Jul 12, 1971 - A .F.Ioffe Physico-Technical Institute of the Academy of. Sciences of the USSR, Leningrad, USSR. Received 12 June 1971. An electrical current ...
Volume 35A, number 6
PHYSICS LETTERS
CURRENT-INDUCED
SPIN ORIENTATION IN S E M I C O N D U C T O R S
12 July 1971
OF
ELECTRONS
M. I. DYAKONOV and V. I. P E R E L
A .F.Ioffe Physico-Technical Institute of the Academy of Sciences of the USSR, Leningrad, USSR Received 12 June 1971 An electrical current in a semiconductor induces spin orientation in a thin layer near the surface of the sample due to spin-orbit effects in scattering of electrons. A weak magnetic field parallel to the current destroys this orientation.
It is well known [1] that s c a t t e r i n g of u n p o l a r ized e l e c t r o n s by an u n p o l a r i z e d t a r g e t r e s u l t s in spatial s e p a r a t i o n of e l e c t r o n s with different s p i n s due to s p i n . o r b i t interaction. F o r this r e a s o n an e l e c t r i c a l c u r r e n t in a s e m i c o n d u c t o r should be accompanied by a spin flow p e r p e n d i c u l a r to the c u r r e n t and d i r e c t e d from the bulk to the surface. It was shown in our p r e v i o u s work [2] that this leads to a c c u m u l a t i o n of spin o r i e n t a t i o n in a thin l a y e r n e a r the s u r f a c e of the sample. In this l e t t e r we c o n s i d e r the influence of a magnetic field on this effect. We also p r e s e n t e x p r e s s i o n s for the phenomenologicai c o n s t a n t s , i n t r o d u c e d in [2], in t e r m s of the s c a t t e r i n g amplitude. Our c a l c u l a t i o n is b a s e d on the following equation for the spin density
asfl/at + aq(~/axot +[~×S]~+s~/-c s = 0,
(1)
where g2= gBgI'I/h, ~B, g a n d / ' / b e i n g the B o h r ' s magneton, g - f a c t o r and magnetic field r e s p e c t i v e ly, 1"s is the spin r e l a x a t i o n t i m e . The density of spin flow t e n s o r qaB may be d e r i v e d by solving the kinetic equatior/for the n o n - d i a g o n a l density m a t r i x ~ m m , ( r , p, t). (m and m' are spin indices.) S p i n - o r b i t effects in s c a t t e r i n g m u s t be included in the c o l l i s i o n t e r m . The q u a n t i t i e s S~ and qotfl are r e l a t e d to ~ in the following way:
S8 = f d 3 SpSfj ~ ( r ' p ' t ) ; qaf3 = fdapvaSPS~ ~ : ( r , p , t ) .
(2)
where ~ are Pauli matrices. We have c a l c u l a t e d q a ~ for a n o n - d e g e n e r a t e s e m i c o n d u c t o r with a c e n t e r of i n v e r s i o n in the c a s e when i m p u r i t y s c a t t e r i n g dominates. Up to
the f i r s t o r d e r in s p i n - o r b i t i n t e r a c t i o n we obtain ~St+ ~ n ~ ^ E +
qolf] = - b E ~ S f l - D ~ + p
eta7+ 7+
s>-
/~SO/
(3) div
s)
where only t e r m s l i n e a r in the e l e c t r i c a l field ~ , S and OS/$x are kept. In eq. (3) b and D a r e the usual mobility and diffusion coefficient, n is the e l e c t r o n c o n c e n t r a t i o n , a s s u m e d to be constant. The coefficients fl and/~1 a r e the r e a l and i m a g i n a r y p a r t s of a single e x p r e s s i o n f3 + i f l l = e N ( 2 ( p ) v . 2 7 r fAB*sin2OdO), (4) 0 where N is the i m p u r i t y c o n c e n t r a t i o n , T (p) is the u s u a l r e ! a x a t i o n t i m e , b r a c k e t s stand for a v e r a g i n g over the Maxwell d i s t r i b u t i o n , A and B a r e the q u a n t i t i e s e n t e r i n g the e x p r e s s i o n for the s c a t t e r i n g amplitude [3]
]
: A + B~[p×p'](p"sinO)
-t
(S)
,
51 = kTfJl/e, which is the E i n s t e i n relation. The equation for the density of e l e c t r o n flow
q = - bnE - f i l E × S ]
- 0rot S
(6)
contains additional t e r m s due to s p i n - o r b i t i n t e r action, 6 = kTfl/e. The second t e r m in eq. (6) is r e s p o n s i b l e for the anomalous Hall effect. Note that the c o n s t a n t s e n t e r i n g eq. (3) a r e the s a m e as in eq. (6). N u m e r o u s t e r m s depending on magnetic field a r e ignored in eqs. (3) and (6). T h e s e t e r m s a r e u n i m p o r t a n t for our p u r p o s e s until G ~"( 1 - a condition we a s s u m e tO be fulfilled. At the s a m e t i m e the value of i2Ts may be l a r g e , s i n c e ~s ) ~. 459
Volume 35A, number 6
PHYSICS Lk:TTERS
Consider a c y l i n d r i c a l sample with c u r r e n t and magnetic field p a r a l l e l to the axis (z-axis). Since the c u r r e n t flow is the only r e a s o n for spin o r i e n t a t i o n , S is p r o p o r t i o n a l to E and n o n - l i n e a r t e r m s in eq. (3), eq. (6) may be neglected. Then solving eqs. (1) and (3) with the boundary conditions qr~ = 0 ([3= z , r , ~9) on the surface, one obtains s:
nfl-~" ~
,
Sz:O
,
o r i e n t a t i o n P : S / n in the spin layer may r e a c h the value ~ Vo/V , v being the t h e r m a l velocity. It is r e m a r k a b l e that the s m a l l constant of spinorbit i n t e r a c t i o n c a n c e l s out in the e x p r e s s i o n for S ~o. As may be seen from eq. (7) the magnetic field r o t a t e s the vector of spin density and r e duces the degree of orientation. A notable dec r e a s e o c c u r s at such magnetic fields that
(7)
~ s ~1.
