CAUSED BY PARAMAGNETIC IMPURITIES SOME NEW EFFECTS IN ZERO BIAS ANOMALIES 1 5

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SOME NEW EFFECTS IN ZERO BIAS ANOMALIES CAUSED BY PARAMAGNETIC IMPURITIES

A. Zawadowski and J. Sólyom

Lecture presented at 11th International Conference on Low Temperature Physics

St.Andrews, Scotland, 21-2d August 1968.

HUNGARIAN ACADEMY OF SCIENCES CENTRAL RESEARCH INSTITUTE FOR PHYSICS

BUDAPEST

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Printed in the Central Research Institute for Physics, Budapest Kiadja a Könyvtár- és Kiadói Osztály. O.v.: dr. Farkas Istvánná Szakmai lektor? Menyhárd Nóra Nyelvi lektor: Hargitai Csaba

Példányszám: 210 Munkaszám: 3951 Budapest, 1968. szeptember 19.

Készült a KFKI házi sokszorosítójában. F.v.: Gyenes Imre

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SOME NEW EFFECTS IN ZERO EIAS ANOMALIES CAUSED BY PARAMAGNETIC IMPURITIES

A. Zawadowaki and J.Sólyom

Institut Max von Laue - Paul Langevin, 8046 Garching, Germany, and Central Research Institute for Physics,

Budapest, Hungary

Recently tunneling diodes /М-М0-М/ containing paramagnetic impurities are investigated experimentally [1] . A possible explanation of some zero bias anomalies observed on these diodes is worked out by the authors [2] • This explanation considers the Kondo scattering of electrons on paramagnetic.impurities. This scattering can depress the electron spectrum around the impurities in the case of inhomogeneous impurity distribution. As the Kondo effect occurs in the energy region of Fermi level, this depression may be observed by tunneling experiments.

Two consequences of this theory are presented here.

1/ Coherence length

Let us consider, as an example, a barrier with infinite heights Í * < o ) an<i an electron gas on its right side (K > 0 ) . The effect of pa

ramagnetic impurities on the local energy spectrum of electrons at the barrier is determined. In tunneling those electrons play an important role whose wave vectors have small components in the plane of the barrier

k . % 0 . W e consider only wave vectors kx perpendicular to the barrier The surface of the barrier is taken to be unit. The Dyson equation for the one electron Green function has to be solved:

Cj."(•') j d K 1 CJ'Jix.x") c(x') - t ( o > ) ( £ ( i )*

where c ( x"j is the distribution function of impurities and ‘t(to)

denotes the scattering amplitude* due to impurities. The free electron Green function can be calculated easily, e.g. at o j s o ± i £

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2

0[ O> (*•*)- — 9il(k**>*"(lt**'>(+ri) + Á { F t*-*')- F (*+*')) (2)

<To±ie w f T V r c J

к

where k F ia the Permi wave number, VP * _£ the Permi velooity,

jC ( t t j a S(l\ (kF u) f f + Sí f

4

ke a j ) ( J j and й k c ia a cutoff in the exchange Hamiltonian, which restricts the interaction to the neighbourhood of the Permi energy, namely

(Ef - vj. Д«с )< £ < ( E F * V f A k c ) . It is supposed that д к с « к р . The function

F ( * • * ' ) can be neglected if

1*1, - \ c ( h )

where |c is the, coherence length, determined by the cutoff. In the opposite limit F ( x + x ' J — Jf which would lead to a cancellation in our final result. It will be supposed that all Impurities are found much nearer to the barrier than j c , The energy spectrum p i кйт0 / ь >) averaged over a small distance ( < f e ) at the barrier can be derived by solving l1) making use of ( 2) and the spectral representation, we get

P ( к * 0 } ы ) ~ О C k,»0> l»r> { ---- -— ---- :--- 7— 7--— j 1 5 )

>

U

У

о

[{+IT **(+•€)\c<)dx J**

107

where is the unrenormalized local density of electron states. The measure of the depression in the energy spectrum caused by the impurities is the following quantity

г « *> ) =

f (kH*o;u>)

f>b (kt * 0 ) 1 + гр0 I I m i t ő l

is )

kF w 2

in-*

where only the imaginary part of i is taken into account, f>0

is the density of states A s (■— J j e e x i d x is the thickness of the impurity layer in atomic distance.

If the scattering amplitude I n n i f v ) exhibits a maximum in the neighbourhood of the Permi energy, this causes a relative decrease of the density of states in this energy region. This change described by (6) can be observed investigating tunneling diodes containing paramag

netic1 impurities at the barrier.

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-

3

The distance of the impurities measured from barrier has to be smaller than |c . The change in the density spreads over a distance, comparable with [c . This effect is similar to the Priedel oscilla

tion which due to the lack of any cutoff Д kc ocour only over atomic distances. The existence of a cutoff follows from the derivation

£3J

of the Kondo Hamiltonian on the basis of the Anderson model. The energy VF & k c is related to the energy of localized "d" level, measured from the Perm! level. The coherence length may be estimated in atomic distances as Ep -w

Vr

дкс

л kc

2/ Dependence on the impurity concentration /selfconsistency/.

The amplitude of the observable zero bias anomaly can be estimated on the basis of (6) , considering the relative increase of the dynamical resistivity R/V/ at zero temperature. We obtain

M a x I

I ж {в})

» Í + 2Д Л/, М а к p 1 + — Nt-

iJ)

I 3

where the scattering amplitude is replaced by [ r p . f 1 which is derived by Hamann

£4J

, and exhibits the maximum at zero energy /zero bias/.

In the case of many impurities the scattering amplitude is modified because the electron spectrum is depressed at the impurity

site in the energy region under consideration.

There are two effects if the average electron spectrum entering into the Kondo problem is changed by a factor s <Ш> < 1

re

a/ the energy width of the anomaly becomes narrower by increasing the impurity concentration because of the effective Kondo tempereture is essentially decreased, namely

k T K « > ) = f c f'r

b/ strong /nonlinear/ dependence of the amplitude on the impurity concentration, namely 11*, j-» -J:— — . and replacing this value

into [ I ) we get

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-

4

^-

г Lé>Í.

Extending experimental investigations of zero bias anomalies in diodes containing magnetic impurities would give information about the cutoff Д k e and the energy dependence of the scattering amplitudes in a large energy region. These informations are not available from other experiments at the present time. The change in the electron spectrum might cause observable effect in NMR experiments if ÍLÍL* «■ 1

kr

We are grateful to Dr. E. MUller-Hartmann for a discussion of the coherence length.

References

+ Address of J.S. and permanent address of A.Z.

x It is called self-energy in Ref. [ 2 ] .

[l] A.F.Q. Wyatt and D.J. Lythall, Physics Letters 25A. 541 /1967/, Phys. Rev.Lett. 20, 1361 /1968/

P. Mezei, Phys. Lett. 2 ^ A , 534 /1967/

L.Y.L. Shen, Bull, of Amer. Phys. Soc. 1Д_, 476 /1968/

[2J A. Zawadowski, Proceedings of 10th International Oonference on Low Temperature Physics, Moscow, Vol. IV p.336.

J. Sólyom and A. Zawadowski, Physics of Condensed Matter I t 325 /1968/ and 2» 342 /1968/.

t3f J. R.Schrieffer and P.A. Wolf, Phys. Rev. 1 4 9 . 491 /1966/

[4] D.R. Hamann, Phys. Rev. 1 ^ 8 , 570 /1967/

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