theoretical modeling of iron-based superconductors
DESCRIPTION
A summer project report by Harsh Purwar, student, Indian Institute of Science Education and Research, Kolkata done in RRCAT Indore through Young Scientist Research Programme - 2010 under the humble guidance of Dr. Haranath Ghosh (Scientific Officer, LASER Physics Applications Division).TRANSCRIPT
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
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Department of Atomic Energy (DAE), Raja Ramanna Centre for Advanced Technology, Indore
THEORETICAL MODELING OF IRON BASED SUPERCONDUCTORS
Project Report
Submitted to Dr. Haranath Ghosh
LASER Physics Applications Division Department of Atomic Energy
Raja Ramanna Centre for Advanced Technology, Indore Indore, Madhya Pradesh, India
Through Young Scientist Research Program - 2010
By Harsh Purwar
3rd Year Student, Integrated M.S. Indian Institute of Science Education and Research, Kolkata
Mohanpur Campus, BCKV, Mohanpur, Nadia, West Bengal, India
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Certificate
It is certified that the work in this project report entitled βTheoretical Modeling of
Iron based Superconductorsβ, by Harsh Purwar student of Indian Institute of Science
Education and Research, Kolkata has been carried out under my supervision from 17th May
2010 to 9th July 2010 and is not submitted anywhere else for publication till date.
Dated:
Place:
(Dr. Haranath Ghosh) Senior Scientific Officer (F) LASER Physics Applications Division Department of Atomic Energy (DAE) Raja Ramanna Centre for Advanced Technology, Indore Indore, Madhya Pradesh, India
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Acknowledgement
I would like to express my gratitude to my mentor honorable Dr. Haranath Ghosh for
his guidance, personal attention and motivation throughout the project. I would also like to
thank Dr. Surya Mohan Gupta and Dr. Kanwal Jeet Singh Sokhey (Coordinators YSRP β 2010)
for giving me this opportunity by selecting me for this program and for all the support and
facilities that they have provided me during my stay in RRCAT, Indore.
I would also like to thank other YSRP students for carrying out long discussions that
helped me in developing critical thinking, understanding and completing my project
successfully.
I thank my parents and sisters for their continued effort, encouragement and moral
support.
(HARSH PURWAR)
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Contents
CERTIFICATE ...................................................................................................................... 2
ACKNOWLEDGEMENT ........................................................................................................ 3
CONTENTS ......................................................................................................................... 4
INTRODUCTION ................................................................................................................. 5
SUPERCONDUCTIVITY β GENERAL DISCUSSION .......................................................................................................... 5 FEW CHARACTERISTIC PROPERTIES OF SUPERCONDUCTORS ......................................................................................... 5 CRITICAL TEMPERATURE ....................................................................................................................................... 6 CLASSIFICATION OF SUPERCONDUCTORS .................................................................................................................. 6 IRON-BASED SUPERCONDUCTORS............................................................................................................................ 7 MECHANISM OF SUPERCONDUCTIVITY .................................................................................................................... 8 LIST OF RECENTLY DISCOVERED SCβS ....................................................................................................................... 8
THEORETICAL MODELING OF FE-BASED SCβS .................................................................... 10
WHY 2 OR 3 BAND MODEL? ............................................................................................................................... 10 TWO ORBITAL PER SITE MODEL ........................................................................................................................... 10 DISPERSION RELATIONS FOR TWO BAND MODEL ..................................................................................................... 10 FERMI SURFACES ............................................................................................................................................... 13 BAND STRUCTURES ............................................................................................................................................ 14 EVOLUTION OF FERMI SURFACE WITH CHEMICAL POTENTIAL ..................................................................................... 15 CALCULATION OF SPIN SUSCEPTIBILITY IN NORMAL STATE ......................................................................................... 18
STUDY OF SPIN DENSITY WAVES ...................................................................................... 21
GENERAL DISCUSSION ON SPIN DENSITY WAVES ..................................................................................................... 21 NESTING OF FERMI SURFACE ............................................................................................................................... 22 DERIVATION OF METALLIC SPIN DENSITY WAVE STATE IN FE β BASED SCβS .................................................................. 23 COUPLED STATE (SDW + SC) ............................................................................................................................. 26 FERMI SURFACES IN THE SPIN DENSITY WAVE STATE ................................................................................................ 27 SOLUTION OF GAP EQUATION IN SDW STATE ........................................................................................................ 27 CALCULATION OF SPIN SUSCEPTIBILITY IN SPIN DENSITY WAVE STATE ......................................................................... 29
WORKS CITED .................................................................................................................. 30
APPENDIX ........................................................................................................................ 32
FORTRAN CODES ............................................................................................................................................ 32 NOTES / CORRECTIONS / COMMENTS ................................................................................................................... 58
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Introduction
SUPERCONDUCTIVITY β GENERAL DISCUSSION Superconductivity is an electrical resistance of exactly zero which occurs in certain materials below a characteristic temperature ππ . It was discovered by Heike Kamerlingh Onnes in 1911. It is also characterized by a phenomenon called the Meissner effect (Figure 1), the ejection of any sufficiently weak magnetic field from the interior of the superconductor as it transitions into the superconducting state. The occurrence of the Meissner effect indicates that superconductivity cannot be understood simply as the idealization of "perfect conductivity" in classical physics.
Figure 1: A magnet levitating above a high-temperature superconductor,
cooled with liquid nitrogen. Persistent electric current flows on the surface of the superconductor, acting to exclude the magnetic field of the magnet
(the Faraday's law of induction). This current effectively forms an electromagnet that repels the magnet.
The electrical resistivity of a metallic conductor decreases gradually as the temperature is lowered. However, in ordinary conductors such as copper and silver, this decrease is limited by impurities and other defects. Even near absolute zero, a real sample of copper shows some resistance. Despite these imperfections, in a superconductor the resistance drops abruptly to zero when the material is cooled below its critical temperature. An electric current flowing in a loop of superconducting wire can persist indefinitely with no power source.
Superconductivity occurs in many materials: simple elements like tin and aluminum, various metallic alloys and some heavily-doped semiconductors. Superconductivity does not occur in noble metals like gold and silver, or in pure samples of ferromagnetic metals.
In 1986, it was discovered that some cuprate-perovskite ceramic materials have critical temperatures above 90 K (β183.15 Β°C). These high-temperature superconductors renewed interest in the topic because of the prospects for improvement and potential room-temperature superconductivity. From a practical perspective, even 90 K is relatively easy to reach with the readily available liquid nitrogen (boiling point 77 K), resulting in more experiments and applications.
