p t i o n e d g e s , Ka p l o t t e d vs. p h o t o n e n e r g y , h e n c e interpreted a s d i r e c t t r a n s i t i o n [137, 139].

,i

.',oo

i"

i r.

#;"

01. I_..

"

L I R I 8 14

,.'5

~"%"

..i®:

t ] Phot o n - E n a r 9 le

Fig. 32. LiAlB14. E l e c t r o a b u o r p t i o n e n e r g y [142].

0.I

0.01

Boride LiAIB14

2

$

Eg opt

4

remarks

= 1.82 eV = : = :

2.05 1.8 3.6 2,8

vs.

photon

Fig. 31. LiAIB14. A b s o r p t i o n c o e f f i c i e n t (a) a n d p h o t o c o n d u c t i v i t y (b) p l o t t e d ve p h o t o n e n e r g y [141] ( t h e n u m b e r s i n d i c a t e c o n s e c u t i v e c o n d u c t i v i t y m e a s u r e m e n t s with a d e l a y of a b o u t 1 h, e a c h ) .

band gap

E~ oat E¢ opt Eg opt Bz opt NaBi5 MgAIB14

11wise

leUJ

eY eV eV eV

indirect transition (level in t h e g a p ) p h o t o c o n d , maximum extrapol, electroabe. p h o t o c o n d , maximum extrapol, electroabs.

Et etectr. --= 0.32 eV e l e c t r i c a l c o n d u c t i v i t y Ez electr. : 0.51 eV e s t i m a t e d f r o m c o n d . ( p u r e material) E~ electr. = 0.47 oV e s t i m a t e d from c o n d . (Ni d o p e d )

references [141] [141] [160] [141] [142] [143] [144] [144]

Optical

properties

197

S t r u c t u r e family of YBC¢ T h e r e s u l t s a v a i l a b l e a r e l i s t e d below. I t i s n o t e w o r t h y t h a t t h e b a n d g a p s in t h e s e m o s t c o m plex c r y s t a l s o f t h e i c o s e h e d r a l b o r o n t y p e do n o t e s s e n t i a l l y d e v i a t e from t h o s e o f m u c h s i m p l e r s t r u c t u r e families. Betide .

.

.

.

.

.

.

.

.

.

.

band gap .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

remarks .

.

.

.

.

YBsLs

E~ .t

SmB~ GdBN

E~ ,i " 0.8 eV F~ a~t = 0.73 eV El e = 0.87 eV E.,.I = 1.27 oV

YbB~,

=

0.8

.

.

.

.

.

.

.

eV

.

.

.

.

.

.

.

references .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

estimated from cond. (T>600 K) (T)600 K)

.

.

.

.

.

.

.

.

[67] [145] [145] [145] [145]

Metal h e x a b o r i d e s t r u c t u r e famil~ B e s i d e s t h e Brillouin z o n e (fig. 33), a s a r e p r e s e n t a t i v e example t h e c a l c u l a t e d b a n d s t r u c t u r e of Oal~ is s h o w n in f i g s . 34 [148, 147]. F o r b a n d s t r u c t u r e c a l c u l a t i o n s of LaBs, Eui~ a n d

,~

Fig.

-----

1.2

33.

Metal h e a b o r i d e s ; Brillouin zone.

o ~

,

~ ~

r

~

x

h z

M

' ~

['

R

S

x

I, A

'

A

r

RH ~;t Ce~l g-

k

Fig. 34. CaBs. Calculated s e l f - c o n s i s t e n t b a n d s t r u c t u r e a n d d e n s i t y of s t a t e s . T h e s y m m e t r y p o i n t s o f k a r e i n d i c a t e d in fig. 33 [146, 147]. StuBs is r e f e r r e d to [148 - 150]. The f o l l o w i n g q u a l i t a t i v e d i f f e r e n c e s in c o m p a r i s o n w i t h t h e c a l c u l a t e d b a n d s t r u c t u r e s of b o r i d e s i s o s t r u c t u r a l to a l p h a - r h o m b o h e d r a l b o r o n a r e o b v i o u s . T h e y seem to b e c h a r a c t e r i s t i c f o r t h e w h o l e g r o u p of m a t e r i a l s : - small o r e v e n z e r o band gaps,

:e%/ :

'

J

I StuB6 Ill

I i

-

1.o

loo 9O

~]|

ti~l

i

i "~,~

~

~,

~1 °

2g

~

~ EuBs

v~ !

*q :

!0 "

1

eV

~0

9

B

12

eV 18

hcu ~

Fig. 35. StuBs. R e f l e c t i v i t y a n d d e r i v e d r e a l a n d i m a g i n a r y p a r t of t h e d i e l e c t r i c f u n c t i o n v s . p h o t o n e n e r g y . T h e Hmita of u n c e r t a i n t y a t low e n e r g i e s a r e c a u s e d b y e x p e r i m e n t a l u n c e r t a i n t y [151, 152] Fig. 36. EuBs a n d LaBs. R e f l e c t l v i t y v s . p h o t o n e n e r g y .

s t r o n g c u r v a t u r e of the bands meaning small e f f e c t i v e m a s s e s of c a r r i e r s .

Indeed, the optical r e flectivity spectra s h o w n in f i g s . 35, 36 [151 153] do n o t show absorption edges but strongly marked plasma edges. I n t h e c a s e o f SuBs was suggested [214] that the band structur e is s t r o n g l y i n f l u e n c ed e v e n b y v e r y small O c o n t e n t s l e a d i n g to a degenerate semiconductor.

198

H. Werheit

Boride

band gap

remarks

references

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

KBe

0.29 eV

el. c o n d .

[143]

CaBs

2.11 0.3 0.4 0.2

eV eV eV eV

calculated calculated el. c o n d . (T>820 K) el. c o n d .

[ 146] [147] [154] [155]

SrBe

3.68 eV 0.38 eV 0.45 eV

calculated el. cond. (T>1250K) el. c o n d . (T> 700K)

[ 146] [154] [156]

LaBs

0

c a l c u l a t e d (metal)

[148]

StuB6

0 eV 2.3 meV 5 meV

c a l c u l a t e d (semimetal) el. cond. el. c o n d .

[149] [157] [69]

EuBs

0 eV 1 eV 0.38 eV 0.3 eV 0.032eV 0.05...0.1 eV

c a l c u l a t e d (semimetal) o p t . , d e r i v e d f r o m refl. el. c o n d . el. c o n d . el. c o n d . el. c o n d .

[ 150] [153] [158] [69, 159]

0.14 eV 0.08lee

el. c o n d . el. c o n d .

[69, 759] [ 160]

YbB6

eV

[160]

The t a b l e s h o w s s t r o n g u n c e r t a i n t i e s in t h e d e r i v e d v a l u e s of t h e a c t i v a t i o n e n e r g i e s . Optical i n v e s t i g a t i o n s o n p u r e m a t e r i a l s would be a s u i t a b l e method to c l e a r u p t h e s e p r o b l e m s . B o r o n d o d e c a b o r i d e s t r u c t u r e family As a l r e a d y m e n t i o n e d a b o v e , d o d e c a b o r i d e 8 w i t h d i v a l e n t metals a r e p r e d i c t e d to be s e m i c o n d u c t o r s o r i s o l a t o r s , t h o s e with t r i v a l e n t metals to be metallic. M o r o v e r it is a s s u m e d , t h a t d o d e c a b o r i d e s w i t h s e m i c o n d u c t i n g o r i n s u l a t i n g c h a r a c t e r would n o t he s t a b l e [72]. B u t S c h w e t z e t al. [71] o b t a i n e d d i f f e r e n t mixed p h a s e s o f d o d e c a b o r i d e s , w h i c h a r e o b v i o u s l y isolating or semiconducting. This can doubtlessly be taken from the fact that these c r y s t a l s are colored, also when q u a n t i t a t i v e optical i n v e s t i g a t i o n s were hithertoo not performed. 4.3 E l e c t r o n i c s t a t e s in t h e b a n d S y s t e m a t i c i n v e s t i g a t i o n s of t h e e l e c t r o n i c s t a t e s in t h e b a n d g a p w e r e h i t h e r t o o o n l y p e r f o r m e d in t h e c a s e of b e t a - r h o m b o h e d r a l b o r o n . These gap states determine decisively the electronic transport properties of this s e m i c o n d u c t o r . T h e most i m p o r t a n t s t a t e s of t h i s k i n d w e r e a l r e a d y m e n t i o n e d in c o n n e c t i o n w i t h t h e a n a l y s i s of t h e a b s o r p t i o n e d g e : A partly occupied valence band edge.

level

of h i g h

density

of s t a t e s

approximately

0.2

eV a b o v e

the

A level of e v e n h i g h e r d e n s i t y of s t a t e s a t a b o u t 0.4 eV below t h e c o n d u c t i o n b a n d e d g e , w h i c h level i8 u n o c c u p i e d in t h e t h e r m a l e q u i l i b r i u m a t low t e m p e r a t u r e s . The e n e r g y b a n d s c h e m e (fig. 28) a l r e a d y c o n t a i n s t h e s e levelej t h e y a c t o n t h e f r e e c a r r i e r s like t r a p p i n g l e v e l s of h i g h c a p t u r e c r o s s s e c t i o n . T h e e s s e n t i a l o u t l i n e of t h i s s c h e m e c a n be t a k e n a s l a r g e l y c o n f i r m e d b y a lot of d i f f e r e n t i n v e s t i g a t i o n s ; it h a s p r o v e d to d e s c r i b e m o s t of t h e e l e c t r o n i c p r o p e r t i e s c o n s i s t a n t l y . Hence f o r a detailed d i s c u s s i o n of d i f f e r e n t e x p e r i m e n t a l r e s u l t s a n d p o s s i b l e a l t e r n a t i v e s to t h i s b a n d s c h e m e is r e f e r r e d to P r u d e n t l a t i [1181. By o p t i c a l p r e e x c i t e t i o n w i t h l i g h t of s u f f i c i e n t p h o t o n e n e r g y ( h ~ ) El) a n d s u b s e q u e n t c a p t u r e of t h e e l e c t r o n s , t h e e l e c t r o n t r a p p i n g level b e c o m e s o c c u p i e d to a v e r y h i g h d e g r e e . S i n c e a n immediate r e c o m b i n a t i o n of t r a p p e d e l e c t r o n s with h o l e s is n o t p o s s i b l e , t h i s

199

Optical properties

,..Li 2S

B 1.2

20

_g ,s

9¸ ~ = ~ %

Io

/ 0

$

0.~.

