3.

1. 2.

L I T E R A T U R E C I T E D

S. Glashow, I. Iliopoulos, and L. Maiani, Phys. Rev. D__2, 1285 (1970). J . D. Biorken and C. H. Llewellyn Smith, Phys. Rev., D7, 887 (1973); E. M. Lipmanov, Dissertat ion, Volgograd- Dubna (1965). 13. M. Ponteeorvo, Usp. Fiz. Nauk, 104, No. 3 (1971).

E L E C T R O N S T A T E S IN P A R A M A G N E T I C

S E M I C O N D U C T O R S

A. A. T r u s h c h e n k o a n d A. L . P i v o v a r o v UDC 537.311.33

The interact ion of a conduction e lec t ron with a discontinuity of the internal p a r a m e t e r in par t ia l ly o rde red or d i sordered sys tems may lead, under cer ta in conditions, to antoloealization of the e lec t ron close to this dis- continuity. Autolocalized e l e c t r o n - p o l a r o n s ta tes were f i rs t investigated in ionic c rys ta l s in [1]. The ex i s - tence of polarons of large radius is a consequence of the long-range Coulomb forces responsible for the in te r - action of the e lec t ron with the lattice polar izat ion. It was shown in [2] that the shor t - range forces in the in te r - action of an e lec t ron with fluctuations of the internal p a r a m e t e r of the sys tem may lead to the formation of stable quas i -par t i c les of large radius - fluctuons. Fluctuons may ar i se , for example, in incompletely o rde red magnets ( fe r ro - and paramagnets) , in solid (or liquid) solutions, and in a number of o ther sys tems . A r e - view of the cur ren t state of fluctuon theory is given in [3].

The formation of fluctuons is favorable , in f r e e - e n e r g y t e r m s , if the change in the f ree e n e r g y / x ~ for localization of an e lec t ron at a fluctuon potential well is negative. In fe r romagne t s , fluctuons may exist as stable quas i -par t i c les only in a specif ic t empera tu re range bounded by the Curie t empera tu re . In pa ramag- netic semiconductors , fluctuons may ar i se only at t em p e ra tu r e s less than a cer ta in cha rac t e r i s t i c t e m p e r a - ture T*. Outside this t empera tu re r ange , /x~ is positive and fluctuons cannot exist in a stable s tate .

The value of the change in f ree ene rgy s of the sys t em must be de termined not only in o r d e r to find the t e mpe ra tu r e region in which fluctuons can exis t but also so as to calculate the rat io of the number of f luc- tuons to the number of conduction e lec t rons (see [3] for details) . In [2, 4, 5], a d i rect var ia t ional method was used to calculate A~ in fe r romagne ts at finite t e m p e r a t u r e s and in ideal paramagnets , For example, for i d e a l paramagnet ic semiconductors with spin S, A~ is de termined f rom the condition that the functional I[r be a minimum

Av h ~ rT j~ I--exp(--~-~ f~(r),-" ) dr.

(2S-k l ) [ I - -exp (--(2S-k l)~-~[*(r)l )]

Here m is the effect ive e lec t ron mass; v is the volume of the e l emen ta ry cell; A is the cha rac te r i s t i c in te r - action energy of the e lec t ron and the spins of the magnetic atoms.

In [2, 4, 51, the t r i a l functions r chosen to descr ibe the autolocalized e lec t ron state close to the region of changed magnetizat ion were the s imples t s i ng l e -pa rame te r approximations

'~6-- e-'r~and ~H -- e-=r"

tn using the di rect var ia t ional method, there a r i ses the question of the accuracy of the obtained resu l t s (their dependence on the choice of the t r i a l function). The present work examines the effect of the fo rm of approximation on the values of the cha rac t e r i s t i c fluctuon p a r am e te r s A6, T*, and E (the energy of the auto- localized electron) determined in [2, 4, 5]. The investigation is c a r r i ed out for ideal paramagnets with spins

Kiev Polytechnical Inst i tute. Trans la ted f rom Izves t iya Vysshikh Uchebnykh Zavedenii , Fizika, No. 5, pp. 132-136, May, 1976. Original ar t ic le submitted November 20, 1975.

ma a, rotec d co y ht g tere i~ ~ Pub,ishin Co.option, 227 West 17ti, S,ree,. Yor , I0011. No pi t I o f this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic mechanical, photocopying, I [microfilming, recording or otherwise, wi thout written permission o f the publisher. A copy o f this article is available irom the publisher for $ 7.50.

657

r 03

TA

BL

E

1

B.

