Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://lizard.phys.msu.su/home/science/Masyukov-Dmitriev-09-FPM-InN.pdf Äàòà èçìåíåíèÿ: Wed Feb 9 10:20:12 2011 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:05:09 2012 Êîäèðîâêà:

. . , . .
. . . e-mail: dmitriev@lt.phys.msu.su 538.935+517.958+537.84 : , , , , . . . , . Abstract N. A. Masyukov, A. V. Dmitriev, A new numerical method for the solution of the Boltzmann equation in the semiconductor nonlinear electron transport problem, Fundamentalnaya i prikladnaya matematika, vol. 15 (2009), no. 6, pp. 77--97. A new iterative method for solving the Boltzmann transport equation in the space-uniform case is introduced. The method is based on the use of a mesh in the momentum space to represent the distribution function. In intermediate points the function is found with the help of interpolation. The method is used to study the hot-electron transport in bulk n-In N, which is a promising material for optoelectronic applications.

1.
, [9--11], [1, 2, 4, 7, 8]. , , , , .
, 2009, 15, 6, . 77--97. c 2009 , « »


78

. . , . .

, -, , , [1, 2, 7, 8]. , , . , [3], . , . , , , , . . . , , , . InN GaN AlN. - . , , , , [16]. . . , , E eE f (k) f (k) =- + St(f ), t k (1)




79

k -- . f f (k) = NV ,
k

(2)

V -- , N -- . St(f ) =
k

{W (k k)f (k )[1 - f (k)] - W (k k )f (k)[1 - f (k )]},

(3)

W (k k ) -- k k . - , . - , . - [17], . , . [13, 15], . , , . . , . . 1 - f . InN , , . InN . , [12, 14, 19--22, 24] -, [23] -- . (. [12, 14, 20--24]) N = 1017 -3 , . : n-InN 1018 -3 [6]. , [26] N = 9 · 1018 -3 T = 77 K.


80

. . , . .

, , . n- . , . [26].

2.
InN n- N = 9 · 1018 -3 , , T = 77 K. , Ni , . . . , ( ) - .

2.1.
, . , [25] (. 1). . InN . k = (1 + ), 2m
22

(4)

. 2 1 m , (5) = 1- g m0 m0 -- . A . : = k , 2mA
22

(6)

k A, -- A-.




81

1. InN

- A- - A- LO- TO-

m mA g -A LO TO 0 sLA sTA DA DO e14

m0 m0

0,12 1,0 2,0 2,2 73 57 8,4 15,3 6,24 2,55 7,1 109 0,375 6,81

105 /c 105 /c / /2 /3

2.2.
[5]. , . . Wi (k k ) = 2 N V
i

4e2 0

2

(k, k )S (|k - k |) (k) - (k ) . |k - k |4

(7)

(k, k ) . ( , - ) (k, k ) =
cell

d3 ru (r)uk (r) , k

2


82

. . , . .

u (r) u (r) -- k k , . - (4) [1 + (k )][1 + (k )] + (k, k ) = (k )(k )
kk kk 2

[1 + 2(k )][1 + 2(k )]

,

(8)

- . S (|k - k |) : S (q ) = 2 q 2 1+ 2 q
2 2

,

-- , 1 = 4e2 0 f (k) . (k) (9)

k

, (9) -- . ( ) W± (k k ) = â
j

2 1 (k, k )S (|k - k |) â V 11 Bj (|k - k |) Nj (k - k )+ ± (k) - (k ) j (k - k ) 22

,

«+» , «-»-- , j , . q:
LA

(q) = sLA q,



TA

(q) = sTA q,

, , , : LO (q) = LO , TO (q) = TO . j - Nj (q) T .




83

B (q ) : BDA (q ) = BPA BDO BPO
2 DA q, 2s 2 e2 DPZ , (q ) = 2sq 2 DO , (q ) = 2 2e2 (q ) = , q2 ¯

DPZ = 4e14 /0 -- , 1 1 = ¯


-

1 . 0

, DA- PO- , PA- DO- -- , . . (AP) W
AP

(k k ) =

2 1 (k, k )S (|k - k |) (k) - (k ) â V Bj (|k - k |)[2Nj (k - k )+1], â
j

. - (i), (DA) (PA) , (PO) . A- DO-. DO- DA- q . T = 77 K DA- , DO-, . , S (q ) 1 q , . , , . , . .


