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Interband transitions and band structure of a BiSb alloy
Volume 10, number 3
PHYSICS LETTERS
n(e) I
,~~N CLiEC DT U C O ITN N R O S -,?
-9
-3,
3
1,81 9
~,~m,o,,,
f3d ELECTRONS Fig. 2. Expected difference angular correlations from spherically symmetric distributions of unpgired 3d. and conduction electron spins. The angles eF {+) = kF(+)/mc correspond to the Fermi radii for the conduction electrons with spins up and down respectively. the curve l a b e l l e d "3d e l e c t r o n s " in fig. 2. The m e a s u r e d d i f f e r e n c e , however, b e c o m e s p o s i t i v e at s m a l l 0. T h i s fact m a y be explained in t e r m s of an s - d i n t e r a c t i o n c a u s i n g a p o l a r i s a t i o n of the conduction e l e c t r o n s . A s s u m i n g a s i m p l e model of two s p h e r i c a l F e r m i s u r f a c e s with r a d i i kF(+-) for the conduction e l e c t r o n s with + and - spin, one o b t a i n s u c o n t r i b u t i o n f r o m the conduction e l e c t r o n s r e p r e s e n t e d by the t r u n c a t e d i n v e r t e d p a r a b o l a (not n o r m a l i s e d with r e s p e c t to the 3d curve) in fig. 2. In case this p o l a r i s a t i o n i s a n t i p a r a l l e l to that of the 3d e l e c t r o n s t h i s c o n t r i b u -
INTERBAND
TRANSITIONS
AND
15 June 1964
tion has to be added to the 3d e l e c t r o n c u r v e , in the opposite case it should be subtracted. The e x p e r i m e n t a l r e s u l t s can only be explained if one a s s u m e s an a n t i p a r a l l e l p o l a r i s a t i o n of the conduction e l e c t r o n s , i. e . , the sign of the s - d i n t e r action is negative. A s i m i l a r r e s u l t was obtained by Shull and Yamada 4) f r o m diffraction of p o l a r i s . ed n e u t r o n s . An a l t e r n a t i v e explanation in t e r m s of a spin dependent r a d i a l p a r t of the 3d wave functions 5) without a s s u m i n g a conduction e l e c t r o n p o l a r i s u t i o n s e e m s u n l i k e l y as computations show that, with a m a g n e t i c m o m e n t for iron of 2,2 ~B, one would need on the o r d e r of 10 to 15 3d e l e c t r o n s to achieve a sign r e v e r s a l of n ( Q . The a u t h o r s a r e indebted to Dr. T. R i s t e for suggesting this p r o b l e m and showing continuous i n t e r e s t , and to Mr. E. F r i k k e e for valuable d i s cussions.
References 1) S. S. Hanna and R. S. Preston, Phys. Rev. 109 (1958) 716. 2) G. Trumpy, Phys. Rev. 118 (1960) 668. 3) J. Ldvseth, Phys. Norvegiea 1 (1961-63) 145. 4) C.G.Shull and Y.Yamada, J. Phys. Soc. Japan 17, Suppl. B-III (1962) 1. 5) J.H.Wood and G.W. Pratt J r . , Phys. Rev. 107 (1957) 995.
