36

TRANSITION TEMPERATURE AND SPECIFIC HEAT OF SINGLEBAND SUPERCONDUCTING SmOFeAs

Shamin Masih, Piyush Masih and Sarita Khandka

ABSTRACTIron pnictides are unconventional systems with multiband nature. The itinerant electrons play a crucial role in studying the thermodynamicalproperties. Considering the multiband structure of pnictides, expressions for TC and specific heat are derived using model Hamiltonianand Green's Function technique. The order parameter vs temperature curve gives TC for the single band model. The TC is found to be 11Kfor the compound SmOFeAs which is less than the actual TC of SmOFeAs. This low value of TC may result because only single band isconsidered. Considering multiband may result in increase in TC. In the specific heat vs temperature curve, jump is seen at 11K. The jumpis found to be 1.5x10-5 eV/atom K which is not in accordance with the BCS model, thus signifying that pnictides are unconventional innature.

Keywords: Iron pnictides, thermodynamical properties, superconductivity, specific heat.

Ph.D. Scholar, Assistant Professor, Associate ProfessorDepartment of Physics, Faculty of Science

Sam Higginbottom University of Agriculture, Technology And Sciences, Allahabad – 211007 (U.P.)

INTRODUCTIONThe phenomenon of complete dissappearence of resistivity in

a material is termed as superconductivity and was discovered in1911 by Heike Kamerlingh Onnes in mercuryat 4.2 K. Thischaracteristic temperature termed as the critical temperature TC isunique for a material. Ongoing research in this field is trying tobring superconductivity to room temperature. For this, variouskinds of superconductors are being developed and studied.Extensive research has been able to answer several queries,though many remain still unanswered.

The decade old class of superconductors are iron basedsuperconductors called iron pnictides with the general formulaLnOFeAs (Ln=La, Ce, Sm, Gd, Nd, Pr). These are reportedly thenewest class of superconductors discovered in the year 2008 byHideo Hosono at the Tokyo Institute of Technology. LaO1-xFxFeAs was found to be superconducting below 26K. Uponreplacing lanthanum in this compound with samarium has shownto superconduct at 55 K.

With the discovery of copper based superconductors in 1986and several other compounds over the years, one wouldn't haveanticipated a compound containing iron could also showsuperconductivity. Meissner effect, discovered in1933, became aprerequisite for a material to be superconducting, thereby makingone to believe that superconductivity and magnetic fields arehostile to each other. And with iron as the most magnetic material,this class of iron pnictides has amused scientists worldwide. Likethe cuprates, the pnictides are layered in nature and showcoexistence of superconductivity and antiferromagnetism intheir phase diagram. The d electrons in pnictides play asignificant role. Configuration of Fe atom shows that there arefour unpaired d electrons that hybridize with the three unpairedp electrons of arsenic demanding a multiband description.

Pnictides stand second after the cuprates to show high valuesof TC. The maximum TC seen so far in pnictides is 56K. Such highvalues of TC in this newly discovered class of superconductors

has gained the interest of physicists in understanding themechanism behind these materials.

Thus pnictides being multiband structures, various bands arefound at the Fermi surface due to overlapping of orbitals.Studying pnictides as a function of the number of bands helpsin studying various properties of the compound.

In view of this, the present study is carried out. In this paper,a model Hamiltonian is proposed for the compound SmOFeAsbased on the multiband approach. The model Hamiltonian usesitinerant nature of electrons to study the thermodynamicalproperties of the compound. The paper aims to study theoreticallythe transition temperature TC and specific heat of single bandsuperconducting SmOFeAs.

MATHEMATICAL TECHNIQUE ANDFORMULATION

In the present theoretical work, double-time retarded Green'sfunction technique given by Zubarev is used as a mathematicaltool for investigation of superconducting order parameter andspecific heat. The model Hamiltonian for the multiband ironpnictide compound under study is described as:

In equation (1), the first term represents energy of itinerantelectrons. The second term denotes the intraband interaction term.Vmm is the intraband potential. The third term is the interbandinteraction term. It represents pair hopping between hole andelectron pockets, that is tunnelling between the bands. Vmn isthe interband potential. m is band index, k is wave vector and ?is spin index for fermions.

The Allahabad Farmer, Vol. LXXIV, No. 3, July - September, 2018

Received 04-09-2018Accepted 14-09-2018

37

In the present analysis, considering the two Green's functions

Here r and s are the band index and q denotes wave vector.Using the first Green function (2a), the equation of motion is

expressed as:

…(3)

Using the second Green function (2b), the second equationof motion is written as:

where the order parameters are defined as:

is number factor representing number of charge per unitvolume.

Solving for one band model, using the two equations of motion(3) and (4) and substituting r = s = 1, two equations are obtained:

…….(5a)

…….(5b)

According to the law of conservation of energy, = and taking and and rewriting bothequations,

……(6a)

....…(6b)

Solving the two equations (6a) and (6b), G2 is obtained:

……(7)

Where,

The superconducting order parameter is defined as

….(8)

The correlation function is related to G2 as:

…..(9)

Where, = -1 for fermions, k is Boltzmann constant and T isabsolute temperature in kelvin.

Equation (9) is rewritten as:

The expression of superconducting order parameter isobtained by substituting equation (10) in equation (8):

...(11)

Taking 11 = V11 11 Converting summation into integration with cut off

energy from the Fermi level and substituting as :

....(13)

Tc can be expressed as:

...(14)

For one band model, the electronic specific heat is defined as:

...(15)

Correlation function is related to G1 as:

To find G1, solving simultaneously equations (6a) and (6b),G2 is obtained:

……. (17)

Using expression of G2 and substituting in equation (6a), G1 isobtained.

