Received 4 January 2011; revised 7 April 2011; accepted 24 June 2011

Transcription

1 Idia Joural of Pure & Applied Physics Vol. 49, eptember 0, pp pecific heat jump ad trasitio temperature for La x Ba x uo 4, Bi a r u O +3 ad l a Ba u O +3(+4) supercoductors P W Otieo Nyawere * & KM Khaa, Departmet of Mathematics & omputig cieces, Kabarak Uiversity, Private Bag 057, Kabarak, Keya Departmet of Physics, Moi Uiversity, P.O.Box Eldoret, Keya * potieo@kabarak.ac.ke Received 4 Jauary 0; revised 7 April 0; accepted 4 Jue 0 he trasitio temperature ad the specific heat jump / i La x Ba x uo 4, Bi a r u O +4 ad l a Ba u O +3(+4) are calculated usig exotic pairig model. hese values are calculated at both bucklig mode ad breathig mode. he values calculated are compared with kow experimetal values. If is the gap i the allowed eergy states, the the jump i the specific heat is. hese results show that the calculated values of the ratios ad compare well with experimetal values. Keywords: pecific heat, upercoductors, Desity of states Itroductio he bulk properties of solids, primarily itegral characteristics of the excitatio spectrum of electroic, phooic ad magetic degrees of freedom have bee ivestigated by kowledge of specific heat. I particular, the observatio of the specific heat jump occurrig at the supercoductig phase trasitio i oxide supercoductors has cofirmed the bulk ature of high temperature supercoductivity. he specific heat of YBO ad LMO has bee best studied, while that of Bi ad l compouds has bee studied less 3-7. he paret compoud of the La-Ba-u-O supercoductor, La uo 4, is ati-ferromagetic isulator. he uo plaes i this case do ot have ay metallic characteristics. As Ba, r or a are added to the compoud, electros are removed from the uo plae leavig behid vacacies (holes) i the bad. Evetually, there are eough holes to make the uo layers metallic ad supercoductig. Bi a r u O +4 is a bismuth based supercoductor where =,, 3 are the umber of immediately adjacet uo plaes. he higher the umber of uo plaes, the higher the trasitio temperature up to saturatio. his compoud has immediately adjacet uo plaes with a calcium plae i betwee r-o ad two Bi-O plaes which separate these immediately adjacet plaes before the ext u-o plae. Its high- is attributed to log periodic modulated superstructure i x-y plae, existece of several closely related structures of r ad Bi-O ad stabilized modified perovskite structure. he compouds l a Ba u O +3(+4) ad l a Ba u O +3(+3) cotai immediately adjacet u-plaes with a a plae betwee each immediately adjacet u-o plae. he plaes Ba-O, l-o ad aother Ba-O separate these uo plaes. his compoud has three differet types of oxyge atoms that is O p oxyge atoms i the u-o plae, O z apical oxyge atoms directly above ad is part of Ba-O plae ad O oct oxyge atoms that are part of plaes ad i octahedral surrouded by l ad Ba atoms. he existece of several equivalet positios of oxyge atoms causes a strog aharmoic perturbatio, which ca icrease the electro-phoo couplig 4,5 leadig to icrease i trasitio temperature. pecific heat discotiuity at due to the secod order trasitio of the ormal state to supercoductig state ad the electroic specific heat coefficiet γ (ommerfeld gamma) are importat properties of all these three supercoductig materials. his specific heat coefficiet is proportioal to the desity of electroic states at the Fermi surface ad is oe of the parameters which specify the iteractios of the electros ad hece, used to determie the trasitio temperature. he kowledge of these electroic properties may lead to uderstadig of high mechaisms. Exotic pairig 5 has bee reported to be cotributig to high i YBa u 3 O 7δ. Applyig the same to

