CHAPTER V

LATTICE DYNAMICLAL STUDY ON ELECTRON SUPERCONDUCTORS

ND zCU04, PR2CU04 AND G D ~ C U O ~

5.1 STRUCTURE AND SYMMETRY

Zone centre lattice dynamlcal calculat~c,ns on the

tetragonal crystals Na2Cu04. PrCu04 and Gd2Cu04 are

reported in this chapter. In contrast to the Yittirium

based and Lanthanum based superconductors. which have

electron vacancies or holes as carriers. the Nd2CuS and

Pr2Cu04 supercondtlctors which are relatively new I451 have

electrons as charge carriers. The tetraoonal phnse belongs

to the space group I 4 / m and differs frc,m the LazCu04

structure (belonging to the same group) in the pnsitic~n of

the apical oxygen atoms of La2Cu04 which in this case

occupy the P, site symmetry. These structures. called the T'-strucutres do not have the Cu-O octahedra Experimental

results show that the undoped Nd2CuOI 1s a typical

semiconductor. and doping it wlth ce4+ (Nd2_xCexCu0 1 4-Y

transforms it to a semimetal for x = 0.15, which huwever

shows a drop in resistivity at 9k indicating

superconductivity in a small portion of the sample. It

Table 5-1: S i t e s y m m e t r i e s and p o s ~ t i o n c o o r d i n a t e s f o r

LnpCuOl (Ln-Nd. P r . a d ) . space g r o u p I 4 / m . T' s t r ~ l c t u r e .

S i t e Wycoff No. o f P a s i t i n r ~ atom s y m e t r y N o t a t i o n p o s i t i o n s + f i , z , L - Z , W Z )

transforms into a superconductor rTc = 24kr on annealing

l~nder certain condit iuns . Super conductivity is also

evident in the T'-st.ructure Pr2-yCexCu0 iTcr 24 K). The 4-Y

structure is shown in figure 5-1.

There are 7 atoms per unit cell. The Cu atoms

occupy sites with 4 / m symmetry. The oxygen atoms in

plane with the copper atoms denoted hy O(l) occupy the

site wit,h m synanetry, the Ln (Ln = Nd.Pr,Gd) occupy the

4mm site symmetry and the 0(2) atoms havp the JmZ point.

symmetry, details of which are qlven i r ~ tahle 5-1 The

lattice parametres employed in these calculations ere ( ~ n

A ) a = 3.94. c = 11.9776 for Nd2?i.1O4 1461, a = 3 9 6 6 . c - 12.248 for Pr2C!104 (171 and a = ? 1398, r = 1 1 . 4 9 8 for

Gd,f1~0~ 1471. The?-? are 21 viht-at~onhl frequencies. the

optical frequencies distributed as

5.2 PREVIOUS EXPERIMENTAL AND THEORETICAL S T U D I E S

Tajima et a1 [461 have performed optical phonon

spectra measurements on the cuprntes with T' structures

Ln2Cu04 ( for Ln - Pr. Nd, Srn, Eu and Gd ) as well as

(NdCe)2Cu@4. These rnmpclunds have a romlrlon spectral

profile with four peaks. These four peaks have been

identified as four infrared activp modes 4 ( E u ) for E C ,

Their Eu mode frequencies are in close agreement with

Heyen's observed frequencies 1481. Crawforrl st al. 1491

have repor ted IR measurements on NdzxCeXC~~O4, but the

assignement in terms of vihrat.iona1 eigen vectors are

incomplete. Except for these spectra none have been

rnros~.~r~d sr-1 far. Sugni et a1 have den~onstrated that

Nd2Cu04 is an anti f errornagnet hy measuri ng two magnon

Raman scattering. Crawford et al. 14P1 have published an

analysis of the infrared act ive phonons in Pt-2-xCeyCu04.

Heyen et, al. 1481 have measured fa1 infrared reflection

and Reman spectra on single crystals of Nd2-ycexCu04.

Using the Kramers-Kronig analysis. they have determined

the TO and LO of the IR reflection spectrn. To calculate

eiaen val~les and eigen vectors of the optical modes. they

used a shell model that employs a Born-Mayer po+,ential

along with the long range coulomh potential.