…....(18)

Where

Equation (16), correlation function is calculatedas:

....(19)

The electronic specific heat for one band model, defined as inequation (15) comes out as:

..(20)

Converting summation into integration,

....(21)

RESULTS AND DISCUSSIONThe superconducting order parameter is a measure of the

binding energy of Cooper pair. Its variation is studied withtemperature.

Shamin Masih, Piyush Masih and Sarita Khandka

38

Figure 1: Superconducting order parameter 11 (meV) versustemperature T (K) for 1 band model

Using equation (14), the variation of superconducting orderparameter 11 versus temperature T, for one band model is shownin figure 1. The figure shows that with increasing temperature,the order parameter decreases and at = 0, T =Tc. This is theusual behaviour of vs T and hence justified. From figure, theTc for one band model comes out to be 11 K. During calculation,the values of D = 0.05, N = 1027, NoV11= 0.24.

The specific heat is the amount of heat per unit mass requiredto raise the temperature by one unit. Its variation is studied withtemperature.

in TC. Studying pnictides provides scholars the much neededimpetus to carry research in this highly promising field. Thepresent research work can be extended to study a multibandmodel and other thermodynamical properties of pnictides. Thespecific heat jump in this compound does not show BCSbehaviour and hence can be explained by another mechanism.

REFERENCESAswathy, P.M., Anooja, J.B., Sarun, P.M. and Syamaprashad, U.(2010) An overview on iron based superconductors.

Superconductor science and technology. 23 (2010)073001 (20pp) doi: 10.1088/0953-2048/23/7/073001.

Andersen O.K. and Boeri L. (2011) On the multi-orbital bandstructure and itinerant magnetism of iron-basedsuperconductors. Annalen der physiks. doi.org/10.1002/and p.201000149.

Bednorz, J. G. & Müller, K. A. (1986) Possible high Tcsuperconductivity in the Ba-La-Cu-O system. CondensedMatter Zeitschrift for Physik B for 9 Springer- Verlag. Z.Phys. B - Condensed Matter 64, 189-193 (1986).

Cvetkovic, V. and Tesanovic, Z. (2009) Valley density- wave andmultiband superconductivity in iron-based pnictidesuperconductors. Physical Review B 80, 024512 (2009)

Ding, L., He, C., Dong, J. K., Wu, T., Liu, R. H., Chen, X. H., 2and Li1, S. Y. (2008) Specific heat of the iron-based high-Tc

superconductor SmO1-x FxFeAs.Phys. Rev.B77,180510(R) (2008): DOI:10.1103/PhysRevB.77.180510.

Hosono H. (2011) Iron Pnictide Superconductors: Discovery andCurrent Status. Material Matters Volume 4 Article 2.

Ishida, Kenji., Nakai, Yusuke., and Hosono, Hideo.(2009) ToWhat Extent Iron-Pnictide New Superconductors HaveBeen Clarified: A Progress Report. J. of the PhysicalSociety of Japan. J. Phys. Soc. Jpn. 78, 062001 (2009)

Johnston David C. (2010) The puzzle of high temperaturesuperconductivity in layered ironpnictides andchalcogenides. Advances in Physics, 59:6, 803-1061.Vol.59, No. 6, November, 803-1061. doi.org/10.1080/00018732.2010.513480.

Laad, M. S. & Craco, L. (2009) Theory of Multi-BandSuperconductivity in Iron Pnictides. Vol. 103, Issue. 1 -3 July DOI:10.1103/PhysRevLett.103.017002.

Meissner W., Oschenfeld R. (1933) Einneuer Effektbei Eintrittder Supraleitfähigkeit.Naturwiss, 21 (44): 787-788.

Maple, M.B. (1995) Interplay between superconductivity andmagnetism. Physica B 215(1995) 110.

Podolsky, D. (2012) Untangling the Orbitals in Iron-BasedSuperconductors. Phys. Rev. X 2, 021009 (2012) doi:10.1103/Physics.5.61.

Singh, D. J (2012) Magnetism and Superconductivity in IronPnictides. acta physica polonica a. the EuropeanConference Physics of Magnetism 2011 (PM'11), Poznan,June 27-July 1, 2011 Vol. 121 (2012) No. 5-6.

Wang Z., Luo H., Koshelev A., Xie R., Welp U., Kwok W.K.,Cheng P., Gang M. (2009) "Specific heat and phase diagrams of

single crystal iron pnictide superconductors". Physica C Superconductivity (Impact Factor:1.11). 05/2009;469(9):575-581.

Zubarev, D. N. (1960) Double-Time green functions in statisticalphysics. American Institute of Physics. Soviet PhysicsUspekhi, Volume 3, Number 3. Uspekhi 3:320-345.

Figure 2: Specific Heat (eV/atom-K) versus temperatureT (K) for 1 band model

Using equation (21), the variation of specific heat withtemperature T for one band model is shown in figure 2. Initiallythe specific heat for superconductor is less than the specific heatfor normal conductor, it then suddenly increases and then dragsdown at a particular temperature. This temperature at which thespecific heat drags down is the Tc of the system which is 11 K asseen in the figure. At Tc, the curve shows a jump. The jump isfound to be 1.5 * 10-5 eV/atom K here which is contrary as seenin the BCS model (Johnston 2010), thus signifying that pnictidesare unconventional superconductors.

SUMMARY AND CONCLUSIONThe two thermodynamical properties, superconducting order

parameter and specific heat are studied using one band modelapproach. The low value of TC is a result of consideration ofsingle band only. Considering multiband may result in increase

The Allahabad Farmer, Vol. LXXIV, No. 3, July - September, 2018