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VOLUME DEPENDENCE OF THE ELECTRONIC
STRUCTURE OF Cu2Mg
J. Schirber, A. Switendick, W. Gordon
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C6, supplkment
au
no
8, Tome 39, aotit 1978, page
~6-1101
VOLUME DEPENDENCE
OF
THE ELECTRON1 C STRUCTURE OF cu 2 ~ g tJ . E . S c h i r b e r
,
A.C. ~ w i t e n d i c k * and W.L. Gordonm.t .t
' ~ a n d i a
Laboratories, AZbuquerque, New
Mexico
87185,U.S.A.
fZ
Case Western Reserve University, CZeveZand, Ohio
44106, U.S.A.R&sumB.- Nous avons mesurd p l u s i e u r s s e c t i o n s de l a s u r f a c e de Fermi de Cu2Mg en f o n c t i o n d e l a p r e s s i o n e t nous comparons l e s r d s u l t a t s 5 des c a l c u l s de s t r u c t u r e d e bandes oii l ' o n f a i t v a r i e r l e volume.
Abstract.- We have made measurements of t h e p r e s s u r e dependence of s e v e r a l c r o s s s e c t i o n s of t h e Fermi s u r f a c e of Cu2Mg and compare them w i t h a volume dependent band s t r u c t u r e c a l c u l a - t i o n .
The pseudobinary Laves systems a r e t h e c l a s s i c a l examples of a l l o y s i n which t h e s t r u c t u r e
i s widely accepted t o b e determined by t h e e l e c t r o n c o n c e n t r a t i o n . Since t h e s e systems c o n s t i t u t e t h e most common s t r u c t u r e s assumed by b i n a r y combina- t i o n of m e t a l s (over 220 known examples t o d a t e ) / l / i t
is
of c o n s i d e r a b l e i n t e r e s t t o understand t h e i r e l e c t r o n i c p r o p e r t i e s and t h e r e l a t i o n of t h e s e p r o p e r t i e s t o t h e s t r u c t u r a l d e t a i l s and pha- s e diagrams. Cu2Mg i s t h e p r o t o t y p e c u b i c Laves s t r u c - t u r e . A s t h e e l e c t r o n c o n c e n t r a t i o n e / a i s i n c r e a - sed by a d d i t i o n of h i g h e r v a l e n c e a d d i t i v e s r e p l a - c i n g t h e Cu, t h e d e n s i t y of s t a t e s determined by h e a t c a p a c i t y / 2 / and by s u s c e p t i b i l i t y / 3 / measu- rements i s observed t o r i s e s h a r p l y t o a maximum n e a r e / a=
1.45 and t h e n f a l l p r e c i p i t o u s l y u n t i l the phase t r a n s i t i o n (near e / a=
1.75) from c u b i c t o t h e MgNi2 type s t r u c t u r e . Since t h e i n i t i a l work of Laves and W i t t e / 4 / over f o r t y y e a r s ago, i t has been more o r l e s s accepted t h a t t h e peak i n t h e den- s i t y of s t a t e s occurred when t h e Fermi s u r f a c e c o n t a c t s t h e (311) B r i l l o u i n zone f a c e / 2 / . The subsequent drop i n t h e d e n s i t y of s t a t e s a s e / a i n c r e a s e s toward t h e phase change i s s a i d t o cor- respond t o f i l l i n g zones which i s i n t e r r u p t e d by a n o t h e r peak corresponding t o c o n t a c t i n g t h e (222) f a c e . Heine and Wearie /5/ have argued from pseudo- p o t e n t i a l c o n s i d e r a t i o n s t h a t t h e s e i d e a s a r e i n-
t
T h i s work was supported by t h e U.S. Dept.of Energy, under C o n t r a c t AT(29-1)-789.tt
A.U.S. Dept. of Energy f a c i l i t y .d i f f i c u l t y .
I n t h i s s t u d y we r e p o r t d i r e c t measurements of t h e Fermi s u r f a c e a s a f u n c t i o n of h y d r o s t a t i c p r e s s u r e and an APW band s t r u c t u r e a s a f u n c t i o n of i n t e r a t o m i c s p a c i n g which g i v e 60r t h e f i r s t time a d i r e c t measurement of d e t a i l s of t h e e l e c - t r o n i c s t r u c t u r e a t t h e Fermi s u r f a c e which can be r e l a t e d t o t h e experimental anomalies.