BAND
STRUCTURE
OF A BiSb
ALLOY
L. C. HEBEL and G. E. SMITH Bell Telephone Laboratories, Murray Hill, New Yersey Received 20 May 1964 We wish to r e p o r t on m e a s u r e m e n t s of i n f r a r e d m a g n e t o - r e f l e c t i o n f r o m a single c r y s t a l B i 8 8 . 6 S b l l . 4 alloy. The m e a s u r e m e n t s w e r e m a d e at liquid helium t e m p e r a t u r e s u s i n g f r e q u e n c i e s between 80 cm -1 and 400 cm -1 with m a g n e t i c fields up to 20 kG. Reflection s i n g u l a r i t i e s a r e o b s e r v e d which a r e i n t e r p r e t e d in t e r m s of d i r e c t i n t e r b a n d t r a n s i t i o n s 1) between Landau l e v e l s of a conduction band and those of a band lying beneath it. The p a r a m e t e r s which d e s c r i b e the two bands in the alloy a r e found to be e s s e n t i a l l y i d e n t i c a l to those which d e s c r i b e the e l e c t r o n bands in p u r e bismuth. The b i s m u t h - a n t i m o n y s y s t e m f o r m s a corn-
plete solid solution; it has been c u s t o m a r y to a s sume that the alloys have a band s t r u c t u r e which g~es continuously f r o m that of p u r e b i s m u t h to that of pure antimony. Galvanomagnetic m e a s u r e m e n t s 2) have shown that in the r e g i o n of 12% antim o n y in b i s m u t h the alloys become s e m i c o n d u c t ing. The i n f r a r e d r e f l e c t i o n was m e a s u r e d f r o m a s a m p l e containing 11.4% antimony~ which was , grown by a zone levelling technique ~). The c o m position was d e t e r m i n e d by an e l e c t r o n m i c r o probe and found to be u n i f o r m to the l i m i t of a c c u r a c y of the i n s t r u m e n t (+ 0.2 at %). R e f l e c t i v i t y of the s a m p l e was m e a s u r e d by a doublebeam technique d e s c r i b e d e l s e w h e r e 4). 273
Volume ]0, number 3
PHYSICS LETTERS
BL88.6 ~'b[I.4
~'e=203Cm-1
15 June 1964
4.0 X 10.3
/ S~'88.6 Sb1~4
30 10 Fig,
2,0
]_. A p h o t o g r a p h
3.0 4 .O 5.0 MAGNETIC FIELD (kO) o f 8. r e c o r d e r
trace
6,0
7.0
8.0
R e f l e c t i o n p e a k s , such as the one at 3 kG in fig. 1, w e r e o b s e r v e d and studied as a function of f r e quency and m a g n e t i c field. The p eak s a r e i n t e r p r e t e d as o n s e t s of allowed interband, t r a n s i t i o n s b e t w e e n Landau l e v e l s of the two bands f o r #z = 0 and a r e a n a l y z e d using the s a m e type of two-band m o d e l as w a s i n t r o duced by Lax 1) f o r p u r e bismuth. In the Lax m o d e l the o r b i t a l and spin s p l i t t i n g s a r e taken to be equal 4). The e n e r g i e s at k z = 0 of the Landau l e v e l s of the two bands a r e then functions of E G , the gap between bands at z e r o f i e l d , and a l s o ¢0c, a c y c l o t r o n f r e q u e n c y with m a s s m* a p p r o p r i a t e to the bottom of the conduction band. T h e Landau l e v e l e n e r g i e s r e l a t i v e to the c e n t r e of the gap a r e then given by
E n : +[(½EG) 2 + nEGliWc]½,
(1)
w h e r e the plus (minus) sign is used f o r the conduction (valence) band. Using the s e l e c t i o n r u l e An = + 1, the 0 - 1 i n t e r b a n d t r a n s i t i o n at #z = 0 has a f r e q u e n c y given by 1£ ECO: [½EG)2 + Ec~coe]½ + (5 G) • (2) The 0 - I interband transition is the one most sensitive to the value of E G and the non-parabolicity of the bands *. In terms of the cyclotron frequency for free electron mass, O)co = eB/rnoc, one can show directly from eq. (2) that C0co/C0 = ( e B / mo Cco) = ( m* / mo)( 7ico- E c O / E G.
(3) T h u s , a plot of COco/W v e r s u s ~co should give a s t r a i g h t line whose i n t e r c e p t i s EG, the gap e n e r gy, and whose slope is ( m * / m o E G ) . The e x p e r i m e n t was p e r f o r m e d with the m a g n e t i c f i el d p e r p e n d i c u l a r to the s u r f a c e and along the b i s e c t r i x axis. To t e s t the l i n e a r r e l a t i o n s h i p p r e d i c t e d by eq. (3), the v a l u e s of O)co/COv e r s u s w w e r e plotted in fig. 2 f o r the m o s t p r o m i n e n t r e f l e c t i o n p eak , the one at 3.07 kG in fig. 1. The fit of the points to the s t r a i g h t line in fig. 2 shows that the p e a k s have been p r o p e r l y identified as 274
/ t
B // 81SECTRIX AXIS
m,
/
E~ (15,4~O.a)x meV
of refleetivity
versus magnetic field for fixed frequency. The magnetic field is parallel to the bisectrix axis.
