HAL Id: jpa-00218095
https://hal.archives-ouvertes.fr/jpa-00218095
Submitted on 1 Jan 1978
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
POINT-CONTACT SPECTROSCOPY OF PHONONS
IN METALS
I. Yanson, I. Kulik
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C6, supplkment au
no
8, Tome 39, aofit 1978, page
C6-
1564POINT-CONTACT SPECTROSCOPY O F PHONONS I N METALS
I.K. Yanson and 1.0. Kulik
Physico-Technical Institute of L a , Temperatures of the Ukr SSR Academy of Sciences, Lenin's av.
47,
Kharkov 310164, U.S.S.R.
Rdsum6.- On prdsente l'dtude expdrimentale et thdorique d'un type nouveau d'une spectroscopie depho- nons dans des mdtaux normaux, appelds "spectroscopie par contacts par points
".
On montre que la deu- xieme ddrivEe du courant par rapport 3 la tension, en contact ponctuel, est proportionnelleB
la "fonction de transport de l'interaction electron-phonon" G(w) = ti2(w) F(w), ofi F(w) est la densit.6 des dtats de phonons, ti2(w) est la valeur moyenne du carrd de l'dldment de matrice de l'interaction electron-phonon moyenn&sur la surface de Fermi du mdtal. On prdsente les rdsultats de l'dtude des effets de l'anisotropie et du rBle des processus de collision Elastique sur des spectres par contact par points d'un certain nombre de mdtaux, dont Pb, Sn, In, Al, Cu, Ag, Au, Zn, Cd et aussi Na. ti2(w) est pratiquement constante pour les mdtaux polyvalents, tandis que dans le cas des mdtaux nobles cette grandeur diminue considdrablement avec l'augmentation de l'dnergie. Pour Na, ti2(w) augmente avec l'augmentation de w.Abstract.- Both experiment and theory of new type of spectroscopy of phonons in normal metals, called "point-contact spectroscopy", is presented. The second derivative of current over voltage for the point-contact junction is shown to be proportional to the "transport function of electron-phonon in- teraction" G(w) = ti2(w) F(w), where F(w) is the density of phonon states and ti2(o) the squared matrix element of electron-phonon interaction averaged over the Fermi surface of the metal. The data revea- ling the anisotropy effects, and the role of the elastic scattering processes, are presented anddis- cussed for the number of metals, including Pb, Sn, In, Al, Cu, Ag, Au, Zn, Cd, and partly Na. For the polyvalent metals, the ti2(w) is nearly constant quantity, whereas in the noble metals, it is a strongly decreasing function of the energy. For Na, ti2(w) increases as o increases.
In the experiments described in /I/, for the first time the non-Ohmic I-V dependences for shorted tunnel junctions of normal metals were revealed, and close resemblance of d21/dv2 vs V dependences with the phonon density of states, F(w) at w = eV had been established. The observed I-V-characteristics were related to the nonequilibrium phonon generation by "hot" electrons transiting through a small (d %
50
i)
orifice connecting two metals (figure 1) /2,3/. The constriction resistance within the Knudsen limit d << li (li is the impurity scattering length, d the orifice diameter) is given by the formula
which by the order of magnitude coincides with the Sharvin formula / 4 / Ro % pl/d2. Here, S = nd2/4 a d
S is the Fermi surface area. This expression results F
from integrating over the trajectories of electrons, which represent the exact solution of the collision- less B o l t ' m a ~ equation. The trajectories fall into transit type ( I ) , and screen-reflected ones (2) (see figure 1). The distribution of electostatic po- tential in the vicinity of the orifice is
v
= (1
-
n(;)/~n). (2)Fig. 1 : The model of the point contact : a circu- lar orifice of the diameter d = 2a in the nontrans- parent screen C separating two'metals.