0.8

1.2 h~ ,, . =

1.5

eV

2.0

100 Fig. 37. B e t a - r h o m b . boron. P h o t o a b s o r p t i o n s p e c t r u m a t 113 K, excitation of trapped electrons into the conduction band [1, 105].

200

200

400

TiK) Fig. 38. B e t a - r h o m b . b o r o n . A b s o r p t i o n c o e f ficient vs. temperature at characteristic photon energies showing the low-temperature phase transition. Results obtained after t h e s a m p l e w a s k e p t in c o m p l e t e d a r k n e s s a t 78 K f o r 9 h. P h o t o n e n e r g i e s : (1) 0.9, (2) 0.84, {3) 0.574, (4) 0.4 eV [161, 162].

non-equilibrium state remains largely unalt e r e d f o r a lot o f h o u r s . T h e r e f o r e i t s optical excitation spectrum can be obtained as the difference between the absorption s p e c t r a o f t h e e x c i t e d a n d t h e u n e x c i t e d s t a t e (fig. 37) [1, 106]. T h e a d a p t e d c a l c u l a t i o n l e a d i n g to a n a c t i v a t i o n e n e r g y o f : 0.45 eV p r e s u m e d t h a t t h e n a r r o w a b s o r p t i o n b a n d a t 0.4 eV i s to b e a t t r i b u t e d to a n e x c i t e d s t a t e o f t h e t r a p p i n g level. B u t t h i s p r o v e d n o t to be c o r r e c t . R e c e n t i n v e s t i g a t i o n s [161, 162] s h o w e d t h a t t h i s n a r r o w a b s o r p t i o n b a n d is l a r g e ly i n d e p e n d e n t o f t h e o c c u p a t i o n o f t h e e l e c t r o n t r a p p i n g level, a n d i t s a b s o r p t i o n is o n l y s u p e r i m p o s e d to t h e e x c i t a t i o n s p e c t r u m o f t h e t r a p p e d e l e c t r o n s . M o r e o v e r i t w a s s h o w n [161, 162] to e l e c t r o n i c t r a n s i t i o n s b e t w e e n r a t u r e e q u i l i b r i u m s t a t e (T < 180 a t a b o u t 180 K a p h a s e t r a n s i t i o n absorption spectrum, showing at continuum.

that this narrow absorption band, which can be attributed n a r r o w e l e c t r o n i c l e v e l s , i s c h a r a c t e r i s t i c f o r t h e low t e m p e K) o f b e t a - r h o m b o h e d r a l b o r o n . With i n c r e a s i n g t e m p e r a t u r e , t a k e s p l a c e (fig. 38), o p t i c a l l y i n d i c a t e d b y a c h a n g e o f t h e higher temperatures obviously an excitation into an energy

T h e p h y s i c a l b a c k g r o u n d of t h i s p h a s e t r a n s i t i o n h a s r e m a i n e d u n c l e a r . B u t t h e p h a s e t r a n s ition s e e m s to b e r a t h e r e f f e c t i v e , b e c a u s e a c r i t i c a l e x a m i n a t i o n o f o t h e r p h y s i c a l p r o p e r t i e s of b e t a - r h o m b o h e d r a l b o r o n e x h i b i t e d d i s c o n t i n u o u s b e h a v i o u r a t t h i s t e m p e r a t u r e ; b e c a u s e of t h e v e r y l o n g r e l a x a t i o n time, t h e p h a s e t r a n s i t i o n m a y b e o v e r l o o k e d in t h e s e e x p e r i m e n t s before: - P h o t o a b s o r p t i o n s p e c t r a c h a n g e s i g n i f i c a n t l y [93, 105]. - The thermally stimulated electrical conductivity after optical excitation exhibits a minim u m o r a t l e a s t a s a d d l e p o i n t [163]. - The slope of the photoconductivity with constant spin density is distinctly different [I16]. - An i n t e r n a l f r i c t i o n r e s o n a n c e p e a k w a s o b t a i n e d [164] T h e d i e l e c t r i c l e a s h a s a d i s t i n g u i s h a b l e maximum [165]. - The sound attenuation increases steeply towards higher temperatures [166]. T h e m a g n e t i c s u s c e p t i b i l i t y h a s a m i n i m u m [167]. T h e m a g n e t o r e s i s t a n c e c h a n g e s b y o r d e r s o f m a g n i t u d e [168]. -

-

-

F r o m fig. 39 c a n b e t a k e n t h a t t h i s p h a s e t r a n s i t i o n c a n b e i n f l u e n c e d b y o p t i c a l e x c i t a t i o n . Therefore in a certain temperature range a kind of optical bistability exists. Additionally to fig. 26, where in the photoconductivity spectrum at room temperature two m a x i m a w e r e o b t a i n e d , w h i c h w e r e a t t r i b u t e d to e l e c t r o n i c t r a n s i t i o n s i n c o n n e x i o n w i t h g a p s t a t e s , in fig. 40 [1, 93, 106] t h e p h o t o c o n d u c t i o n a t a low t e m p e r a t u r e a f t e r o p t i c a l p r e e x c i t a t i o n i s s h o w n . B e s i d e s t h e m a x i m u m e v o k e d b y b a n d - b a n d g e n e r a t i o n , t h e e x c i t a t i o n of t r a p p e d e l e c t r o n s (maximum a t 1.3 tin.m) d e p e n d i n g o n p r e e x e i t a t i o n , a n d t h e e x c i t a t i o n o f h o l e s (maximum a t 6.9 (M m) i s v i s i b l e .

F'CGC-N

H. Werheit

200

( .,,--\

l

,,w=O.,e,'

i-\

"" SO 'E

......

~n

'i

10

"%:.. I [

~o ~00

200

10-6 ~

T(K)

~

_

400

300

Fig. 39. Beta-rhomb. boron. Absorption coefficient at 0.4 eV v s . t e m p e r a t u r e a f t e r 6.5 h a t 78 K in c o m p l e t e d a r k n e s s a n d s u b s e q u e n t o p t i c a l excitation; with r e s p . w i t h o u t f u r t h e r optical excitation d u r i n g the heating p r o c e s s {161, 162].

3.0

I0-8

units

B

aa

timeotter precedingillumination 20rain 40rain o, 60rain

___

1 __._~

I .~._~-

2

6

~

B I/T.103(,K -I)

Fig, 41. B c t a - r h o m b . b o r o n with d i f f e r e n t C c o n t e n t . Electrical c o n d u c t i v i t y v s . r e c i p r o c a l t e m p e r a t u r e ; (1, 2, 3) w i t h o u t , {1=, 2=, 3=) a f t e r 1 h optical excitation. C co~ltent: (1, l =) .100 ppm, (2, 2=) 1600 ppm, (3, 3~) 3200 ppm

•

1.5- ~

0

.

| 169].

r~--~r

I

2

3

L ~5

~

5

~

7

r-

8pro 9

Fig. 40. B e t a - r h o m b . b o r o n . P h o t o c u r r e n t 100 K v s . w a v e l e n g t h a t d i f f e r e n t r a t e s p r e c e e d i n g o p t i c a l excitation ( I , 93, I05].

at of

Valuable i n f o r m a t i o n on the i n f l u e n c e o f t h e t r a p p i n g l e v e l s a n d of t h e c a r b o n i m p u r i t i e s on t h e t r a n s I x ) r t p r o p e r t i e s w e r e o b t a i n e d from s i m u l t a n e o u s m e a s u r e m e n t s of o p t i c a l a b s o r p t i o n a n d e l e c t r i c a l c o n d u c t i v i t y of b e t a - r h o m b o h e d r a l b o r o n with d i f f e r e n t C c o n t e n t at i n c r e a s i n g t e m p e r a t u r e s without and after optical p r e e x c i t a t i o n (fig. 41, 42) [169J. I t could be c o n c l u d ed t h a t t h e a s s u m p t i o n of some a u t h o r s [170-174] t a k i n g t h e c o n d u c t i o n i n c r e a s i n g with t e m p e r a t u r e s i m p l y a s glow c u r v e is n o t p e r m i s s i b l e , b e c a u s e t h e main maximum d o e s not coincide with t h e s t e e p e s t p a r t of t h e t r a n s m i s s i o n glow c u r v e , w h i c h i n d i c a t e s t h e g r e a t e s t r a t e of t h e r m a l excitation o f t r a p p e d electrons into the conduction band, Contrary, this c o n d u c t i v i t y maximum c o i n c i d e s w i t h t h e f i r s t b e g i n n i n g of i n c r e a s i n g t r a n s m i s s i o n . U s i n g t h e e n e r g y b a n d s c h e m e , t h e following i n t e r p r e t a t i o n w a s g i v e n :

I i

.... .....

1.0

'*"

t~=0.9eV

0B 06 }

:

0AI

13,~.. °.~ °

0.2 0

I 0

tOO

200

•

, J,o ..... "?"~:f° #V

300

600 T(°K)

Fig. 42. B e t a - r h o m b . b o r o n with d i f f e r e n t C c o n t e n t . Relative c h a n g e of optical t r a n s m i t t a n c e a t 0.9 vV vs. t e m p e r a t u r e a f t e r o p t i c a l excitation at 123 K ( s i m u l t a n e o u s l y o b t a i n e d with r e s u l t s in fig. 41) [169].

