PHONONS IN METALS

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PHONONS IN METALS

J. Ashkenazi, M. Dacorogna

To cite this version:

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JOURNAL DE PHYSIQUE

ColZoque C6, suppZe'ment au n o 12, Tome 42, de'cembre 1981 page C6-355

PHONONS I N METALS

J. ~shkenazi* and M. Dacorogna

D6partement de Physique de Za MatiBre Condense'e, Universite' de GenBve, 24 Quai Ernest-Ansermet, 121 1 GenBve 4 , SwitzerZad.

Abstract.- By using stationarity properties within the finite temperature density functional approach, it is shown that the calculation of phonons and electron-phonon coupling processes can be based on the coupling between renormalized "bare" phonons

and the H K S one-electron system, under the condition that the

potential remains "frozen". This explains the success of calculations of phonon anomalies and electron-phonon coupling parameters in transition metals which were based on rigidly moving poteltials. The approach is rigorously applied using the

LPITO-ASA band method. It is found successful in ab initio

calculations of shear moduli in transition metals, and promising for complete phonon and electron-phonon coupling calculations.

1. Introduction.- The success of fixst principles microscopic calculations of phonon spectra in metals, was limited in the past to free- electron like cases. Following the pseudopotential technique, one can start there from the jellium model assuming that the pseudopotential is weak, and expanding the energy change in powers of it. The obtained expression is separated into electronic and ionic parts /I/, and has been extensively applied for simple metals /2-4/.

This type of calculations does not succeed in transition metals and their compounds and alloys, which exhibit interesting properties (not found in general in simple metals) such as anomalies correlated with instabilities and high superconducting transition temperatures. In the last few years there have been many attempts to develop the microscopic theory of phonon-spectra and electron-phonon coupling in these metals /5-14/. Xowever, to our knowledge, there has yet Seen no

completely first-principles (parameterless) calculation of phonon

spectra in transition metals, for general

q

points. In addition, recent

*

P r e s e n t a d d r e s s : P h y s i c s D e p a r t e m e n t , T e c h n i o n , H a i f a , I s r a e l

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C6-356 JOURNAL DE PHYSIQUE

explanations of phonon anomalies looked contradictory: some were based on one-electron theory, and other one many-body screening effects.

In this work we develop further the ideas initiated in ref. 14 showing that many-body effects (including non-local exchange-correlation) could be represented, under certain conditions, as one-electron effects. So the problem of screening could be avoided, simplifying first principles calculations of phonon spectra and electron-phonon

coupling in transition metals for general 9 points.

2.

-

Stationarity Properties.- Let us generalize to finite temperatures

the ideas of ref. 14, using Elermin's generalization /15/ to the HKS

approach /16-17/. One introduces a free energy functional F I v , ~ ~ of

the variables: v(g) the external potential to the electrons system,

and n(r)

-

a "permissible" electronic density. We consider an external

potential which has a smooth dependence on a parameter B: v(B,g), and

denote by n(B,r) the corres?onding thermally averaged density. The free

energy F(B), corresponding to this potential, is equal to F{v(p),n(B)].

In addition, this functional satisfies a minimum property under

density variations, which introduces the stationarity property /14/ :

In order to calculate P I one introduces /17/ an effective system

of non-interacting electrons (the "one-electron system") with the same thermally averaged density n(r). In this system one uses the one-

electron Hamiltonian Hn = .h2v2/2m

+

vn (g)

,

where the effective one-

electron potential vn is given by /17/:

vc(g) is the Coulomb potential due to the electrons, and vxc(g) is the so called "exchange-correlation" potential. Hn satisfies the secular equation: IinIbk> = Eb(k)(bk>, yielding a spectrum of eigen-values and eigen-states, obtained by a band calculation. The one-electron free energy Fn can be expressed as a functiondaf vn, and is given by:

rn{vn} = nEF

+

TI

log [l-f (Eb

( k )

) 1 ( 3 )

(4)

where n is the number of electrons, Eq is the Fermi level, T is tempe-

rature, and f (E) is the Fermi function.

Let us consider a one-electron potential which has a smooth dependence on a parameter a: vn(a,g) and denote by n(a,g) the thermally averaged density related to it through:

n (l) =

1

f ( E b (k) _-1

1

$b (&,g)

!

2 ( 4 bk _-

where $ (k,r) is the wave-function corresponding to Ib&>. The free

b - -

energy functional of the real electrons system is given by /17/:

where Uc is the total Coulomb energy of the electron-nuclei system, Fxc is the so called "exchange-correlation" free energy, and Gn is given by:

On the basis of the same arguments that led to Eq. (l), and the properties of a Legendre transformation /14/ we get the following stationarity property (with respect to variations in vn):

~,{n(a) 1 = P ~ C V ~ ( C ~ ~ )

-

/vn(aOlr)n@g)d3r + 0 (a-ao) 2 (7)

BY combining Eqs'. (I)/, ( 5 )

,

(7)

,

and denotinq:

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C6-358 JOURNAL DE PHYSIQUE

This expression allows us to calculate F on the basis of approximate

-

n and vn, which could bespecially chosen to simplify the calculation. Let us consider for simplicity monoatomic crystals with closely packed

structures. They are composed of Wigner-Seitz (:JS) cells of volume Q

1/3 and "radius" s=(3Q/4n)

.