7-
/
~ ~o
/
/
[/
/
/
1.0
/
/ / / / /
0
20
80
120
I;0 2;0 2z~O 280 320
~(CM-') Fig. 2. A plot of ~co/~ versus frequency for the 0 - 1 transitions.
360
a r i s i n g f r o m the 0 - 1 t r a n s i t i o n as d e s c r i b e d by the two-band m o d e l . T r a n s i t i o n s involving higher quantum n u m b e r s o r t r a n s i t i o n s c a l c u l a t e d with a p a r a b o l i c m o d e l would r e s u l t in a d i f f e r e n t functional r e l a t i o n s h i p and a p o o r fit. The p r i n c i p a l r e s u l t c o n c e r n i n g the band s t r u c t u r e of this alloy is the equality of band gap and e f f e c t i v e m a s s found h e r e with t h o se m e a s u r e d in p u r e bismuth. The m a g n e t o r e f l e c t i o n e x p e r i m e n t s on p u r e b i s m u t h 1) f o r B along the b i s e c t r i x axis yield an e n e r g y gap of E G = 15 MeV and an e f f e c t i v e c y c l o t r o n m a s s f o r the light e l e c t r o n of m * / m o = 2.12 × 10 -3. The p r e s e n t e x p e r i m e n t on the alloy f o r the s a m e o r i e n t a t i o n y i e l d s E G = (15.4±0.8) MeV and m * / m o = (2.17~-0.11)x 10 -3. Such equality of p a r a m e t e r s i n d i c a t e s that the interband t r a n s i t i o n s in the al l o y a r e a s s o c i a t e d with the s a m e band e x t r e m a which gave the light e l e c t r o n c y c l o t r o n m a s s in p u r e bismuth. T h i s c o n c l u s i o n is f u r t h e r s t r e n g t h e n e d by the peak in fig. 1 at about 6 kG, which can be a s s o c i a t e d with the h eav y e l e c t r o n m a s s , The a s s u m p t i o n that the e l e c t r o n band d o e s not change upon alloying has been used p r e v i o u s l y in i n t e r p r e t i n g both g a l v a n o m a g n e t i c m e a s u r e m e n t s 2) and c y c l o t r o n r e s o n a n c e e x p e r i m e n t s 5) on an a l * Actually, allowed interband transitions for B perpendicular to the surface are not very sensitive to a small inequality in the orbital and spin splittings. The assumption is thus used in this paper for c o n v e n i e n c e .
Volume 10, number 5
PHYSICS LETTERS
loy c o n t a i n i n g 5% antimony. Our i n f r a - r e d m e a s u r e m e n t s show that at l e a s t the e l e c t r o n band gap and light m a s s c o m p o n e n t s a r e l a r g e l y unaffected in alloys c o n t a i n i n g up to 12% antimony. Such a c o n c l u s i o n is also in a g r e e m e n t with a t i g h t - b i n d ing band s t r u c t u r e c a l c u l a t i o n of Mase 6~. In h i s work it is a s s u m e d that the p r i n c i p a l change upon adding a n t i m o n y to b i s m u t h i s a r e d u c t i o n in s p i n o r b i t coupling, a t o m i c a n t i m o n y having a s m a l l e r s p i n - o r b i t coupling than b i s m u t h . He found that as the s p i n - o r b i t coupling is r e d u c e d , the change in the e l e c t r o n bands i s s m a l l
15 June 1964
The a u t h o r s wish to thank M. A. Short for growing the m a t e r i a l used in this work and J. S t r a u t i n s for a s s i s t a n c e in m a k i n g the i n f r a - r e d m e a s u r e m ents. 1} R.N.Brown, J.G.Mavroides and B. Lax, Phys. Rev. 129 (1963) 2055. 2) A, L. Jain, Phys. Rev. 114 (1959) 1518. 3) M.A. Short and J. J. Schott, to be published. 4} L.C.Hebel and P.A.Wolff, Phys. Rev. Letters 11 (1963) 517. 5) G.E. Smith, Phys. Rev. Letters 9 (1962} 487. 6} S. Mase, J. Phys. Soe. Japan 14 (1959} 584.