2 68 INDIAN J PURE & APPL PHY, VOL 49, EPEMBER 0 La x Ba x uo 4, Bi a r u O +4 ad l a Ba u O +3(+4) shows that aharmoic perturbatio of phoos with quadratic temperature depedece sigificatly icreases the trasitio temperature. he shapes of the specific heat graphs for supercoductig phase ad ormal phase as fuctio of temperature 8 idicate shape of specific heat jumps typical of supercoductig state givig support to exotic pairig theory due to aharmoic perturbatio as cotributig to the electro-phoo couplig. he graphs are liear for but for it has two sectios. First, is a expoetial term for idicatig existece of eergy gap i the electroic eergy levels ad secodly, the liear term for 0.7. his is a further proof that the properties of the specific heat discotiuities ad the specific heat coefficiet ad their depedece o the may be essetial to uderstad the ature of supercoductig trasitio i high- ceramic oxides. I ormal state, the specific heat is composed of the lattice cotributio ad the electroic cotributio give as = + 3 where 3 is associated with Debye vibratios of lattice. At very low temperatures, the liear term domiates which arises from the kietic eergy of the heat motios of the electro gas i the supercoductig state. he phoo specific heat es αexp( /) domiates at low temperatures ad vaishes expoetially i the limit of very low temperatures. I supercoductig state =3γ / 3 where 3γ/ is the gradiet of the liear part of the curve. At very low temperature, phoo cotributio will be egligible i supercoductig state ad the specific heat will be due to the electroic motio. I YBO, the ratio / is.43 at = where is the specific heat i the supercoductig state ad is specific heat i the ormal state. he specific heat jump is the sudde rise i specific heat at trasitio temperature whe specific heat values are plotted agaist temperature. I this paper, the trasitio temperature ad specific heat jump are calculated. heory he well kow B theory is ot able to explai the properties of high- supercoductors. However, pairig of electros does occur ad the said pairig is assumed to be exotic 5. I this theory, it is assumed that there are three electros that take part i supercoductig curret ad these electros iteract with each other through harmoic forces. wo of these electros form a boud pair while the third oe is a polarizatio electro which hops from oe lattice site to aother site of similar symmetry. Photoiduced Rama scatterig studies have cofirmed that there exists strog aharmoic ature of apical oxyge vibratios. I fact, whe spectral fuctio of electro-phoo iteractios is compared with phoo spectrum i bismuth compouds, it shows that both low frequecy vibratios (bucklig mode) ad high frequecy vibratios (breathig mode) cotribute to the electro-phoo couplig 6. hus, polarizatio electro causes perturbatio with respect to apical oxyge vibratios leadig to cotractio of u p -O z bod. his perturbatio is of the form: 3 4 β x + γ x () where β ad γ are perturbatio parameters. At low temperatures, the phoo cotributio to specific heat is : 3 = + () Whe phoo cotributio is eglected, the the term 3 does ot cotribute to the specific heat. Eq. () becomes: = γ (3) where γ replaces ad is the specific heat coefficiet (ommerfeld gamma). I supercoductig state, the specific heat is: 3 3γ = (4) which ca be assumed to be of the form αexp( /k) before the heat jump because of the existece of the eergy gap i the temperature rage If the specific heat jump is / the at =, = γ = = at = ; = γ. hus, Eq. (5 ) is: = γ (5)

3 NYAWERE & KHANNA: PEIFI HEA JUMP AND RANIION EMPERAURE OF UPERONDUOR 69 Hece = γ (6) k k β =, γ = aħω aħω 3 4 o o From Eq. (8), eergy ε ca be expressed as: () where = is the specific heat jump. he perturbatio is of the form of Eq. () ad costats γ ad β represet the asymmetry of the mutual repulsio of the atoms ad softeig of the vibratios at large amplitudes, respectively. hey may or may ot be temperature depedet. he eigevalues ad eigefuctios of the uperturbed harmoic oscillator are determied from Hamiltoia H 0 give as: H >= E > (7) 0 0,0 Whe the system is perturbed the Eq. (7) becomes: H >= H + H > E > (8) 0 ( 0 ),0 3 4 where H = β x + γ x. he total eergy of the system 4 is of the form: 3γħ ε = + ω ħ µ ω 5ħ 3 4 4µ ω + + exp β 30 θ 3 exp = A ( A A ) E k + γ + β (9) 3γħ = + ħω = + + where A, A, µ ω 5ħ 3 4 E A3 = 4µ ω + + β 30 ad θ =. k ε he specific heat =. Now, three cases arise depedig upo how β ad γ deped o the temperature. () β ad γ are liear fuctios of temperature; () β ad γ are quadratic fuctios of temperature; (3) β ad γ are idepedet of temperature. I this work, we assume that β ad γ are quadratic fuctios of temperature. he calculatios are doe for both bucklig modes which are perturbatios that occur at low temperature 580 K ad breathig modes are calculated at high temperatures of 60 K. he parameters β ad γ are defied as: θ 4 ε = A + ( Ar + A )exp () where A k Ar = = a A k 4 3, A 4 6 oħω aoħ ω ad 3 { r r } = A θ + A + A θ + 4A exp () 3 Results 3. Breathig Mode (i) La Ba u O 4 From Ref. (8) ad Eq. (), the expressio for specific heat is foud to be: 60 = exp { } (3) Figure shows the variatio of specific heat versus temperature. he value of is read from the graph at the poit coicidig with the liear graph. Volume of states V(O) is determied as i Ref. [7]. Whe the curve ad are compared at the specific heat jump, the specific heat coefficiet γ = J/K. From Eq. (5), the specific heat jump ad other ratios ca easily be determied. Here = JK ad = 94 K. θ Fig. s ad versus temperature for La (=)