By optical e x c i t i n g a n d s u b s e q u e n t t r a p p i n g of e l e c t r o n s t h e o c c u p a t i o n d e n s i t y of t h e level n e a r t h e v a l e n c e b a n d e d g e is r e d u c e d , which l e a d s to a n i n c r e a s i n g h o p p i n g p r o b a b i l i t y . When t h e t e m p e r a t u r e i n c r e a s e s t t h e h o p p i n g p r o c e s s e s become t h e r m a l l y a c t i v a t e d a n d t h e c o n d u c t i v i t y i n c r e a s e s , too. But w h e n a t s o m e w h a t h i g h e r t e m p e r a t u r e s t h e t h e r m a l e n e r g y b e c o m e s s u f f i c i e n t to excite t r a p p e d e l e c t r o n s , w h i c h s u b s e q u e n t l y r e c o m b i n e , t h e o c c u p a t i o n

Optical properties

201

d e n s i t y a n d h e n c e t h e h o p p i n g c o n d u c t i v i t y of h o l e s d e c r e a s e s while t h e c o n d u c t i v i t y of e x cited e l e c t r o n s i n c r e a s e s a n d a r e a l glow maximum o r i g i n a t e s a t a s o m e w h a t h i g h e r t e m p e r a t u r e . P r o b a b l y t h i s i n t e r p r e t a t i o n is o v e r s i m p l i f i e d ; c o n t r i b u t i o n s of b a n d c o n d u c t i o n a n d c o n d u c t i o n w i t h i n t h e e l e c t r o n t r a p p i n g level seem to be p o s s i b l e , too [169]. I n 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 t h e c h a n g e of o p t i c a l t r a n s m i t t a n c e (fig. 42) [169] a f t e r p r e e x c i t a t i o n a t low t e m p e r a t u r e s a n d s u b s e q u a n t h e a t i n g , f i r s t a s t r o n g d e c r e a s e is s e e n a c c o r d i n g to t h e e x h a u s t i o n of t h e e l e c t r o n t r a p p i n g level, b u t a t h i g h e r t e m p e r a t u r e s a s t e p - w i s e i n c r e a s i n g a b s o r p t i o n o c c u r s , w h o s e i n t e n s i t y d e p e n d s o n t h e C c o n t e n t of t h e sample. It is a s s u m e d t h a t t h i s a b s o r p t i o n is d u e to e x c i t e d s t a t e s o f c a r b o n i m p u r i t i e s w h i c h become o c c u p i e d w h e n t h e Fermi level p a s s e s them d u r i n g i n c r e a s i n g t e m p e r a t u r e s . A v a l u a b l e m e t h o d to o b t a i n i n f o r m a t i o n method. E l e c t r o n - h o l e p a i r s a r e g e n e r a t e d a n d a h i g h e l e c t r i c field d r i v e s e l e c t r o n s s i g n o f t h e aplied v o l t a g e . From t h e f l i g h t

on t h e mobility of c a r r i e r s is t h e t i m e - o f - f l i g h t b y o p t i c a l e x c i t a t i o n o n o n e s u r f a c e of t h e sample, or holes to the opposite surface depending on the time t h e mobility c a n be d e r i v e d .

I n b s t a - r h o m b o h e d r a l b o r o n t h e s i t u a t i o n is c o m p l i c a t e d b y t r a p p i n g p r o c e s s e s c a u s i n g s t r o n g l y s p r e a d i n g f l i g h t times. But m o r e o v e r t h e c o m p l e t e l y u n u s u a l s i t u a t i o n w a s f o u n d t h a t t h e t i m e - d e p e n d e n t d i s t r i b u t i o n o f c u r r e n t is i n d e p e n d e n t of t h e s i g n of t h e a p p l i e d v o l t a g e a n d h e n c e t h e time o f f l i g h t m u s t be a s s u m e d to be t h e same f o r e l e c t r o n s a n d holes [174 177]. Uno e t al. [174, 176] t r i e d to i n t e r p r e t s t h i s b e h a v i o u r b y a s s u m i n g v a r i o u s t r a p p i n g levels, w h o s e e n e r g i e s c o n t r a d i c t to t h e r e s u l t s o b t a i n e d f r o m o t h e r , much more r e l i a b l e m e t h o d s . I t s e e m s m o r e likely t h a t t h e c a r r i e r t r a n s p o r t t a k e s place b y a k i n d o f a m b i p o l a r d i f f u s i o n . In a n y w a y a n i m p r o v e d t h e o r y is n e c e s s a r y to d e s c r i b e t h e e x p e r i m e n t a l r e s u l t s . T h e s e e x a m p l e s may d e m o n s t r a t e t h a t in c ~ n n e c t i o n w i t h t h e g a p s t a t e s in b e t a - r h o m b o h e d r a l b o r o n a lot of q u e s t i o n s r e m a i n s o p e n . E s p e c i a l l y m u s t be r e m a r k e d a g a i n t h a t t h e i m p o s s i b i l i t y of immediate t r a n s i t i o n s b e t w e e n v a l e n c e b a n d a n d e l e c t r o n t r a p p i n g level a r e not completely elucidated. I n o t h e r m o d i f i c a t i o n s a n d c o m p o u n d s of b o r o n n o s y s t e m a t i c i n v e s t i g a t i o n s of t h e e l e c t r o n i c s t a t e s in t h e b a n d g a p a r e k n o w n . N e v e r t h e l e s s , t h e s i m i l a r i t y of t r a n s p o r t m e c h a n i s m [2-4] seem to i n d i c a t e t h a t c o m p a r a b l e c o n d i t i o n s of t h e e l e c t r o n i c s t r u c t u r e i n c l u d i n g g a p s t a t e s may to be e x p e c t e d . With r e g a r d to o p t i c a l i n v e s t i g a t i o n s it is p o i n t e d to t h e e d g e tails in beta-AlBll a n d BesAI~BI= [139] (fig. 30) a n d to t h e e l e c t r o a b s o r p t i o n s p e c t r u m of LiAlB14 (fig. 32) [142]. 4.,4 _La.tfii9o- _vibratip_np_, T h e lattice v i b r a t i o n s r e p r e s e n t t h e m o s t immediate r e l a t i o n b e t w e e n c r y s t a l s t r u c t u r e a n d o p t i c a l p r o p e r t i e s of a solid, b e c a u s e t h e p h o n o n r e s o n a n c e f r e q u e n c i e s a r e d e t e r m i n e d b y t h e m a s s e s of v i b r a t i n g a t o m s o r atomic g r o u p s , b o n d i n g d i s t a n c e s a n d a n g l e s , c h a r g e d i s t r i b u t i o n s a n d b o n d i n g s t r e n g t h s . S t r u c t u r a l d i s o r d e r t a k e s e f f e c t o n t h e d a m p i n g . Hence from t h e i n v e s t i g a t i o n of lattice v i b r a t i o n s immediate i n f o r m a t i o n o n t h e c r y s t a l s t r u c t u r e c a n be o b t a i n e d a n d m o r e o v e r , t h e lattice v i b r a t i o n s p e c t r u m c a n be immediately u s e d a s a f i n g e r - p r i n t s y s t e m to i d e n t i f y c r y s t a l s t r u c t u r e s . T h e p o s s i b l e o s c i l l a t i o n s of t h r e e - d i m e n s i o n a l c r y s t a l s t r u c t u r e s a r e r e p r e s e n t e d b y t h e a c o u s t i c a n d o p t i c a l b r a n c h e s in t h e f i r s t Brillouin zone. N a t o m s p e r u n i t cell lead o n p r i n ciple to 3 * n d i s p e r s i o n c u r v e s w ( q ) , t h r e e o f w h i c h a r e a c o u s t i c b r a n c h e s a n d t h e r e m a i n i n g 3 * n - 3 a r e t h e o p t i c a l b r a n c h e s . From t h i s it b e c o m e s o b v i o u s t h a t in b o r o n a n d i t s b o r o n - r i c h b o r i d e s w i t h l a r g e n u m b e r s of a t o m s in t h e u n i t cell c o m p l e t e t h e o r e t i c a l c a l c u l a t i o n s a r e e x t r e m e l y difficult. T h e s e p r o b l e m s d e c r e a s e o n l y i n s i g n i f i c a n t l y b y t h e n u m b e r o f b r a n c h e s b e i n g r e d u c e d b e c a u s e of s y m m e t r y r e a s o n s . To d e t e r m i n e t h e p h o n o n d i s p e r s i o n c u r v e s c o m p l e t e l y b y e x p e r i m e n t , t h e p h o n o n s m u s t become excited with momenta b e t w e e n 0 in t h e c e n t e r a n d ~ l~/a a t t h e b o u n d a r i e s o f t h e f i r s t Brillouin zone, w h i c h c a n n o t be r e a l i z e d b y o p t i c a l e x c i t a t i o n b e c a u s e o f k ~ w a ~ 0. I n p r i n c i p l e , t h i s excitation c a n be r e a l i z e d b y inelastic X-ray s c a t t e r i n g and inelastic n e u t r o n scattering, but s u c h r e s u l t s are p r e s e n t l y o n l y a v a i l a b l e f o r L a ~ [178, 179, 180] ( s e e fig. 69). To o b t a i n n e u t r o n s c a t t e r i n g s p e c t r a is d i f f i c u l t b e c a u s e of t h e g r e a t a b s o r p t i o n c r o s s s e c t i o n of b o r o n . Elastic p r o p e r t i e s a n d s o u n d v e l o c i t y yield i n f o r m a t i o n o n t h e a c o u s t i c v i b r a t i o n s c h a r a c t e r ized b y a p p r o x i m a t e l y i n - p h a s e m o v i n g o f o p p o s i t e l y c h a r g e d a t o m s o r atomic g r o u p s . On t h e o t h e r h a n d , in o p t i c a l v i b r a t i o n s t h e c e n t e r o f m a s s r e m a i n s fixed, a n d o p p o s i t e l y c h a r g e d a t o m s o r atomic g r o u p s move i n p h a s e - o p p o s e d . B u t o n l y if a p e r m a n e n t dipole moment e x i s t s , on w h i c h t h e E v e c t o r o f t h e l i g h t - w a v e c a n act, t h e v i b r a t i o n is r e p r e s e n t e d in t h e IR a b s o r p t i o n a n d r e f l s o t i v i t y s p e c t r u m , a n d a c c o r d i n g l y it is called IR a c t i v e .

