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ELECTRIC DOMAINS IN METALS AT LOW TEMPERATURES
A. Slutskin, A. Kadigrobov
To cite this version:
A. Slutskin, A. Kadigrobov. ELECTRIC DOMAINS IN METALS AT LOW TEMPERATURES.
Journal de Physique Colloques, 1978, 39 (C6), pp.C6-1115-C6-1116. �10.1051/jphyscol:19786494�.
�jpa-00217977�
JOURNAL DE PHYSIQUE
Colloque
C6,supplkment au
no8, Tome 39, aotit 1978, page
C6-1115ELECTRIC DOMAINS I N METALS AT LOW TEMPERATURES
A.A. S l u t s k i n and A.M. Kadigrobov
Physico-Technical Institute of Low Temperatures, UkrSSR Academy of Sciences, 47, Lenin Prospect, Kharkov, 310164, USSR
RBsum6.- On montre que l e s c a r a c t d r i s t i q u e s c o u r a n t - t e n s i o n d e s mBtaux, soumis b un champ magndtique a s s e z i n t e n s e pour provoquer une r u p t u r e c o h s r e n t e , p r g s e n t e n t d e s i n s t a b i l i t g s . On d t u d i e l e s domaines q u i s e ddveloppent e n conssquence d e c e t t e s i t u a t i o n e t on dgtermine l e s paramstres d e s s t r u c t u r e s s t a b l e s d e domaines. On p r d v o i t un nouveau t y p e d e domaines
"cycliques" p a r t i c u l i e r s aux mdtaux.
A b s t r a c t . - It i s shown t h a t c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s of m e t a l s under "coherent"
magnetic break-down a t low temperatures can be N-shaped. E l e c t r i c domain i n s t a b i l i t i e s de- v e l o p i n g i n t h i s s i t u a t i o n a r e s t u d i e d and s t a b l e domain s t r u c t u r e parameters found. A new t y p e of "cyclic" domains p e c u l i a r t o m e t a l s i s p r e d i c t e d .
The Gunn e f f e c t i s now w e l l known. I t o c c u r s i n semi-conductors w i t h N-shaped c u r r e n t - v o l t a g e c h a r a c t e r i s t i c / l / and i s due i n s t a b i l i t y of inhomogeneous d i s t r i b u t i o n s of t h e e l e c t r i c f i e l d g over t h e I-V c h a r a c t e r i s t i c segment w i t h
aj/ag<o
(j i s t h e c u r r e n t d e n s i t y ) a n d c o n s i s t s i n spontaneous f o r m a t i o n of e l e c t r i c domains moving a l o n g -t J. T h i s paper i s t o show t h a t i n m e t a l s some s i m i l a r phenomena can t a k e p l a c e a t low tempera- t u r e s and t o d i s c u s s t h e i r g e n e r a l f e a t u r e s .Our p r i n c i p a l a t t e n t i o n w i l l be paid t o t h e mechanisms r e s p o n s i b l e f o r t h e N-type I-V c h a r a c t e r i s t i c a s s o c i a t e d w i t h t h e coherent magne- t i c break-down (CMB) /2/. T h i s i s t h e term f o r t h e s i t u a t i o n where t h e
UHT*>
1 i n e q u a l i t y i s v a l i d , where wH i s t h e c h a r a c t e r i s t i c Larmor frequencyand r* t h e e l e c t r o n l i f e t i m e i n t h e s t a t i o n a r y s t a t e of t h e magnetic break-down spectrum. (Owing t o an e l e c t r o n i n t e r a c t i o n w i t h d i s l o c a t i o n d e f o r - mation f i e l d s and phonons, i n g e n e r a l may be l e s s
t h a n T t h e e l e c t r o n - i m p u r i t y r e l a x a t i o n time.) imp
'
We s h a l l c o n s i d e r a common c a s e where a l l t h e c l a s s i c a l e l e c t r o n o r b i t s i n magnetic f i e l d -+ H = (O,O,H~ a r e closed and t h e magnetic break-down combines them i n t h e $-space i n t o a c u r r e n t "magne- t i c break-dom layer" (MB l a y e r , whose t h i c k n e s s i s 6P) of p e r i o d i c magnetic break-down configura- t i o n s open along one f o t h e b a s i c v e c t o r s of t h e
+
tr e c i p r o c a l l a t t i c e , b = (O,b,O] ( c u r r e n t j= j ,O,O)).
