Phonon dispersion in hcp metals

Physica 90B (1977) 223-236 © North-Holland Publishing Company

PHONON DISPERSION IN hcp METALS

J. SINGH and R. SINGH Department of Physics, Panjab University, Chandigarh-160014, India

and

S. PRAKASH Laboratoire de Chemie Physique, Betiment 350, Avenue Jean Perrin, 91405 Orsay, France

Received 2 December 1976

The phonon dispersion curves for hcp metals Zn, Zr, T1, Tb and Ho are calculated in the pseudopotential approach. Animalu transition metal model potential for the bare-ion potential is used in conjunction with the Hartree dielectric function. The exchange and correlation corrections due to Singwi et al. are applied to the dielectric screening. The calculated phonon frequencies are all found to be real and in reasonably good agreement with the neutron inelastic scattering measurements which establishes the validity of the Animalu transition metal model potential for the hcp structure.

1. Introduction

The major problem in the calculation of phonon dispersion in metals is to provide an appropriate description for the role played by conduction electrons. The phonon dispersion relations of simple hcp metals Be, Mg and Zn have been calculated by many authors [1-7] . Prakash and Joshi [8] calculated the phonon spectra of Be, Mg and Zn from first principles. Zn is inbetween the simple metals and group B transition metals. Therefore it can be treated both in the simple metal limit and in the transition metal limit. It is interesting to see the effect of d bands in Zn when it is considered in the transition metal limit. Animalu [9] has given a transition metal model potential (TMMP) which he has applied for a simple class of fcc and bcc transition metals. It has been found that the calculated phonon frequencies agree reasonably well with the results obtained from the inelastic neutron scattering experiments. Animalu has not applied the TMMP to calculate the lattice dynamical properties of the hop structure, which is of lower symmetry than the fcc and bcc structures. Therefore, before performing the complicated numerical calculations it is worthwhile to see how the Animalu TMMP works in the case of 3d, 4d and 5d series of the hcp metals, in general. Once the potential is known for a metal one can study a variety of thermodynamical properties particularly the specific heat of a metal, electrical resistivity of solid and liquid metal [10], the transition temperature for superconductivity [ 11 ] and many aspects of the electronic structure of metals.

The phonon dispersi6n curves of a number ofhcp metals Zn [12], Sc [13], Y [14], Zr [15], T1 [16], Tb [17] and Ho [18] have been measured by the neutron inelastic scattering experiments. In one of our papers [19] we reported the calculations of the phonon dispersion curves for Sc and Y. Because of the highly anisotropic nature of Zn little attention has been paid to it. The lattice dynamics of T1 is also of much interest because it is super- conducting below 1.37 K. Recently there has been much interest in the study of lattice dynamics of heavy rare earth metals. Therefore, in this paper the Animalu TMMP has been applied to calculate the phonon dispersion relations of Zn, Zr, T1, Tb and Ho in order to see how the TMMP works for hcp transition and rare earth metals. The theory is given in section 2 and the results are presented and discussed in section 3.

223

224 J. Singh et al./Phonon dispersion in hcp metals

2. Theory

The hcp structure, with space group D4h, consists of two interpenetrating hexagonal lattices. Ifa is the distance between nearest neighbours in any plane and c is the distance between alternate planes, the hcp lattice may be described as a Bravais lattice with a basis where the unit cell has the edges

= 1 .* 1%/c~af , a 3 eft. (1) a l=a[, a 2 --sat+ =

Here [, ] and k are the unit vectors along the cartesian coordinates x,y, z, respectively. The unit cell contains two 1 2 1 atoms, one at the origin and other at 5a 1 + 5a 2 + ia 3. The volume of the unit cell is given as

=1 2 a ~ V % c . (2)

Atomic volume ~20 is half of the unit cell volume.