The Fermi s u r f a c e of CunMg h a s been i n v e s t i - g a t e d u s i n g pulsed f i e l d and t o r q u e de Haas-van Alphen techniques by Wagner and Gordon
161.
These i n v e s t i g a t o r s compared t h e i r r e s u l t s with a valen- ce-eight f r e e - e l e c t r o n Fermi s u r f a c e and found some s i m i l a r i t y b u t no g e n e r a l correspondence. They observed an anomalously l a r g e e f f e c t i v e mass f o r a small s h e e t a t t r i b u t e d t o a second band h o l e s u r f a c e p r e d i c t e d t h a t t h i s s h e e t might be impor- t a n t i n a l l o y i n g and p r e s s u r e s t u d i e s . We used samples w i t h a nominal r e s i s t a n c e r a t i o of 25. De Haas-van Alphen f r e q u e n c i e s were determined u s i n g t h e f i e l d modulation technique / 7 / i n a 55 kOe superconducting s o l e n o i d . P r e s s u r e s were generated i n He and d e r i v a t i v e s of Permi s u r f a c e c r o s s sec- t i o n s determined i n b o t h f l u i d181
and s o l i d/9/
He by s h i f t i n g i n d i v i d u a l o s c i l l a t i o n s o f a given frequency. The p r e s s u r e d e r i v a t i v e dRnF/dP i s t h e n given by B-~AH/AP where B i s t h e magnetic induction and AH i s t h e s h i f t i n f i e l d of t h e o s c i l l a t i o n w i t h a change of p r e s s u r e AP.
The p r e s s u r e r e s u l t s a r e summarized i n Table I. We have employed t h e o r b i t nomenclature of Wagner and Gordon. F r e e - e l e c t r o n s c a l i n g , t h e f i l l i n g of t h e same f r a c t i o n of t h e B r i l l o u i n zone a s i t i n c r e a s e s i n s i z e w i t h i n c r e a s i n g p r e s s u r e ,
c o n t r i b u t e s a f a c t o r given by 2 / 3 t h e volume com- p r e s s i b i l i t y
/
10/.Table I
P r e s s u r e d e r i v a t i v e s of c r o s s s e c t i o n s of t h e Fermi s u r f a c e of Cu2Mg. The o r b i t d e s i g n a t i o n s a r e from P,eference 5. The p r e s s u r e d e r i v a t i v e s a r e i n u n i t s of I O - ~ kbar-l. Free e l e c t r o n s c a l i n g v a l u e i s 7.1 i n t h e s e u n i t s . 1 1 I Cross
:
F i e l d 1 1 Frequency:
d&nF/dP s e c t i o n:
O r i e n t a t i o n;
I I l l l l l I F2j
u l l ]j
6.7 X 105 G ~ - 3 0 0 ( ~ 1 0 0 ):
1 0 1 7.1 X l o 5 G : - 3 0 0 ( ~ 50) F4,7j
~ 0 0i 1 . 4 5 ~ 1 0 ~ ~ j - 2 ( ~ ~ 1) D 0 0 1 7.1 x lo7 G 8 (+ 1) F9;
l l l l l For CugMg t h i s c o n t r i b u t i o n i s 7.1 X ~ o - ~ k b a r - l . With t h e very l i m i t e d d a t a we have s o f a r , we s e e t h a t o n l y t h e l a r g e s t c r o s s s e c t i o n o b s e r v e d,
F9, behaves even q u a l i t a t i v e l y a s p r e d i c t e d by f r e e - e l e c t r o n c o n s i d e r a t i o n s . 0 f p a r t i c u l a r s i g n i f i c a n c ei s the v e r y l a r g e n e g a t i v e d e r i v a t i v e observed f o r t h e F2 c r o s s s e c t i o n . As pointed o u t by Wagnerand Gordon t h i s s h e e t has a s s o c i a t e d w i t h i t a remarkably l a r - ge e f f e c t i v e mass (mX/m %.35) w i t h r e s p e c t t o i t s
s i z e which i s * l ~ - ~ t i m e s t h e a r e a of t h e maximum c r o s s s e c t i o n of Fermi sphere.
We have c a l c u l a t e d t h e e l e c t r o n i c energy band s t r u c t u r e of CupMg a t both normal volume and reduced volume u s i n g t h e AFW method I l l / . The po- t e n t i a l ~ were d e r i v e d from a s u p e r p o s i t i o n of atomic p o t e n t i a l s / 1 2 / .