FEW CHARACTERISTIC PROPERTIES OF SUPERCONDUCTORS The characteristic properties of metals in the superconducting state appear highly anomalous when regarded from the point of view of the independent electron approximation. The most striking features (1) of a superconductor are:
A superconductor can behave as if it had no measurable DC electrical resistivity. Currents have been established in superconductors which, in the absence of any driving field, have nevertheless shown no discernible decay for as long as people have had the patience to watch.
A superconductor can behave as a perfect diamagnet. A sample in thermal equilibrium in an applied magnetic field, provided the field is not too strong, carries electrical surface currents. These
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currents give rise to an additional magnetic field that precisely cancels the applied magnetic field in the interior of the superconductor.
A superconductor usually behaves as if there were a gap in energy of width 2Ξ centered about the Fermi energy, in the set of allowed one-electron levels. Thus an electron of energy ν can be accommodated by (or extracted from) a superconductor only if ν β νπΉ exceeds Ξ. The energy gap Ξ increases in size as the temperature drops, leveling off to a maximum value Ξ 0 at very low temperatures.
CRITICAL TEMPERATURE The transition to the superconducting state is a sharp one in bulk specimens. Above a critical temperature ππ the properties of the metal are completely normal; below ππ , superconducting properties are displayed, the most dramatic of which is the absence of any measurable DC electrical resistance. Measured critical temperatures range from a few milli-degrees Kelvin up to a little over 50 K. The corresponding thermal
energy ππ΅π, varies from about 10β7 eV up to a few thousandths of an electron volt.
CLASSIFICATION OF SUPERCONDUCTORS Superconductors can be classified in accordance with several criteria that depend on our interest in their physical properties, on the understanding we have about them, on how expensive is cooling them or on the material they are made of. Based on their physical properties:
Type I superconductors: Those having just one critical field, π»π , and changing abruptly from one state to the other when it is reached.
Type II superconductors: Having two critical fields, π»π1 and π»π2
, being a perfect superconductor
under the lower critical field (π»π1) and leaving completely the superconducting state above the
upper critical field (π»π2), being in a mixed state when between the critical fields.
Based on the understanding we have about them:
Conventional superconductors: Those that can be fully explained with the BCS theory or related theories.
Unconventional superconductors: Those that failed to be explained using such theories like iron based superconductors.
This criterion is important, as the BCS theory is explaining the properties of conventional superconductors since 1957, but on the other hand there have been no satisfactory theory to explain fully unconventional superconductors. In most of cases type I superconductors are conventional, but there are several exceptions as Niobium, which is both conventional and type II. Based on their critical temperatures:
Low-temperature superconductors or LTS: Those whose critical temperature is below 77K.
High-temperature superconductors or HTS: Those whose critical temperature is above 77K. This criterion is used when we want to emphasize whether or not we can cool the sample with liquid nitrogen (whose boiling point is 77K), which is much more feasible than liquid helium (the alternative to achieve the temperatures needed to get low-temperature superconductors). Based on the Material:
Some Pure elements, such as lead or mercury (but not all pure elements, as some never reach the superconducting phase).
Some allotropes of carbon, such as fullerenes, nanotubes, diamond or intercalated graphite.
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Most superconductors made of pure elements are type I (except niobium, technetium, vanadium, silicon and the above mentioned carbons).
Alloys, such as Niobium-titanium (NbTi), whose superconducting properties where discovered in 1962.
Ceramics, which include the YBCO family, which are several yttrium-barium-copper oxides, especially YBa2Cu3O7. They are the most famous high-temperature superconductors. Magnesium di-boride (MgB2), whose critical temperature is 39 K (2), being the conventional superconductor with the highest known ππ .
IRON-BASED SUPERCONDUCTORS Iron-based superconductors (sometimes misleadingly called iron superconductors) are chemical compounds (containing iron) with superconducting properties. In 2008, led by recently discovered iron pnictide compounds (originally known as oxypnictides also called ferropnictides), they were in the first stages of experimentation and implementation. (Previously most high-temperature superconductors were cuprates and being based on layers of copper and oxygen sandwiched between other (typically non-metal) substances). This new type of superconductors is based instead on conducting layers of iron and a pnictide (typically arsenic) and seems to show promise as the next generation of high temperature superconductors. Much of the interest is because the new compounds are very different from the cuprates and may help lead to a theory of non-BCS-theory (Bardeen, Cooper and Schrieffer Theory) superconductivity. It has also been found that some iron chalcogens super-conduct; for example, doped πΉπππ can have a critical temperature ππ of 8 K at normal pressure, and 27 K under high pressure. A subset of iron-based superconductors with properties similar to the oxypnictides, known as the 122 Iron Arsenides, attracted attention in 2008 due to their relative ease of synthesis. The oxypnictides such as πΏπππΉππ΄π are often referred to as the β1111β pnictides. (3)
Crystal structure of LaOFeAs (left) and FeAs layer (right) (4)
The crystalline material πΏπππΉππ΄π , stacks iron and arsenic layers, where the electrons flow, between planes of lanthanum πΏπ and oxygen π . Replacing up to 11 percent of the oxygen with fluorine improved the compound β it became superconductive at 26 K, the team reports in the March 19, 2008 Journal of the American Chemical Society. Subsequent research from other groups suggests that replacing the lanthanum in LaOFeAs with other rare earth elements such as cerium, samarium, neodymium and praseodymium leads to superconductors that work at 52 K. Compounds such as ππ2πππΉπππ3 (discovered in 2009) are referred to as the β22426β family. (as πΉππππ2πππ3).
Fe
La
O
As
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Also in 2009, it was shown that un-doped iron pnictides had a magnetic quantum critical point deriving from competition between electronic localization and itinerancy.
MECHANISM OF SUPERCONDUCTIVITY Modern conventional superconductors work at temperatures between absolute zero and 100 K; by comparison, certain cuprates became superconducting at temperatures exceeding 163 K. Some people have speculated that in cuprate superconductors the electrons are paired due to spin fluctuations that occur around the copper ions however other models have also been proposed and at this time there is no consensus on the actual mechanism for cuprate superconductivity. There are claims that in iron based superconductors orbital fluctuations are far more essential. However, as in the cuprates the mechanism for high temperature superconductivity remains unknown at this time. On the other hand, the spin fluctuations that could glue together cuprate electrons might not be enough for those in the iron-based materials. Instead orbital fluctuations β or variations in the location of electrons around atoms β might also prove crucial, Haule speculates. In essence, the iron-based materials give more freedom to electrons than cuprates do when it comes to how electrons circle around atoms. Orbital fluctuations might play important roles in other unconventional superconductors as well, such as ones based on uranium or cobalt, which operate closer to absolute zero, Haule conjectures. Because the iron-based superconductors work at higher temperatures, such fluctuations may be easier to research. However, spectroscopic measurements have shown that Haule's calculations do not describe these materials accurately. In particular the correlated approach that he used predicts a Hubbard band that is not seen. Further work is needed to unravel the properties of these materials. Transition temperatures of some of the recently discovered iron based superconductors (from 2006 to 2009) are listed in the next sub-section.