2.1. Dynamical matrix

The general expression for the phonon frequencies corresponding to a phonon wavevector q is given by [20]

w2aa(]) = E Dc~3(q, Jj')a3(J'), (3)

where aa(]) is the a cartesian component of the amplitude vector a(j) for the jth atom in the unit cell. Da¢(q,ff') is the dynamical matrix and is given as

i '~! r

(4)

. .F l r ..P Dc,~(q,11 ) = ~ t~, 49a#(l,11 ) e x p ( - i q . [ - r ( l ' ) + r ( j ) - r ( ] ' ) l } , forjv~/". (5)

The/th cell is chosen as the origin, r(l') is the position vector of the/th unit cell and r(j') is the position of the ]th atom in the lth cell. ~#(l',]]') is the usual force constant and M is the mass of the ion. In the further discussion we drop the prime over l. The dynamical matrix consists of three parts [21 ],

D,~(q ,n ) D2tj(q,ff') + c .., E .., - ' = Dg,~(q,n ) + Dg, o(q,z ). (6)

~ arises due to the exchange overlap potential between the ion-cores and is neglected in the present calculations. gives the ion-ion Coulomb interaction and is evaluated with the help of Ewald s 0 function transformation

[22, 23] and for the sake of completeness the expression for/)co is given below:

C .., 4~rZ2e2 (q+H)a(q+ ( [ q + HI2] o;o(q,n )- ,Ma~ y H)O exp H Iq +HI2 - 4£-ff ]exp(iH'r//,)

Z2e2 M

I H [3

J. Singh et al./Phonon dispersion in hcp metals 225

and

Here

and

+(. 4~3lr(l, ff')]3 .., ~ 6 n l ~ , )l.)ex p (_n2lr(,,//,)12) ]

- bad [eft (nlr(l,]]')l) Ir(l,j]')l 3

2nlr(l,]]')l exp (-~721r(',//')12)] }

v~

X exp [-iq'r(l, ff')], for/4:]',

C .. 4rrZ2e2[~H(q+H) ,~(q+H)#exp( - lq+HI21 D~43(q'll) : nMf2------O lq + nl 2 4~ I/

. ~ o ~ (-InI21] zV - - E [exp (iq "r(l)} - 1] exp ~,4--~-/J M l.o

! Ir(/)l 5 ~ q

X exp [-r/2lr(1)12]]- Sag [erf [rflr(l)l] + ~ ]r(l)] 3

2wlr(/)l v~-

Xexp[_~72]r(/)[21] } _ r 4nZ 22 H~Hg

l - IH I 2 ~ Z2e 2 . . . . . . X expt 4-~-joxp (~ - - , . . o - . Z { [~(" ~]oLr(,,,, ~]~

Ir(/d/')l 5

+(4,731r(l,]]')1 6rtlr(l,]]')X [_nZlr(l,#,)l] ] X[3erf[~lr(l,]]')l] \ ~ q ~ )exp

- 6a~ [eft [nlr(l,]]')l] Ir(l,]]')l 3

2rflr(l,H')l v~

exp [-n2lr(l,]]')[2] ] } ] .

Y 2 orf(, =l- foY. o

(7)

(8)

(9)

r(l, jj') = r(j) -- r(j') -- r(Z) = rff, -- r(l). (10)


Z is the valence of the ion, H is the reciprocal lattice vector and e is the electronic charge. 7? is a parameter for rapid convergence, and n is the number of atoms in a unit cell.

DEt3 is the electronic or band structure contribution arising from the ion-elect ron- ion interaction.

2.2. Electronic or band structure contribution

The electronic contribution D~,E~ is the most important part in the lattice dynamical studies and is given as

E .. 47rZ2e_____~ 2 (q+H)a(q+H)~G(q+H)- Z i--~ G(H) Dg¢(q'11) = nMf20 Iq +HI 2 H~0

- ~ X ~ exp(iH'r#,)] jq:j' HeO [ HI2 G(H)

E .., 4rrZ22 Z ( q + H ) a ( q + H ) # G ( q + H ) e x p ( i H . r / j , ) ' D~(q,11 ) nM£2o H [q +HI2

for j 4:j ' .

(11)

(12)

Here G(q) is the energy wavenumber characteristic function and is defined as [9]

[47rze2(l + aeff)] -2 e(q) -- 1 G(q) . . . . IVb(q)l 2

k f20q 2 J e(q)[1 - f(q)] " (13)

Here Vb(q ) is the Fourier transform of the bare-ion model potential and e(q) is the dielectric screening function. 0tef f is a parameter taking into account the orthogonalization correction and is discussed by Animalu and Heine [24].