* * * * *
THE
DE HAAS-VAN ALPHEN EFFECT IN C H R O M I U M THE MAGNETIC STRUCTURE
AND
B. R. WATTS Royal Society Mond Laboratory, University of Cat, bridge
Received 25 May 1964 D u r i n g the c o u r s e of an i n v e s t i g a t i o n of the De H a a s - V a n Alphen effect in single c r y s t a l s of c h r o m i u m in p u l s e d m a g n e t i c fields it h a s proved p o s s i b l e to draw some c o n c l u s i o n s about the a n t i f e r r o m a g n e t i c s t r u c t u r e by c o m p a r i n g r e s u l t s obtained f r o m m a t e r i a l which had no s p e c i a l t r e a t ment with those f r o m m a t e r i a l which had been t r e a t e d in a way s i m i l a r to that d e s c r i b e d by Montalvo and M a r c u s 1). The t r e a t m e n t involves cooling through the N~el t e m p e r a t u r e , TN (about 40°C), in a m a g n e t i c field of 65 kG p a r a l l e l to one of the (100) d i r e c t i o n s . As O v e r h a u s e r and A r r o t t 2) have pointed out, the study of such t r e a t e d m a t e r i a l could help to d i s t i n g u i s h between d i f f e r ent m o d e l s for the a n t i f e r r o m a g n e t i c s t r u c t u r e . Neutron diffractior, s t u d i e s 3) show that the a n t i f e r r o m a g n e t i c s t r u c t u r e changes in cooling through about 120OK (the spin flip t e m p e r a t u r e TF) in such a way that in any d o m a i n the spin dir e c t i o n s flip f r o m b e i n g t r a n s v e r s e to longitudinal with r e s p e c t to the r e l e v a n t (1007 d i r e c t i o n 2 along which the m a g n e t i c s t r u c t u r e has a long wavelength modulation (wavelength about 25 unit cells). Thus the m a g n e t i c s t r u c t u r e below T F (the De H a a s - V a n Alphen effect i s m e a s u r e d at 1.2OK which is of c o u r s e below TF) h a s t e t r a g o n a l s y m m e t r y with 2 a s the t e t r a g o n a l axis. The r e l e v a n t a s p e c t s of the r e s u l t s of the p r e s e n t study of the De H a a s - V a n Alphen effect a r e shown in figs. 1 and 2. Fig. 1 shows that the cooling field t r e a t m e n t p r o d u c e s complete t e t r a g o n a l s y m m e t r y in
the De H a a s - V a n Alphen f r e q u e n c i e s in the s e n s e that none of the f r e q u e n c i e s for the m a g n e t i c field H along z (the (100) cooling field d i r e c t i o n ) coincides with any for H along x or y (these a r e the other two (100) d i r e c t i o n s and behave i d e n t i c a l l y , as they should). It follows t h e r e fore that below TF the t r e a t e d m a t e r i a l m u s t have only a single domain d i r e c t i o n 2 and this coincides with the direction* of the field in which the m a t e r i a l was cooled through T N. If we a s s u m e that the d i r e c t i o n 2 is unchanged by the spin flip at T F it follows that between T N and T F the spins a r e a r r a n g e d t r a n s v e r s e l y to the d i r e c t i o n of the cooling field and that 2 is p a r a l l e l to the cooling field. This conclusion i s c o n s i s t e n t with the m a g netic s t r u c t u r e p r o p o s e d by Kaplan 4) in which the spins lie in the ~100} plane n o r m a l to ~ and s p i r a l about 2 to p r o d u c e the long p e r i o d m o d u l a t i o n , but, in the light of the d i s c u s s i o n s by O v e r h a u s e r and A r r o t t 2) and O v e r h a u s e r 5) of the effect of cooling in a field, i n c o n s i s t e n t with v a r i o u s a l t e r n a t i v e s t r u c t u r e s c o n s i d e r e d by .them. The complete t e t r a g o n a l s y m m e t r y produced in the De H a a s - V a n Alphen effect by the cooling field i m p l i e s that all the o b s e r v e d e x t r e m a l o r b i t s * The alternative possibility, that there are equal proportions of two kinds of domains in which ,~ is r e spectively x and y, is excluded by the fact that no De Haas-Van Alphen frequency is common to the two cases H along x or y and H along z (i. e., the fact of complete tetragonal symmetry). 275
PHYSICS LETTERS
n(e) I
,~~N CLiEC DT U C O ITN N R O S -,?