LIST OF RECENTLY DISCOVERED SCβS Following is a list of some of the recently discovered iron β based superconductors (2006 β 2009) with their reported critical temperatures and the group that discovered it.
S. No.
Compound (Powder/Single
crystals)
Critical Temperature π»π
References
1 πΏπππΉππ ~5 K Y. Kamihara et al., J. Am. Chem. Soc. 128,
10012 (2006)
2 πΏπππππ ~3 K T. Watanabe et al., Inorg. Chem. 46, 7719
(2007)
3 πΏπ π1βπ₯πΉπ₯
β πΉππ΄π π₯ = 0.05 β 0.12
26 K Y. Kamihara et al., J. Am. Chem. Soc. 130,
3296 (2008)
4 πΏπ π1βπ₯πΆππ₯2+ πΉππ΄π 0 K
Y. Kamihara et al., J. Am. Chem. Soc. 130, 3296 (2008)
5 πΏπ π1βπ₯πΉπ₯ πππ΄π 2.75 K π₯ = 0 3.8 K π₯ = 0.1
Z. Li et al., arXiv: 0803.2572
6 πΏπ1βπ₯πππ₯ ππππ΄π 2.75 K π₯ = 0
3.7 K π₯ = 0.1 β 0.2 L. Fang et al., arXiv: 0803.3978
7 πΏπ1βπ₯πππ₯ ππΉππ΄π 25 K π₯ = 0.13 H.H. Wen et al., EPL 82, 17009 (2008)
8 πΆπ π1βπ₯πΉπ₯ πΉππ΄π 41 K π₯ = 0.2 G.F. Chen et al., arXiv: 0803.3790
9 ππ π1βπ₯πΉπ₯ πΉππ΄π 52 K π₯ = 0.11 Z.-A. Ren et al., 0803.4283
10 ππ π1βπ₯πΉπ₯ πΉππ΄π 52 K π₯ = 0.11 Z.-A. Ren et al., 0803.4234
11 πΊπ π1βπ₯πΉπ₯ πΉππ΄π 36 K π₯ = 0.17 P. Cheng et al., arXiv: 0804.0835
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12 ππ π1βπ₯πΉπ₯ πΉππ΄π 55 K π₯ = 0.1 β 0.2 Z.-A. Ren et al., 0804.2053
R.H. Liu et al., arXiv: 0804.2105
13 πΈπ’,ππ π1βπ₯πΉπ₯ πΉππ΄π No stable πππΆπ’πππ΄π
structure G.F. Chen et al., arXiv: 0803.4384
14 π΅ππΉπ2π΄π 2 ~5 K (SDW at 140 K) M. Rotter, Marcus Tegel et al., PRB
78,020503 (2008)
15 π΅π1βπ₯πΎπ₯πΉπ2π΄π 2
π₯ = 0.4 ~38 K, πβπΆπ2ππ2
M. Rotter, Marcus Tegel, Dirk Johrendt, PRL 101, 107006 (2008)
16 πππΉπ2π΄π 2 C.Krellner et al., arxiV 0806.1043
17 πππ₯πΎ1βπ₯πΉπ2π΄π 2 π₯ = 0.5 β 0.6
36.5 K Kalyan Sasmal et al., arXiv 0806.1301
18 πππ₯πΆπ 1βπ₯πΉπ2π΄π 2 π₯ = 0.5 β 0.6
37.2 K Kalyan Sasmal et al., arXiv 0806.1301
19 πΎπΉπ2π΄π 2 3.8 K
20 πΆπ πΉπ2π΄π 2 2.6 K
21 πΆππΉπ2π΄π 2 170 K (Structural,
Magnetic Transition) N. Ni,S. Nandi et al ., PRB 78, 014523 (2008)
22 πΉπππ 18 K Y. Mizuguchi, F. Tomioka, S. Tsuda, T.
Yamaguchi and Y. Takano, arXiv 0807.4315
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Theoretical Modeling of Fe-Based SCβs
WHY 2 OR 3 BAND MODEL? The high transition temperatures and the electronic structures of the Fe-pnictide superconductors suggest that the pairing interaction is of electronic origin (5). First principle band structure calculations suggest that the superconductivity in these materials is associated with the Fe-pnictide layer, and that the density of state near the Fermi level gets its maximum contribution from the Fe β 3d orbitals namely - 3ππ₯π¦ , 3ππ¦π§ ,
3ππ§π₯ , 3ππ§2 and 3ππ₯2βπ¦2 .
Several tight binding models for the band structure have been proposed. Cao et al. (6) used 16 localized Wannier functions to construct a tight binding effective Hamiltonian. Kuroki et al. (7) have used a 5 orbital tight binding model to fit the band structure near the Fermi energy. Others have introduced generic two band models. However the relationship of these latter models to the multiple Fermi surface electron and hole pockets found in LDA calculations was unclear until S. Raghu et al. (8) suggested a minimal 2-band model that exhibits a Fermi surface which looks similar topologically to that obtained from these calculations.
TWO ORBITAL PER SITE MODEL This model has two orbitals ππ¦π§ ,ππ₯π§ per site on a two dimensional square lattice of iron. As mentioned
earlier in iron arsenide layer iron atoms and arsenide atoms are not on the same plane. All iron atoms form a plane and the arsenide atoms lie slightly above and below the plane as shown below.
DISPERSION RELATIONS FOR TWO BAND MODEL The tight binding part of the Hamiltonian for two orbital per site model can be written as,
π»0 = ππ ,πβ π+ π β π π + πβ π π3 + ππ₯π¦ π π1 ππ ,π
π ,π
where the wave function ππ ,π is given by,
ππ ,π = πΆππ¦π§ ,π ,π
πΆππ₯π§ ,π ,π
ππ ,π is the annihilation operator for spin π electrons in the two orbitals and similarly ππ ,πβ is the creation
operator for spin π electrons in the two orbitals. And,
πΉπ atoms (all in a plane)
π΄π atom (slightly below the plane of πΉπ atoms)
π΄π atom (slightly above the plane of πΉπ atoms)
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π+ π = β π‘1 + π‘2 cosππ₯ + cos ππ¦ β 4π‘3 cos ππ₯ cos ππ¦
πβ π = π‘1 β π‘2 cosππ₯ β cosππ¦
ππ₯π¦ π = β4π‘4 sin ππ₯ sinππ¦
1 is the 2 Γ 2 identity matrix, π3 and π1 and the standard Pauliβs matrices given by,
π1 = 1 00 β1
, π3 = 0 11 0
π‘π βs are the various tight binding parameters as illustrated in Figure 2.