3. Application to Free Energy Differences.- Simple applications of

Eqs. (8),'for free energy differences between two crystals with the

same volume, are achieved /14/ by using the "frozen potential" and the "frozen density" conditions. Namely vn(g) and

n(r)

are taken in (8b,c)

the same for both crystals in the inner parts of the W S cells, and at

least on the average close to their boundaries. Considering for example the influence of an additional external potential V(r) (which changes v

(z)

in (8a) to v (r)

+

V ( 2 ) )

.

It turns out, by using these conditions in (8d), that the free energy change is given to the first order in V by:

This expression can be applied for the effect of electric fields on the electrons. Its importance is that exchange-correlation effects do not

appear explicite1y.B~ introducing spin and current, one can apply a

similar expression also to magnetic fields.

Another application is for the free energy change 6 F with tempe-

T

rature, assuming that the volume change is negligible, which is valid

at low temperatures (the thermal expansion coefficient is connected

with anharmonicity, and vanishes T

-

=OK). In this case v(2) remains

fixed, and for v (r) and n(g) we use the above conditions. By (ad)

n

-

we get to thefirst order in 6T:

-

6 F fv 1 is a one-electron term, while (YTFxc { n 1 is due to thermally

T n n

activated low lying collective excitations (their calculationsare under

the frozen potential and density conditionsrespectively).

Another application /14/ is for the free energy change 6 F under a Y

volume conserving lattice strain (characterized by a strain parameter

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C6-359

-.,

case also the condition that n ( g ) is a muffin-tin (PIT) density. Namely

it is spherically symmetric around the W S cell centres, and flat

"enough" on their boundaries. By n(a,r) we denote the spherically

averaged density n(a,r) around the WS cell centre, and introduce the

quantity Z(a) =On(a,s). The band-structure value is denoted by

Zo = Z(a ) . The corresponding density n(ao,r) has the first r-deriva-

tive n(lP(ao,s) close to zero by symmetry. In order to satisfy the MT density condition, we require that also the second r-derivative n(')(s)

will be zero, and denote the corresponding Z(a) value by Zeff.

The result of the three conditions (frozen potential and density, FIT

density) on (ad) is /14/:

where UM representsthe Madelung energy of unit charge point ions in a

uniform neutralizing background. 6yFxc has vanished here, including the leading non-local terms, which can be expressed through an ex-

pansion in density gradients /16/ at the IS cells boundaries, and

fall by the flatness of the MT density there. In the casethat y represents a symmetry removing strain (such as a frozen-in phonon) it

can be shown /14/ that for a suitably chosen Zeff, the error in Eq.(lU

is reduced to order y3, and thus it fLl)'is valid within the harmonic

approximation. This value of Z is given by /14/:

ef f

where the parameter o has been chosenas : a = n(') ( c t , s ) , and it represents the density deviation from the flatness condition.

The accuracy of Eq. (11) (to order y2) is then linked with the vali- dity of a linear approximation for non-flatness energy effects (such as non-local exchange-correlation). The density a-dezivative in Eq.

(12) is obtained by perturbation expansion of the wave-functions. As was shown in ref. 14, this restricts the permissible variations, obtaining a sharply determined value of Zeff for simple and transition metals.

In table I we represent the values of Zo and Zeff calculated (at

T=O) for the bcc and fcc metals of the 4th-6th rows, on the basis of

self-consistent LMTO-ASA band calculations /18/ (neglecting core

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JOURNAL DE PHYSIQUE

Table I: The calculated values of Zo,Zeif and t h e r e l a t i v e difference

between them in percents.

Table I1 : The calculated tetragonal shear nmduli of fcc transition metals CLalc = C' + C ' at T = OK in comparison with experiment ( C ' )

b M exp

Units c u R h ~d A g I r P t AU

Ryd/atm

CL

-0.25tO.OS t0.02fP.OS -0.38*0.04 -0.20tO.OS tO.lZfO.05 -0.5120.05 -0.SOfO.OS

=ti

t0.39t 0.04 to-68tO.07 +0.68t0.07 +0.41t 0.04 +0.85to.09 t0.88to.09 t0.6520.06 +0.14t0.09 t0.70t 0.12 +0.30'0.12 t0.21f 0.0s t0.97t 0 . 1 1 t0.37f 0.14 t0.15f 0.11

10" ~ / m ~

C;alc 2.6f1.6 ll.Ot2.0 4.4tl.8 2.721.1 14.7t2.1 5.322.0 1.9tl.b

2.56 11.5 2.9 1.71 17.2 5.22

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between Zo and Zeff. This change represents energy effects which are included in (14) indirectly, through the variation procedure, concerning among other, non-local exchange-correlation effects. In the free electron limit, Zeff coincides with Zo, and Eq. (11) with similar expressions applied using the pseudopotential technique /1,2/. From

table I it turns out that, within the transition metal series, the

fifth column is the closest to free-electrons concerning Zeff

-

Zo,

while the tenth column is the farthest.