-9
-3,
3
1,81 9
~,~m,o,,,
f3d ELECTRONS Fig. 2. Expected difference angular correlations from spherically symmetric distributions of unpgired 3d. and conduction electron spins. The angles eF {+) = kF(+)/mc correspond to the Fermi radii for the conduction electrons with spins up and down respectively. the curve l a b e l l e d "3d e l e c t r o n s " in fig. 2. The m e a s u r e d d i f f e r e n c e , however, b e c o m e s p o s i t i v e at s m a l l 0. T h i s fact m a y be explained in t e r m s of an s - d i n t e r a c t i o n c a u s i n g a p o l a r i s a t i o n of the conduction e l e c t r o n s . A s s u m i n g a s i m p l e model of two s p h e r i c a l F e r m i s u r f a c e s with r a d i i kF(+-) for the conduction e l e c t r o n s with + and - spin, one o b t a i n s u c o n t r i b u t i o n f r o m the conduction e l e c t r o n s r e p r e s e n t e d by the t r u n c a t e d i n v e r t e d p a r a b o l a (not n o r m a l i s e d with r e s p e c t to the 3d curve) in fig. 2. In case this p o l a r i s a t i o n i s a n t i p a r a l l e l to that of the 3d e l e c t r o n s t h i s c o n t r i b u -
INTERBAND
TRANSITIONS
AND
15 June 1964
tion has to be added to the 3d e l e c t r o n c u r v e , in the opposite case it should be subtracted. The e x p e r i m e n t a l r e s u l t s can only be explained if one a s s u m e s an a n t i p a r a l l e l p o l a r i s a t i o n of the conduction e l e c t r o n s , i. e . , the sign of the s - d i n t e r action is negative. A s i m i l a r r e s u l t was obtained by Shull and Yamada 4) f r o m diffraction of p o l a r i s . ed n e u t r o n s . An a l t e r n a t i v e explanation in t e r m s of a spin dependent r a d i a l p a r t of the 3d wave functions 5) without a s s u m i n g a conduction e l e c t r o n p o l a r i s u t i o n s e e m s u n l i k e l y as computations show that, with a m a g n e t i c m o m e n t for iron of 2,2 ~B, one would need on the o r d e r of 10 to 15 3d e l e c t r o n s to achieve a sign r e v e r s a l of n ( Q . The a u t h o r s a r e indebted to Dr. T. R i s t e for suggesting this p r o b l e m and showing continuous i n t e r e s t , and to Mr. E. F r i k k e e for valuable d i s cussions.
References 1) S. S. Hanna and R. S. Preston, Phys. Rev. 109 (1958) 716. 2) G. Trumpy, Phys. Rev. 118 (1960) 668. 3) J. Ldvseth, Phys. Norvegiea 1 (1961-63) 145. 4) C.G.Shull and Y.Yamada, J. Phys. Soc. Japan 17, Suppl. B-III (1962) 1. 5) J.H.Wood and G.W. Pratt J r . , Phys. Rev. 107 (1957) 995.
