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P,

3.
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-

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6

17

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1. Baliga J. // Semicond. International. 1997. No. 4. P. 64. 2. . ., . ., . . // . 1998. 5. . 34. 3. . ., . // . . -. 517.958:621.372.8

.

. 1999. 1. . 12 (Moscow University Phys. Bull. 1999. No. 1. P. 8). 4. ., . . .: , 1970. 5. . ., .. . .: . - , 1992. 6. .. .: . - , 1963.
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8 > > > > < > > > > :

1@H? r @' z i Hr ? @@r 1 @ (r"E r r @r 81 @ >E > r @'

i H' + ik"Er = 0; Hz + ik"E' = 0;
) + 1 @@' ("E' ) + i "Ez = 0; r z ? i E' ? ikHr = 0; (1)

1.

> > < > i Er ? > >1 @ > (r : r @r

@E ? @r z Hr) + 1 r

ikH' = 0; @ @ ' H' + i Hz = 0:

(2)

. , . -

Oz
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. -

,

"(r; '; z) = "(r) , "1 "2 > 0 (r; '; z ) 1 .
.

E ; H exp ?i!t+i z+in'
, . :

0 @1@ 2 r @ r r @ r r + k "r inr @@r 1 ?k"n B r B B B in 1 @ r ? n2 + k2"r ?ik"r @ B B r @r r @r B @ @ @ kn" ik @ r r" @@r r" @ r ? " r 01 1 = 2r @ 1 A X;
:

X = (Hr ; H'; Ez)T = (H?Ez )T : (1) (2) Er , E' ',

Hz

n2

1 C C C CX = C C C A
(3)

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H? 2 W 1

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X1(1) = 0

X3(1) = 0; jX (0)j < 1
:

H? = grad + rot ;
grad ' (r) = @@r '; in ' r
T T

?ik"X3 = 1 @@r (rX2) ? 1 inX1; r r
..

; :
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1 ?ik"Ez = r @@r (rH') ? 1 inHr: r

@ rot ' (r) = in ' ? @ r ' r

2.

,

W
(X; Y ) = r dr (x1 y 1 + x2 y2 + "x3 y 3 );
0

V

1 div rot = r @@r r in ? in @@r = 0; r r
2 @ + 1 @r r (rot rot ; ez ) = in r r @ (rot grad ; ez ) = in @ r ? 1 @@r r r

Z1

(X; Y )1 = r dr div H? div H? +
0

Z1

W1

@= ; @r r in = 0; r

+ "r dr grad Ez ; grad Ez + (X; Y ) ;
0

Z1

?

8 < = div H?; : @ jr=1 = 0; j (0)j < 1; @r 8 = (rot H ; e ) = ?ik"E ; < ?z z :(rot ; n) = @ @ r=1 = 0; j (0)j < 1;
).

1@ div H? = r @ r rHr + in H' r
( (3) 0 1. (3):

Y

T

r

V
-

? (X; Y )1 ? b (X; Y ) = ? 2 ? 1 (X; Y ) 8 Y 2 V ; (4) ? b (X; Y ) = dr k2 "r Y1X1 + Y2 0 @ +kn"Y3 X1 ? ik"rY2 @ r
,

8 < = (rot H?; ez ) = ?ik"Ez ; : r=1 = 0; j (0)j < 1:
W1, 0 H? = grad + rot
div H? (rot H? ; ez ) = ?ik"E H? 2 W 1
. . 00 0 H? = H? ? H? :

z

-

Z1 n

X2 ? k"nY1 X3 X3 ? ik"rX2 @@r W1
.

+

Y3 :
-

o

W

1. .

