EFFECT OF STRONG LATTICE-DISORDER ON THE SUPERCONDUCTING TRANSITION TEMPERATURE AND ON FERROMAGNETISM

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EFFECT OF STRONG LATTICE-DISORDER ON THE SUPERCONDUCTING TRANSITION TEMPERATURE AND ON FERROMAGNETISM

K. Bennemann

To cite this version:

K. Bennemann. EFFECT OF STRONG LATTICE-DISORDER ON THE SUPERCONDUCTING

TRANSITION TEMPERATURE AND ON FERROMAGNETISM. Journal de Physique Colloques,

1974, 35 (C4), pp.C4-305-C4-307. �10.1051/jphyscol:1974458�. �jpa-00215649�

JOURNAL DE PHYSIQUE

Colloque C4, suppliment au no 5, Tome 35, Mai 1974, page C4-305

EFFECT OF STRONG LATTICE-DISORDER ON THE SUPERCONDUCTING TRANSITION TEMPERATURE AND ON FERROMAGNETISM (*)

K. H. BENNEMANN Institut fiir Theoretische Physik, Freie Universitat Berlin, Berlin 33, Germany

Rksumk.

Nous pressentons une theorie pour expliquer dans le cas des metaux simples et des transitions metalliques, comment varie la temperature de transition de superconductivite

avec les changements du spectre de frequence des phonons, de la densite electronique d'etat et des elements de la matrice electronique qui provient d'un reseau fortement desordonn6. Nous montrons aussi dans le cas du Nickel qu'un rkseau fortement desordonne rCduit la polarisation du spin du d-Clectron.

Abstract.

-

theory is presented to explain for simple metals and transition metals the dependence of the superconducting transition temperature Tc on the changes in the phonon frequency spectrum, electronic density of states and in the electronic matrix elements which result from strong lattice-disorder. Also, we show for Ni, that strong lattice-disorder reduces the d-electron spin polarization.

Recently, many experiments have shown that TABLE I1

strong lattice-disorder affects the properties of super- conductors [1, 21 and ferromagnets [3]. The experi- mental situation can be characterized as follows.

For simple metals like, for example, Al, Sn and G a one observes as shown in table I that strong lattice- disorder favors superconductivity and in particular enhances Tc [I, 41. For transition-metals, however, one observes as shown in table I1 that strong lattice-

Experimental (T,""') and theoretical results (T,""") of the szlperconducting tramition temperature of simple amorphous metals. (Note, for G a and Bi J. Carbotte calculated Tc

6 K and Tc

8 K, res- p ectiuely.)

Metal - A1 Bi Hg Jn Pb Sn T1 Zn Cd G a

T F P [ K ] T:" [ K ]

-

-

- ^{4 3.8}

- ^{6 8 (bcc)}

6.8 4.1 7.2

6.7 6.5

- 3.9 ^10.0

- ^8.0

- ^8.4 - ^11.0

Calculated and experimental TC-value~ of amorphous transition metals. The Griineisen constant yi is obtained from specific heat data and yi from the observedpressure dependence of Tc. Tc, refers to the crystalline metal.

We use AQ

AQ, where ASZ, is the atomic volume increase upon melting.

Metal T Z P [K]

- -

Ti .39

V 5.30

/3-La 6.10

Zr .50

Nb 9.23

Mo .92

Tc 8.22

Ru .50

H f .09

Ta 4.48

W .012

Re 1.70

0 s .67

Ir .14

T,C""

[K]

YG

YG Y G

YG TFP [K]

-

-

-

- - _-

^{3.5 6.0}

- 8.5 ^8.0

< 2.0

< 1.0 1.5-2.0

- ^3.5

- ^7.5

2.5-3.0

< 1.0 disorder may weaken as well as strengthen super- conductivity [2]. The most striking feature of the dependence of the superconducting transition tempe-

(*)

Supported

DFG under SFB

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974458

04-306 K. H. BENNEMANN

rature T, on the number of electrons per atom (eta)

observed for amorphous transition metals is that it does not follow the

Matthias rule

as in the crys- talline case, but rather exhibits a triangular peak around (e/a)

7 for the 4d- and 5d-transition metal series [2]. Photoemission-experiments and magneti- zation measurements have shown [3] that strong lattice-disorder reduces the magnetization in Ni, Co and Fe, for example. Results are given in table 111.

Experimental and calculated valzres for the magne- tization M and the Curie-temperature Tc of amorphous Ni, Co a n d ' ~ e . Mo and T,, rgfer to the crystalline state.