5 .3 PRESENT WORK

A . THE SHORT RANGE FORCE MODEL

The definition of valence short range forces in

terms of bond lenrlths and angle bcnding constants is

carried not as usual for bonds shnrter than 3.5 A and for

angles defined in t-he copper-oxygen plane Initially, the

stretching constants Cu-Ofl), Ln-O(2). Ln-O(1) were

considered for building the short range parameters. Later

it was discovered that introducing the Cu-Ln bond (around

3.2 A) t.o the short range interact-ions gave hetter

agreement between the experimental and theoretical values.

Cu and Ln are most unlikely to form a bond pair therefore

included only in thls n~odel and not cons~dered in the

Rigid Ion model and the polarizable ion model wher-e the

coulomb interactions are included along with the short

range forces Interaction constants were also defined

between stretches Cu-O(1):Cu-O(1). Ln-O(2),Ln-O(2).

Ln-0(2):Ln-O(1) and Ln-O(1):Ln-O(1): stretch-bend

interaction constant defined for Cu-O(l):O(l)-Cu-n(1): and

bend-bend interact-ion for O(1)-Cu-O(l):O(l)-Cu-O(1)

resulting in a total of eleven force constants. The bond

stretching coordinates that are d p f ined a?-e aivpn in table

5-2 and the angle hending coordinates are glven in table

5-3. The hond lengths are calulated frarn the aforesaid

values of lattice parnmetres. Initially. the forcc

constant values were taken on a rough comparision with the

fnrce ronstant valt~e~ of s~rnilar stretrlles thnt were

obtained in La2Su04. as these compounds belong to the same

space qroup and dn nnt differ very much in their

structures. In course of the cnlculati~n the force

constants were adjusted to fit with the reported values of

measured frequency data of Heyen e t a1 . 1481 fc~r Nd2Cu\.

For Pr2Cu0 and Od CuO . the only available data were the 2 4

Eu values reported hy Tajima et al. [461 nnd the fits of

the parametres correspond to these values. The short range

constants that were finally obtained after several trjals

in the valence force field model are given under the

column (a) for Nr12Cu04. Pr2Cu04 and Gd2Cu04 in tables 5-4.

5-5, and 5-6 respectively The computed frequency values

are shown in tables 5-12. 5-13 and 5-14 respectively.

Table 7 - 2 : Bond stretching coordinates fnr trtranonal

Ln2CuCl, ILn - i1)Nd. f i i ) Pr & (iii) Gdl

In the valence Short Range Force model

N1.1mher Atoms Bond ripe of Name denoting the involved length 1 bond index coord~nate I i i iii

table 5-2 ( c o n t d . )

13 3-16

14 3-17 i 2.375 2.403 2.353 Ln-0(2) rZ 15 3-18

16 3-19

17

18 3-21 2.581 2.615 2.558 Ln-O(l) r,

19 3-22

20 3-23

21

3-20 ! 1-24

2 2 1-25

23 1-26

24 1-27 3.247 3.282 3.216 Cu-Ln

25 1-28 r'

26 1-2s

27 1-30

28 1-31 ,

Table 5-3: Angle bending conrdinates for tetragonal

Ln2Cu0, (Ln - Nd. Pr. Fd) Numbers Atr?ms angle Type nf Narrlp denot lng the involved ang 1 e 1 ndtx c n n r d ~ n a t e

Tahle 5-4: Force constant val1.1es for tet.ragona1 Nd2ru04

a1 Short. Range Valence force Model b) Rigid Iorl Mndrl

c ) Polarizable Ion Model

Code Type Atoms Code Vnl~le of involved speci f i- forcr

cation constant

1 Bond Cu-O(1) f , 1 453 1 350 1 90C stretch

3 Nd-Of 1 ) 3

fl 75C 0 506 fl 900

4 Cu-Nd 0.250 - - f. 5 Angle 0fl)-Cu-O(l)

f5 0 1 8 0 0 . 3 0 0 0 . 4 0 0

hending

6 Stretch- Cu-O(ll,ru-O(ll f 0 145 0 130 @ 170 stretch

8 Stretch- Nd-O(2) :Nd-O(21 fZ-Z 0 130 -0 085 0 200 stretch

Table 5 - 5 : Force constant values for tetragonal Pr2Cu0,

o Short Range Valence Force Model b Rigid Ion Model c Polarizable Inn Model

Code Type Atoms Code Value of involved speclfi- f nrce

cat ion constant

1 Bond Cu-0(l) i

1 25 1 25 1 75 stretch

2 Pr-0(2) 2 n ~2 o 82 1 22

4 Cu-Pr f4

0 30 - -

9 Angle O(~)-CU-O(~) f5 n . 2 9 n 35 0.40 bending

6 Stretch- Cu-O(1):Cu-O(1) f i - , 0 1 5 n 15 0.15 stretch

7 Stretch Cu-O(l). f i - s D o e o o e o . 0 e - bend @(l)-Cu-O(l)

8 Stretch- Pr-012) :Pr-012) fZ-Z -0.04 -0 .04 -0.04 stretch

Table 5 - 6 : Force cnns t an t vnlnes f o r t e t r a g o n a l Rd,Cul - - a Shor t Range Valence Force Model b Rlnld Ion Modal c P o l a r ~ z a h l e Inn Model