(I)
-
tran- sit trajectories of electrons, (2)-
trajectories of electrons reflected at the screen. R(?) is the solid angle at which the orifice is seen from the point $where
n(:)
is the solid angle at which the orifice-P
is seen at the point r. Taking into account the nonelastic collisions of electrons transiting through the orifice, with phonons one gets at T = 0
8
where 0 =
-
a3 is the effective volume of gene- eff 3ration of phonons which are responsible for the ob-
s e r v e d n o n l i n e a r b e h a v i o r . G(w) t h e " t r a n s p o r t " F(w) shows t h a t , f o r p o l y v a l e n t m e t a l s (Pb, Sn, I n , s p e c t r a l f u n c t i o n of t h e p o i n t c o n t a c t : A l ) t h e e n e r g y dependence of t h e mean squared EPI
m a t r i x e l e m e n t i s n o t pronounced. F o r t h e n o b l e me- t a l s (Cu, Ag, Au), t h e La-phonons maxima a r e of es-
N(O) P-P
'
G(w) =
-
2 s e n t i a l l y lower i n t e n s i t y a s compared t o t h e TA-pho- n o n s , f i g u r e 2. T h i s means t h a t g2(w) i s a d e c r e a -
v
I s i n g f u n c t i o n o f e n e r g y . F o r t h e a l k a l i m e t a l s (Na)
,
(4) somewhat u n e x p e c t e d l y , t h e o p p o s i t e s i t u a t i o n o c c u r s :
+ -+ t h e TA-maxima a r e s u p p r e s s e d a s compared t o LA-ones. Here, K ( v , v l ) i s t h e s t r u c t u r a l f a c t o r
making a l l o w a n c e f o r t h e c o n s t r i c t i o n geometry. G(w) = c2(w) F(w) d i f f e r s from t h e known f u n d a m e n t a l e l e c t r o n - p h o n o n i n t e r a c t i o n (EPI) f u n c t i o n
g(w) = a2(w) F(w) i n t h e K - f a c t o r , t a k i n g i n t o ac- c o u n t a momentum change i n t h e e l e c t r o n - p h o n o n scat- t e r i n g p r o c e s s . It should b e n o t e d t h a t f i n d i n g t h e G(w) f u n c t i o n from t h e p o i n t - c o n t a c t s p e c t r o s c o p y ( 4 ) , i . e . by m e a s u r i n g d21/dv2 v s V dependence, i s a l s o p o s s i b l e i n t h e Maxwell l i m i t 1 << d . I n t h e + -+ i l a t t e r c a s e , t h e f o r m u l a f o r K ( v , v l ) i s d i f f e r e n t from ( 5 ) , and t h e e f f e c t i v e volume
neff
i n (4) i s a 2 1 r a t h e r t h a n a3. The i n t e n s i t y o f "point-contacti spectrum" i s weaker i n t h e d i r t y l i m i t t h a n i n t h e c l e a n one. It i s n e c e s s a r y t h a t u n e q u a l i t y d < A s h o u l d h o l d , where A = ( 1 . 1 ) i s t h e d i f f u s i o n 1 Ph l e n g t h o f e n e r g y r e l a x a t i o n .
The e x p e r i n e n t s were c a r r i e d o u t on t h e num- b e r of m e t a l s , i n c l u d i n g Pb, Sn, I n , A l , Cu, Ag, Au, Zn, Cd, Na. The p o i n t c o n t a c t s were formed by punc- t u r i n g a t h i n i n s u l a t i n g l a y e r s e p a r a t i n g two poly- c r y s t a l l i n e m e t a l l i c f i l m s /1,5/, o r b y "needle-an- v i l " t y p e p r e s s u r e j u n c t i o n / 6 / made o f A l , Cu, Zn s i n g l e c r y s t a l s o f v a r i o u s o r i e n t a t i o n s . The depen- d e n c e of second harmonic o f s m a l l m o d u l a t i n g v o l t a g e , V 2 , on t h e c o n t a c t v o l t a g e , V , was measured. I n t h e c a s e of p u r e j u n c t i o n s (d << 1 . ) t h e p o s i t i o n s o f t h e maxima i n t h e p o i n t - c o n t a c t s p e c t r u m V (eV) Q 2
g 2 ( e v ) F(eV) were i n good agreement w i t h t h e maxima i n phonon d e n s i t y of s t a t e s , known from t h e n e u t r o n s c a t t e r i n g e x p e r i m e n t s . I n c a s e o f d i r t y j u n c t i o n s
(d >> li)., t h e s p e c t r u m V (eV) had s a n e t i m e s s t e p s 2
( r a t h e r than.maxima) superimposed o v e r t h e monotonic
r
background. The l a t t e r i s b e l i e v e d t o b e d u e t o t h e n o n e q u i l i b r i u m d i s t r i b u t i o n of phonons ( n o t accoun- t e d f o r i n ( 4 ) ) .
The ccmparison between t h e e x p e r i m e n t a l l y measured f u n c t i o n G(w) and phonon d e n s i t y o f s t a t e s
F i g . 2 : Comparison o f phonon s p e c t r u m F(w),
( a ) w i t h t h e p o i n t - c o n t a c t s p e c t r a c o r r e s p o n d i n g t o (b) p o l y c r y s t a l l i n e f i l m s , ( c ) and ( d )
-
Zn s i n g l e c r y s t a l s o r i e n t e d a l o n g t h e (0001) and (1 120) d i r e c - t i o n s , r e s p e c t i v e l y . T = 1.5 KJOURNAL DE PHYSIQUE
References
/I/ Yanson, I.K., Zh.
Exp.Th. Phys. 66 (1974) 1035
;Proceedings of LT-14,
3
(1975) 5 0 r
/2/
Kulik, I.O., Shekhter, R.I., Omelyanchouk, A.N.,
Sol. State Comm.
23
(1977) 301
/3/
Kulik,
I.O., Omelyanchouk, A.N., Shekhter,R.I.,
F i z .
Nizk. Temp.
(LowTemp. Phys. Kharkov)
3
(1977) 1543
/4/