BAND
STRUCTURE
OF A BiSb
ALLOY
L. C. HEBEL and G. E. SMITH Bell Telephone Laboratories, Murray Hill, New Yersey Received 20 May 1964 We wish to r e p o r t on m e a s u r e m e n t s of i n f r a r e d m a g n e t o - r e f l e c t i o n f r o m a single c r y s t a l B i 8 8 . 6 S b l l . 4 alloy. The m e a s u r e m e n t s w e r e m a d e at liquid helium t e m p e r a t u r e s u s i n g f r e q u e n c i e s between 80 cm -1 and 400 cm -1 with m a g n e t i c fields up to 20 kG. Reflection s i n g u l a r i t i e s a r e o b s e r v e d which a r e i n t e r p r e t e d in t e r m s of d i r e c t i n t e r b a n d t r a n s i t i o n s 1) between Landau l e v e l s of a conduction band and those of a band lying beneath it. The p a r a m e t e r s which d e s c r i b e the two bands in the alloy a r e found to be e s s e n t i a l l y i d e n t i c a l to those which d e s c r i b e the e l e c t r o n bands in p u r e bismuth. The b i s m u t h - a n t i m o n y s y s t e m f o r m s a corn-
plete solid solution; it has been c u s t o m a r y to a s sume that the alloys have a band s t r u c t u r e which g~es continuously f r o m that of p u r e b i s m u t h to that of pure antimony. Galvanomagnetic m e a s u r e m e n t s 2) have shown that in the r e g i o n of 12% antim o n y in b i s m u t h the alloys become s e m i c o n d u c t ing. The i n f r a r e d r e f l e c t i o n was m e a s u r e d f r o m a s a m p l e containing 11.4% antimony~ which was , grown by a zone levelling technique ~). The c o m position was d e t e r m i n e d by an e l e c t r o n m i c r o probe and found to be u n i f o r m to the l i m i t of a c c u r a c y of the i n s t r u m e n t (+ 0.2 at %). R e f l e c t i v i t y of the s a m p l e was m e a s u r e d by a doublebeam technique d e s c r i b e d e l s e w h e r e 4). 273
Volume ]0, number 3
PHYSICS LETTERS
BL88.6 ~'b[I.4
~'e=203Cm-1
15 June 1964
4.0 X 10.3
/ S~'88.6 Sb1~4
30 10 Fig,
2,0
]_. A p h o t o g r a p h
3.0 4 .O 5.0 MAGNETIC FIELD (kO) o f 8. r e c o r d e r
trace
6,0
7.0
8.0
R e f l e c t i o n p e a k s , such as the one at 3 kG in fig. 1, w e r e o b s e r v e d and studied as a function of f r e quency and m a g n e t i c field. The p eak s a r e i n t e r p r e t e d as o n s e t s of allowed interband, t r a n s i t i o n s b e t w e e n Landau l e v e l s of the two bands f o r #z = 0 and a r e a n a l y z e d using the s a m e type of two-band m o d e l as w a s i n t r o duced by Lax 1) f o r p u r e bismuth. In the Lax m o d e l the o r b i t a l and spin s p l i t t i n g s a r e taken to be equal 4). The e n e r g i e s at k z = 0 of the Landau l e v e l s of the two bands a r e then functions of E G , the gap between bands at z e r o f i e l d , and a l s o ¢0c, a c y c l o t r o n f r e q u e n c y with m a s s m* a p p r o p r i a t e to the bottom of the conduction band. T h e Landau l e v e l e n e r g i e s r e l a t i v e to the c e n t r e of the gap a r e then given by
E n : +[(½EG) 2 + nEGliWc]½,
(1)
w h e r e the plus (minus) sign is used f o r the conduction (valence) band. Using the s e l e c t i o n r u l e An = + 1, the 0 - 1 i n t e r b a n d t r a n s i t i o n at #z = 0 has a f r e q u e n c y given by 1£ ECO: [½EG)2 + Ec~coe]½ + (5 G) • (2) The 0 - I interband transition is the one most sensitive to the value of E G and the non-parabolicity of the bands *. In terms of the cyclotron frequency for free electron mass, O)co = eB/rnoc, one can show directly from eq. (2) that C0co/C0 = ( e B / mo Cco) = ( m* / mo)( 7ico- E c O / E G.
(3) T h u s , a plot of COco/W v e r s u s ~co should give a s t r a i g h t line whose i n t e r c e p t i s EG, the gap e n e r gy, and whose slope is ( m * / m o E G ) . The e x p e r i m e n t was p e r f o r m e d with the m a g n e t i c f i el d p e r p e n d i c u l a r to the s u r f a c e and along the b i s e c t r i x axis. To t e s t the l i n e a r r e l a t i o n s h i p p r e d i c t e d by eq. (3), the v a l u e s of O)co/COv e r s u s w w e r e plotted in fig. 2 f o r the m o s t p r o m i n e n t r e f l e c t i o n p eak , the one at 3.07 kG in fig. 1. The fit of the points to the s t r a i g h t line in fig. 2 shows that the p e a k s have been p r o p e r l y identified as 274
/ t
B // 81SECTRIX AXIS
m,
/
E~ (15,4~O.a)x meV
of refleetivity
versus magnetic field for fixed frequency. The magnetic field is parallel to the bisectrix axis.