- -

Code Type Atom.? Code Value of i nvn lved speclf 1- f n r ce

c a t i o n cons t an t

1 Rnnd ~ u - n ( l ) fs 1 2 5 1 2 5 1 75 s t r e t c h

5 Angle @( l ) -Cu-O( l ) f5

0 . 2 8 0 3 5 0 . 4 hending

7 ~ t r a t r h CII-1311). f l - 5 n no a 118 o (18 - bend 0 ( 1 ) - C 1 ~ - 0 ( 1 )

8 S t r e t c h - Gd-O12):Gd-0(2) f Z r 2 -0 .04 -0 04 - 0 . 0 4 s t . re tch

' u n l t s : S t r e t c h and stretch-slr: t .ch f o r c e constant.a a r e

i n d y n / A , stret .ch-band c o n s t a n t s a r e i n mdyn hand and

bend-bend f o r c e c o n s t a n t s a r e mdyn A

R . 1 HE RIGID I O N MODEL

The Rigid ion model considers the coulombic

interaction hetween the ions of t.he lattice (considered as

point. charqes) alorlg wi t.11 ?he sliort. range int.eractions

arising from valence force fields. The internal

coordinates defined hy the bond stretches and angle

bending conrdinates are slightly different from the

coordinates defined in the short. range model, where an

additional stretching coordinate viz..Cu-Nd bond (around

3.25 A ) was included to give hetter agreement with

experimentally observed frequency values. This reduces the

nurrlher of internal coordinates hy eight: The same set of

stretching and hending r.oordinat~s ~.tsed for the Rigid Ion

I I I L W I ~ I i n (15~11 ill 1 1 1 ~ Prtlnr ~ ~ . ? I I I I ~ > 1r!11 r~r~,Iml 7 T~IP + ~ I > I P R

5-7 and 5-8 give thr str~t.ching and the bending

coordinates defined for the short range part. Fur the long

rynoe coulornh interactlnns the val~t~s of the rh3rg~s were

taken from those charpe values rthtnined as bes t fits in

the tetragnnal phase of Ln CuO . The charqe values were 2 4

then adjusted over several trials maintaining the charge

neutrality of the cell to give better aareement with

vibrational data. The four additional porametres for the

R i u j r l Tor1 rnc~rlrl rille t r , charg~s rNrl CII. 1.11 fT.r t - NII. Pr

Gd). O(1) and O(2) are g ~ v e n in tahle 5-9 The same values

crf charges were assigned for the three crystals Nd2Cu04.

Pr CuO and Gd2Cu04 and found to give goc~d agreement The 2 4

Bravais unit cell is of the hody centred type nnd was

transformed into a primit ive lattice hefore the colllornhic

contribution to the dynamic81 matrix was calculated The

transformat ion from the hody centred cell tr:, the p r l m ~ t ~ v e

cell given by equation 3-2 in chapter 111 holds good for

this structure as well. The LO-TO frequenc~es were

calculated by changing the direction of the wave vector

from the (100) or (010) direction to the (001) direction.

In the a or b direction of propogation, the degeneracy in

the Eu modes is lifted and the LO-TO splittings are

identified. The APU modes vibrate with TO frequencies in

the a or b direction of propogation i e . when E is

parallel to the c-axis: Whereas when E is perpendicular to

C , the Azu modes vibrate with a higher frequency helnnging

to the LO mode of vibration while the degeneracy in the Eu

modes are manifested. The calculated frequency for

Nd2Cu04, were compared with experimental values reported

hy Heyen et al. l4Rl and Most.oller e t al. 131. For

PrzCl104 and Rd2C~104 the LO-Tc! splittings in t.he fr~qurncy

was not available in literature srl t h p only 3>.:a1lahle Fu

mode frpqr~encies rpportpd hy Tn~lma et al. I 461 were

employed for fittina the results. The force const-ants for

the short range part were taken from the Short ran:+? force

model and then modified to get results in conformity with

exper~rnental frequencies. The flnal values of the short

range force const.ants used in the Rigid Ion model are

tabulated in table 5-4. 5-5 and 5-6 for NdZSuO4. Pr2Cu04

and GdzCu04. The calc~~lated valups of freql~encjes are

given under column fh) in t.able 5-12. 5-13 and 5-14 for

NdzCuO,. Pr2Cu04 and Gd2Cu04 respectively.