2.3. Phonon frequencies along symmetry directions

The secular equation can be factorized along symmetry directions, as was done by Czachor [25], and the expressions for the phonon frequencies are given below:

(i) [0001 ] direction, z polarization, longitudinal mode:

c°2(qz) = Ozz(q,lJ ) + IOzz(q,]]')l (qx = qy = O; qz = qz); (14a)

(ii) [0001 ] direction, x or y polarization, transverse modes:

co2(qz) = Oxx(q,lJ ) + IOxx(q,JJ')l (qx = qy = O ; qz = qz); (14b)

(iii) [01T0] direction, y polarization, longitudinal mode:

2 ( q y ) = Dyy(q,jj) + [Dyy(q,jj')l (qx = qz = 0; qy = qy); (14c)

(iv) [0170] direction, x polarization, transverse mode II to basal plane:

6o2(qy) = Oxx(q,JJ) + [Oxx(q,J]')l (qx = qz = O; qy = qy); (14d)

J. Singh et al./Phonon dispersion in hep metals

(v) [0110] di rect ion, z polar izat ion, transverse mode I to basal plane:

c°2(qy) = Ozz(q,/J) +- IOzz(q,//')l (qx = qz = O; qy = qy).

The plus sign is for the optical mode and the minus sign is for the acoustic mode .

227

(14e)

3. Results and discussion

The physical parameters o f Zn, Zr, T1, Tb and Ho used in this calculat ion are given in table I. The Cou lomb

coupl ing coeff ic ients o f all the metals are calculated using the eqs. (7) and (8). Because the Cou lomb coupl ing

coeff icients may be o f some interest in the lat t ice dynamical studies, these are tabula ted in tables I I - V I for a

selected set o f phonon wavevectors .

Table I The physical parameters for Zn, Zr, T1, Tb and Ho in atomic units

Parameters Zn Zr T1 Tb Ho

Ionic mass M 65.37 91.22 204.39 159.20 164.94 Lattice constant a 5.0358 6.1059 6.5406 6.8034 6.7618 Lattice constant c 9.3480 9.7353 10.4348 10.7675 10.6163 Ratio c/a 1.8565 1.5944 1.5954 1.5827 1.5700 Atomic volume ~o 102.6513 157.1614 193.2967 215.8076 210.1832 Valency Z 2 4 3 3 3 Fermi momentum k F 0.8340 0.9100 0.7716 0.7439 0.7504

Table II Coulomb coupling coefficients for Zn [in units of (47rZ2e2/nM~20)]

[0001] direction

q/qmax Dxx(q, ff) Dyy(q,jj) Dzz(q, jj) Dxx(q,jj') Dyy(q,jj') Ozz(q,ff')

0.0 0.02349 0.02349 0.95301 -0.02349 -0.02349 -0.95301 0.1 0.02348 0.02348 1.95302 -0.02320 -0.02342 0.04641 0.2 0.02347 0.02347 1.95305 -0.02234 -0.02234 0.04467 0.4 0.02345 0.02345 1.95310 -0.01900 -0.01900 0.03800 0.6 0.02342 0.02342 1.95318 -0.01381 -0.01381 0.02761 0.8 0.02338 0.02338 1.95324 -0.00726 -0.00726 0.01451 1.0 0.02337 0.02337 1.95326 0.00000 0.00000 0.00000

[0110] direction

q/qmax Dxx(q,]]) Dyy(q,]]) Ozz(q,]]) Dxx(q,]]' ) Dyy(q,]]') Dzz(q, ff')

Re Im Re Im Re Im

0.0 0.02349 0.02349 0.95301 -0.02349 0.00000 -0.02349 0.00000 --0.95301 0.00000 0.1 0.02789 1.03888 0.93322 -0.02377 0.00802 0.95737 -0.00813 -0.93360 0.00015 0.2 0.04067 1.08230 0.87703 -0.02456 0.01595 0.90273 -0.01693 -0.87818 0.00105 0.4 0.08587 1.22158 0.69257 -0.02735 0.03108 0.71897 -0.03917 -0.69162 0.00821 0.6 ~0.14213 1.36821 0.48967 -0.03074 0.04412 0.49883 -0.07254 -0.46810 0.02859 0.8 0.18797 1.46940 0.34265 -0.03314 0.05335 0.29349 -0.12408 -0.26044 0.07090 1.0 0.20556 1.50444 0.29003 -0.03306 0.05727 0.11684 -0.20238 -0.08389 0.14529


Table III Coulomb coupling coefficients for Zr [in units of (4nZ2e2/nMI20)]