Figure 2: A schematic showing the hopping parameters of the two-orbital π ππ β π ππ model on a square lattice.
Here ππ is a near neighbor hopping between π β orbitals, ππ is a near neighbor hopping between π β orbitals, ππ is the second neighbor hopping between similar orbitals and ππ is the second neighbor hopping between different
orbitals.
So the Hamiltonian can be rewritten as,
π»0 = πΆππ¦π§ ,π ,πβ πΆππ₯π§ ,π ,π
β
π+ π β π 0
0 π+ π β π +
πβ π 0
0 βπβ π
π ,π
+ 0 ππ₯π¦ π
ππ₯π¦ π 0
πΆππ¦π§ ,π ,π
πΆππ₯π§ ,π ,π
= πΆππ¦π§ ,π ,πβ πΆππ₯π§ ,π ,π
β
π+ π + πβ π β π ππ₯π¦ π
ππ₯π¦ π π+ π β πβ π β π
πΆππ¦π§ ,π ,π
πΆππ₯π§ ,π ,π
π ,π
= ππ ,πβ π πππ ,π
π ,π
Using Bogoliubov transformation we have,
ππ ,π = πΆππ¦π§ ,π ,π
πΆππ₯π§ ,π ,π =
π’π βππ£πππ£π π’π
πΌπ ,π
π½π ,π
β πΆππ¦π§ ,π ,π = π’ππΌπ ,π β ππ£ππ½π ,π & πΆππ₯π§ ,π ,π = ππ£ππΌπ ,π + π’ππ½π ,π
β πΆππ¦π§ ,π ,πβ = π’ππΌπ ,π
β β ππ£ππ½π ,πβ & πΆππ₯π§ ,π ,π
β = ππ£ππΌπ ,πβ + π’ππ½π ,π
β
Now rewriting the Hamiltonian above in terms of the transformed operators πΌπ ,π and π½π ,π we get,
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π»0 = π+ π + πβ π β π π’ππΌπ ,π β β ππ£ππ½π ,π
β π’ππΌπ ,π β ππ£ππ½π ,π
π ,π
+ π+ π β πβ π β π ππ£ππΌπ ,πβ + π’ππ½π ,π
β ππ£ππΌπ ,π + π’ππ½π ,π
π ,π
+ ππ₯π¦ π ππ£ππΌπ ,πβ + π’ππ½π ,π
β π’ππΌπ ,π β ππ£ππ½π ,π
π ,π
+ ππ₯π¦ π π’ππΌπ ,π β β ππ£ππ½π ,π
β ππ£ππΌπ ,π + π’ππ½π ,π
π ,π
Simplifying and collecting terms we have,
π»0 = π+ π + πβ π β π π’π2 + π+ π β πβ π β π π£π
2 + 2πππ₯π¦ π π’ππ£π πΌπ ,πβ πΌπ ,π
π .π
+ π+ π + πβ π β π π£π2 + π+ π β πβ π β π π’π
2 β 2πππ₯π¦ π π’ππ£π π½π ,πβ π½π ,π
π ,π
+ β2ππβ π π’ππ£π + ππ₯π¦ π π’π2 β π£π
2 πΌπ ,πβ π½π ,π + π½π ,π
β πΌπ ,π
π ,π
Now we demand the coefficient of the cross terms (last term) equal to zero to get, 2ππβ π π’ππ£π = ππ₯π¦ π π’π
2 β π£π2
And from the normalization condition we already have, π’π
2 + π£π2 = 1
Solving these two equations for π’π and π£π yields,
π’π 2 =
1
2 1 Β±
πβ π
πβ2 π + ππ₯π¦2 π
π£π 2 =
1
2 1 β
πβ π
πβ2 π + ππ₯π¦2 π
Also,
2π’ππ£π = ππ₯π¦ π
πβ2 π + ππ₯π¦2 π
Substituting these values back into the Hamiltonian we get a diagonalized Hamiltonian,
π»0 = π+ π + πβ2 π + ππ₯π¦2 π β π πΌπ ,πβ πΌπ ,π
π ,π
+ π+ π β πβ2 π + ππ₯π¦2 π β π π½π ,πβ π½π ,π
π ,π
Representing these eigenvalues as,
πΊπΒ± = π+ π Β± πβπ π + ππππ π β π
β π»0 = νπ+πΌπ ,π
β πΌπ ,π + νπβπ½π ,π
β π½π ,π
π ,π
= πΌπ ,πβ π½π ,π
β νπ
+ 00 νπ
β πΌπ ,π
π½π .π
π ,π
This result can also be interpreted in a slightly different manner; we can label the two energy eigenvalues or dispersion relations as electron and hole dispersions. We select νπ
+ and label it as νππ and similarly νπ
β as
νπβ . Following is a three dimensional surface plot showing the electron and hole like dispersions in the two
directions ππ₯ and ππ¦ in momentum space.
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Figure 3: Electron and hole like dispersions as a function of π in momentum space for our choice of π = π.ππ.
Upper curve represents πΊππ and lower πΊπ
π.
FERMI SURFACES Fermi surfaces are constant energy surfaces. For two dimensions these can regarded as constant energy contours or plots. Following are the Fermi contours plotted using the dispersions for the normal state derived in the previous section,
νππ = νπ
+ = π+ π + πβ2 π + ππ₯π¦2 π β 0 {Red dots}
νπβ = νπ
β = π+ π β πβ2 π + ππ₯π¦2 π β 0 {Blue dots}
taking the constant energy close to zero (may also be called zero energy plots) in the full (top left), reduced or folded (top right) and reduced & rotated (bottom center) Brillouin zone with chemical potential π = 1.45 and other tight binding parameters π‘1 = β1.0 eV, π‘2 = 1.3 eV and π‘3 = π‘4 = β0.85 eV. Blue colored lines (dots) corresponds to the hole pockets and red corresponds to electron pockets.