7-
/
~ ~o
/
/
[/
/
/
1.0
/
/ / / / /
0
20
80
120
I;0 2;0 2z~O 280 320
~(CM-') Fig. 2. A plot of ~co/~ versus frequency for the 0 - 1 transitions.
360
a r i s i n g f r o m the 0 - 1 t r a n s i t i o n as d e s c r i b e d by the two-band m o d e l . T r a n s i t i o n s involving higher quantum n u m b e r s o r t r a n s i t i o n s c a l c u l a t e d with a p a r a b o l i c m o d e l would r e s u l t in a d i f f e r e n t functional r e l a t i o n s h i p and a p o o r fit. The p r i n c i p a l r e s u l t c o n c e r n i n g the band s t r u c t u r e of this alloy is the equality of band gap and e f f e c t i v e m a s s found h e r e with t h o se m e a s u r e d in p u r e bismuth. The m a g n e t o r e f l e c t i o n e x p e r i m e n t s on p u r e b i s m u t h 1) f o r B along the b i s e c t r i x axis yield an e n e r g y gap of E G = 15 MeV and an e f f e c t i v e c y c l o t r o n m a s s f o r the light e l e c t r o n of m * / m o = 2.12 × 10 -3. The p r e s e n t e x p e r i m e n t on the alloy f o r the s a m e o r i e n t a t i o n y i e l d s E G = (15.4±0.8) MeV and m * / m o = (2.17~-0.11)x 10 -3. Such equality of p a r a m e t e r s i n d i c a t e s that the interband t r a n s i t i o n s in the al l o y a r e a s s o c i a t e d with the s a m e band e x t r e m a which gave the light e l e c t r o n c y c l o t r o n m a s s in p u r e bismuth. T h i s c o n c l u s i o n is f u r t h e r s t r e n g t h e n e d by the peak in fig. 1 at about 6 kG, which can be a s s o c i a t e d with the h eav y e l e c t r o n m a s s , The a s s u m p t i o n that the e l e c t r o n band d o e s not change upon alloying has been used p r e v i o u s l y in i n t e r p r e t i n g both g a l v a n o m a g n e t i c m e a s u r e m e n t s 2) and c y c l o t r o n r e s o n a n c e e x p e r i m e n t s 5) on an a l * Actually, allowed interband transitions for B perpendicular to the surface are not very sensitive to a small inequality in the orbital and spin splittings. The assumption is thus used in this paper for c o n v e n i e n c e .
Volume 10, number 5
PHYSICS LETTERS
loy c o n t a i n i n g 5% antimony. Our i n f r a - r e d m e a s u r e m e n t s show that at l e a s t the e l e c t r o n band gap and light m a s s c o m p o n e n t s a r e l a r g e l y unaffected in alloys c o n t a i n i n g up to 12% antimony. Such a c o n c l u s i o n is also in a g r e e m e n t with a t i g h t - b i n d ing band s t r u c t u r e c a l c u l a t i o n of Mase 6~. In h i s work it is a s s u m e d that the p r i n c i p a l change upon adding a n t i m o n y to b i s m u t h i s a r e d u c t i o n in s p i n o r b i t coupling, a t o m i c a n t i m o n y having a s m a l l e r s p i n - o r b i t coupling than b i s m u t h . He found that as the s p i n - o r b i t coupling is r e d u c e d , the change in the e l e c t r o n bands i s s m a l l
15 June 1964
The a u t h o r s wish to thank M. A. Short for growing the m a t e r i a l used in this work and J. S t r a u t i n s for a s s i s t a n c e in m a k i n g the i n f r a - r e d m e a s u r e m ents. 1} R.N.Brown, J.G.Mavroides and B. Lax, Phys. Rev. 129 (1963) 2055. 2) A, L. Jain, Phys. Rev. 114 (1959) 1518. 3) M.A. Short and J. J. Schott, to be published. 4} L.C.Hebel and P.A.Wolff, Phys. Rev. Letters 11 (1963) 517. 5) G.E. Smith, Phys. Rev. Letters 9 (1962} 487. 6} S. Mase, J. Phys. Soe. Japan 14 (1959} 584.