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] n t h e sasw¢~ w a y , a s t h e b o n d i n g p r o p e r t i e s of t h e c r y s t a l s d e t e r m i n e e s s e n t i a l l y t h e i n d i v i dual vibration spectra, they are also relevant for several macroscopically measurable integral p r o p e r t i e s of t h e s o l i d s , a s e.g. m e l t i n g p o i n t , h a r d n e s s , t h e r s ~ d e x p a n s i o n a n d c o m p r e s s i b i l i t y a n d t h e y c a n be q u a l i t a t i v e l y o b t a i n e d f r o m s u c h data. But a s c a n be s e e n f r o m t a b l e 1, f o r t h e b o r o n - t y p e m a t e r i a ) s t h e a v a i l a b l e s c o p e of t h e s e q u a n t i t i e s is v e r y limited a n d m o r e o v e r in some c a s e s t h e y seem to be u n c e r t a i n to some e x t e n d . N e v e r t h e l e s s m o s t o f t h e d a t a a r e a t l e a s t q u a l i t a t i v e l y c o r r e l a t e d a s e x p e c t e d , l~ut n o t e w o r t h y i s t h e g r e a t h a r d n e s s of SiBe c o n n e c t e d w i t h a r a t h e r low m e l t i n g p o i n t .

As d i s c u s s e d a b o v e , Briliouin s c a t t e r i n g is t h e o n l y o p t i c a l m e t h o d to o b t a i n i n f o r m a t i o n o n t h e a c o u s t i c b r a n c h e s o f t h e lattice v i b r a t i o n s a n d b e s i d e s , t h i s i n f o r m a t i o n is limited to t h e c e n t e r of t h e Brillouin Zone. N e v e r t h e l e s s , it may be u s e f u l t o i n c l u d e t h e a c o u s t i c b r a n c h e s in t h e s y s t e m a t i c d i s c u s s i o n o f o p t i c a l v i b r a t i o n s , Los, a s will be s h o w n b y c o m p a r i n g t h e s p e c t r a of a l p h a - a n d b e t a - r h o m b o h e d r a ] b o r o n . 400[ ~300 ......

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T h e a c o u s t i c b r a n c h e s of t h e v i b r a t i o n s p e c t r u m r a n g e f r o m z e r o to a c u t o f f f r e q u e n c y d e p e n d i n g o n t h e c o n d i t i o n s of t h e c r y s t a l s t r u c t u r e . Slack e t el. [181] e s t i m a t e d t h i s a c o u s t i c - p h o n o n c u t o f f f o r s e v e r a l b o r o n - t y p e s t r u c t u r e s from s o u n d velocity, Debye t e m p e r a t u r e a n d l a t t i c e c ~ n s t a n t ( s e e t a b l e l ) . At f i r s t s i g h t , t h e g r e a t d i f f e r e n c e b e t w e e n t h e c u t o f f f r e q u e n c i e s of a l p h a - a n d b e t a - r h o m b e h e d r a l b o r o n is s t r i k i n g , b u t t h i s c a n b e q u a l i t a t i v e l y u n d e r s t o o d b y c o m p a r i n g t h e c r y s t a l s t r u c t u r e s . C o n s i d e r i n g t h a t t h e u n i t cell of b e t a r h o m b o h e d r a l b o r o n c~n b e imagined a s a p p r o x i m a t e l y a s s e m b l e d o u t of e i g h t u n i t c e l l s of a l p h a - r h o m b o h e d r a l b o r o n , t h e f i r s t Briliouin 7~nc of b e t a - r h o m b o h e d r a ] b o r o n c a n be a p p r o x i m a t e l y o b t a i n e d b y b i s e c t i n g t h a t o n e of a l p h a - r h o m b o h e d r a l b o r o n [7, 182]. T h e n a c c o r d i n g to fig. 43, t h e a c o u s t i c p h o n e s c u t o f f of a l p h a - r h o m b o h e deal b o r o n is t r a n s f e r r e d i n t o a n o p t i c a l v i b r a t i o n in t h e c e n t e r of t h e BriUouin zone of b e t e - r h o m b o h e d r a ] b o r o n , w h o s e a c o u s t i c - p h o n e s c u t o f f is c o n s e q u e n t l y a p p r e c i a b l y l o w e r e d .

Optical v i b r a t i o n s of t h e i c o s a h e d r o n As m e n t i o n e d , i c o s a h e d r a with ordy s l i g h t l y d i f f e r e n t atomic s p a c i n g a r e t h e g e n e r a ] b a s i c s t r u c t u r e u n i t s of t h e m o d i f i c a t i o n s a n d o f t h e m o s t c o m p o u n d s of b o r o n . T h e r e f o r e t h e v i b r a t i o n m o d e s of t h e s e i c o s a h e d r a , w h i c h of c ~ u r s e a r e modified w i t h i n t h e a c t u a l c r y s t a l field b e i n g mainly d e t e r m i n e d b y t h e a v e r a g e i n t e r i c o s a h e d r a l b o n d i n g s t r e n g t h , form t h e main p a r t of the. v i b r a t i o n s p e c t r a of t h e JR a c t i v e a n d Ramae a c t i v e p h o n o n s . I n d e e d , t h e r e a r e no c a l c u l a t i o n s o f t h e v i b r a t i o n a l f r e q u e n c i e s from f i r s t p r i n c i p l e s available, b u t s y m m e t r y a n d g r o u p t h e o r e t i c a l c o n s i d e r a t i o n s yield a t l e a s t t h e d i f f e r e n t v i b r a t i o n s p e c i e s a n d t h e allied n u m b e r s of v i b r a t i o n modes [182, 183, 184]. M o r e o v e r , a s s u m i n g b o n d i n g o n l y to n e a r e s t n e i g h b o u r s w i t h i n a r e g u l a r i c o s a h e d r o n , r e p r e s e n t e d b y 30 identical Hcoke's law s p r i n g s , Beckel a n V a u g h a n [185] c a l c u l a t e d t h e e i g h t d i f f e r e n t f r e q u e n c i e s . T h e s e d a t a a r e l i s t e d in t a b l e 2. Table 2: V]BPJ',T1ON SPECIES OF THE 1COSAHEDRON [184, ]85] symmetry 1'14 Ft* l's* F*' I's" r=I'="

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Optical properties

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T h e r e s o n a n c e f r e q u c n ( - y o f t h e I R - a c t i v e v i b r a t i o n (805.1 cm -t) d i f f e r s ~ o m e w h a t f r o m 750 cm -t derived by Becher [186] from the experimentally obtained (Bt~H~)'" vibration. A detailed description of the actual movements of atoms in the icsahedron attributed to the different vibration modes is given by Beckel and Vaughan [185]. The calculation of the vibration frequencies demands the knowledge of the coupling constants. Weber e n d T h o r p e [184] s o l v e d t h i s p r o b l e m b y c a l c u l a t i n g t h e p h o n o n f r e q u e n c i e s b a s e d o n a s s u m p t i o n s o n c e r t a i n c ~ m b i n e t i o n s o f v a l e n c e f o r c e f i e l d p o t e n t i a l s i n c a s e of (BtsHtffi)-°, (BttDtffi)-- a n d ( B t i C l n ) - . T h e n t h e y c o m p a r e d t h e r e s u l t s w i t h d a t a o f t h e I R - a c t i v e a n d R a m a n - a c t i v e p h o n o n s e x p e r i m e n t a l l y o b t a i n e d . T h e b e s t f i t l e d to t h e i r b o n d m o d e l c h a r a c t e r ized b y s t r o n g l y d a l o c a l i z e d , i.e. metallic b o n d i n g o f 2 s e l e c t r o n s (1 p e r atom). For some modifications of crystalline boron and their isostructural borides the vibrations were c a l c u l a t e d b y g r o u p t h e o r y [182, 183]; t h e I R - a c t i v e a n d R a m a n - a e t i v e modes~ w h i c h a r e r e l e v a n t f o r t h e o p t i c a l i n v e s t i g a t i o n s a r e s h o w n in t a b l e 3. Of c o u r s e i d e a l i z e d s t r u c t u r e s m u s t be p r e s u m e d , w h i c h m e a n s t h a t d e f o r m a t i o n s o f t h e i c o s a h e d r a in 1~0~ the crystal structures a n d small d e v i a t i o n s o f t h e p o s i t i o n s of cm-; 8 single atoms from crystallographic planes were neglected. Differences between experimental and theoretical results may 130C a d d i t i o n a l l y a r r i s e f r o m a c c i d e n t a l d e g e n e r a c y o r c a u s e d b y too low oscillator strengths. 1200- The i n f l u e n c e of e x t e r n a l b o n d i n g of t h e i c o s a h e d r o n on its v i b r a 1100 t i o n s c a n b e q u a l i t a t i v e l y e s t i m a t e d . I n (BttHlt)-- t h e d e g e n e r a t e d i // I R - a c t i v e v i b r a t i o n of t h e i c o s a h e d r o n i s o n l y s l i g h t l y s h i f t e d / t o w a r d s lower f r e q u e n c i e s , b e c a u s e t h e m a s s o f t h e h y d r o g e n a t o m ~------7 is a d d e d to e a c h b o r o n atom. B u t i n c r y s t a l s t h e c o u p l i n g o f t h e ~ vibrating icosahedra by intericosahedral bonding becomes important 900 ~ a n d t h e degenerate v i b r a t i o n o f t h e i c o s a h e d r o n s p l i t s a p p r o x i m a t e ~ ' ~ /" --~'~ ly s y m m e t r i c a l l y in t h e c r y s t a l field. T h e a m o u n t o f s p l i t t i n g ~ . J d e p e n d s on t h e s t r e n g t h o f i n t e r i c o s a h e d r a ] b o n d i n g , i.e. o n t h e ~. ~-"--'d e g r e e o f s a t u r a t i o n o f t h e m u l t i p l e - c e n t e r b o n d s . T h e lower t h e 700 - C ~ _ - ~ . . . . ~ number of multiple-center bonds and the greater instead the num.... bar of saturated outward directing icosahedral bonds, the stronger .I" _ ~ _ is the crystal field and hence the frequency splitting. Since this 60o~ . _ ~ c r y s t a l field a c t s o n t h e v i b r a t i o n f r e q u e n c i e s a n d o n m a c r o s c o p i c re" properties like thermal extension, compressibility and hardness as 500 0 025 050 o75 ~00 well, a n a c c o r d i n g c o r r e l a t i o n is e x p e c t e d . RhornbohedroI bond strengln

Fig. 44. A l p h a - r h o m b o h . boron. Variation of the Raman-octivc frequencies with the intericosahcdra] coupling strength {dashed lines Y'2*, solid l i n e s l's., points: Raman data []84, ]87].