8div H 00 = 0; >? <(rot H 00 ; e ) = 0; > 00 ? z :(H?; n) = 0; r=1
, ,

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W1

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W

H? = grad + rot :


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8 > > < > > :

u

n

|

un = jun(0)j @u @r n r

fn ; < 1;
=1

A un r=1 = 0:
(5)

,

n

. |

Xn 2 W 1
,

19
| -

=0

( grad div H = H ; n ?n ?n
n " Ez n

= "Ez

n
,

(7)

Wr2 , fu (r) : 1 0; 1] ; @ un 1 0; 1] u (r) 2 C C0 @ r r=1 = 0g 1 R kuk2 = r dr du 2 , L2 , r dr 0 1 0; 1] u (r) 2 C R1 r dr juj2 . kuk2 = 0
, (5) . . 5].

,

, :

Hrn (1) = Ezn (1) = 0

A L2 r

un = Afn ; un 2 Wr2;
|

fun g
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, ,

-

"

Wr2

ffng

= ?ikEm ; ' = div (" grad ') . f ng , f mg , fEzm g
m
, ..

8 E = "E ; > m " zm zm >E < zm = 0; > (r) r=10; > :

8 n n = n; >@ < > @ r n r=1 = 0; : n = Ezn = 0;
(1) = 0;

(8)

(9)
m

, . -

kXn k2 6 C , 1 2 kEzn k2 2 6 C , kgrad EznkL2 6 C . L 1 fXng 2 W f ng , f ng , fEzn g Wr2 f k g , f kg L2 r fEzk g . , fH?k g = fgrad k + rot k g f Ezk g L2 . " (r) r L2 W , "1 > " (r) > "2 , r W1 W : , .

fXng 2 W 1 fEzm g kdiv H?n k2 2 6 C , L

; ; Ez
,

f ng , f mg , fEzm g
, . (8) (9) , , 8], ,

|

, W1, f ng , f mg , W1,

L2 r

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" (r)
(9) ,

1

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? O n2
W1
3. .

. 1 . 2

(6)
4.

W (c ., ., 6]) B A, b (X; Y ) = ? (Y ; BX )1 ; (X; Y ) = (Y ; AX )1 :

1

W1

W

-

ker A = 0
(6)

,

n

G = A?1 , n2 .

|

,

? X + B X = ? 2 ? 1 AX; X 2 W 1 :
3.

(4)

(I + B ) GX = X; (6)

(6) , . | . 2. . 7]. -

A W 1 ker A = 0
,

BG pp = 1=2 , . . c > 0, p kBGxk1 6 c kGxk1 kxk1?p : 1 x y = Gx = A?1x

9,

6],

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G

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n
, 6

n?2

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A

kByk1 6 c kykp kAyk 1

1?p 1

6 b kykp kyk1 10

?p

10

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20

,B| kBk1 = b , kBy k2 6 (By; By)1 6 b j(y; By)1j = b jb (y)j : 1 . , .. . 2,

y 2 W1

.
4]. (

3.

.

. 2000.

6
-

00-01-00111).

jb (X )j 6 C kX k1 kX k0 ;
2 kBy k1 6 b jb (y)j 6 C b kyk1 kyk0 ; p( p = 1=2 ) G.

1. 2. 3. 4.

BG N (r)

,

G

| ,

n

n2

,

5. 6. 7. 8.
,

r

,|

const N 2 = r
,

1 lim p N (r) < 1: r=1 r
.

9.

. ., . 5. . 1123. . . // . . ., . . 1999. 369, 4. . 1. . ., . . 1999. 39, 11. . 1869. .A. . .: , 1997. .. . .: , 1973. .. 1985. ., . . 1. .; .: , 1951. .. . :

. // ., .,

. 1982. . 1991.
27

264

, .

,

. . // . . //

1. . 140.

. . .: , . , 1986.
p 29.03.00

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4. (6)

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530.145

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l = 0; 1; : : :
( , 1]. ( ,

-

A V (r) = arK ? Z + r2 ; a > 0; K = 1; 2; : : : ; (1) r
, -

|

1=r2

) , -

1=r2

. , )( (1): 2{4].

2m = 1;

~

=1

)

.

d2 ? ar dr2

K

1) ? l(l +r2 + A + Z + E = 0; (2) r