* * * * *
THE
DE HAAS-VAN ALPHEN EFFECT IN C H R O M I U M THE MAGNETIC STRUCTURE
AND
B. R. WATTS Royal Society Mond Laboratory, University of Cat, bridge
Received 25 May 1964 D u r i n g the c o u r s e of an i n v e s t i g a t i o n of the De H a a s - V a n Alphen effect in single c r y s t a l s of c h r o m i u m in p u l s e d m a g n e t i c fields it h a s proved p o s s i b l e to draw some c o n c l u s i o n s about the a n t i f e r r o m a g n e t i c s t r u c t u r e by c o m p a r i n g r e s u l t s obtained f r o m m a t e r i a l which had no s p e c i a l t r e a t ment with those f r o m m a t e r i a l which had been t r e a t e d in a way s i m i l a r to that d e s c r i b e d by Montalvo and M a r c u s 1). The t r e a t m e n t involves cooling through the N~el t e m p e r a t u r e , TN (about 40°C), in a m a g n e t i c field of 65 kG p a r a l l e l to one of the (100) d i r e c t i o n s . As O v e r h a u s e r and A r r o t t 2) have pointed out, the study of such t r e a t e d m a t e r i a l could help to d i s t i n g u i s h between d i f f e r ent m o d e l s for the a n t i f e r r o m a g n e t i c s t r u c t u r e . Neutron diffractior, s t u d i e s 3) show that the a n t i f e r r o m a g n e t i c s t r u c t u r e changes in cooling through about 120OK (the spin flip t e m p e r a t u r e TF) in such a way that in any d o m a i n the spin dir e c t i o n s flip f r o m b e i n g t r a n s v e r s e to longitudinal with r e s p e c t to the r e l e v a n t (1007 d i r e c t i o n 2 along which the m a g n e t i c s t r u c t u r e has a long wavelength modulation (wavelength about 25 unit cells). Thus the m a g n e t i c s t r u c t u r e below T F (the De H a a s - V a n Alphen effect i s m e a s u r e d at 1.2OK which is of c o u r s e below TF) h a s t e t r a g o n a l s y m m e t r y with 2 a s the t e t r a g o n a l axis. The r e l e v a n t a s p e c t s of the r e s u l t s of the p r e s e n t study of the De H a a s - V a n Alphen effect a r e shown in figs. 1 and 2. Fig. 1 shows that the cooling field t r e a t m e n t p r o d u c e s complete t e t r a g o n a l s y m m e t r y in
the De H a a s - V a n Alphen f r e q u e n c i e s in the s e n s e that none of the f r e q u e n c i e s for the m a g n e t i c field H along z (the (100) cooling field d i r e c t i o n ) coincides with any for H along x or y (these a r e the other two (100) d i r e c t i o n s and behave i d e n t i c a l l y , as they should). It follows t h e r e fore that below TF the t r e a t e d m a t e r i a l m u s t have only a single domain d i r e c t i o n 2 and this coincides with the direction* of the field in which the m a t e r i a l was cooled through T N. If we a s s u m e that the d i r e c t i o n 2 is unchanged by the spin flip at T F it follows that between T N and T F the spins a r e a r r a n g e d t r a n s v e r s e l y to the d i r e c t i o n of the cooling field and that 2 is p a r a l l e l to the cooling field. This conclusion i s c o n s i s t e n t with the m a g netic s t r u c t u r e p r o p o s e d by Kaplan 4) in which the spins lie in the ~100} plane n o r m a l to ~ and s p i r a l about 2 to p r o d u c e the long p e r i o d m o d u l a t i o n , but, in the light of the d i s c u s s i o n s by O v e r h a u s e r and A r r o t t 2) and O v e r h a u s e r 5) of the effect of cooling in a field, i n c o n s i s t e n t with v a r i o u s a l t e r n a t i v e s t r u c t u r e s c o n s i d e r e d by .them. The complete t e t r a g o n a l s y m m e t r y produced in the De H a a s - V a n Alphen effect by the cooling field i m p l i e s that all the o b s e r v e d e x t r e m a l o r b i t s * The alternative possibility, that there are equal proportions of two kinds of domains in which ,~ is r e spectively x and y, is excluded by the fact that no De Haas-Van Alphen frequency is common to the two cases H along x or y and H along z (i. e., the fact of complete tetragonal symmetry). 275