In the case o f a l p h a - r h o m b o h e d r a l boron, Weber and T h o r p e [184] calculated the actual phonon frequencies by taking the intra-icosahedral force constants unchanged from the best model o f (Bl=/hs)-". T h e s t r ~ m g i n t e r i c o s a h e d r a l b o n d s a l o n g t h e e d g e s o f t h e u n i t cell w e r e t a k e n i n t o a c c o u n t b y a s s u m i n g b o n d - s t r e t c h i n g and bond-bending potentials. The multiple-center bonds within the u n i t call w e r e a s s u m e d to be s i g n i f i c a n t l y w e a k e r a n d t h e r e f o r e omitted. The coupling constants were obtairmd by adjusting the two highest R a m a n - a c t i v e f r e q u e n c i e s to c o r r e s p o n d i n g experimental r e s u ] t s [184, 187]. T h e g o o d a g r e e m e n t is s h o w n in f i g . 44.

I n fig. 46 t h e b e s t a v a i l a b l e MIR a n d FIR s p e c t r a o f r e p r e s e n t a t i v e s o f t h e a l p h a - r h o m b o h e d r a l s t r u c t u r e g r o u p a r e p r e s e n t e d []37, 182, ]83, 188]. T h e v i b r a t i o n s o f t h e i c o s a h e d r o n , w h i c h a r e e x p e c t e d to s p l i t n e a r l y s y m m e t r i c a l l y r e l a t i v e to t h e v i b r a t i o n f r e q u e n c y o f t h e i s o l a t e d i c o s a h e d r o n (75o cm ~l) [186] c a n e a s i l y b e d i s c e r n e d . I n a l p h a - r h o m b o h e d r a l b o r o n , t h e y r a n g e f r o m a b o u t 600 to 900 em'~, a n d in t h e c o m p o u n d s f r o m a b o u t 1100 to a b o u t 400 c:m-L T h e r e f o r e t h e b o n d s t r e n g t h s m u s t b e d i s t i n c t l y h i g h e r t h a n i n a l p h a - r h o m b o h e d r a l b o r o n b u t it is o b v i o u s l y r o u g h l y i n d e p e n d e n t f r o m t h e k i n d o f a t o m s s a t u r a t i n g t h e t h r e e c e n t e r b o n d a n d f r o m t h e c h a i n b o e i n g t w o - o r t h r e e - a t o m i c . By c o m p a r i n g BI=C~ with BsO t h i s is c o n f i r m e d a l s o b y h a r d n e s s m e a s u r e m e n t s ( t a b l e I), w h o s e r e s u l t s a g r e e w i t h i n t h e a c c u r a c y o f m e a s u r e m e n t . T h e s p l i t t i n g i n t h e c a s e o f Bee i s s o m e w h a t g r e a t e r i n d i c a t i n g accordingly a stronger intericosahedral coupling strength. Both v i b r a t i o n s in b o r o n c a r b i d e , w h i c h a r e d i s t i n c t J y s e p a r a t e d t o w a r d s h i g h e r f r e q u e n c i e s (1580 a n d 1370 cm-l), c a n be a t t r i b u t e d to t h o s e b e l o n g i n g to t h e t h r o e - a t o m i c c h a i n (cp. t a b l e 3). F r o m r a e a s u r e m e n t s o n s i n g l e c r y s t a l s [182] w a s o b t a i n e d t h a t t h e s t r o n g e r o n e

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Fig. 45. A l p h a - r h o m b o h e d r a l b o r o n s t r u c t u r e g r o u p , Optical t r a n s m i s s i o n r e s p . r e f l e c t i v J t y s p e c t r a in t h e latt.ice v i b r a t i o n r a n g e . a) A l p h a - r h o m b o h e d r a l b o r o n [137]; b) BIePe [188]; c) BigOt [183]; b o r o n c a r b i d e [182, 183] ( t h e ordinate: v a l u e d of the; s p e c t r a a r e s o m e w h a t u n c e r t a i n bec:ause of e x p e r i m e n t a l r e a s o n s ) .

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(1580 cm -l) b e l o n g s t o t h e s p e c i e s A h (E II e); it o b v i o u s l y depends on a rather strong dipole-moment within the threea t o m i c c h a i n . A c c o r d i n g to s y m m e t r y c o n siderations, in a two-atomic chain there s h o u l d be n o d i p o l e - m o m e n t , a n d a c c o r d i n g ly t h i s v i b r a t i o n i s a b s e n t i n t h e s p e c t r a o f B60 a n d BiP, h o w e v e r o n l y a t f i r s t s i g h t . A better resolution of the reflection spectrum o f I~O (fig. 62) s h o w s a w e a k v i b r a t i o n a t a b o u t 1470 em-L T h u s s o m e i o n i c i t y s e e m s to e x i s t a l s o in a t w o - a t o m i c c h a i n , p o s s i b ly because of unsymmetric_~fl s t r u c t u r a l d i s t o r t i o n s o f t h e i c o s a h e d r a [183].

In boron carbide a composition-dependent metal-insulator transition was found near t h e c a r b o n - r i c h limit o f t h e h o m o g e n e i t y DO# r a n g e [29, 29]. T h e p h y s i c a l r e a s o n of t h i s 1100 ttO0 1400 1200 W&VENUMBER5 CM-I transition is not yet clarified. But optics] i n v e s t i g a t i o n s o f s e l e c t e d p h o n o n s {fig. 47) l e d to t h e c o n c l u s i o n t h a t t h e t r a n s i t i o n i s Fig. 46. B40. V i b r a t i o n o f t h e O - 0 a t o m i c accompanied by a structural change arrangement on the main diagonal of the unit [7,189]. While t h e v i b r a t i o n o f t h e i c o s a cell e v o k e d b y w e a k i o n i c i t y [183]. edron changes its frequency distinctly, the vibration of the chain remains unaltered. Hence the atomic c~mposition of the C-B-C chain is not changed at the metal-insulator transition. The change of the icosahedral vibration frequency can be explained by the assumption that with increasing C content a growing number o f i c o s a h e d r a i n c o r p o r a t e s u b s t i t u t i o n a l C a t o m s , b u t t h i s s u b s t i t u t i o n d e c r e a s e s a b r u p t l y to zero at the transition. The refiectivity increasing b e d i s c u s s e d below.

t o w a r d s low w a v e n u m b e r s

is attributed

The Ranmn spectrum of alpha-rhombohedral boron obt s i n e d b y R i c h t e r e t al. [187] w a s a l r e a d y t h e o r e t i c a l l y interpreted i n f i g . 44. S h e l n u t t e t al. [187] r e p o r t e d recently a comparing study on the Ramn spectra of a l p h a - r h o m b o h e d r a l b o r o n a n d i s o s t r u c t u r a l t n ) r i d e s (fig. 48). A g a i n o n l y in c o m m o n b o r o n c a r b i d e v i b r a t i o n s a t high frequencies were obtained. Contrary to these authors assuming that these Raman peaks were caused by graphite contamination, the precise coincidence with s t r o n g [ R - a c t i v e p h o n o n s m a k e s it m u c h m o r e p r o b a b l e that by structural deformation a certain amount of R a m a n a c t i v i t y m a y b e e v o k e d . T h e low R a m a n i n t e n s i t i e s o f BoAs a g r e e w i t h t h e w e a k 1R a b s o r p t i o n bands r e p o r t e d b y B e c h e r a n d T h e v e n o t [188]. B u t t h e s t r o n g l i n e a t low f r e q u e n c i e s i s s t r i k i n g , P o s s i b l y t h e r e e x i s t s a correlation w i t h low f r e q u e n c y I R - a b s o r p t i o n b a n d s in o t h e r b o r i d e s c o n t a i n i n g m e t a l a t o m s o f h i g h atomic number. Beta-rho_m.b.~hvdral s t r u c t u r e K r O u ~ The anisotropic absorption and r e f l e c t i o n of s i n g l e crystals of beta-rhombohedral boron in the spectral r a n g e o f l a t t i c e v i b r a t i o n s a r e p l o t t e d in fig. 49 a~ b [182, 191]. T h e e x p e r i m e n t a l d a t a w e r e f i t t e d a c c o r d i n g to a d i s p e r s i o n a n a l y s i s u s i n g 24 m o d e s A2= (]~ I I c) a n d 25 m o d e s E= (E I e). T h e d i f f e r e n c e to t h e g r o u p t h e o r e t i c a l c a l c u l a t i o n (31 A h , 52 El) is a t t r i b u t e d to a c c i d e n t a l d e g e n e r a c i e s a n d too s m a l l o s c i l l a t o r s t r e n g t h s of the lacking modes. From these results the refraction a n d t h e a b s o r p t i o n i n d e x w e r e d e r i v e d (fig. 50). A p a r t from the range of splitted vibrations of the icosahedra a s t r o n g d o u b l e - v i b r a t l o n o f t y p e A h (E II c) i s d i u t i n c l y