Table 5-7: Bond stretching coordinates for tetrngonal

LnPCuOI ILn - (i) Nd. (ii) Pr h ( l i i ) Gd]

N~rmher Atoms Bond TYF'P of Name denoting the involved I m g t h honrl index coordinat.? i i i i i i

14

15

16

cont inl~ed

Tahle 5 - R : Anate bending coordinates for tetragonal

Ln2Pu0, (Ln = Nd. Pr . Gd\ -- - - - -- -

N~urnbers At,orns angle Type of Name dcnotlnp the lnvnlved angle index cnnrdinat.c

Table 5-9 : Charges assigned to atoms of Ln2CuOl

(Ln - Nd.Pr.Gd) bl R i a l d Ion Model

c ) Polarizable Ion Model

A t urn Charge

b c

C . POLARIZABLE I O N MODEL

In addition to the short range forces and char-ges

defined in the Rigid Ion model. six additional parametres

airse due to the inclusion of the polarizablity of the

ions into the potential of the Polarizable Ion

model .Initially, the short. range force cnnst.ant.s and

charge values were adopted from the R ~ g i d Ion model. The

values of the polarizablit~es were taken from the values

used in the case of tet.ragona1 La2Cu04 and RS these

crystals are closely related in structure, etc, we do not

expect the polarizahlity nf the ions to differ very much

As nn rnliahle data on the h ~ g h frequency dielectric

tensor ( 6 measurements were available in literature.

the values of polarizahlity used for these crystals could

not be further modified However. It could he expected

that the calculated values of k oiven in table 5-11 are \ J -

not very different from t-heir actual values. Th? values of

the short range const.ants and the charges were adjtusted

over a few trials to get the best fit with experimentally

determined frequency values. The same charge values used

in the Rigid ion model were found to give good agreement

i n t h e p o l a r i z a b l e ion tnodel a l s o . The f i n a l v a l u e s of t h e

f o r c e c o n s t a n t s f o r Nd2Cu04, PrpC~~C14 and G(12C~u04 a r e

r e p o r t e d i n t a b l e s 5-4. 5-5 and 5-6 respec t . ive ly u~nder t h e

colllrnn ( c ) . The computed f r e q u e n c i e s a r e g iven i n t a b l e s

5-12. 5-13, and 5-14. For Nd2C!1O, t h e d a t a r e p o r t e d by

Heyen e t a1 . I481 and Mostol l e r e t a1 . 131 have been used

f o r cornpar i s l o n . For PrzCu04 and Gd2Ct.104, t h e Eu mode

f r e q u e n c i e s r e p o r t e d by Tajima e t a1 1461 were a l o n e

a v a i l a b l e . Hence t h e c a l c u l a t e d v a l u e s may prove a s

gu idance v a l u e s f o r f u t u r e s t u d i e s .

Table 3-10 : Polarizablity values Fnr tetrngonal Ln2Cu04

(where Ln = Nd. Pr, G d )

Atom Polarizablity 2 X X YY z z

Table 5-11: High Frequency Dielectric tensctr- valuca For

tetragnnal Ln2Cu04 (Ln = Nd, Pr. Gd)

Crystal Calculated

XX YY - & ',

Table 5-12 Calculated and observed frequencies TOfLo) in cm-

for NdZCu'),

Observed Calculated frequencips Mode frequencies

present work 1481 I31

a b c 1491

Tnhle 5-12 Calculated ,rind otlserved frequenclec T ( I ( L ~ ? I I ~ I cnl-

tor Pr,CuO,

Of~served C n l culated f r e q ~ l e n c ~ e s Mode frequencies

present work (461

n h c

Table 5-14 Calculated and observed frequencles TOILO) In cm*

for Gd,CuOl

Observed Cnlculnted frequ~ncles Mode frequencles

prcsont work 1461

a b c

5. 4 RESULTS AND DISCUSSIONS

The availahlity of high qua1it.y ?inole rrystal

enabled Heyen et a1 . I481 to measl.trP the phclr1c111 frequenriee

for Nd2Cu04 crystal. The values reported by them are in

close ag~-eemerit to the experimental values reported by

Mostoller et al. 131. It can he seen from the tahles 5-12.