[0001 ] direction

q/qmax Dxx(q,]]) Oyy(q,]]) Dzz(q,]]) Dxx(q,]]') Dyy(q,]]') Dzz(q,]]')

0.0 0.05105 0.05105 0.89789 -0.05105 -0.05105 -0.89789 0.1 0.05103 0.05103 1.89792 -0 .05042 -0 .05042 0.10084 0.2 0.05098 0.05098 1.89800 -0.04855 -0.04855 0.09711 0.4 0.05082 0.05082 1.89833 -0 .04129 -0 .04129 0.08260 0,6 0.05061 0.05061 1.89874 -0 ,03000 -0 .03000 0.06001 0.8 0.05045 0.05045 1.89907 -0 .01577 -0 .01577 0.03155 1.0 0.05039 0.05039 1.89919 -0 .00000 -0 .00000 0.00000

[0110] direction

q/qmax Dxx(q,]]) Dyy(q,]]) Dzz(q,]]) Dxx(q,]]') Dyy(q, ff') Dzz(q,fi')

Re Im Re Im Re Im

0.0 0.05105 0.05105 0.89789 -0.05105 0.00000 -0.05105 0.00000 -0 .89789 0.00000 0.1 0.05483 1.05979 0.88535 -0 .05146 0.01556 0.93449 -0 .01572 -0 .88302 0.00016 0.2 0.06585 1.08469 0.84944 -0 .05268 0.03089 0.89262 -0.03223 -0 .83993 0.00133 0.4 0.10477 1.16670 0.72850 -0 .05689 0.05988 0.74479 -0.07061 -0 .68790 0.01073 0.6 0.15324 1.25690 0.58985 -0.06178 0.08449 0.55112 -0.12111 -0 .48933 0.03661 0.8 0.19273 1.32172 0.48554 -0 .06472 0.10185 0.35068 -0.18946 -0 .28596 0.08761 1.0 0.20788 1.34470 0.44741 -0.06325 0.10956 0.16218 -0.28091 -0 .09892 0.17135

Table IV Coulomb coupling coefficients for T1 [in units of (47rZ2e2/nMS2o)]

[0001] direction

q/qmax Dxx(q,]]) Dyy(q,]]) Dzz(q,#) Dxx(q,j]') Oyy(q,]]') Dzz(q,]]')

0.0 0.05090 0.05090 0.89824 -0.05090 -0 .05090 -0.89824 0.1 0.05089 0.05089 1.89825 -0.05028 -0 .05028 0.10050 0.2 0.05084 0.05084 1.89835 -0.04841 -0.04841 0.09677 0.4 0.05068 0.05068 1.89867 -0 .04118 -0 .04118 0.08222 0.6 0.05048 0.05048 1.89908 -0 .02992 -0 .02992 0.05981 0.8 0.05031 0.05031 1.89941 -0.01573 -0.01573 0.03144 1.0 0.05025 0.05025 1.89954 0.00000 -0.00000 0.00000

[0110] direction

q/qmax Dxx(q, ff) Dyy(q,]j) Dzz(q,j]) Dxx(q,]]') Dyy(q, jj') Dzz(q,jj')

Re Im Re Im Re Im

0.0 0.05090 0.05090 0.89824 -0 .05090 0.00000 -0.05090 0.00000 -0 .89824 0.00000 0.1 0.05470 1.05967 0.88567 -0 .05132 0.01552 0.93462 -0.01569 -0 .88334 0.00016 0.2 0.06572 1.08464 0.84969 -0.05254 0.03083 0.89270 -0.03215 -0.84021 0.00132 0.4 0.10469 1.16691 0.72853 -0.05675 0.05975 0.74475 -0.07046 -0 .68804 0.01070 0.6 0.15321 1.25738 0.58963 -0 .06163 0.08431 0.55096 -0.12089 -0.48935 0.03656 0.8 0.19276 1.32238 0.48515 -0 .06458 0.10163 0.35049 -0.18919 -0.28591 0.08753 1.0 0.20792 1.34542 0.44696 -0.06311 0.10932 0.16199 -0 .28060 -0 .09887 0.17125

J. Singh et al./Phonon dispersion in hcp metals

Table V Coulomb coupling coefficients for Tb [in units of (47rZ2e2/nM~20)]