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The last figure above is the Fermi surface mapped experimentally using the data obtained from Angle Resolved Photo-Electron Spectroscopy (ARPES) which matches the FS obtained from the theoretical calculations shown above. The πΌ, π½ and πΎ surfaces indicated in the ARPES image are the hole (two concentric circles at the center) Fermi pockets (blue) and the electron Fermi pockets (red) shown in the adjacent figure of the FS in the reduced + rotated BZ.
BAND STRUCTURES The corresponding (all parameters remain same) band structures for normal state in the full (top) and reduced (bottom) Brillouin zone are as follows,
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EVOLUTION OF FERMI SURFACE WITH CHEMICAL POTENTIAL As mentioned earlier Fermi surface is a constant or zero in our case energy surface and it is very clear from
the explicit form of the obtained dispersions for νππ and νπ
β that the topology of the FS is very much dependent on the chemical potential π which gives an indirect measure of the amount of electron or hole doping in the system. Following are the Fermi surfaces and band structures for the proposed two band model for various values of chemical potential π .
Fermi surfaces in full Brillouin zone for π = (top left to right) π.ππ, π.ππ, π.ππ; (bottom left to right) π.ππ, π.ππ
and π.ππ. Blue colored lines (dots) correspond to hole pockets πΊππ β π and red to electron pockets πΊπ
π β π .
And these fold to give,
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Fermi surfaces in folded or reduced Brillouin zone for π = (top left to right) π.ππ, π.ππ, π.ππ; (bottom left to
right) π.ππ, π.ππ and π.ππ. Color convention remains same. Corresponding band structures does not change topologically but due to change in the chemical potential the Fermi level changes and it appears as if the whole band structure shifts down (up) as π is increased (decreased).
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Band structures in the full Brillouin zone for same values of chemical potential as mentioned above. Color codes
also remain same.
Similarly in the folded Brillouin zone,
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Band structures in the reduced or folded Brillouin zone for same values of chemical potential as mentioned above.
Color codes also remain same.
CALCULATION OF SPIN SUSCEPTIBILITY IN NORMAL STATE Spin susceptibility π for the tight binding model (8) can be written as,
ππ π‘ π, πΞ© = ππππΞ©Ο TΟSs βq, Ο . St q, 0 π½
0
Due to the existence of two degenerate orbitals in the model, the spin susceptibility also has two orbital indices π and π‘. Here,
ππ π =1
2 ππ πΌ
β π + π π πΌπ½ππ π½ π
π
is the spin operator for the orbital labeled by π . The physical spin susceptibility is given by,
ππ π, πΞ© = ππ π‘ π, πΞ©
π ,π‘
The one loop contribution to the spin susceptibility can be obtained as,
ππ π, πΞ© = βπ
2π Tr πΊ π + π, πππ + πΞ© πΊ π, πππ
π ,ππ
ππ π, πΞ© = β1
2π
π + π, π π, πβ² 2
πΞ© + πΈπ ,π+π β πΈπ β² ,π ππΉ πΈπ ,π+π β ππΉ πΈπ β² ,π
π ,π ,π β²
Here πΈπ ,π , π = Β±1 is the eigenvalue of the upper and lower band of the dispersions derived above
and π, π is the corresponding normalized eigenvector. π is the total no. of integration or summation
points and ππΉ πΈπ ,π is the Fermi distribution function given by,
ππΉ πΈ =1
exp π½πΈ + 1
with π½ = 1 ππ΅π where ππ΅ is the Boltzmann constant and π is the temperature. The eigenvalues have already been calculated in the above sections,
πΈ+1,π = νπ+ = π+ π + πβ2 π + ππ₯π¦2 π β π
πΈβ1,π = νπβ = π+ π β πβ2 π + ππ₯π¦2 π β π
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The eigenvectors were calculated using Wolfram Mathematica and then were normalized to give,
π, +1 = π+1 πβ π + πβ2 π + ππ₯π¦2 π
ππ₯π¦ π , 1
π,β1 = πβ1 πβ π β πβ2 π + ππ₯π¦2 π
ππ₯π¦ π , 1
with π+1 and πβ1 being the normalization constants given by,
π+1 = ππ₯π¦2 π
πβ π + πβ2 π + ππ₯π¦2 π 2
+ ππ₯π¦2 π
πβ1 = ππ₯π¦2 π
πβ π β πβ2 π + ππ₯π¦2 π 2
+ ππ₯π¦2 π
Following plot shows the variation of real part of static π = 0 spin susceptibility with the wave vector π for π = 1.45 at 100 deg. K, where one can see the structure associated with the various nesting points and density of state features. For the particular choice of parameters, the largest value of ππ π, 0 occurs around π = π, 0 , which suggests a transition to an anti-ferromagnetic ordered phase at some critical interaction strength. This is also in agreement with the band structure calculations (9) (10).
Figure 4: Showing variation of real part of static spin susceptibility ππΊ π,π as a function of wave vector (q) for
π = π.ππ at πππ deg. K.
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Figure 5: Variation of Real part of SSS ππΊ with π for three different wave vectors π as indicated in the figure.
As could be seen in the figure above there is a sudden change in the static spin susceptibility for π = 0,0 at π close to 1.2 which corresponds to the onset of the electron Fermi pockets. When the chemical potential is increased further, the Fermi level gets closer to the Van Hove singularity and the SSS increases gradually for all the three wave vectors shown.
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Study of Spin Density Waves
GENERAL DISCUSSION ON SPIN DENSITY WAVES The spin density wave is an anti-ferromagnetic ground state of metals for which the density of the conduction electron spins is spatially modulated. In conventional anti-ferromagnets like πππΉ2 the magnetic moments have opposite orientation and are located at two crystallographic sub-lattices. On the contrary the SDW is a many particle phenomenon of an itinerant magnetism which is not fixed to the crystal lattice. SDW are observed in metals and alloys; most prominent is chromium and its alloys. The SDW also occurs as ground state in strongly anisotropic systems, for example the one dimensional organic conductors. In analogy to the magnetic order of anti-ferromagnets below Neil temperature, the electron gas becomes unstable for temperatures below an ordering temperature πππ·π and enters a collectively ordered ground state of an itinerant anti-ferromagnet. The reason of the instability of the electron gas at the transition to the SDW ground state is the so-called nesting of the Fermi surface. In a metal the density of the conduction electrons with spin β and with spin β is the same everywhere; the spatial variation of the total charge density is given by,
π π = πβ π + πβ π and only reflects the periodicity of the crystal lattice. The development of a SDW violates the translational invariance; now the charge density πβ β has a sinusoidal modulation,
πβ β π =1
2π0 π 1 Β± π0 cosπΈ. π
with π0 the amplitude and πΈ the wave vector of the SDW. The wavelength π = 2π π of SDW is determined by the Fermi surface of the conduction electrons and in general not a multiple of (i.e. commensurate with) the lattice period πΏ. In fact, the ratio ππ πΏ can change with temperature, external pressure, doping and other parameters.
