Ge- and Pb- associated hole-traps in ZnTe crystals

Volume 26A, number 11

P H Y SI C S L E T T E R S

We now define, for a given state of an a r b i t r a r y N - f e r m i o n s y s t e m , the m e a n s q u a r e d m o m e n t u m /~ and t w o : p a r t i c l e distance d by N~2 = ( ~ i p / 2 ) and ½Y(N- l)d-2 =( ~ ~ (r i - r j ) 2 ) . According to the

22 April 1968

n o n - s a t u r a t i o n of g r a v i t a t i o n a l forces [4]: .~. d' >t ½ N ~

(3-dim. f e r m i o n s )

w h e r e 3' = ~N( l N - 1 ) < ~ ~ l r ~ - rjl-1,

(5)

> -1 is the

i
½N~2 + :~w2N(N - 1)d-2 = (H> >~Eo(N) where Eo[N) is the g r o u n d - s t a t e e n e r g y of the H a m i l t o n i a n H studied above. When u s e is made of the e s t i m a t e s (1) or (2), the quadratic i n e q u a lity with r e s p e c t to w so obtained, i m p l i e s the validity of the following generalized uncertainty

re lations f o r N-fermions: /~. t~ >I ½Nt/

(1-dim. f e r m i o n s )

(3)

~. d >t aN½~

(3-dim. f e r m i o n s )

(4)

The m e a n i n g of these i n e q u a l i t i e s is v e r y s i m p l e : the a v e r a g e length (resp. volume) of a l - ( r e s p . 3) d i m e n s i o n a l N - f e r m i o n s y s t e m is l a r g e r than N t i m e s the a v e r a g e d e B r o g l i e length ~//~ (resp. volume (ti/p)3) within a n u m e r i c a l coefficient. Or, s t i l l , one cannot put m o r e than one f e r m i o n in a p h a s e - s p a c e cell with volume of o r d e r ~ ( r e s p . ~/3). This is, of c o u r s e , a s t a n d a r d h e u r i s t i c i n t e r p r e t a t i o n of the P a u l i exclusion p r i n ciple. The p r e s e n t work p r o v e s it r i g o r o u s l y for afinite n u m b e r of p a r t i c l e s * . Let u s still m e n t i o n a s i m i l a r inequality, obtained in a study of the

h a r m o n i c a v e r a g e distance. R i g o r o u s i n e q u a l i t i e s such as (3), (4), (5), exp r e s s i n g the p h y s i c a l n a t u r e of the exclusion p r i n c i p l e and independent of the f o r c e s operating between the p a r t i c l e s , might prove useful in studies of m a n y - f e r m i o n s y s t e m s . It is a p l e a s u r e to thank B. J a n c o v i c i for a stimulating correspondance. * For an infinite system of fermions with density p, it may be shown that ,~ >~fb where Po oc p} is the average momentum for a free Fermi gas. This is proved for instance in ref. 2 but results at once from the variational principle [3] and is certainly known to many people.

~eferences 1. A.de Shalit and I.Talmi, Nuclear shell theory, (Academic Press, New York, 1963) p. 3. 2. R.M.Mazo, J.Math.Phys.8 (1967) 1546. 3. E. Lieb, private communication. 4. J. -M. Levy-Leblond, to be published.

* * * * *

Ge- AND P b - A S S O C I A T E D

HOLE-TRAPS

IN Z n T e

CRYSTALS

T. IIDA and H. WATANABA

Department of Physics, Osaka City University, Osaka, Japan Received 20 March 1968

This letter reports that a Ge or Pb ion in ZnTe crystals traps a hole localized mainly on the four Te ions around the Ge or Pb ion.