229

[0001 ] direction

q/qmax Dxx(q,]]) Dyy(q,]]) Ozz(q,]]) Dxx(q,]]') Dyy(q,]]') Dzz(q,]]')

0.0 0.04484 0.04484 0.85546 -0.04484 -0 .04484 -0 .85546 0.1 0.04482 0.04482 1.85548 -0.04429 -0.04429 0.14275 0.2 0.04477 0.04477 1.85556 -0 .04264 -0.04264 0.13746 0.4 0.04459 0.04459 1.85586 -0 .03627 -0 .03627 0.11693 0.6 0.04438 0.04438 1.85623 -0.02635 -0.02635 0.08495 0.8 0.04420 0.04420 1.85653 -0.01385 -0.01385 0.04466 1.0 0.04413 0.04413 1.85664 -0 .00000 -0 .00000 0.00000

[0110] direction

q/qmax Dxx(q,]]) Dyy(q,]]) Dzz(q,]]) Dxx(q,]]' ) Dyy(q, ff') Dzz(q,]]')

Re Im Re Im Re Im

0.0 0.04484 0.04484 0.85546 -0 .04484 0.00000 -0.04484 0.00000 -0.85546 0.00000 0.1 0.04860 1.05331 0.84321 -0 .04526 0.01432 0.94069 -0.01449 -0 .84120 0.00015 0.2 0.05953 1.07747 0.80812 -0.04649 0.02844 0.89877 -0.02981 -0 .79992 0.00123 0.4 0.09818 1.15711 0.68982 -0 .05077 0.05518 0.75035 -0.06626 -0 .65434 0.00996 0.6 0.14630 1.24486 0.55394 -0.05583 0.07795 0.55494 -0.11573 -0.46455 0.03413 0.8 0.18552 1.30801 0.45156 -0.05913 0.09399 0.35167 -0.18422 -0 .27072 0.08204 1.0 0.20056 1.33042 0.41410 -0.05828 0.10096 0.15991 -0 .27699 -0.09308 0.16123

Table VI Coulomb coupling coefficients for Ho [in units of (47rZ2e2/nMs20)]

[0001 ] direction

q/qmax Dxx(q,]]) Dyy(q,]]) Dzz(q,]]) Dxx(q,]]') Dyy(q,]]') Dzz(q,/]')

0.0 0.05476 0.05476 0.89047 -0.05476 -0.05476 -0 .89047 0.1 "'0.05474 0.05474 1.89049 -0.05408 -0.05408 0.10817 0.2 0.05468 0.05468 1.89060 -0.05208 -0 .05208 0.10416 0.4 0.05449 0.05449 1.89099 -0 .04430 -0 .04430 0.08860 0.6 0.05425 0.05425 1.89147 -0 .03219 -0 .03219 0.06437 0.8 0.05406 0.05406 1.89186 -0 .01692 -0 .01692 0.03384 1.0 0.05398 0.05398 1.89201 -0 .00000 -0 .00000 0.00000

[0110] direction

q/qmax Dxx(q,]]) Dyy(q,]1) Dzz(q,]]) Dxx(q,]]') Dyy(q,]]') Dzz(q,]]')

Re Im Re Im Re Im

0.0 0.05476 0.05476 0.89047 -0.05476 0.00000 -0.05476 0.00000 -0 .89047 0.00000 0.1 0.05849 1.06295 0.87853 -0 .05519 0.01651 0.93118 -0.01668 -0.87599 0.00017 0.2 0.06934 1.08632 0.84431 -0.05645 0.03278 0.89043 -0.03414 -0.83398 0.00136 0.4 0.10769 1.16346 0.72882 -0 .06080 0.06351 0.74595 -0.07446 -0.68515 0.01095 0.6 ,0.15544 1.24860 0.59593 -0 .06582 0.08958 0.55517 -0.12685 -0.48935 0.03727 0.8 0.19435 1.30999 0.49563 -0 .06877 0.10795 0.35591 -0.19685 -0 .28714 0.08888 1.0 0.20928 1.33179 0.45889 -0 .06706 0.11616 0.16703 -0.28931 -0.09995 0.17313

230 J. Singh et al./Phonon dispersion in hep metals

In the calculation of the electronic contribution to the phonon frequencies the Animalu TMMP [9] is used for the bare-ion potential and the screening function used is the Hartree dielectric function defined as

[ 4rtZe2(1 + o%ff)] e(q) = 1 + [ 1 - f(q)] [ ~ - 0 ~ X(2@F)' (15)

where

3 1 X(X)=(.~F)[_~+l[l--x21 l + x ] 4 \ x / l n l - - x ] '

E F =h2k2/2m*.