Based on t h e s e energy band c a l c u l a t i o n s , we confirm t h e assignments of Wagner and Gordon of F6 t o t h e s i x t h band c i g a r s . S i m i l a r l y we concur w i t h t h e a s s o c i a t i o n of F2 t o second band h o l e s c e n t e r e d a t
r
d e r i v e d from t h e empty F 15 s t a t e s . T h i s s t a t e i- t r i p l y d e g e n e r a t e and i s bonding p - l i k e on the magnesium and copper. From t h e calcu- l a t i o n s of Cuqmg w i t h reduced l a t t i c e c o n s t a n t we f i n d a l a r g e energy s h i f t f o r t h i s s t a t e of + . 3 3 Ry/a.u. compared w i t h a f r e e - e l e c t r o n v a l u e o f + . l 2 Ry/a.u. The measured small s i z e of t h i s o r b i t means t h a tr
15 i s c l o s e t o t h e Fermi energy, 2.005 Ry, i n c o n t r a s t t o t h e band p r e d i c t i o n of .02 Ry. T h i s d i f f e r e n c e i s n o t unreasonable i n view of the approximations involved i n t h e band c a l c u l a t i o n s . I f we a d j u s t t h e s t a t e s a t F 15 down by %.015 Ry, t h esecond band h o l e s u r f a c e w i l l be of c o r r e c t topolo- gy and s i z e . S i m i l a r l y , a d j u s t i n g t h e s t a t e s a l o n g
< I l l > by -0.025 Ry one can g e t r i d of both t h e t h l r d band a n s (necks) and f o u r t h band arms due t o t h e f l a t n e s s of t h e s e bands. N e i t h e r we nor Wagner and Gordon were a b l e t o observe t h e s e necks which should g i v e r i s e t o prominent dH-vA o s c i l l a - t i o n s i f a n a l o g i e s w i t h AuGa2 /13/ and t h e f r e e - e l e c t r o n model hold. These a r e i n f a c t t h e s t a t e s a s s o c i a t e d w i t h t h e (311) p l a n e s i n t h e f r e e - e l e c - t r o n p i c t u r e which a r e presumed t o account f o r t h e anomalies i n t h e s u s c e p t i b i l i t y and h e a t c a p a c i t y . The i n c r e a s e i n t h e d e n s i t y of s t a t e s upon i n c r e a - s i n g e / a from 1.33 t o 1.45 seems more l i k e l y t o be a s s o c i a t e d w i t h t h e 5 f i f t h band maximum a t zone c e n t e r r 2 ' and with t h e secondband maximum a t
r.
The o b s e r v a t i o n of o r b i t s o n t h e f o u r t h band s u r f a c e , F7 and F4, w i t h weak a n g u l a r v a r i a t i o n , i n p a r t i c u l a r along <110>
,
i s s t r o n g l y i n d i c a t i v e of l a c k of < I l l > arms. These o r b i t s t o o would have n e g a t i v e p r e s s u r e d e r i v a t i v e s which would be much s m a l l e r due t o t h e f a c t t h a t t h e i n t e r s e c t i o n of t h e Fermi energy w i t h t h e bands a l o n g < I l l > and <110> i s f u r t h e r fromr
r e f l e c t i n g t h e l a r g e r a r e a . The band s t r u c t u r e r e s u l t s p r e d i c t both f i f t h and s i x t h band e l e c t r o n s u r f a c e s c e n t e r e d a t X although t h e o r i g i n of F6> F5 i n some d i r e c t i o n s / 6 / i s a s y e t u n c l e a r and a w a i t s more d e t a i l e d c a l c u l a t i o n s . The o r i g i n of F9 and hence i t s p r e s s u r e dependence i s u n c e r t a i n b u t s i n c e i t i s l a r g e and s c a l e s w i t h t h e B r i l l o u i n zone w i t h p r e s s u r e , i t is n o t a c r u c i a l t e s t of a band s t r u c u r e model. Although more a n a l y s i s i s needed we b e l i e v e t h e above in-t e r p r e t a t i o n i s t h e most c o n s i s t e n t w i t h t h e band c a l c u l a t i o n s and t h e p r e s s u r e d e r i v a t i v e r e s u l t s .
Summarizing, o u r model w i t h v e r y small adjustments i n t h e p o s i t i o n s of t h e c a l c u l a t e d bands y i e l d s i ) closed 2nd band h o l e s a t