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Fig. 49. B e t a - r h o m b . boron structure group. a, b) Beta-rh. B: Absorption coefficient rasp. Reflectivity in the lattice vibration range vs. wave n u m b e r [182, 191L c) Cua,sAh.~lhos; r c flectivity vs, wave n u m b e r [ 194],

s e p a r a t e d t o w a r d s h i g h e r f r e q u e n c i e s . T h i s v i b r a t i o n is a t t r i b u t e d to t h e B10-B-BJ0 s u b u n i t in t h e u n i t cell. By t h i s t h e cited r e l a t i o n s h i p b e t w e e n b e t a - r h o m b o h e d r a l b o r o n a n d b o r o n carbide for symmetry r e a s o n s becomes obvious. The o p t i c a l lattice v i b r a t i o n f r e q u e n c i e s d i s c u s s e d a b o v e a r e o n e - p h o n o n p r o c e s s e s . Besides, m u ] t i - p h o n o n p r o c e s s e s with s i g n i f i c a n t l y l o w e r excitation p r o b a b i l i t i e s exist. The c o r r e s t ~ n d ing a b s o r p t i o n in b e t a - r h o m b o h e d r a l b o r o n , w h i c h is l a r g e l y u n s t r u c t u r e d b e c a u s e of t h e g r e a t v a r i e t y of p o s s i b l e c o m b i n a t i o n s of o n v - p h o n o n p r o c e s s e s p can be r e c o g n i z e d in t h e a b s o r p t i o n s p e c t r u m (fig. 51} [192]. I t r a n g e s t o w a r d s a m u l t i - p h o n e n c u t o f f a t a b o u t 2.000 to 2500 cm-* (4 to 5t~m). F u r t h e r i n f o r m a t i o n on t h e p r o p e r t i e s of p h o n o n e in b e t a - r h o m b o h e draI boron were o b t a i n e d from t h e i r t e m perature dependence. While t h e p h o n o n f r e q u e n c i e s s h i f t to l o w e r v a l u e s with i n c r e a s i n g t e m p e r a t u r e , t h e osci}l a t e r s t r e n g t h s remain Fig, 50. B e t a - r h o m b . u n a l t e r e d a n d t h e d a m - boron, a) Re,frat:tire p i n g c o n s t a n t s i n c r e a s e index and b) absorp[182]. Fig. 52 s h o w s tJc,n ind(-~'.x vs. wFJVe the experimental r e - n u m b e r in t h e lattice s u ] t s of t h e f r e q u e n c y v i b r a t i o n range s h i f t of s e v e r a l p h o - [1821. none compared with the calculated Gr0neisen shift, which desc r i b e s t h e i n f l u e n c e of the themaal lattice dilatation. From the good agreement at higher temperatures c a n bc c o n c l u d e d t h a t

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t h e i n f l u e n c e of t h e b o n d i n g a n h a r m o n i c i t y , w h i c h is e x p e c t e d to v a r y a s T ~, is below t h e d e t e c t i o n The e x e m p l a r y t e m p e r a t u r e d e p e n d e n c e o f t h e d a m p i n g c o n s t a n t s of phonons in beta-rhombohedral b o r o n is d e m o n s t r a t e d in fig. 53 [182]. The t h e o r e t i c a l c u r v e , w h i c h fits the experimental r e s u l t s s a r i s f e c t o r i ] y , is c o m p o s e d b y a s s u m i n g three-phonon and four-phonon processes for energy dissipation.

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I n t h e s p e c t r a l r a n g e from 10 to 100 cm -~ b e t a - r h o m b o h e d r a l b o r o n l 2 3 ,~ 5 5 ? 8 9 10 II 72 13 is l a r g e l y t r a n s p a r e n t . T h e r e f o r e it w a s p o s s i b l e to o b t a i n t h e a v e r a g ed a n i e o t r o p i c r e f r a c t i v e i n d e x a n d Fig. 5]. B e t a - r h o m b . b o r o n , e n i e o t r o p y of t h e o p t i c a l the dielectric c o n s t a n t from i n t e r a b s o r p t i o n . A b s o r p t i o n c o e f f i c i e n t v s . w a v e l e n g t h [192]. f e r e n c e s in t h e t r a n s m i s s i o n s p e c t r u m of p l a n e - p a r a l l e l s a m p l e s [182]: E II c :

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of p o ] y c r y s t a l l i n e , b e t a - r h o m b o h e d r a l b o r o n r e p o r t e d b y L a g r e n e u d i [193]. Hence no e s s e n t i a l d i s p e r s i o n p r o c e s s e s in t h e s p e c t r a l r a n g e < 10 cm "~ a r e to be

norm v s . t e m p e r a t u r e c o m p a r e d with the calculated Gr~neisen shift [182]. The d e n s i t y of s t a t e s of t h e H a m a n - a c t i v e p h o n o n s of b e t a - r h o m b o h e d r a ] b o r o n w a s c a l c u l a t e d b y Weber a n d T h o r p e [184]. T h e b o n d s of e q u a t o r i a l a t o m s , w h i c h in b e t a - r h o m b o h e d r a l b o r o n a r e m u c h s t r o n g e r t h a n in a i p h ~ - r h o m b o h e d r a l boronp w e r e h e r e c o n s i d e r e d . Fig. 54 shows the satisfactory qualitative agreem e n t w i t h t h e e x p e r i m e n t a l r e s u l t s [181]. expected,

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Fig. 53. B e t a - r h o m b . b o r o n . D a m p i n g constant of the 1250/1275 cm -1 phonon vs. temperature. Theoretical curve composed of t h r e e - p h o n o n proeesses (long-dashed line) a n d four-phonon processes (shortd a s h e d line) f o r e n e r g y d i s s i p a t i o n [ 182].

f.

, i

500

1000 E c m "1-1

Fig. 54. B e t e - r h o m b . b o r o n . Ramen s p e c t r u m ( d a s h e d line.) c o m p a r e d with t h e t h e o r e t i c a l l y c a l c u l a t e d p h o n e n d e n s i t y (solid line) [181, 184].

H. Werheit

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F i g . 55, A l p h a - t e t r a g o n a l s t r u c t u r e g r o u p . Opticzll t r a n s m i s s i o n a n d r e f l e x i o n s p e c t r ~ i n t h e Jatticc. v i b r u t i o n r a n g e .

a) Alpha-totragonal boron [195]; b) BeBll [195}; c) CsAI~Bu [196]; BsxCI [183]; Bs0C, [183]; BsC [196] (the ordlrmlx; values of the sp(:ctr+i are somewhat reaso as),

un¢.:erttdrl becautqe of t:xpcrJmt;nt~l]

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Fig. 56. 13eta-tetragonal s t r u c t u r e g r o u p . Optical r e flexion s p e c t r a in t h e .Lattice v i b r a t l o n r a n g e . a) ALpha-AIBtt [194]; ALl~Be0~Btt [194]; (^iBe)Blt [196] {the o r d i n a t e v a l u e s a r e s o m e w h a t u n c e r t a i n b e c a u s e of e x p e r i m e n t a l r e a s o n s ) . US

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Fig. 57. O r t h o r h o m b i c s t r u c t u r e g r o u p . Optical t r a n s m i s s i o n a n d r e f l e x i o n s p e c t r a in t h e h i g h f r e q u e n c y p a r t of t h e lnttice v i b r a t i o n r a n g e , a) LiAIBt4 [194]; MgAIBt4 194] ( t h e o r d i n a t e v a l u e s a r e s o m e w h a t u n c e r t a i n b e c a u s e of e x p e r i m e n t a l r e a s o n s ) .

212

H. Werheit

T h e o n l y i s o a t r u c t u r a l b e t i d e , w h o s e IR r e f l e x i o n s p e c t r u m c a n b e s h o w n , i s Cua.BAl2.sBl05 (fig. 49 c) [194]. Main f e a t u r e s of t h e s p e c t r u m s e e m to b e r e l a t e d to b e t a - r h o m b o h e d r a l boron, but certainly the comparison is difficult, because the only measurements were p e r f o r m e d w i t h u n p o l a r i z e d l i g h t on u n o r i e n t a d s a m p l e s . A d i s t i n c t d i f f e r e n c e c a n be s e e n a t low w a v e n u m b e r s , w h e r e , a g a i n i n a b e t i d e w i t h i n t r o d u c e d m e t a l a t o m s r a t h e r s t r o n g v i b r a t i o n s c a n be s e e n .

T h e o p t i c a l s p e c t r a of t h e r e p r e s e n t a t i w ; s of t h i s g r o u p (fi g. 55) [183, 194 - 196] e x h i b i t s u c h a low s i m i l a r i t y t h a t t h e u n i f o r m c l a s s i f i c a t i o n in o n l y o n e s t r u c t u r e g r o u p m u s t be e m p h a t i c a l l y c a l l e d i n q u e s t i o n . Only t h e h i g h - f r e q u e n c y l i m i t of t h e s p l i t t a d v i b r a t i o n s of t h e i c o s a h e d r a , w h i c h c h a n g e s from 980 cm -1 in a l p h a - t e t r a g o n a l b o r o n to a b o u t 1100 em -1 in t h e b o r i d e s s e e m s to s h o w some s y s t e m a t i c i n t e r r e l a t i o n , b u t t h i s l i m i t i s f o u n d a t s i m i l a r f r e q u e n c i e s a l s o in o t h e r s t r u c t u r a l g r o u p s a n d t h e r e f o r e i t i s n o t c h a r a c t e r i s t i c f o r t h i s g r o u p o n l y . A g a i n i n C2AbB~s (cf. r e m a r k i n s e c t i o n 2.3) a s a m e t a l b o r i d e a s t r o n g low-frequency v i b r a t i o n b a n d is c o n s p i c u o u s , w h i c h c a n h a r d l y be a t t r i b u t e d to i n n e r v i b r a t i o n s of t h e i c o s a h e d r a . b e t a - t e t r a g o n a l s t r u c t u r e jg_rQu~ T h e s p c ; c t r a (fig. 56) [194, 196] m u s t b e s e p a r a t e l y d i s c u s s e d i n d i f f e r e n t s e c t i o n s . I n t h e s p e c t r a l r a n g e < 500 cm -l t h e s t r u c t u r a l r e l a t i o n s h i p o f t h e c r y s t a l s b e c o m e s o b v i o u s . I n d e e d t h e r e i s so me c h a n g e i n t h e o s c i l l a t o r s t r e n g t h s of c e r t a i n p h o n o n s a n d s ome s h i f t s of r e s o n a n c e f r e q u e n c i e s c a n be s e e n ; n e v e r t h e l e s s b y t h e q u a l i t a t i v e s i m i l a r i t y of t h e p h o n o n s p e c t r a t h e s t r u c t u r a l r e l a t i o n s h i p of t h e s e c r y s t a l s i s c o n f i r m e d . T h e s t r o n g i n f l u e n c e of k i n d a n d c o n c e n t r a t i o n of m e t a l a t o m s o n t h e t h e q u a n t i t a t i v e p a r a m e t e r s of l a t t i c e v i b r a t i o n offers an interesting aspect to investigate these interactions. [n t h e s e lower.