(16)

(17)

k F is the Fermi momentum and m* is the effective mass of the electrons. The factor f(q) takes care of exchange and correlation corrections. The exchange and correlation correction factor f(q) is taken from Singwi et al. [26] and is given as

f(q) = A[1 - exp {-B(q/kF)2}]. (18)

Here the constants A and B depend upon the interelectronic distance. The summation in eqs. (11) and ( t2) is carried up to 501 reciprocal lattice vectors. The phonon frequencies for

Zn and Zr, along the symmetry directions [0001] and [0110], in units of

2 2 } = 1 [ 47rZ"e'] PP 2rr[ nMaoJ ' (19)

are shown in figs. 1 and 2. The calculated phonon frequencies are compared with the experimental results of Borgonovi et al. [12] and Bezdek et al. [15] for Zn and Zr, respectively. It is evident from the figs. 1 and 2 that the results obtained by using the parameters tabulated by Animalu are higher than the experimental values. In order to obtain a good fit with the experiment, parameters A0, A 1 and R m are renormalized, in the same way as did Animalu [9] by matching the phonon frequencies at the symmetry point A in the longitudinal branch along the [0001] direction. The phonon frequencies with the present parameters, tabulated in table VII, are also shown in figs. 1 and 2. In the case of Zn, with the Animalu parameters the transverse branches are in reasonable agreement with the experiment but the longitudinal branches are higher. With the present parameters the longitudinal branches are in better agreement with the experimental values but the transverse branches are lower by about 29% at the symmetry point I'. In Zr, with the present parameters in the [0001] direction all the branches are in reasonably good agreement with the experimental values except in the LO branch. The LO branch shows good agreement near the zone boundary but is about 25% higher near the centre of the Brillouin zone. In the [0110] direction the TOL and TOll deviate from the experiment by about 25% near the symmetry point P and near the zone boundary, respectively. The TO± branch in the present calculations of the phonon frequencies of Zn shows oscillating behaviour. This oscillating behaviour is not surprising because Iyengar et al. [27] have observed the same type of behaviour in the neutron scattering measurements. Such a behaviour in Zn is also found by Bajpal [7], but the calculations with the Animalu parameters do not show such an oscillating behaviour. It is interesting to note that the quantity 23 k co~(q) [28, 29] where ~o k is the angular frequency of the kth polarization branch, is constant within the variation of 4.5% in the [0001 ] direction while it is strongly dependent on q in the [01]-0] direction. This different behaviour in the two symmetry directions of the calculated phonon frequencies supports the strong anisotropy in Zn. In the present calculations LA and LO branches, in the [01]-0] direction, are parallel over a wide range of q/qmax which is also found experimentally in Zn [12] and indicates certain

J. Singh et al./Phonon dispersion in hcp metals 231

. I ¢ -

¢ -

0 . 8

0 . 7

O.E

0 . 5

i i

. . . . . . [o 0To] -...-.,,

- -.,-~..,

! !

• TO.L L~TAI I

o T A I • LO

• TOll o LA

' I I 1

[ooo0 I

o T O

O TA

• LO

a LA

0 . 4 ', • """

0.3 ~ , & L'" " "', . f " Q

s" • .11 o r • . . . . . . . . . . . . . .

O . . . . . . . . . . . . . • 0.2 . .... . o o ,-'" O O

0.1 , ',. . .~ , , , / o

0 o 0 ~ - •

M 0.8 0.6 0 .4 0.2 r ' 0.2 0.4 0.6 0 . 8 A

q / q m a x

Fig. 1. Phonon dispersion curves for Zn along the symmetry directions [0001] and [0110]. The solid lines show present results while the dashed lines those using the Animalu parameters. • [] • o &• show the experimental values.

rigidity of the basic vector with respect to stress parallel to the basal plane. A similar trend is also found for the TAll and TOll branches.