The energy spectrum i n s i d e t h e MB l a y e r c o n s i s t s of magnetic bands En(PZ,Px) = En(PZ,Px+APx)
,
which a r e broadened Landau l e v e l s where n i s t h e numberof t h e magnetic band, P Z t h e momentum p r o j e c t i o n on t h e d i r e c t i o n of -f H , p e r i o d APx = 2reH H/cb 'L
( 1 0 - ~ - 1 0 - ~ ) b . For t h e magnetic break-down poobabi- l i t y U(H)*I, t h e c h a r a c t e r i s t i c magnetic band width and separation%(wH, and t h e average e l e c t r o n v e l o - c i t y along t h e X - a x i s , < v > = aEn/aPx, i n s i d e t h e
X
MB l a y e r i s of o r d e r of t h e Fermi v e l o c i t y
V (<V > = 0 ) . O u t s i d e t h e MB l a y e r , < v > = 0 , and
F Y X 9Y
hence t h e MB l a y e r c o n t r i b u t i o n t o j dominates a t
B
l (b/6P) ( l / $ r* rimp)<< I .a ) One of t h e CMB mechanisms of t h e non-linear n a t u r e of>(;) i s based on e l e c t r o n l o c a l i z a t i o n i n f i e l d
E
={$,0,03 i n s i d e quantum " t r a p s " of a width bx*%wHle& due t o t h e energy c o n s e r v a t i o n law E x + En(PZ,P ) = c o n s t . I f t h e parameterX L
'dyn = AX/!L* (!LY = vF, ) , then, f o r B < < ] , we have E w l / ~ . Thus, t h e N-type I-V i s t h e c a s e , i f jwyd yn
'dyn
2
1. The "dynamic" mechanism d e s c r i b e d can be r e a l i z e d e x p e r i m e n t a l l y , i f t h e a n g l e between -+ H and t h e p l a n e p e r p e n d i c u l a r t o b i s about . 1 0 - ~ - 1 0 ~ r adn.
b) I f Tdyn>> l
,
t h e n t h e n o n - l i n e a r i t y of j(8
may b e caused by e l e c t r o n d i f f u s i o n on e n e r g i e s En(PZ, Px) due t o f i e l d&.
We f o r m u l a t e t h e p r i n c i p a l i d e a of t h e " d i f f u s i o n " mechanism. assuming t h a t t h e e n e r g i e s r a n g e s , An =[E, ( m i d (mm?land
En
An+l do n o t o v e r l a p (where En (max) and En(min) a r e t h e a b s o l u t e maximum and minimum of En(PZ,P )
X i n s i d e t h e MB l a y e r ) . T h i s may b e t h e c a s e i n low 6P m e t a l s (6P;
m)
i fif
i s p a r a l l e l t o t h e symmetry a x i s ( e . g . , Be, Ru, A l ) . I n t h i s c a s e , t h e main c o n t r i b u t i o n t o j ( w i t h B<<]) i s due t oArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786494
e l e c t r o n s from An. Since t h e d i f f u s i o n flow a c r o s s t h e boundaries of An i s small i n
8 ,
t h e n i f Ydif T~~~ / r E < < l (idiff =
( b / 6 ~ ) 6 ~ 3 . 1 (e%?)' i s t h e c h a r a c t e r i s t i c time of t h e d i f f u s i o n change of E by an amount of 6En = E?)-rE
i s t h en n
c h a r a c t e r i s t i c time of t h e electron-phonon energy r e l a x a t i o n ) , t h e d i f f u s i o n d i s t r i b u t e s e l e c t r o n s i n s i d e A almost uniformly, and c u r r e n t j becomes
-8
Y:fff n LW%-'. The " d i f f u s i o n " mechanism causest h e N-type I-V c h a r a c t e r i s t i c s throughout t h e range Ydiff
'
Ycr I n c o n t r a s t t o semiconductors, i n m e t a l s 1 ,B 5 Bcr