Metal (M/Mo)exp (MIMo)calc (TelTco)exp (Tc/Tco)calc

- - - -

Ni 0.6 0.5

-

Co 0.7 0.65 0.7

Fe 0.8 0.9 0.8

In the following we present a simple theory which attempts to explain the above mentioned experimental results obtained for superconductivity or ferromagne- tism in strongly disordered respectively amorphous metals as due to the increase in the atomic volume resulting from lattice-disorder. As a consequence of the increase AQ in the atomic volume there results a change in the (1) phonon density of states [4], (2) the electronic density of states [4] and (3) the d- electron matrix elements which are important for the electron-phonon coupling 151 as well as for the occur- rence of ferromagnetism in Ni, Co and Fe, for exam- ple [6]. The increase in the atomic volume by AQ causes in general a lattice softening and then a decrease A@, in the Debye temperature. It follows easily from the theory of superconductivity [4, 5, 81 that this lattice softening tends to enhance the superconducting transition temperature. Strong lattice-disorder tends to smear and uniformize the density of states. This leads in general to a decrease of N(0) the electronic density of states at the Fermi-energy

if in the crystalline metal N(0) > 1 [stateslev-at.] and to an increase of N(0) otherwise. Note, an increase in N(0) enhances Tc. This increase of Tc is in general larger for simp!e metals than for transition metals. As indicated by the Stoner criterium for ferromagnetism, UN(0) 2 1, a decrease of N(0) tends to weaken ferro- magnetism for a given intra-atomic Coulomb inter- action U. While the s-electron matrix elements rele- vant for superconductivity are relatively insensitive [4]

with respect to the atomic volume change AQ, the d-electron matrix elements important for supercon- ductivity and ferromagnetism, respectively, depend sensitively on the atomic volume. The increase in the atomic volume resulting from lattice-disorder causes as can be most easily seen by using the tight- binding approximation [6] or the renormalized-atom- theory [9] (a) a narrowing of the d-band width W

and (b) in general due to the decrease in the Coulomb interaction between the s- and d-electrons an increase of (ed - E ~ ) . These changes in the d-electron-band tend to enhance the superconducting transition temperature. However, these changes in the electron d-band tend to reduce the magnetization of Ni, since AQ causes an increase in &(TI), the bottom of the s-band, and a lowering of the d-electron energies

and thus a filling of the minority d-electron spin band by s-electrons [lo]. The situation is similar for Co and Fe [lo].

The effect of strong lattice-disorder on the super- conducting transition temperature Tc is determined by using [4, 5, 81

Here, < w2 > gives the squared phonon energies averaged over the phonon spectrum. The constant A may depend weakly on the crystal structure and is in the following approximated by 0.96 [4].

denotes the Coulomb pseudopotential which, in principle, can be determined by the isotope effect measurements or by using [5, 81

where W is the d-band width. The electron-phonon coupling constant A can be written as [5, 81

where M is the ionic mass of the unit cell and

the atomic volume. For transition metals < ^{J 2} > ^N(0)

can be written as [5, 81

Here, the function Fis found empirically to vary slowly with (e/a) and with N(O), if N(0) 2 1 [stateslev-at.] [5].

However, for N(0) < 1 [stateslev-at.] one finds from the experimental T, values F cc N(0) [5]. The parame- ter y is essentially determined by the matrixelements [5]

J 1 = < 1 ( V V 1 ( 2 > and J2 = < 11 VV2 1 1 >,

where I i > denotes a d-electron state in Wannier representation at the atomic site i and Vi is the ionic potential felt by the d-electron at the atomic site i.

Note, J, and J2 are related to the volume dependence of the center of gravity and width of the d-band, respectively. Clearly, both J , and J2 become smaller if the atomic volume increases. The changes in < w2 >,

l and

resulting from lattice-disorder are calculated by using

and

STRONG LATTICE-DISORDER ON THE SUPERCONDUCTING TRANSITION TEMPERATURE '24-307 Here, y, is the Griineisen constant and No(0) refers For the d-electron density of states Ifda(&) one finds to the crystalline metal. For simple metals we use [4] now

and for transition metals we use [5]

with [I I ] a In y/d ln Q

go

Here, a is the nearest neighbour distance in the crystalline metal and go the Slater coefficient of the d-electron wave function [5].

The results obtained [4, 121 are given in tables I and 11.

For Ni we determine the change in UN(0) and the magnetization. M - ^{(ndt -} n d J ) by using the Heine Hamiltonian [13]

with

cd6

zd + U n d - @ , nda

Determining the changes A&,,, Ay and A E ( ~ , ) with the help of the renormalized atom theory [9] we obtain for the magnetization the results [ l o ] shown in table I11 and for example for Ni the result N(0)/No(O) rn 0.6 for A Q / Q

0.1.

In summary, it follows that the change in the super- conducting transition temperature Tc due to lattice- disorder results essentially from AOD in the case of simple metals, but from A@, and Ay in the case of transition metals. Thus, for simple metals T, is enhanc- ed in amorphous metals. For transition metals, however, T, is enhanced or suppressed depending on the interplay of A@, and Ay and T,

Tc(e/a) reflects the dependence of

on e/a [5].

It follows that the magnetization in Ni, Co and Fe is reduced in the amorphous metal since AQ causes cd

zd - A E ~ , c ( r l )

& ( r l ) ^f A & ( r l ) and

W - t W - A W

Here, s,(k) is the unhybridized s-electron energy as a and thus a filling of the d-band holes by s-electrons.

function of wave number k and y is the hybridization

Acknowledgments.

We are grateful to G. Kerker, constant which is approximately given by

K. Aoi, J. W. Garland and K. Levin for useful dis-

Y

B(&d

G ) ) . cussions.

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