84

. . , . .

3.
, : fj (k) eE fj (k) =- + St(fj )+ t k
l

StIV (fj ,fi ),
i=j

j = 1,...,l,

(10)

l -- . . :
l

fi (k) = NV .
i=0 k

(11)

- A-, l = 2.

3.1.
, , , (3) , . , , , f (k) = f (k ,k ). ( ) , St = Sti + StOP + StAP + StIV (1), (12)

, . k cos + k sin cos , k q 2 (k ,k ,k ,,) = k 2 + k 2 - 2kk (k ,k ,,), 1 D(k ) = 2 dk k 2 (k ) - (k ) . (k ,k ,,) = , .




85

k 2 (k ,k ), k 2 = k 2 + k . k (k ,,), k -- , -- k z , -- k k , z . (k ,k ,,) k k , q (k ,k ,k ,,) -- k k k-, D(k ) -- . (4) D(k ) = m k 2 2 1+ 2 2 k 2 . m

(8):
2

[1 + (k )][1 + (k )] + (k ,k ,k ,,) =

(k )(k )(k ,k ,,) .

[1 + 2(k )][1 + 2(k )]

Sti = 4e2 0
2

ND(k ) 2

1

d cos [f (k cos, k sin ) - f (k ,k )] â
-1 2

â
0

d

(k ,k ,k ,,)S q (k ,k ,k ,,) . q 4 (k ,k ,k ,,)

Sti , 0 , . , , , :
2

d
0

2k 2 - 2k k cos + -2 S q (k ,k ,k ,,) = 2 . 4 (k ,k ,k , ,) 2 - 2k k cos + -2 )2 - (2k k sin )2 ]3/2 q [(2k

, , , . StOP = 1 2 StOP,j + StOP,j ,
j + -

,


86
1 +

. . , . .

StOP,j = D(kj,+ )
-1

d cos {f (kj,+ cos , kj,+ sin )[1 - f (k ,k )](Nj +1) -

- f (k ,k )[1 - f (kj,+ cos , kj,+ sin )]Nj }â
2

â
0 - StOP,j

d (k ,k ,kj,+ ,,)S q (k ,k ,kj,+ ,,) Bj q (k ,k ,kj,+ ,,) ,
1

= D(kj,- )
-1

d cos {f (kj,- cos , kj,- sin )[1 - f (k ,k )]Nj -

- f (k ,k )[1 - f (kj,- cos , kj,- sin )](Nj +1)}â
2

â
0

d (k ,k ,kj,- ,,)S q (k ,k ,kj,- ,,) B q (k ,k ,kj,- ,,) ,

kj,

±

(kj,± ) = (k ) ± j ,
-

Nj . , (k ) - j < 0 StOP,j . 0 , . , , , , , . Stqel
AP 2

D(k ) = 2

1

d cos [f (k cos , k sin ) - f (k ,k )] â
-1

â
0

d (k ,k ,k ,,)S q (k ,k ,k ,,) â Bj q (k ,k ,k ,,) 2Nj q (k ,k ,k ,,) +1 ,
j

â

, , . , , , , , . ,




87

. , , . , q (k ,k ,k ,,), , Stqel
AP

=

1 2

Stqel
j

+

AP,j

+ Stqel

-

AP,j

,

, ,
1 + Stqel AP,j 2

= D(kj,+ )
-1

d cos
0

d â

â S q (k ,k ,k ,,) Bj q (k ,k ,k ,,) (k ,k ,kj,+ ,,) â â f (kj,+ cos , kj,+ sin )[1 - f (k ,k )] Nj q (k ,k ,k ,,) +1 - - f (k ,k )[1 - f (kj,+ cos , kj,+ sin )]Nj q (k ,k ,k ,,)
1 - Stqel AP,j 2

,

= D(kj,- )
-1

d cos
0

d â

â S q (k ,k ,k ,,) Bj q (k ,k ,k ,,) (k ,k ,kj,- ,,) â â f (kj,- cos , kj,- sin )[1 - f (k ,k )]Nj q (k ,k ,k ,,) - - f (k ,k )[1 - f (kj,- cos , kj,- sin )] Nj q (k ,k ,k ,,) +1 . kj,± (k ,k ,,) j - -1 kj,± (k ,k ,,) = k ± sj q (k ,k ,k ,,), k k-, . , , , , . , . , StIV (1) = 1 2 2 BDO,j Stj (1) + Stj (1) ,
j + -

DO-,


88
1 +

. . , . .