4 630 INDIAN J PURE & APPL PHY, VOL 49, EPEMBER 0 = = 0.33, γ = = 8. mjk mol γ =.8 mjk g at. V(O)(states/ev.atom)=3.75(states/ev.u.atom),.3 = pecific heat jump is the give as: =.83 γ = mjk mol (ii) Bi a r u O +4 I Ref. (8), the specific heat for breathig mode for the compoud Bi a r u O +4 is give as: 60 = exp { } (4) Agai at jump (Fig. ), the values for trasitio temperature, specific heats ad specific heat coefficiet γ ca be obtaied. =.89 4 J/K, = J/K ad = 3 K. = =.9γ = = 3.53 mjk mol γ = 0.76 mjk g at. V(O)(states/ev.atom)=0.487(states/ev.u.atom), ( / )=.9 ad the specific heat jump is 6.75 mjk mol. (iii) l (=3) l aba (uo 3 ) 3 pecific heat equatio for l (=3) for the breathig mode is here derived as foud i Ref. (8). Fig. 3 shows the variatios of specific heat with temperature. 60 = exp { } (5) =.94 4 JK, = JK ad = 3 K = =.83γ = = 3.64 mjk mol γ = 0.4 mjk g at. V(O)(states/ev.atom) = 0.50(states/ev.u.atom), / =83. he, specific heat jump is 6.66 mjk mol. 3. Bucklig Mode 3.. La (=) La Ba u O 4 pecific heat equatio for supercoductig phase is obtaied from the Eq. (6): 580 = exp { } (6) Fig. 4 is used to get the relevat values. = JK, = JK ad = 6 K. = =.085γ = = 3.06 mjk mol γ = 0.4 mjk g at. Fig. s ad versus temperature for Bi (=3) Fig. 3 s ad versus temperature for l (=3)

5 NYAWERE & KHANNA: PEIFI HEA JUMP AND RANIION EMPERAURE OF UPERONDUOR 63 Fig. 4 s ad versus temperature for bucklig mode for La (=) Fig. 6 s ad versus temperature for bucklig mode of l (=3) able ummary of results. La Ba u O 4. Bi a r u O l aba (uo 3 ) 3 4. La x Ba x uo 4 5.Bi a r u O l aba (uo 3 ) 3 / γ (mjk gat) V(O) (states/ / / ev.u.atom) Fig. 5 s ad versus temperature for bucklig mode for Bi(=3) V(O)(states/ev.atom) = 0.634(states/ev.u.atom), / =.085. he specific heat jump is 3.3 mjk mol 3..: Bi ( = 3) Bi a r 3 u 3 O he supercoductig phase of Bi ( = 3) for its bucklig mode state is give i Eq. (7): 580 = exp { } (7) From Fig. (5), = JK, =.07 4 JK ad = 5 K. = = 3.03γ = = 0.33 mjk mol γ = 6.3 mjk g at. V(O)(states/ev.atom) = 0.87(states/ev.u.atom), 4.0 = Hece, specific heat jump = = 9. mjk mol. 3..3: l ( = 3) l a Ba (uo 3 ) 3 Eq. (8) is the derived equatio for the supercoductig specific heat for l ( = 3) for bucklig mode: 580 = exp { } (8) From Fig. (6), = JK, =.4 4 JK ad = 9.8 K. = =.083γ = =.04 mjk mol γ = mjk g at. V(O)(states/ev.atom) =.66(states/ev.u.atom),.08 =. pecific heat jump is 3.03 mjk mol.

6 63 INDIAN J PURE & APPL PHY, VOL 49, EPEMBER 0 4 Discussio able presets the summary of the calculatios. he electroic specific heat 9 decreases expoetially at temperature < ad vaishes at << without ay residual specific heat. his characteristic is displayed i both low ad high frequecy modes. From the preset study, it is clear that oly La Ba u O 4 has the highest desity of state of 3.75, the rest of the compouds have lower desity of states compared to as give i Ref. (7). his low desity of states 9 idicates few paired carriers close to the Fermi surface i the uo plaes. La Ba u O 4 has desity of states withi the experimetal values of YBO. his may be as a result of two u carriers istead of three i YBO. I B theory, specific heat jump A=.43. For three compouds, we have 0.33<A<.9. Large A is associated to strog couplig via high frequecy phoos ad A< is low frequecy modes of vibratio. It ca the be cocluded that these compouds have both high ad low modes of vibratios. rasitio temperature is proportioal to either bucklig mode or breathig mode. High correspods to high frequecy mode of vibratio. Refereces Mihailovic D, Physica : upercoductivity, B4(-3) (99) Haze R M, Physical Properties of High-c supercoductors, World cietific, igapore, A Juod Physical Properties of High emperature upercoductors II, World cietific, igapore M KArap Kirui Aharmoic apical oxyge vibratio i high-c superco-ductors DPhil thesis Moi Uiversity, Eldoret, Keya, Khaa KM & Kirui M Karap, Idia J Pure & Appl Phys, 40() (00) Kozowski A, arawski Z, Koodziejczyk A, hmist J, et al. Physica : upercoductivity, 84(98) (99) Plakida N M, High-temperature supercoductivity: experimet ad theory, priger-verlag, 995, pp NyawerePWO pecific Heat Jump i High-c upercoductors MPhil hesis Moi Uiversity, Eldoret, Keya, Khaa K M, Karap M, Kirui W, akwa P K, orogey K Y, Rotich Ay-odo & Nyawere P W O, Idia J Pure & Appl Phys, 45() (007) 99. heg A M & Herma Z Z, Nature, (-3) (998) 33.