10~

--

--tr

�9 10

3 A

8,21

--

0,88

6 8,

13

0,55

2 8,

06

1,99

7,

68

9,44

7,

32

17,1

6,

97

24,9

6,

64

32,6

6,

34

40,1

6,

05

47,5

5,

79

54,5

5,

54

61,3

5,

32

67,8

5,

11

74,0

4,

92

79,9

4,

75

85,5

= o,

25

Aq~

n 10

3 B

. 102

A

8,12

--

0,47

8,

04

2,72

7,

88

5,94

7,

72

9,18

6,

99

25,3

6,

37

40,6

5,

83

55,1

5,

37

68,6

4,

96

81,4

4,

61

93,6

4,

29

105

= o

,5

~ =

0,7

5

B. 1

02

8,38

8,

18

7,99

7,

07

6,28

5,

61

5,04

4,

56

4,15

3,

80

3,60

AcD

II Ac

~JI 1

--

.1

03 !

B . 1

02

--

,1

03

A

A

- 3,

78

8,25

0,

159

8,00

4,

17

7,75

24

,8

7,50

45

,1

7,27

64

,5

7,04

82

,6

6,02

99

,2

5,20

II

4 4,

56

128

4,05

14

1 3,

63

3,29

3,

01

2,76

--2,

32

3,03

8,

53

14,1

19

,7

25,4

52

,9

78,1

10

0 12

0 13

7 15

2 16

5 17

8

B. 1

0 "~

8,18

8,

01

7,54

7,

23

5,90

4,

92

4,19

3,

64

3,20

2,

85

2,55

A6P

ll

�9 10

3 B

-102

A

--2,

79

0,72

1 11

,6

19,1

55

,3

87,1

11

4 13

7 15

7 17

4 19

0

__

A('D

II I

. 10

~ A

~G

10:

1,66

--

1,3

2

19,9

16

,2

36,4

33

,2

51,2

49

,4

68,1

65

,6

83,2

80

,6

96,5

96

,0

121

120

133

131

144

142

155

153

196

195

204

202

211

210

217

215

TABLE 2

B beg h~m -- A -- -A"-

3,,34 - -0 ,162 0,184 4,02 --0,300 .0,175 4,23 - -0 ,364 .0~227 3,94 - -0 ,224 ~,t55 3,57 --0,022 0,048~ 3,19 0,258 0,300 0,628 3,82 3,83 0,543 4,06 4,09 0,478 4,26 4,29 0,425 4,43 4,45 0,317 4,83 4,86 0,292 4,93 4,98 0,270 5,02 5,09

B

3,88 3,57 3,26 2,97 2,73 2,34 2,04 1,64 1,372

_ M,_..H

A

--0,144 -0,051

0,097 0,299 0,425 0,750 1,030 1,450 1,790

S =1/2 mud S =15/2. As shown in [2], the analyt ical express ions for the case S =1/2 complete ly coincide (after the appropriate change of var iables) with the express ions for ideal solid solutions with equal concentrat ions of both components . Calculations of A,I~ T*, E, and 6M (the fluctuon magnetic moment) were ca r r i ed out in the following approximations

~t~(l§ +n~( l+ar -~a2r2)exp(- -ar ) ; Sffi ~t"

. . . . . . . , r ~ a , r

'~I|I ~ exp (--x" r) - - , r > ~ a ,

r

Here a , fl, u , and u t are the p a r a m e t e r s of the approximation in which the var ia t ion of the functional in Eq. (1) is c a r r i e d out. The value of a in r is de termined by the condition M(a) =M(0)/2, where M(r) is the mag- net izat ion in a local f e r romagne t ic mic ro reg ion (cf. [6]) and cor responds to the width of a rec tangular sphe r - ical ly s y m m e t r i c potential well (for which, as is known, this function is an exact solution of the SchrSdinger equation).

The resu l t s obtained in numer ica l calculat ions of A~h/A for spin S=1/2 are shown in Table 1. The sub- scr ip t on s indicates the approximation in which the calculation was c a r r i e d out. B in Table t cor responds to the express ion

9,~ 3e h ~ [xT'~a:~ aE {~r~:i3

charac te r iz ing the re la t ion between the p a r a m e t e r s of a r ea l semiconductor - t h e width of the conduction band AE, A, and the t empera tu re . Resul ts of a calculation for CH are given explici t ly in [7]. F r o m Table 1 it fot- lows that, for the approximations I, tI, and IH, T* is higher than in the Gaussian approximation by 3.8, 4.5 (for ~ =0.5), and 2.5%, respec t ive ly . Here, B~ =0.0796, B~ =0.0816, B*II=0.0819, B~I I =0.0810 (if B >B *, the exis tence of fluctuons in a stable s tate is impossible) . For t e m p e r a t u r e s T ~T* (i.e., fo r B->0.08), the lowest value of 2x ~ is obtained for the t w o - p a r a m e t e r approximation CH (and for $I among the var ious s ing le -pa ram- e t e r t r i a l functions considered) . With reduction in t empera tu re (T <T*), i .e . , with decrease in B, the lowest value of Lx$ cor responds to the approximation HI. The physical explanation for this is that , as the t e m p e r a - tu re falls , the fluctuon potential well broadens, and comes to r e semble a rec tangula r spher ica l ly symmet r i ca l well. F o r B < 0~ be t t e r resu l t s are obtained with the t r i a l function CG than with r and @II* However, fo r 0.08 > B~0,05 , the use of ~G to determine A,b/A gives an e r r o r in the range 20-3% as compared with the use of ~bHt. For B <0.05, the e r r o r becomes of the o r d e r o f 1%.