Stj (1) = D(kj,

+ ,2

)
-1

d cos {f (kj,
,1 1

+ ,2

cos , kj,
+ ,2

+ ,2

sin )[1 - f (k
+ ,2

,1

,k,1 )] â

â (Nj +1) - f (k Stj (1) = D(kj, - f (k kj,
±, 2 ,1 - -, 2

,k,1 )[1 - f (kj,
-, 2

cos , kj,
-, 2

sin )]Nj },
,1

)
-1

d cos {f (kj,
-, 2

cos , kj,
-, 2

sin )[1 - f (k

,k,1 )]Nj -

,k,1 )[1 - f (kj,

cos , kj,

sin )][Nj +1]},

j (kj,
±, 2

) = 1 (k1 ) - 12 ± j ,

Nj . , j - 1 (k1 ) - 12 ± j < 0, . , , , 1 2. kj,±,1 : (kj,
±, 1

) = 2 (k2 )+12 ± j .

3.2.
l = 1. k i = -kmax + i +
j k = j 0 ,

1 2

0 , i = 0, 1,...,

2kmax - 1, 0 kmax j = 0, 1,... , . 0

kmax . , , . , kmax . , «» kmax , , . , , . - [18]. , -




89

. , (10). (10). 0 t = 0 fij = j = f 0 (k i ,k ). n- (n +1)- . (k ,k )-. f n (k ,k ) (9) (n) = (f n ) . Stij 3.1. , , . (10),
n fij+1 = f n

k-

eE

t, k

+t Stij ,

(11), t -- . ( , , . .). . , l . l l . , , (. 3.1). . , , . F (k, T , µ0 ) , , , .

4.
4.1. 15 /
[26] 15 / T = 77 .


90

. . , . .

15 / , [26]. , kmax = = 1,2 â 107 -1 . (. 1).

. 1. , E = 15 /, T = 77 K. 15 â 30 t = 10-15 . tsim = 0,5 â 10-12 . , , k

(. 2). E = 15 / t . t = 10-15 15 â 30. . 2 , . , , . , . , , F
n/eq

(k ) = F (k, T , µ0 ) + cos(10

-6

k ),

, (. 2, n/eq).




91

. 2. . vd ( ) . E = 15 /, T = 77

. 2 , , (q/el) ( , 3.1). , . . 2 Duo . 2 , tCPU (Intel Core T72500, 2 GHz, ) tsim = 5 · 10-13 .

. 3 vd , [26]. ( 15 â 30, t = 10-15 ), (. 2). , .


92

. . , . .

. 3. vd [26]. vd 15 â 30, tsim = 0,5 â 10-12 , t = 10-15

4.2. 15 /
. , N = 9 · 1018 -3 , T = 77 K. . kmax , 15 / , , , . kmax . , . , . 15 / . , - , , , . -




93

, , . , , , . , , , . , - , . 2 , .
2. , . tsim -- ,

E , / kmax , -1 (kmax ), kmaxA , -1 A (kmaxA )+-A , - A- t, 10-15 tsim , 10-12

15 1,2 0,4

30 50 1,7 2,5 0,72 1,32 -- -- 15 â 30 -- 1,0 0,5

> 50 4,4 2,89 4,0 2,80 20 â 40 10 â 20 0,5 8,0

. 4 , k f (k ) = 2 dk k f (k ,k )

. f (k ) , f (k ,k ), . , , , . . 5 70 /. , . . , , . , -, ,


94

. . , . .

. 4. f (k ) . E = 15 /, -- 45 /, -- 70 /. , . 2

, A-, , .


InN N = 9 â 1018 -3 T = 77 K. . , , . , , , . , . -




95

. 5. : . : . N = 9 â 1018 -3 , T = 77 K. . 2

, , , . , . « - 2009--2013 » ( -2312).


[1] . . . -- .: , 1987. [2] ., . . -- .: , 1979. [3] . ., . . . -- .: , 1975.


96

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