The parameters of potential for T1, Tb and Ho are not tabulated by Animalu, but these are constructed in the same way as by Animalu [9]. For T1 the ion-core radius R c is taken from Kittel [30]. The ion-core radii of Tb and Ho are not available to us. But the ion-core radii of the trivalent ions of the lanthanide series contract fairly smoothly as we go through the group from 1.11 A at cerium to 0.94 )~ at ytterbium. This is the famous lanthanide contraction [30]. Therefore the ion-core radius of Tb can be taken as the average of the ion-core radii of Gd and Dy and that of Ho can be taken as the average of the ion-core radii of Dy and Er which is a fairly good approxi- mation. The correlation energy L c has been taken from Hubbard [31 ]. For TI, Tb and Ho the parameters A 0, A 1 and R m are again normalized so as to match the calculated and the experimental results at the same point as in Zn and Zr. The parameters of the Animalu TMMP for T1, Tb and Ho are also given in table VII. The calculated phonon frequencies for T1, Tb and Ho are shown in figs. 3 -5 a:~d are compared with the experimental results due to Worlton and Schmunk l16], Hounmann and Nicldow [17] and Nicklow et al. [18], respectively. The form factors for all the metals are shown in fig. 6.

In the case of T1, in the [0001 ] direction all the branches are in reasonably good agreement with the experimental values near the zone boundary. At the centre of the Brillouin zone the LO branch is lower by about 15% but the TO branch is higher. It has been pointed out by Worlton and Schmunk [16] that at small values ofq the experimental results are rather unreliable because of extremely high background particularly in the transverse


0.8

0.7

0.6

F'~ 0.5

o.4 I

C ~ 0.3

0.2

0.1

0 . 0

- L O ATA. [0 I TO] . o TA.L & T O l l . . . - . . . . . . . . . . . . . . .

_ /.--<

[.0 0 0 I] a LA a L O

. . . . . -- o TA

" ' """ • T O

4 4 .

jjs

s"

s"

• /'" • ~ ~ * w ~

/ (

I~1 0.8 0.6 0.4 0.2 [" 0.2 0.4 0.6 0.8 A

q / q m a x

Fig. 2. Phonon dispersion curves for Zr along the symmetry directions [0001] and [0110]. The description of the figure is the same as in fig. 1.

Table VII The renormalized Animalu TMMP parameters. All the quantities are measured in atomic units except LEcl which is in Rydberg

Parameters Zn Zr TI Tb Ho

A o 1.105 2.000 1.450 1.320 1.550 A 1 1.140 2.100 1.350 0.900 0.800 A 2 0.980 1.500 1.700 1.700 1.790 R m 2.200 1.600 2.600 1.950 2.000 m* 1.100 1.000 1.000 1.000 1.000 R c 1.570 1.493 1.985 1.890 1.843 nef f 0.079 0.044 0.085 0.066 0.062 IEcl 0.091 0.095 0.093 0.091 0.092

optical phonons along the [0001 ] direction. The force constant analysis of the experimental results gives values of the force constants which are in poor agreement with their experimental values which may be due to the high background. Our calculated phonon frequencies of TI are in better agreement with the experimental values than the calculations of Sharan and Bajpai [32]. Moreover, in the calculations of Sharan and Bajpai the TOll and LA

0

C

0.7

0.6

[ '0 ITO] I T O J . ATAII

o TAJ- • LO

ATOl l o LA

[ 0 O 0 I] • TO

OTA

• LO

a LA 0.5

0.3

0.2

O.I

0 .0 M 0.8 0.6 0.4 0.2 p 0.2 0.4 0 .6 0.8 A

q / q m a x

Fig. 3. Phonon dispersion curves for TI along the symmetry directions [0001] and [01i0]. The solid lines show the calculated phonon frequencies. • [] • © • A show the experimental results.

233

0.7

0.6

[oETo] I E o o o O

Q.

O

¢:

C

0.~

0.3

0,2

o a%,= ~ I •

0.I oTAJ_ • LO • T O • L O -

A T O l l a LA o o TA a LA

0.0 , , , M O.S 0.6 0.4 O.Z p o.z 0.4 o.s o.e A

q / q m o x

Fig. 4. Phonon dispersion curves for Tb along the symmetry directions [0001] and [0110]. The description is the same as that of fig. 3.

234

C .E

0,7 J i [o ITo]

0.6

0.5

0.4

0.3 (

0.2

0.1

I I I I I i I

[ooot]

: i o "

0 . 0 M 0 .8 0 .6 0 .4 0 .2 r' o .2 0 . 4 0 .6 o.B A

q / q m a x

Fig. 5. Phonon dispersion curves for Ho along the symmetry directions [0001] and [01i0] . The description is the same as that of fig. 3.