metal b e t i d e s ,

too, t h e p h o n o n s p e c t r u m e x t e n d s

to w a v e n u m b e r s of 100 cm "L a n d

T h e c o n t r a s t i n t h e f r e q u e n c y r a n g e > 500 cm -l i s r e r a a r k a b l e , w h e r e d i s t i n c t d i f f e r e n c e s i n t h e s p e c t r a a p p e a r . E v e n s m a l l d i f f e r e n c e s in t h e m e t a l c o n t e n t s of Be-A1 b o r i d e s h a v e a s t r o n g i n f l u e n c e on t h e s p e c t r u m . O r t h o r h o m b i c s t r u c t u r e grout~ T h e I R - a c t i v e p h o n o n s p e c t r a of LiAIBI4 a n d MgAIB14 (fi g. 57) [194] a r e l i m i t e d t o f r e q u e n c i e s b e t w e e n 600 a n d 2000 cm -l. N e v e r t h e l e s s t h e s t r u c t u r a l r e l a t i o n s h i p i s c o n f i r m e d . O b v i o u s l y t h e r e i s a s t r o n g i n f l u e n c e of t h e m e t a l a t o m s o n c e r t a i n l a t t i c e v i b r a t i o n s . The s i n g l e p h o n o n b a n d in LiAIBI~ a t a b o u t 950 cm -~ is s p l i t t a d in MgAIBI4. At h i g h e r f r e q u e n c i e s b e t w e e n a b o u t 1700 a n d 1800 cm -1 a w e a k r e s o n a n c e i s i n d i c a t e d i n t h e r e f l e x i o n s p e c t r u m a n d c o n f i r m e d b y t r a n s m i s s i o n . At p r e s e n t it c a n n o t be d e c i d e d , w h e t h e r i t i s a n a t o m i c vibration or an electronic transition.

O)i~

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t~ Ck4-1

Fig. 58. YB66. O p t i c a l r e f l e x i o n s p e c t r u m in t h e l a t t i c e v i b r a t i o n r a n g e [197].

Optical properties

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< 120 cm -1, t h e a b s o r p t i o n

lated at 80 cm" (tab. 1)[181].

Fig. 59. LaBs; c a l c u l a t e d phonon d i s p e r s i o n c u r v e s (solid l i n e s ) , a l a s t i c c o n s t a n t s ( d a s h e d lines) a n d Raman a n d n e u t r o n s c a t t e r i n g r e s u i t s ( d o t s ) [180]. For s y m m e t r y

Metal h e x a b o r i d e s LaBs a n d YbBe a r e t h e o n l y b o r i d e e , w h o s e p h o n o n d i s p e r s i o n c u r v e s a n d e l a s t i c c o n s t a n t s a r e cornp l e t e l y c a l c u l a t e d b y t h e o r y [180]. The r e s u l t s o f LaBs s h o w n in fig. 59 a r e in good -" a g r e e m e n t with t h e ~mm

points33, in t h e Brillouin zone s e e fig.

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No i n v e s t i g a t i o n s o n t h e l a t t i c e v i b r a t i o n s o f d o d e c a b o r i d e s a r e known. In c a s e s o f h i g h e l e c t r i c a l c o n d u c t i v i t y t h e y will be l a r g e l y m a s k e d b y t h e a b s o r p t i o n o f free carriers.

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1 ~_~-~.~_ ~_~_ __~_ _ ......

,

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I

. . ._ _. . .

~

veloci-

l e n t - a n d t r i v a l e n t - m e t a l h e x a b o r i d e e w e r e p e r f o r m e d [199, 200]. Fig. 60 s h o w s t h e s p e c t r a o f EuBs a n d GdBs f o r cornp a r i s o n . The s y s t e m a t i c r e l a t i o n s h i p is o b v i o u s . [n fig. 61 s h o w s t h e s h i f t o f t h e l{uman f r e q u e n o i e s d e p e n d i n g o n t h e l a t t i c e c o n s t a n t is s e e n . It was c o n c l u d e d t h a t t h e m e t a l b o r o n i n t e r a c t i o n in t r i v a l e n t - m e t a l h e x a b o r i d e e (metallic) is smaller t h a n in d i v a l e n t - m e t a l h e x a b o r i d e s ( i s o l a t i n g ) , w h i c h is a t t r i b u t e d to t h e s h i e l d i n g e f f e c t o f t h e c o n d u c t i o n e l e c t r o n s [199, 200].

i

range

cm -j. Hence, in s p i t e o f t h e b i g a g g r e g a t e s o f icos a h e d r a in t h e s t r u c t u r e , no f u r t h e r d i s p e r s i o n p r o c e s s is e x i s t e n t in t h i s r a n g e [198]. This is in a g r e e m e n t with t h e a c o u s t i c - p h o n o n c u t o f f c a l c u -

,~

200 0

T h e p h o n o n s p e c t r u m (fig. 58) i s only w e a k l y s t r u c tured. Obviously the vibrations are strongly d a m p e d . This r e s u l t a g r e e s w i t h t h e i n t e r p r e t a t i o n of the thermal conductivity by strong phonon scatt e r i n g [197]. The g r e a t a g g r e g a t e s o f b o r o n a t o m s in t h i s material a r e p r o b a b l y t h e r e a s o n f o r t h e s t r o n g v i b r a t i o n s a t low f r e q u e n c i e s . I t c a n b e a s s u m e d t h a t l i b r a t i o n a l v i b r a t i o n s a r e f o u n d in t h i s s p e c t r a l r a n g e . This l e a d s to t h e c o n c l u s i o n t h a t t h e low f r e q u e n c y v i b r a t i o n s in t h e o t h e r metal b o r i d e s may b e e v o k e d b y t h e metal a t o m s l e a d i n g to b i g g e r aggregates by strong bonding between icosahedra.

~

~ ~

~

Fig. 61. Metal h e x a b o r i d e s . Raman w a v e n u m b e r s o f metal h e x a b o r i d e s with b i v a l e n t a n d t r i v a l e n t metals vs. l a t t i c e c o n s t a n t [196, 197].

FR~'E CARRIERS T h e r e a r e o n l y few r e s u l t s o f o p t i c a l i n v e s t i g a t i o n s o f f r e e c a r r i e r s in t h e m o d i f i c a t i o n s o f b o r o n a n d i t s b o r o n - r i c h c o m p o u n d s . In t h e c r y s t a l s c o n t a i n i n g i c o s a h e d r a , t h e main r e a s o n may b e t h e n a r r o w e n e r g y b a n d s c a l c u l a t e d in some c a s e s a n d a s s u m e d in t h e o t h e r s . T h e n t h e e f f e c t i v e m a s s of c a r r i e r s a c c o r d i n g to l/m* -- 1/~ ~ *~sE/ ~k ~ is g r e a t a n d h e n c e t h e plasma r e s o n a n c e f r e q u e n c y is e x p e c t e d a t low f r e q u e n c i e s a n d m o r e o v e r its s l o p e to b e s t r o n g l y i n f l u e n c e d b y s c a t t e r i n g ( s e e s e c t i o n 3.1).

214

H. Werheit

Fig. 45e s h o w s t h e p l a s m a e d g e s of s e v e r a l b o r o n c a r b i d e s a m p l e s in t h e FIR r e g i o n [183]. By c o m p a r i n g t h e f r e q u e n c y d e p e n d e n c e m e a s u r e d w i t h t h e c a l c u l a t e d c u r v e s in fig. 9, w h o s e p a r a m e t e r s a r e r o u g h l y a d a p t e d to t h e c o n d i t i o n s in b o r o n c a r b i d e , o n e c a n s e e t h a t p l a s m a r e s o n a n c e f r e q u e n c y a n d collision f r e q u e n c y a r e n e a r l y of t h e same o r d e r . But t h e collision f r e q u e n c y of t h e c o m p a c t - g r a i n e d s i n t e r e d b o r o n c a r b i d e is d i s t i n c t l y h i g h e r t h a n t h a t of c o a r s e - c r i a t a l l i n e molten material. T h i s c o n f i r m s t h e a s s u m p t i o n , t h a t t h e e l e c t r o n i c t r a n s p o r t p r o p e r t i e s in c o m p a c t - g r a i n e d m a t e r i a l a r e s t r o n g l y i n f l u e n c e d b y t h e g r a i n b o u n d a r i e s [201]. A c c o r d i n g to s e c t i o n 3.1, with nh = 4...9 1019 cm -t ( e s t i m a t e d b y Geiat e t al. [202] in

comparable boron carbide)

b/p " 3 1014 s "1 E . : 10

t h e e f f e c t i v e m a s s of t h e h o l e s c a n b e e s t i m a t e d to mb*/ma : 3 ... 5 T h i s v a l u e q u a l i t a t i v e l y a g g r e e s with t h e n a r r o w b a n d s c a l c u l a t e d b y A r m s t r o n g [80]. F o r a m o r e p r e c i s e d e t e r m i n a t i o n of t h e e f f e c tive m a s s it is n e c e s s a r y to make s y s t e m a t i c i n v e s t i g a t i o n s on s t r u c t u r e , t r a n s p o r t a n d o p t i c a l p r o p e r t i e s of b o r o n c a r b i d e of d i f f e r e n t chemical c o m p o s i t i o n a n d p r e p a r a t i o n .

100~

700

In the reflectivity spectrum of t h e structure-related BeO ( c o m p a c t - g r a i n e d , h o t - p r e s s e d ) (fig. 45d), too, a s t r o n g l y s m e a r e d p l a s m a e d g e s e e m s to be i n d i c a t e d a t low f r e q u e n c i e s .