Z n T e c r y s t a l s doped with Ge or Pb ions have b e e n r e p o r t e d to show e l e c t r o n spin r e s o n a n c e (ESR) a f t e r the c r y s t a l s a r e i r r a d i a t e d by light with the band gap e n e r g y [1,2]. Since the ESR s p e c t r a have h y p e r f i n e s t r u c t u r e s due to 73Ge o r 205Pb and s u p e r h y p e r f i n e s t r u c t u r e s due to 125Te, the ESR s i g n a l i s a t t r i b u t e d to e i t h e r the e l e c t r o n o r hole wich is t r a p p e d by Ge o r Pb. The

e l e c t r o n o r hole spends the t i m e p a r t l y on Ge o r Pb and p a r t l y on the s u r r o u n d i n g T e ions. We suggest that the p a r a m a g n e t i c c e n t e r is a hole which spends m o s t of the t i m e on the four Te ions a r o u n d the Ge o r Pb. The suggestion is obtained by c o m p a r i n g the o b s e r v e d v a l u e s of the h y p e r fine p a r a m e t e r A and s u p e r h y p e r f i n e p a r a m e t e r s As and A~ with those calculated. The o b s e r v e d

541

Volume 26A, number 11

PIIYSICS

a n d c a l c u l a t e d v a l u e s a r e l i s t e d in t a b l e 1. N o t e that the observed values are those for the elect r o n o r h o l e s p e n d i n g p a r t of t h e t i m e in t h e r e s spective orbitals. Af is the hyperfine parameter f o r a s i n g l e e l e c t r o n o r h o l e s p e n d i n g all t h e t i m e in t h e 4 s o r b i t a l of Ge +++ o r in t h e 6 s o r b i t a l of P b ÷++ f r e e ion. C a l c u l a t e d v a l u e s o f A s a n d Act are those for a single electron or hole spending a l l t h e t i m e in t h e 5 s a n d 5p o r b i t a l s of a neutral Te atom. In o u r p r e v i o u s p a p e r [4], t h e f o l l o w i n g r e l a tions are established:

I. E T T E I I S

Table 1 Observed and calculated values of A .As and Ao-, in era-1. Z n T e :Ge A of 73Ge or 205Pb

0.0219 [1]

0.5230 [2]

0.0889 [3]

2.6O

A s of 125Te

0.0159 [ 1]

(}.0125 [2]

1.257

1.274

(1)

whereAs ° andAa° are the hyperfine parameters i s d o u b l y n e g a t i v e s t a t e of T e ion. N',t s a n d N~tcr a r e c o e f f i c i e n t s in t h e l i n e a r c o m b i n a t i o n f o r t h e a p p r o x i m a t e w a v e f u n c t i o n [4] of t h e t r a p p e d e l e c tron or hole

a 1 = N ( ~ s - XsX s - XcrX(~) .

(2)

~ s i s t h e 4 s o r b i t a l of Ge o r t h e 6 s o r b i t a l of lab. Xs i s a n a p p r o p r i a t e l i n e a r c o m b i n a t i o n , i n v a r i a n t u n d e r t r a n s f o r m a t i o n s of t h e t e t r a h e d r a l p o i n t g r o u p , of 5 s o r b i t a l s of f o u r T e i o n s a r o u n d t h e Ge o r lab ion; X(~ i s s i m i l a r to ×s e x c e p t f o r 5per i n s t e a d of 5 s o r b i t a l s . O n e s h o u l d b e a b l e to f i n d t h e v a l u e s of (N~ts)2 a n d (NX(y)2 if t h e v a l u e s o f A s O a n d A c r o w e r e known; however, no values for the negatively i o n i z e d s t a t e a r e a v a i l a b l e . We h a v e , t h e r e f o r e , calculated the corresponding values using numeri c a l w a v e f u n c t i o n s , of T~ in t h e n e u t r a l a t o m i c s t a t e [5]. I n s e r t i n g t h e o b s e r v e d a n d c a l c u l a t e d v a I t t e s f o r A s a n d A(x i n t o (1), we o b t a i n t h e l o w e r limit values

Z n Te : P b

Af

A (5 of 125Te A s = ~ - ( N X s ) 2 A s ° a n d A c r = ~0(NXcr)2Acr ° ,

'22 April 1968

(calc.)

[3]

(calc.)