Figure 6: The change of magnetic moment with distance along x direction for the
commensurate anti-ferromagnetism πΈ = π ,π .
For the understanding of a SDW, the nesting of the Fermi surface is essential. This describes the property of the reciprocal (momentum) space to map parts of the Fermi surface with electron or hole character on top of each other by translation with the wave vector π. The case is most obvious for one dimension, where the Fermi surface consists of two parallel planes at Β±ππΉ . In two or three dimensions a complete
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nesting by just a single π-vector is not possible any more, but different parts of the Fermi surface can be mapped by different π vectors in a more or less perfect way. The spatial modulation of the electron spin density leads to a superstructure, and an energy gap 2Ξ π opens at the Fermi energy; the gap value increases with decreasing temperature the same way as the magnetization. The electrical resistivity exhibits a semiconducting behavior below πππ·π . A typical example of a one-dimensional metal which undergoes a SDW transition is the Bechgaard salt πππππΉ 2ππΉ6.
NESTING OF FERMI SURFACE We say that the Fermi surface is completely nested if translating all the points in the Full BZ by the nesting vector, π, 0 here, does not change the FS topologically that is to say, that the translational symmetry of the FS is maintained when it is translated by the nesting vector. This could be restated in terms of electron and hole pockets as, translating the hole pocket by the nesting vector maps it completely on the electron pocket and vice versa thus preserving the symmetry. It can easily be seen that the nested points on the FS are the points which satisfy the following condition with π as nesting vector π, 0 for the 2 band model,
νπ+ππ = νπ
β nested hole pockets
νπ+πβ = νπ
π nested electron pockets
Also these points satisfy, νπβ β 0 or νπ
π β 0 so as to lie on the FS (or zero energy surface).
Showing nesting of Fermi surface for π = π.ππ & π«πΊπ«πΎ = π.ππ eV. Red dots represent the normal FS as
shown in figures above in the FBZ and Blue dots show the translated points of the FS by the nesting vector π,π .
Actual nested points of the Fermi surface in FBZ for π = π .ππ & π«πΊπ«πΎ = π.ππ eV. Red dots represent
the nested electron pockets and blue dots represent the nested hole pockets.
Folding or to be very specific translation of the points outside the 1st BZ boundary by an appropriate vector so as to transfer all of them inside the BZ, which always is not same as folding results in the following pattern of the nested points.
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Figure 7: Nested points of the FS in the RBZ for π = π.ππ and π«πΊπ«πΎ = π.ππ eV. Again blue dots represent the
nested hole pockets and red represent the nested electron pockets.
DERIVATION OF METALLIC SPIN DENSITY WAVE STATE IN FE β BASED SCβS The Hamiltonian for the two bands per site model in spin density wave (SDW) state can be written as,
π» ππ·π = νπππΆπ ,π
β πΆπ ,π + νπβππ ,π
β ππ ,π + Ξππ·π πΆπ ,πβ ππ+π,π + ππ+π,π
β πΆπ ,π
π ,ππ ,π
+ ν πππΆπ+π,π
β πΆπ+π,π + ν πβππ+π,π
β ππ+π,π
π ,π
+ ΞSDW πΆπ+π,πβ ππ ,π + ππ ,π
β πΆπ+π,π
π ,π
where πΆπ ,π , πΆπ+π,π , ππ ,π and ππ+π,π are the annihilation operators. πΆπ ,π destroys an electron with
the wave vector π and spin π while ππ ,π destroys a hole with wave vector π and spin π. Similarly we have
four creation operators which are essentially the complex conjugate of the annihilation operators. νππ and
νπβ are as derived earlier in the previous section,
νππ = νπ
+ = π+ π + πβ2 π + ππ₯π¦
2 π β πβ¦β¦β¦β¦ 1
νπβ = νπ
β = π+ π β πβ2 π + ππ₯π¦2 π β πβ¦β¦β¦β¦ 2
where,
π+ π = β π‘1 + π‘2 cosππ₯ + cos ππ¦ β 4π‘3 cos ππ₯ cos ππ¦
πβ π = β π‘1 β π‘2 cosππ₯ β cos ππ¦
ππ₯π¦ π = β4π‘4 sin ππ₯ sinππ¦
π‘π βs are the tight binding parameters (real and constant) as stated earlier.