./+ I

1

L 9

~0

11

12 llm

1~

Fig. 62. B e t a - r h o m b o h . boron. Free-carrier abs o r p t i o n a t 600 K v s . w a v e l e n g t h [92, 93].

The o p t i c a l a b s o r p t i o n of f r e e c a r r i e r s w a s d e t e c t e d in b e t e - r h o m b o h e d r a l b o r o n a t h i g h t e m p e r a t u r e s . When b y t h e r m a l e x c i t a t i o n a t h i g h t e m p e r a t u r e s t h e c a r r i e r c o n c e n t r a t i o n o r , if small p o l a r o n a a r e a s s u m e d , t h e c a r r i e r mobility s u f f i c i e n t l y i n c r e a s e s , t h e i n t e r a c t i o n w i t h e l e c t r o m a g n e t i c r a d i a t i o n b e c o m e s m e a s u r a b l e in t h e MIR r e g i o n [92, 93]. Fig. 61 s h o w s t h e i r c o n t r i b u t i o n to t h e total a b s o r p t i o n obtained as the difference between the a b s o r p t i o n s p e c t r a a t 600 a n d 400 0C. T h e i n c r e a s e with w a v e l e n g t h a s T a indicates scattering by charged defects. Since, similar to b o r o n c a r b i d e a n d BsO, t h e r e f l e c t i v i t y s p e c t r a of the beta-tetragonal structure group exhibit a certain increase t o w a r d s l o w e r f r e q u e n c i e s , it s e e m s p o s s i b l e , t h a t in t h e s e c a s e s , too, some c o n t r i b u t i o n of s m e a r e d p l a s m a e d g e s c a u s e d b y s t r o n g l y scattered free c a r r i e r s are seen.

6. C O N C L U S I O N

AND

OUTLOOK

T h e aim of t h i s w o r k w a s to collect t h e main r e s u l t s of t h e o p t i c a l p r o p e r t i e s of t h e modific a t i o n s of b o r o n a n d i t s b o r o n - r i c h c o m p o u n d s , a n d to d e m o n s t r a t e t h e i m p o r t a n c e of o p t i c a l i n v e s t i g a t i o n s in t h e d i f f e r e n t r a n g e s of p h o t o n e n e r g y to g e t i n s i g h t in t h e p r o p e r t i e s of s u c h complex c r y s t a l s . I n s p i t e of t h e f a c t t h a t in m a n y c a s e s t h e i n f o r m a t i o n s a r e still i n c o m p l e t e a n d u n c e r t a i n t i e s w i t h r e s p e c t to s t r u c t u r a l d e t a i l s a n d p u r i t y r e m a i n , it is p o s s i b l e to e x h i b i t i n t e r r e l a t i o n s o f p h y s i c a l p r o p e r t i e s in a n a l o g y to s t r u c t u r a l r e l a t i o n s h i p s . T h e o n l y e x c e p t i o n is t h e a l p h a - t e t r a g o n a i s t r u c t u r e g r o u p . Here t h e o p t i c a l p h o n o n s p e c t r a of t h e c r y e t a l a a v a i l a b l e a r e h a r d l y r e l a t e d a n d t h e r e f o r e t h e c o n c l u s i o n s e e m s a d m i s s a b l e t h a t t h e p r e s e n t i n f o r m a t i o n o n t h e s t r u c t u r e of t h i s g r o u p is n o t s u f f i c i e n t A p a r t f r o m s u c h b a s i c r e s e a r c h , t h e q u e s t i o n of p o s s i b l e t e c h n i c a l a p p l i c a t i o n is i m p o r t a n t . L o o k i n g f i r s t f o r a p p l i c a t i o n s of t h e o p t i c a l p r o p e r t i e s , it s h o u l d b e m e n t i o n e d t h a t b e c a u s e of t h e s t r o n g l y e n e r g y - d e p e n d e n t a b s o r p t i o n , b o r o n films a n d in t h e same w a y films of t h e b o r i d e s a r e s u i t a b l e f o r u s e a s e d g e f i l t e r s in t h e s p e c t r a l r a n g e of h i g h p h o t o n e n e r g i e s [75]. R e c e n t l y T a n a k a e t el. [203] r e p o r t e d o n a n o t h e r a s p e c t of u s i n g YB~ s i n g l e c r y s t a l s : B e c a u s e o f t h e l a r g e l a t t i c e c o n s t a n t (e.g. Y I ~ : a0 = 23.445 A, d+00 = 5.86 A), t h e y a r e convenient monochromator c r y s t a l s for soft X - r a y s and s y n c h r o t r o n radiation.

Optical properties

215

I n t h e c a s e o f b e t s - r h o m b o h e d r a l b o r o n , t h e o p t i c a l b i s t a b i l i t y in a c e r t a i n t e m p e r a t u r e r a n g e a n d t h e l o n g - p e r s i s t e n t o p t i c a l l y i n d u c e d t h e r m a l n o n - e q u i l i b r i u m s t a t e s (cp. s e c t i o n 4.3) m a y suggest technical application. Moreover it should be mentioned that the position of the a b s o r p t i o n e d g e is s i m i l a r to t h a t o n e o f Si a n d h e n c e s u i t a b l e f o r e f f e c t i v e p h o t o - e l e c t r i c energy conversion. Also in o t h e r r a n g e s o f t e c h n i c a l a p p l i c a t i o n b o r o n a n d i t s c o m p o u n d s h a v e b e c ~ m e s u b j e c t o f i n c r e a s i n g i n t e r e s t . T r a d i t i o n a l l y , t h e t e c h n i c a l a p p l i c a t i o n s o f b o r o n c a r b i d e w e r e c o n f i n e d to such uses as abrasive material, for shielding of neutrons and for light-weight armour plates [204]. B u t r e c e n t l y a p p l i c a t i o n f o r i n d u s t r i a l c e r a m i c s a n d c o a t i n g o f metallic tools w e r e d e v e l o p e d . F o r all t h e s e a p p l i c a t i o n s t h e g r e a t h a r d n e s s , t h e h i g h m e l t i n g p o i n t a n d t h e e x t r a o r d i n a r y c h e m i c a l r e s i s t a n c e a r e e s s e n t i a l [205]. Now in p a r t i c u l a r c a s e s , t e c h n i c a l a p p l i c a t i o n s o f t h e s p e c i a l e l e c t r o n i c p r o p e r t i e s b e c o m e d e v e l o p e d , too. A g a i n b o r o n c a r b i d e , whose high thermolelectric power depends on the chemical composition and increases p r o p o r t i o n a l t o t e m p e r a t u r e , i s to b e m e n t i o n e d e s p e c i a l l y . B e c a u s e o f t h e r e l a t i v e l y h i g h e l e c t r i c a n d low t h e r m a l c o n d u c t i v i t y o f b o r o n c a r b i d e t h e a p p l i c a t i o n f o r t h e r m o e l e c t r i c energy conversion presents i t s e l f [28 - 32, 206]. M o r e o v e r i n t h e b e t e - r h o m b o h e d r a l modification of elementary boron a deliberate reversal of the conductivity character was s h o w n to b e p o s s i b l e b y i n t e r s t i t i a l d o p i n g ; w h i c h i s a d e c i s i v e p r e r e q u i s i t e f o r a p p l i c a t i o n s in s e m i c o n d u c t o r t e c h n o l o g y [131]. C e r t a i n l y t h e s e c o n c r e t e e x a m p l e s a r e r e s t r i c t e d to s p e c i a l c o m p o u n d s o r s t r u c t u r e s , but because of the structural relationships one can presume that there are other boron c o m p o u n d s , w h i c h p o s s i b l y f u l f i l t h e n e c e s s a r y c o n d i t i o n s f o r s u c h a p p l i c a t i o n s to a n e v e n h i g h e r d e g r e e . H e n c e f r o m t h i s v i e w p o i n t , too, a s y s t e m a t i c c o m p a r a t i v e i n v e s t i g a t i o n o f t h e modifications and compounds of boron is useful, and as was shown, especially the optical i n v e s t i g a t i o n s p r o m i s e i n s i g h t in t h e p r o p e r t i e s o f s o l i d s w i t h s u c h c o m p l e x s t r u c t u r e s , though there is scarcely an investigation whithout unexpected and surprising results. 7. 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THE AUTHOR

HELMUT WERHEIT H e l m u t W e r h e i t is p r o f e s s o r of e x p e r i m e n t a l p h y s i c s a t t h e U n i v e r a i t / i t - G e s a m t h o c h s c h u l e - D u i s b u r g in t h e F e d e r a l R e p u b l i c o f G e r m a n y . He is h e a d of t h e s c i e n t i f i c g r o u p w o r k i n g o n b o r o n a n d b o r o n - r i c h c o m p o u n d s in t h e s o l l d - s t a t e p h y s i c s l a b o r a t o r y . His s c i e n t i f i c w o r k o n b o r o n a n d r e l a t e d s o l i d e s b e g a n a t t h e U n i v e r s i t y of C o l o g n e w i t h h i s t h e s i s p r e s e n t e d in ]968 a n d w a s c o n t i n u e d t h e r e till h i s i n a u g u r a l d i s s e r t a t i o n p r e s e n t e d in 1972. A f t e r w o r k i n g in t h e c h e m i c a l i n d u s t r y f o r s e v e r a l y e a r s , i n 1975 h e s t a r t e d a g a i n w i t h s c i e n t i f i c i n v e s t i g a t i o n s o f t h e s e m i c o n d u c t o r a n d s o l i d - s t a t e p r o p e r t i e s of b o r o n a n d b o r o n - r i c h b o r i d e s a t t h e U n i v e r s i t y of D u i s b u r g and he has been extending a n d i n t e n s i f y i n g t h i s w o r k in c o o p e r a t i o n w i t h i n d u s t r i a l s c i e n t i s t s a n d f o r e i g n s c i e n t i f i c i n s t i t u t e s . At p r e s e n t h e is v i c e - c h a i r m a n of t h e I n t e r n a t i o n a l S c i e n t i fic C o m m i t t e e o n Boronp B o r i d e s a n d R e l a t e d C o m p o u n d s .