Calculation of the e lec t ron ene rgy E in the fluctuon potential well shows that , even for AS >0, E may be less than zero , the value of E being l a rge r for r andCHI t-ban f ~ CG. ForB.~ 0.05, the lowest value of the ene rgy is obtained for CG (lower by 1-2%than that for r ), while fo r B <0.05 the best resu l t s are obtained for r (2-4%below the resu l t s fo r ~bG).

659

The fluctuon magnetic moment in paramagnets with spin S=1/2 may take values both higher and lower than the results obtained in [4] for a Gaussian t r ia l function fin both cases, the difference may reach 15%).

The results obtained for the energy and the magnetic moment are not inconsistent with the theory of the direct variational method, since it is not these values that are minimized, but &@/A. Therefore, in general, it cannot be asserted that the electron energy is less than the value obtained using r

It is of interest to confirm the conclusions drawn above for spin S =1/2, by carrying out analogous cal- culations with a different limiting value of spin, S=15/2. Table 2 shows vaIues of A@/A calculated for the Gauss[an (~G) and hydrogen-like (r approximations and for r According to these results, B* H=3.46, B* G =3.54, and B'HI =3.66. Using ~G and ~H to determine T* gives e r r o r s of 5 and 8%, respeet[vely, as com- pared with @IIt" Of the considered t r ia l functions for determhfing AlgA, the best is r although with reduc- tion in B (decrease in temperature) the difference between A@III/A and _A@G/A decreases.

In conclusion, the authors would like to express their sincere gratitude to Prof. M. A. Krivoglaz for reading the manuscript and for valuable comments.

I.

2.

3.

4.

5.

6.

7.

LITERATURE CITED

S. I. Pekar, Investigations in the Electronic Theory of Crystals [in Russian], Gostekhizdat (1951). M. A. Krivoglaz, Fiz. Tverd. Tela, 11, 2230 (1969). M. A. Krivoglaz, Usp. Fiz. Nauk, 111, 617 (1973). M. A. Krivoglaz and A, A. Trushchenkc, Fiz. Tverd. Tela, 11, 3119 (1969). M. A. Krivoglaz and A. A. Trushchenko, Ukr. Fiz. Zh., 15, 1940 (1970). M. A. Krivoglaz and A. I. Karasevskii, P is 'ma Zh. Eksp. Teor . Fiz., 19, 454 (1974). A. A. Trushchenko, Ukr. Fiz. Zh., 17, 2016 (1972).

S C A T T E R I N G OF F R E E E L E C T R O N S

Y u . I . K l i m e n k o , V. V. K u l i s h , and O. S. P a v l o v a

ON A FORCE CENTER

UDC 530.145

We consider the scattering of free electrons having anomalous magnetic Pi and electr ic P2 moments at a force center of charge e 0 and magnetic moment go. The interaction potential for the electron and the center is given by the expression

1 ~ e-~'~rd3------~ [eeol--ie(~[,A~x])--ieo~ p~(~z)~ieoi~pa(ox)] (t) U= ~ ~+r~.2

where r 0 is the effective radius of action of the electr ic force of the center (as r0~ % we obtain the Coulomb potential); I is the unit matrix; a , p, and e are Dirae matrices. The wave function of the free electron is writ- ten in the form

I l / ~ - + ~ o / 1 .~w_. ,~+, , , ~

\ "~-~--~o /

Here and below, ~ is a Paul[ matrix, and ~cK is the energy and ~k the momentum of the electron. The spinor V satisfies the equation (crl) V = Uv', where l is the unit vector in the direction of the electron spin; ~ =~ 1 cor- responds to the two possible directions of the spin component: ~ =1 along l , and [ ----1 in the opposite direc- tion.

V. D. Kuznetsov Siberian Physicotechnical Institute at Tomsk State University. Translated from tzvestiya VysshikhUchebnykh ZavedenH, Fizika, No. 5, pp. 136-137, May, 1976. Original article submitted December 2, 1975.

I Tbis material is protected by copy,qght registered in rl . . . . me o f Plenum'Publishing Corporation. 227 Wes~17tt, Street, N e w York. N. Y. 10011. No p a r t ] o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, ] microfilm ing, recording or otherwise wi thout written permission o f the publisher. A copy o f this article is available from the publisher for $ 7. 50. ]

660