0.2

0.0

- 0 . 2

~' - 0 . 4 n" = -

, ~ - o . 6 > 0.0

- 0 . 2

- 0 . 4

y / o

. •

~...,-/ i [ =

b - 0 . ! I

0.0 0.5 1.0 1'.5 Z.O

q / 2 k F

Fig. 6. In (a) the solid, dashed and dot-dash lines show the form factors of Zn, TI and Zr, respectively. In (b) the solid and the dashed lines show the form factors of Tb and Ho, respectively.

J. Singh et aL/Phonon dispersion in hcp metals 235

branches in the [01 ]-0] direction do not cross, which is in disagreement with the experimental results, and also the LA branch in the [0001 ] direction is in serious disagreement with the experiment.

In the rare earth metal Tb, in the [0001] direction, the longitudinal branches are in good agreement with the experimental values while the transverse branches are lower by about 22% at the symmetry point P. In the [01 TO] direction the maximum deviation is 21% in the TOll branch. In Ho also similar trends of the calculated phonon frequencies are found with a deviation of 20% in the transverse branches in the [0001] direction. In the present results of Tb the slopes of the TA branch in the [0001 ] direction and the TAJ_ branch in the [01 TO] direction are approximately the same which is in accordance with the experimental results [17]. It is seen from figs. 4 and 5 that although the qualitative trends of the phonon frequencies of Tb and Ho are the same their direct Coulomb interaction contribution and the ion-electron- ion contribution are significantly different. In other words the electronic structure of Tb and Ho are quite different which is also indicated by the force constant analysis of the experimental results of these metals [ 17, 18]. In the present calculations no Kohn anomaly is found. But both Tb and Ho possess a spiral magnetic structure [33] and this spiral is believed to be caused by the nesting of the Fermi surface [34, 35] in the c direction which may lead to a Kohn anomaly. Very little evidence is found experimentally for the Kohn anomaly in the LA branch in the [0001] direction but as yet it has not been confirmed.

Earlier the Animalu TMMP was applied by us [19] to study the phonon frequencies of Sc and Y and the results were found to be in good agreement with the experimental results. Recently Kulshrestha and Upadhyaya [36], using the parameters tabulated by Animalu, have calculated the phonon frequencies of Sc and Y using the Hartree dielectric function with Hubbard-Sham modification for the exchange and correlation corrections. They report imaginary phonon frequencies for Sc but the results should not differ so much with a slight modification of the screening function. Our results for the dynamical matrices show the symmetry of the hcp crystal structure. It seems that Kulshrestha and Upadhyaya have made some mistake in the calculation of phonon frequencies. Using the parameters of Animalu our results for the phonon frequencies of Zn and Zr are also real though not in good agreement with the experiment. But the agreement can be improved by varying the parameters slightly. So in all the hcp metals Zn, Sc, Y, Zr and T1 thet Animalu TMMP gives reasonable results within the limitations of the theory. Therefore, it is concluded that the Animalu TMMP is, in general, valid for all the 3d, 4d and 5d hcp metals. Here the Animahi TMMP is applied to the rare earth metals Tb and Ho for the first time and the agreement is reasonably good except in the transverse branches which are lower. This may be due to the fact that the ion-cores in the rare earth metals are large and therefore DRt3 may be significant. In the present calculations the Hartree dielectric function, which is a scalar, is used. But to improve upon the results one must take into account the local field corrections, which are noncentral in nature, in the dielectric screening in the transition metals. In the rare earth metals one must take account of the effect of both the d and f bands in the dielectric screening as well as in the model potential. In a subsequent paper the calculation of the phonon frequencies ofhcp rare earth metals, which takes into account the effect of the d as well as f bands, will be reported.

Acknowledgements

The authors are highl, y thankful to Prof. S. K. Joshi for the suggestions and enlightening discussions. We are also thankful to Prof. H. S. Hans for the encouragement. The financial assistance from the Council of Scientific and Industrial Research, New Delhi, is gratefully acknowledged. Computer facilities from the Delhi University Computer Centre, Delhi, are also acknowledged.

References

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Phonon dispersion in hcp metals

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