5-13 and 5-14 that the agreement in general is reasonably

good in the Valence Short Range model for all the three

crystals Nd2Cu04. Pr2Cu04 and Fd2Cu04. A remarkahle

conformity between theoretical and experimental values is

seen In the case of the Rigid Ion model and the

Po 1 ar i zab 1 e Inn model for Nd2Cu04. Pr2CuC; and Gd2Cu04.

From the table 5 - 3 , it is evident that In all the three

crystals, most of the valence force constants in the Rigid

Ion model are lower than in the other two models, while

those used in the Polarizable Ion model are relatively

higher than the valence force constants used In the Short

Range force model

For all the three crystals, an ldentlcal pattern

of the elgen vectors (with sllght dlfferences In the

amplitudes) in the different modes was observed The

charecterlstic features of some of the modes are depicted

in figure 5-4. The A mode is charecterized by an *g

out-of-phase motion of the Ln (Ln = Nd. Pr. Gd) ions above

and below the copper-oxygen plane of a high amplitude

(0.707) in the z-direction. The B mode involves the ig

motion of the 0(1) ions '4' and '5' that Ile along the

Cu-O plane along the orposit-e z direction with an

anlplitude of 0.707 as sown in the figure. The E ( 1 ) mode

(vibrating at a higher frequency) exhibits a small motion

for the Ln ions in the x and y direction with an arr~plitiide

of 0.05. The direction of motion of the ions of the atoms

are against each other. For the E (2) mode, the 0(2) ions

vibrate with small amplitude (0.05) along the x and y

directions and a large displacement for tha Ln lons : the

atom '2' undergoing displacement. in the - x . y d~rect-ion

and the atom '3' in the x. -y direction. All the three AZu

modes involve in-phase z displacements for all atoms of

the unlt cell. In the Azu(l) mode. the z displacement of

the Or11 and 0(2) ions is more pronounced; In the A2J2)

mode, the z displacement of the Ln. 0 (1 ) . and the 0(2)

ions are relatively prominenent: while in the A2j3) mode a

large 2 displacement (-0.915) of t.he central copper. ion

FIG. 5-2

NORMAL MODES OF ~ d , C u o ,

and a z-displacement of 0.275 for the Ln Ion was observed.

The z displacements of the remainig ions of the unlt cell

are quite feeble. The Eu modes are marked by t.he

displacement of all the ions of the unit cell in the x and

y directions : The Eull) mode differs frorn all !he other

EU modes in the rnotion of the 0(1) ions which have a

relatively large displacement (oppositely directed) as

shown in the figure 5-4. In the EU(2), Eu 13) and E u ( 4 )

modes, all equ~valent atoms exhibit in-phase motion in the

x-y direction. The BZu made is marked by the rr~otion of thr

012) ions 6 and 7 along the z direction

The only available data to compare the cnlculated

results far Pr2Cu04 and GdlCuOI are the Eu mode (TO)

frequencies reported hy Tajima et a1 1461. It can he seen

from the tables 5-4. 5-5 and 5-6 that the force constants

for the bonds Cu-O(1). Nd-012) ancl Nd-O(l) and for the

011)-Cu-011) angle are higher for the Polarizable Ion

model than for the other two models. The values of high

frequency dielectric tensor c reported by Ta~lrr~a et al.

are 6 . 5 . 6 . 5 and 5 . 0 for Nd2Cu04. Pr2Cu0+ and GdzCuOl

which they obtained by deriving it. fron~ reflection data.

However these data could no be relied upon. as their E

values for tetragonal LaPCuOI is also very hlgh (6.0) in

comparision to the measured values quoted by Most-oller

(3.6 for E l l c and 3.0 for E L c ) Therefore as discussed

in the previous section the E values obtained from this

work are reported for academic interest The calculations

that have been done on these T' structured coroyc~u~ids ar-e

very few: And the available vibraticrnal data In literature

are very 6cant.y. Hence the results obtained in thls work

could serve as guidance values for futuy-e exper~mental and

theoretical studies and also provide insight 'o the

strength of force const.ants contributing to t h e potential

in the three potential models discussed.