It is not very important at this point of time to explicitly write the expressions of ν ππ and ν π
β , which are also
functions of wave vector, π. However, it is important to note that ν ππ and ν π
β are such that,
ν ππ = νπ+π
π πππ ν πβ = νπ+π
β
with π being the nesting vector equal to π, 0 or 0,π . The above Hamiltonian can be expressed in a matrix form as,
π» ππ·π = ππ ,πβ π» πππ ,π
π ,π
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with
ππ ,π =
πΆπ ,π
ππ ,π
πΆπ+π,π
ππ+π,π
π» π =
νππ 0 0 Ξππ·π
0 νπβ Ξππ·π 0
0 Ξππ·π ν ππ 0
Ξππ·π 0 0 ν πβ
Now, the electron and hole like dispersions can be rewritten as,
νππ = νππ π + νπ
π π νπβ = νππ π + νβ
π π
where νππ π is the contribution to the energies, νππ and νπ
β , coming from the non-nested
portions/regions of the Fermi surface. Similarly νππ π and νβ
π π are the contributions to νππ and νπ
β from the nested regions of the FS respectively. These are functions of wave vector, π and also depend on the tight binding parameters and may be expressed as follows,
νππ π = β π‘1 + π‘2 cosππ¦ β π
ν+π π = β π‘1 + π‘2 cos ππ₯ β 4π‘3 cos ππ₯ cos ππ¦ + πβ2 π + ππ₯π¦2 π
νβπ π = β π‘1 + π‘2 cos ππ₯ β 4π‘3 cos ππ₯ cos ππ¦ β πβ2 π + ππ₯π¦2 π
And similarly, ν ππ = νππ π β νβ
π π ν πβ = νππ π β νπ
π π Now the above Hamiltonian π» π can be rewritten as,
π» π =
νππ π + νππ π 0 0 Ξππ·π
0 νππ π + νβπ π Ξππ·π 0
0 Ξππ·π νππ π β νβπ π 0
Ξππ·π 0 0 νππ π β νππ π
Diagonalizing the Hamiltonian in the mean field approximation using Bogoliubov transformation,
ππ ,π =
πΆπ ,π
ππ ,π
πΆπ+π,π
ππ+π,π
=
βπ’π 0 0 π£π0 πΌπ π½π 00 βπ½π πΌπ 0π£π 0 0 π’π
πΎπ ,ππ
πΏπ ,ππ
πΏπ ,ππ£
πΎπ ,ππ£
Functions π’π and π£π are real and symmetric in the momentum space π’βπ = π’π , π£βπ = π£π and they fulfill the normalization condition,
π’π2 + π£π
2 = 1 Same goes for πΌπ and π½π . Rewriting we have,
πΆπ ,π = βπ’ππΎπ ,ππ + π£ππΎπ ,π
π£ βΉ πΆπ ,πβ = βπ’π πΎπ ,π
π β
+ π£π πΎπ ,ππ£
β
ππ ,π = πΌππΏπ ,ππ + π½ππΏπ ,π
π£ βΉ ππ ,πβ = πΌπ πΏπ ,π
π β
+ π½π πΏπ ,ππ£
β
πΆπ+π,π = βπ½ππΏπ ,ππ + πΌππΏπ ,π
π£ βΉ πΆπ+π,πβ = βπ½π πΏπ ,π
π β
+ πΌπ πΏπ ,ππ£
β
ππ+π,π = π£ππΎπ ,ππ + π’ππΎπ ,π
π£ βΉ ππ+π,πβ = π£π πΎπ ,π
π β
+ π’π πΎπ ,ππ£
β
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To calculate π’π , π£π , πΌπ and π½π we rewrite the Hamiltonian in terms of the transformed operators to get,
π» ππ·π = νππ π + νππ π βπ’π πΎπ ,π
π β
+ π£π πΎπ ,ππ£
β βπ’ππΎπ ,π
π + π£ππΎπ ,ππ£
π ,π
+ νππ π + νβπ π πΌπ πΏπ ,π
π β
+ π½π πΏπ ,ππ£
β πΌππΏπ ,π
π + π½ππΏπ ,ππ£
π ,π
+ νππ π β νβπ π βπ½π πΏπ ,π
π β
+ πΌπ πΏπ ,ππ£
β βπ½ππΏπ ,π
π + πΌππΏπ ,ππ£
π ,π
+ νππ π β νππ π π£π πΎπ ,π
π β
+ π’π πΎπ ,ππ£
β π£ππΎπ ,π
π + π’ππΎπ ,ππ£
π ,π
+ Ξππ·π π£π πΎπ ,ππ
β + π’π πΎπ ,π
π£ β βπ’ππΎπ ,π
π + π£ππΎπ ,ππ£
π ,π
+ Ξππ·π βπ½π πΏπ ,ππ
β + πΌπ πΏπ ,π
π£ β πΌππΏπ ,π
π + π½ππΏπ ,ππ£
π ,π
+ Ξππ·π πΌπ πΏπ ,ππ
β + π½π πΏπ ,π
π£ β βπ½ππΏπ ,π
π + πΌππΏπ ,ππ£
π ,π
+ Ξππ·π βπ’π πΎπ ,ππ
β + π£π πΎπ ,π
π£ β π£ππΎπ ,π
π + π’ππΎπ ,ππ£
π ,π
Simplifying we get,
π» ππ·π = νππ π + νππ π π’π
2 + νππ π β νππ π π£π
2 β 2ΞSDW π’ππ£π πΎπ ,ππ
β πΎπ ,ππ
π ,π
+ νππ π + νππ π π£π
2 + νππ π β νππ π π’π
2 + 2ΞSDW π’ππ£π πΎπ ,ππ£
β πΎπ ,ππ£
π ,π
+ νππ π + νβπ π πΌπ
2 + νππ π β νβπ π π½π
2 β 2ΞSDW πΌππ½π πΏπ ,ππ
β πΏπ ,ππ
π ,π
+ νππ π + νβπ π π½π
2 + νππ π β νβπ π πΌπ
2 + 2ΞSDW πΌππ½π πΏπ ,ππ£
β πΏπ ,ππ£
π ,π
+ β2νππ π π’ππ£π + Ξππ·π π£π
2 β π’π2 πΎπ ,π
π β πΎπ ,ππ£ + πΎπ ,π
π£ β πΎπ ,ππ
π ,π
+ 2νβπ π πΌππ½π + Ξππ·π π½π
2 β πΌπ2 πΏπ ,π
π β πΏπ ,ππ£ + πΏπ ,π
π£ β πΏπ ,ππ
π ,π
Putting the coefficient of non-diagonal term (last two terms) to zero we have, Ξππ·π π’π
2 β π£π2 = β2νπ
π π π’ππ£π Ξππ·π πΌπ
2 β π½π2 = 2νβ
π π πΌππ½π And from the normalization condition we have,
π’π2 + π£π
2 = 1 πΌπ
2 + π½π2 = 1
Solving for π’π and π£π from the above equations we get,
π’π 2 =
1
2
1 βνππ π
νππ π
2+ Ξππ·π
2
π£π 2 =
1
2
1 Β±νππ π
νππ π
2+ Ξππ·π
2
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Also,
2π’ππ£π =Ξππ·π
νππ π
2+ Ξππ·π
2
And solving for πΌπ and π½π we get,
|πΌπ 2 =
1
2
1 Β±νβπ π
νβπ π
2
+ Ξππ·π2
π½π 2 =
1
2
1 βνβπ π
νβπ π
2
+ Ξππ·π2
Also,
2πΌππ½π =Ξππ·π
νβπ π
2
+ Ξππ·π2
Using these values of π’π , π£π , πΌπ , π½π and rewriting the Hamiltonian we get,
π» ππ·π = νπππ β νπ
π π 2
+ Ξππ·π2 πΎπ ,π
π β πΎπ ,ππ
π ,π
+ νπππ + νπ
π π 2
+ Ξππ·π2 πΎπ ,π
π£ β πΎπ ,ππ£
π ,π
+ νπππ β νβ
π π 2
+ Ξππ·π2 πΏπ ,π
π β πΏπ ,ππ
π ,π
+ νπππ + νβ
π π 2
+ Ξππ·π2 πΏπ ,π
π£ β πΏπ ,ππ£
π ,π
where, νπππ = νππ π = β π‘1 + π‘2 cos ππ¦ β π
νππ π = β π‘1 + π‘2 cos ππ₯ β 4π‘3 cos ππ₯ cos ππ¦ + πβ2 π + ππ₯π¦2 π
νβπ π = β π‘1 + π‘2 cos ππ₯ β 4π‘3 cos ππ₯ cos ππ¦ β πβ2 π + ππ₯π¦2 π
The four obtained eigenvalues or quasi particle energies in spin density wave state can be summarized as,
π¬πΒ±πΊπ«πΎ π = πΊπ
ππ Β± πΊππ π
π+ π«πΊπ«πΎ
π
π¬πΒ±πΊπ«πΎ π = πΊπ
ππ Β± πΊππ π
π
+ π«πΊπ«πΎπ
COUPLED STATE (SDW + SC) So far I had derived the quasi-particle energies using the 2-band model in the normal state and in spin density wave state. But in general in superconducting systems we have either normal state or spin density wave state or the superconducting state depending on the temperature. Whereas it has been observed that in most Iron based superconductors both SDW and SC states can coexist within a temperature range for some specified parameters. So for the two band model one can construct the Hamiltonian for the coupled state. And then the quasi-particle energies and the temperature dependence of the gap could be derived. This part couldnβt be completed due to shortage of time.