where S~ = S ~ + iSr, v o = bE is the e l e c t r o n drift velocity, ~ (1 ~ i ~ 2 T s ) l / 2 ( D ~ s ) - l / 2 , 11 is the modified B e s s e l function of the second kind, R is the r a d i u s of the wire. Thus spin o r i e n t a t i o n exists n e a r the s u r f a c e in a l a y e r with t h i c k n e s s of the o r d e r of (DTs)l/2. in a b s e n c e of magnetic field S r = O, Sq~ = n f l v o ~ / 2 b - l D - 1 / 2 , so that the degree of *
TIME-AVERAGE
12 July 1971
References [1] N. F. Mott and H. S. W. Massey, The theory oi atomic collisions (Clarendon Press, Oxford, 1965). [2] M. I. Dyakonov and V. I. Perel, Zh. Eksp. i Teor. Fiz Pis'ma 13, No. 12 (1971). JETP Letters. to be published. [3] L.D. Landau and E. M. Lifshitz, Quantum mechanics (Pergamon, 1965).
**
*
*
HOLOGRAPHY OF OBJECTS WITH UNIFORM SLOW
VIBRATING DRIFT
SINUSOIDALLY
C. S. VIKRAM and R. S. SIROHI Department of Physics, Indian Institute of Technology, New Delhi- 29, India
Received 19 March 1971
The technique of desensitized holography has been suggested to extract information regarding the amplitude of sinusoidal vibration when the vibrating object is slowly drifting.
T i m e - a v e r a g e holographic i n t e r f e r o m e t r y of a s i n u s o i d a l l y v i b r a t i n g object i n v e s t i g a t e d by Powell and Stetson [1] is now being u s e d in many l a b o r a t o r i e s to m e a s u r e v i b r a t i o n amplitudes. T h e s e m e a s u r e m e n t s a r e b a s e d on the t h e o r e t i c a l c o n c l u s i o n that an object point v i b r a t i n g s i n u s oidally will be r e c o n s t r u c t e d with an i r r a d i a n c e p r o p o r t i o n a l to C = j 2 ( 2 1 r A m / ~ ) , where A m is p r o p o r t i o n a l to the amplitude of v i b r a t i o n ; C is c a l l e d c h a r a c t e r i s t i c function to the t i m e - a v e r age holography. However, object v i b r a t i n g s i n u s o i d a l l y and moving with c o n s t a n t velocity is a p r a c t i c a l c a s e applicable to v i b r a t i o n s of a l a r g e n u m b e r of m e c h a n i c a l i n s t r u m e n t s . C h a r a c t e r i s t i c function of t i m e - a v e r a g e holography for this type of m o tion has r e c e n t l y b e e n d e t e r m i n e d f r o m c o h e 460
r e n c e c o n s i d e r a t i o n s by Zambuto and L u r i e [2]. Due to the c o n t r i b u t i o n of r a m p motion in the c h a r a c t e r i s t i c function, it is very difficult to single out v i b r a t i o n amplitude d i r e c t l y f r o m a single t i m e - a v e r a g e hologram. To o v e r c o m e this difficulty, we p r o p o s e h e r e a technique to s e p a r a t e out the c o n t r i b u t i o n s due to s i n u s o i d a l v i b r a tion and r a m p motion. The method c o n s i s t s of r e c o r d i n g two h o l o g r a m s with slightly different exposure t i m e s and then s u p e r i m p o s i n g r e c o n s t r u c t e d p a t t e r n s . Moire fringe p a t t e r n thus obtained gives i n f o r m a t i o n about r a m p motion of the object. Once c o n t r i b u t i o n due to r a m p motion is known, v i b r a t i o n amplitude can e a s i l y be d e t e r mined. According to Zambuto and L u r i e [2], the c h a r a c t e r i s t i c function of t i m e - a v e r a g e holography