Q, I 't h e spatially inhomogeneous d i s t r i b u t i o n s of
g
occu- r i n g a t jo(F2) < j < j o ( F l ) ( g l 2 a r e t h e extreme p o i n t s of t h e N-type I - V c h a r a c i e r i s t i c j(g);
El E2
) develop under t h e l o c a l e1ectri:al neu- t r a l i t y c o n d i t i o n s o t h a t t h e c u r r e n t d e n s i t y j = j ( t ) (t being t h e time) i s c o n s t a n t along t h e whole sample. I f X = ('E2-
<< 1 t h e e q u a t i o n f o rhas t h e form :
a l = j o t 6 ) / & l (1)
Here X i s a c o o r d i n a t e along -f J , c o n s t a n t s t >O and 9. a r e t h e c h a r a c t e r i s t i c time and s p a c i a l s c a l e s , r e s p e c t i v e l y , of t h e non-local f u n c t i o n a l r e l a t i n g j t o
&;
c o n s t a n t S h a s t h e s e n s e of t h e e l e c t r i c domain v e l o c i t y . E s t i m a t e s f o r t 0 ' L ,S a r e : ( a )Z 0
t h e "dynamic" mechanism : to *r
,
to* fir, S* VF 108 cm/s; (b) t h e " d i f f u s i o n " mechanism :
t T to%vF- (6P/b), ~ * L ~ / r ~ % 1 0 ~ c m / s . For t h e
0 E'
h e a t f l u x from t h e m e t a l q
;
i W/cm and t h e sample thic,cness d;
10-' cm, magnitudes of a r e I O - ~l r 2
-
1 0 - 5 ~ ~ ~ ~ .Equation (1) admits of a s t a b l e automodel s o l u t i o n of t h e s o l i t o n type E ( x , t ) :FsO1(~) ,lim
8
sol+
* ' m(c)
= 0 s i m i l a r t o t h e Gunn domain i n semicoiduc- t o r s i n i t s s t r u c t u r e and e x c i t a t i o n manner. I n m e t a l s where;
a r e l e s s t h a n i n semiconductors by a f a c t o r of ;06-
108 t i m e s , experiments may be arranged i n a new c o n t a c t - f r e e manner : i n an N-type I-V sample shaped a s c y l i n d r i c tube of an a r b i t r a r y c r o s s - s e c t i o n shape (with t h e l e n g t h Lo) t h e f i e l d i s induced w i t h a v a r i a b l e magnetic f i e l d f l u x $ ( t ) . The f l u x may b e produced by any means provided t h a t t h e a d i a b a t i c c o n d i t i o na r c l e n g t h along t h e contour) unambiguously d e t e r - mined by t h e e q u a t i o n (1) and t h e c o n d i t i o n
I n m e t a l s "temperature" e l e c t r i c domains (with S 102 cm/s) can a l s o e x i s t due t o t h e dependenae of t h e c o n d u c t i v i t y on t h e temperature and t h e sample J o u l e h e a t i n g .
Ref e r e n c e s
/ l / Bonch-Bruevich, V.L. Zvyagin, 1.P.and Mironov A.G.,Domennaya e l e k t r i c h e s k a y a n e u s t o y c h i v o s t v poluprovodnikakh, Moscow, Nauka Pu-blishers ( 1972).
/ 2 / S l u t s k i n , A.A., Zh. Eksp. Teor. Fiz.
58
(1970) 1098, S l u t s k i n , A.A., Zh. Eksp. Teor. F i z .65
(1973) 2114.
$ / $ < < s / ~ ~ , t ~ - ~ i s f u l f i n e & I f b0"d t h e r e i s a 2 ~ R o
f i n i t e r e g i o n i n t h e p l a n e ( r , (- )*)where any i n i t i a l d i s t r i b u t i o n of
g
d_evelo_ps i n t o a p e r i o d i c40
automodel s o l u t i o n ~ ( X , ~ ) ~ ( S ) = & S + L ~ ) (X i s an
