A remark on the influence of high frequency electric fields and uniform magnetic fields on electronic conduction

Dingle, R . B . 1956

Physica XXII 701-702

LETTER

TO THE

EDITOR

A remark on the influence of high frequency electric fields and uniform magnetic fields on electronic conduction I t is a well k n o w n c o n s e q u e n c e of t h e B o l t z m a n n e q u a t i o n t h a t t h e influence of h i g h f r e q u e n c y electric fields c a n b e e x p r e s s e d b y t h e r e p l a c e m e n t of t h e r e l a x a t i o n t i m e v b y T/(1 + ioJv) if t h e electric fields are t a k e n as p r o p o r t i o n a l to e t''t. T h e s i m p l i c i t y of t h i s r e s u l t is d u e t o t h e i n t r o d u c t i o n of c o m p l e x e l e c t r i c fields. I t does n o t a p p e a r to b e g e n e r a l l y k n o w n *) t h a t it is also possible to e x p r e s s t h e i n f l u e n c e of a s t e a d y m a g n e t i c field w i t h like s i m p l i c i t y , e v e n i n d e e d w h e n h i g h - f r e q u e n c y electric fields are p r e s e n t . F o r c l a r i t y of e x p o s i t i o n , it will be a s s u m e d here t h a t c o n d i t i o n s are s p a t i a l l y u n i f o r m , t h a t t h e e l e c t r o n s are quasi-free, a n d t h a t a " u n i v e r s a l " r e l a x a t i o n t i m e exists. T h e n t h e B o l t z m a n n e q u a t i o n c a n b e w r i t t e n in t h e s y m b o l i c f o r m d/~t

+ ";'. ~l/~v

= --

(t -- to)It,

(1)

w h e r e t = d i s t r i b u t i o n f u n c t i o n , t0 = d i s t r i b u t i o n f u n c t i o n in t h e a b s e n c e of fields, v = e l e c t r o n velocity, a n d v = e l e c t r o n a c c e l e r a t i o n b e t w e e n collisions. S i n c e / t0 will v a r y w i t h t i m e in t h e s a m e w a y as as t h e electric field Eoce i°t, OtlOt=i e o ( t - to). F u r t h e r , m q = e (K + v ^ H/c) w h e r e e ~- e l e c t r o n i c c h a r g e (negative) a n d m = effective e l e c t r o n i c mass. T a k i n g / / t o be in t h e z - d i r e c t i o n a n d Ez = 0, (1) reduces t o

m/

" ~,

+

o%J

av~--

1

ov,j

w h e r e O = e H / m c = c i r c u l a r f r e q u e n c y of p r e c e s s i o n a l m o t i o n of a n e l e c t r o n . Neglecting t e r m s q u a d r a t i c in E , t in t h e first b r a c k e t of (2) m a y b~ r e p l a c e d b y [0, a n d use m a d e of t h e r e l a t i o n s O/o/Or x = (v./v) (O/o/Or) a n d 9/olgv v = (v~/v) (Oto/OV) w h i c h follow f r o m / 0 = / 0 (vn) w h e r e v n = v2x + v~v + v2z. T h e n (2) reduces to -my ~

t V x - ~-

- -

Vy

Fv~ ( t - - t 0 ) +

+i~

(l--t0)=0.

(3)

T h i s e q u a t i o n is easily s o l v e d b y s u b s t i t u t i n g f - - t 0 = vxtx + v~tv a n d s ~ p a r a t e l y e q u a t i n g t o zero t h e t o t a l coefficients of v x a n d vv. T h e n e - -

mY

(O/olOV)

G - - ~/,, + ( r , + i~)/~ -- 0,

(4)

(~tolOv)

E~ + ~ / , + ( r 1 + i~)/~ = 0.

(S)

e --

my

*) The idea is to some extent actually implicit in earlier work, e.g. S o n d h e i m e r, Phys. Rev. 80, 401, 1950.

- - 701 - -

702

UNIFORM MAGNETIC AND HIGH FREQUENCY ELECTRIC FIELDS

M u l t i p l y i n g (5) b y i a n d a d d i n g to (4), f =

where [ =/x+

J~

i/,and

8~tm3e f

]13

(1

-

-

-

(e~/mv) (~/0/Ov) ~/{1 + (i~ + i o) ~},

(6)

~ = E z + i E ~ . T h e c u r r e n t d e n s i t i e s are

-

10) v z v 2 dv --

h3

f

Ix vZz v d v

~ 3h

f

(7)

/ z v4 (Iv,

and J v --

3h a

/v v4 dv.

(8)

f v 4 dv,

(9)

M u l t i p l y i n g (8) b y i a n d a d d i n g t o (7), -- 3h------T -

f

where ~ = J x + i yu. F i n a l l y , s u b s t i t u t i o n f r o m (6) yields t h e v c r y simple r e s u l t t h a t 8~m2e 2

--

3h ~

f ~

rv3 dl°

( 1O)

1 + (i~o + i .(2) r

Since i is a m e a s u r e of p h a s e lag, a n d i of s p a t i a l r o t a t i o n a r o u n d t h e z-axis, all s u c h o p e r a t i o n s as t a k i n g t h e c o n j u g a t e c o m p l e x or r a t i o n a l i z i n g m u s t be p e r f o r m e d s e p a r a t e l y for p h a s e a n d r o t a t i o n . I n p a r t i c u l a r ,

l+(ico+i-Q) r =½

--½i

1 +co2+r'+

1 +w2_r z

1 + co~r ~

1T~o2__~ 2

1 + co+r

--.~ii

1 +(o2_rz

1 2 ~ g 2 - - r 2)

l+~__r2

,

(11)

w h e r e o~± = ~o ~ -Q. T h e first t e r m is a m e a s u r e of t h a t c o m p o n e n t of c u r r e n t w h i c h h a s t h e s a m e p h a s e a n d s p a t i a l d i r e c t i o n as t h e electric field, t h e second t h a t a l t e r e d in p h a s e b y 90 ° b u t n o t s p a t i a l l y r o t a t e d , t h e t h i r d t h a t u n a l t e r e d in p h a s e b u t s p a t i a l l y r o t a t e d b y 90 °, a n d t h e f o u r t h t h a t b o t h a l t e r e d in p h a s e b y 90 ° a n d s p a t i a l l y r o t a t e d b y 90 ° . I t is p a r t i c u l a r l y i n t e r e s t i n g to n o t e t h a t t h e s i m u l t a n e o u s influence of highf r e q u e n c y electric fields a n d a u n i f o r m m a g n e t i c field can t h u s be expressed in t e r m s of t y p e s of f u n c t i o n s of r i d e n t i c a l w i t h t h o s e iflvolved w h e n e i t h e r influence is alone operative. R. B. DINGLE. Received April 23, 1956. Department of Physics, University of Veestern Australia, Nedlands, Australia