Eq. (11) was applied /14/ to calculate the tetragonal shear moduli

C! = 4(Cll-C12) of the non-magnetic fcc transition metals (using the

LS'ITO-ASA band method) and the results are represented in table 11.

C'calc is given as a sum of two terms: C; and C$f, related to 6 Fn and

Y

6 U respectively. Each of these terms has an error bar estimated from

Y M

the accuracy of the calculation /14/, and the results agree with the experimental ones CAxp within the error bars. For Rhodium the theoreti- cal result preceded the experimental one /19/. For these metals the

difference between Zo and Zeff is large (see Table I), and the trans-

formation (12) is essential.

4. Electron-Phonon Hami1tonian.- The shear results correspond to low-q

limit phonons /8/. For finite g the strain parameter y is represented

by periodic displacement vectors 2 (q). Let us introduce the quantities:

a

-

where the derivatives of Hn are determined by the frozen potential condition. There are two approaches concerning the dCrivatives of lbk>/20/: the Bloch approach, under which these derivatives vanish; and the FrBh- lich approach under which /b& > form a complete set of "displaced" states, and the matrix elements of Hn are derivated directly.

At low temperatures one can take (by Eqs. ( 8 ) and (12)) temperature

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JOURNAL DE PHYSIQUE

(1) ( 2 )

+

E

IL'

gbIba (k.q)Qa (9) ab' (k+qIi ab

(2)

+

t

I

gba. a (k,g)

ab% a'

x Q,,(g) *Qa ( 4 ) ab (k_lf ab (6) !

+

1

< b f k

1 v

1

Bklab, (k) 'ab (&I (14)

bb'k

_-

From a perturbation expansion of Eq. (11) it turns out that this

HamiUOonian determines the phonon frequencies at low temperatures, and therefore also their line-widths. In addition, it turns out by Eqs.(4),

( 9 ) , (10) and (ll), that this Hamiltonian can be also used to describe low temperature physical processes where the (real) electrons are in- fluenced by external fields (characterized by V) and electron-phonon coupling (neglecting the temperature dependent contribution of other collective excitations such as spin and charge fluctuations).

So we could base the electron-phonon problem, within the harmonic

approximation, on the coupling (through 9'') and g(2) )between the D (0)

bare phonons, and the HKS one-electron system. The many-body effects are included indiretly through; (i) the renormalization of Zeff which represents, among other things, non-local exchange-correlation effects;

(ii) the use of the frozen potential condition under which potential and screening corrections are mutually cancelled.

5. Discussion.- Electron-phonon coupling calculations in transition

metals are currently done using /11/ the "rigid muffin-tin approximation" (RMTA), or the "modified tight-binding approximation" (MTBA). The frozen potential condition is approximately used in both methods. So their success, within a one-electron theory, is understood on the basis of the above discussion. The RNTA has succeeded /11/ in cal-

culations of electric resistivity, phonon line-widths, tunneling

spectral functions, and the mass-enhancement parameter X in transition

metals; the small deviations (-10-20%) of the calculated X from the

values obtained by different experiments /11/ could be explained as due to spin-fluctuations/21/. The W4TA however failed /11/ in the low-q

limit and for the anisotropy of X in Nb /22/ which has large low-q

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one calculates ratios between infinitesimal numbers, which are sensitive to inaccuracies introduced by the perturbative "shifting" of the wave- functions within the Bloch approach (used in the RMTA). This problem does not exist within the Fr6hlich approach where the shifting of the wavelunctions is treated rigorously, and indeed calculations using it in the MTBA, succeeded where the RMTA failed. Such calculations in Nb

were successful for the anisotropy in A /22,7,9/, and for shear ano-

malies (low-q limit) in Zr-Xb-Mo alloys. Such calculations were s u c

-

cessful inexplaining A values /9/ and phonon anomalies /lo/ in transi-

tion metals and their compounds. The same anomalies could be also explained as a screening effect /5,6/, and this strengthens our argu- mentation that such effects are included in the Hamiltonian (14).

The NTBA has its limitations being based on fitting. By approa- ching the problem with the LMTO-ASA band method, one can rigorously interpret the frozen-potential condition. The advantages of the MTBA are restored. Furthermore, the algorithms to calculate Zeff (see table I) D(O), g(l) and g(2) can be naturally built within the framework of the method /23/. So complete phonon calculations for general q-values, and also rigorous calculations of electron-phonon coupling processes have become handy.

ACKNOWLEDGEMENT.- The authors acknowledge sitmulating discussions with Professor Martin Peter, and his encouragement and suggestions.

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-

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