0.0064 [ 1]

0.0067 [2]

0.090

0.091

(ealc.)

(calc.)

l i k e l y to t r a p t h e h o l e , r a t h e r t h a n t h e e l e c t r o n , we c o n c l u d e t h a t t h e Ge o r P b i n c o r p o r a t e d into Z n T e c r y s t a l s t e n d s to t r a p a h o l e m o s t l y l o c a l i z e d on t h e f o u r n e i g h b o r i n g T e i o n s . Another evidence for the hole-trapping is obt a i n e d b y t h e f o l l o w i n g o b s e r v a t i o n [6]. No E S R s i g n a l i s o b s e r v e d in I n - a d d e d Z n T e : P b c r y s t a l s ; h o w e v e r , E S R s i g n a l i s o b s e r v e d in S b - a d d e d Z n T e : P b c r y s t a l s in t h e d a r k b e f o r e i r r a d i a t i o n by light with the band gap energy. Indium imp u r i t i e s a r e a s s u m e d to t r a p h o l e s w h i c h lab ++ i o n s s h o u l d t r a p if n o In w e r e a d d e d . A n t i m o n y impurities release holes, which are trapped a r o u n d lab ++ i o n s w i t h no i r r a d i a t i o n . T h e h o l e - t r a p p i n g m o d e l a s s o c i a t e d w i t h Ge a n d lab i o n s in Z n T e c r y s t a l s will a p p l y to o t h e r g r o u p IV e l e m e n t s d o p e d in v a r i o u s k i n d s of I I - V I c o m p o u n d s . A d e t a i l e d p a p e r w i l l b e p u b l i s h e d in an other journal. T h e a u t h o r s a c k n o w l e d g e P r o f e s s o r s M. A o k i a n d H. O h k u r a a n d D r s . K. Suto a n d K. S u g i b u c h i for valuable discussions.

(NXs) 2 >/ 0.05 a n d (NX(y) 2 >I 0.71 f o r Ge,

(3) (NXs)2 >~ 0.04 a n d (NXcr) 2 >~ 0.74 f o r P b . T h e p r o b a b i l i t y of f i n d i n g t h e t r a p p e d e l e c t r o n o r h o l e in t h e 4 s o r b i t a l of Ge o r in t h e 6 s o r b i t a l of lab i s g i v e n b y t h e f o r m u l a ( A / A f ) ( g e / g ) . ge i s t h e f r e e e l e c t r o n g v a l u e , 2.0023. T h e o b s e r v e d v a l u e i s 2.1375 f o r Ge a n d 2.167 f o r lab. T h e probability estimated is 0.23 f o r Ge a n d 0.19 f o r lab,

(4)

A n i n s p e c t i o n of (3) a n d (4) i m p l i e s t h a t t h e p r o b a b i l i t y of f i n d i n g t h e e l e c t r o n o r h o l e in t h e 5p o r b i t a l s of t h e f o u r s u r r o u n d i n g T e i o n s i s a t l e a s t t h r e e t i m e s a s l a r g e a s t h a t in t h e 4 s of Ge o r 6 s of P b , S i n c e t h e f o u r n e g a t i v e T e i o n s a r e

542

References 1. K. Sugibuchi, Annual Meeting of the Physical Society of Japan, Sendal, April, 1967. 2. M. Aoki and K. Suto, P r o c . Int. Conf. II-VI Semiconducting compounds, P r o v i d e n c e , September, 1967, (Benjamin, I n c . , New York) to be published. 3. R e f e r r e d to in W. Knight, Solid state physics, eds. F. Seitz and D. Turnbull, Vol. 2 (Academic P r e s s , New York, 1956). 4. H. Watanabe, Phys. Rev. 149 (1966) 402. 5. F. H e r m a n and S. Skillman, Atomic s t r u c t u r e calculations, ( P r e n t i c e - H a l l , I n c . , New J e r s e y , 1963). 6. M. Aoki and K. Suto, J. Phys. Soc. Japan, April 1968, to be published.