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FERMI SURFACES IN THE SPIN DENSITY WAVE STATE Fermi surfaces or contours in spin density wave state are as shown below for two different values of chemical potentials π = 1.45 & 1.53 in the reduced Brillouin zone taking the spin density wave gap Ξππ·π equal to 0.02 eV. It can be seen from the results derived above that the quasi particle energies in SDW state,
πΈπΒ±ππ·π π = νπ
ππ Β± νππ π
2+ Ξππ·π
2
πΈβΒ±ππ·π π = νπ
ππ Β± νβπ π
2
+ Ξππ·π2
depend on the temperature through Ξππ·π and on the extent of nesting of the Fermi surface. In both the figures below only two of the electron pockets are gapped. This is because only those two are nested in the reduced BZ and so are gapped. The extent of nesting of the FS is as shown in the above section for π = 1.53 in the RBZ. The spin density wave gap is zero in the non-nested regions and it only contributes to the quasi particle energy in SDW state in the nested regions. So the portions of the FS which are nested have higher quasi particle energies (or to be specific the absolute value of energy is higher) than the non nested portions due to the addition of the SDW gap and as a result of this they do not show up on the FS (constant energy contour) and it seems as if the FS is gapped in those regions. This can be clearly seen comparing the following figure with Figure 7.
Fermi surfaces in the spin density wave state for π«πΊπ«πΎ = π.ππ eV and π = π.ππ (left) & π.ππ (right). Red and
blue dots represent the hole and electron pockets respectively. The inset in both figures shows a magnified view of the electron pockets where gaps could be clearly seen. One can compare the figure on the right with the figure
above showing nested regions of the FS in the reduced BZ for π = π.ππ & π«πΊπ«πΎ = π.ππ eV.
SOLUTION OF GAP EQUATION IN SDW STATE The band gap Ξππ·π equation for spin density wave state as derived in (11) can be written as,
ΞSDW = β2ππππ‘ππ Ξππ·π 1
πΈππβ tanh
π½πΈππβ
2 +
1
πΈππ+ tanh
π½πΈππ+
2
π
+ Ξππ·π 1
πΈπββ tanh
π½πΈπββ
2 +
1
πΈπβ+ tanh
π½πΈπβ+
2
π
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Here, in the nested region we have,
πΈππ β
Β±
= Β± νππ β
2+ Ξππ·π
2
whereas in the non nested region we have,
πΈππ β
Β±
= νππ β
These simplified forms of πΈππ β
Β±
are obtained, assuming that there is perfect/complete nesting of the Fermi
surface in which case νπππ = 0 and so νπ
π β = νπ β π π .
The above spin density wave gap equation gives the temperature dependence of the band gap or SDW gap through π½ which is equal to 1 ππ΅π , ππ΅ being the Boltzmann constant in eV/ πK and π is the temperature
in ππΎ. In this equation the summation over π is carried out close to the Fermi surface and in the region which is nested and lies inside the reduced BZ. In the non-nested region Ξππ·π is taken to be zero. The constant outside the summation ππππ‘ππ is inter β orbital interaction potential. Following is the temperature dependence of the spin density wave gap for chemical potential π = 1.53 and ππππ‘ππ = β0.34 eV.
Figure 8: Spin density wave gap versus temperature for π = π.ππ and πΌπππππ = βπ.ππ eV. It can be seen that the
critical temp. for SDW state for such a system is close to 160 deg. K.
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Following graph shows the variation of SDW gap and critical temperature for various values of the chemical potential. Chemical potential changes the extent of nesting of the Fermi surface as explained above due to which the number of integration or summation points change which in turn changes gap as well as ππ
ππ·π .
Figure 9: SDW gap versus Temperature (T) for various values of π as indicated.
CALCULATION OF SPIN SUSCEPTIBILITY IN SPIN DENSITY WAVE STATE The main concept of calculation of static spin susceptibility in the spin density wave state remains same as explained above in the previous section for the normal state. In the formula for spin susceptibility,
ππ π, πΞ© = β1
2π
π + π, π π, πβ² 2
πΞ© + πΈπ ,π+π β πΈπ β² ,π ππΉ πΈπ ,π+π β ππΉ πΈπ β² ,π
π ,π ,π β²
Here π, π is the eigenvector of the 4 Γ 4 Hamiltonian defined above for the SDW state. As we have four eigen vectors here π can take values -2, -1, 1, and 2 (just convention). πΈπ ,π is the corresponding eigenvalue. π and ππΉ remain same β number of summation points and Fermi function. The summation here in SDW
state is carried out only for those values of π which nest. This again makes ππ dependent on the extent of nesting of the FS. Due to the shortage of time this calculation couldnβt be completed and verified to an extent, so that it could be presented here with a proper explanation of the observed phenomena.
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Works Cited 1. Ashcroft, Neil W. and Mermin, N. David. Solid State Physics. [ed.] Dorothy Garbose Crane. College Edition. s.l. : Harcourt College Publisher, 1976. pp. 725-755. ISBN: 0030839939. 2. Superconductivity at 39 K in magnesium diboride. Nagamatsu, Jun, et al. s.l. : Nature, February 5, 2001, Nature, Vol. 410, pp. 63-64. doi:10.1038/35065039. 3. Iron-based superconductor. Wikipedia, the free encyclopedia. [Online] wikipedia.org. http://en.wikipedia.org/. 4. Superconductivity and phase diagram in iron-based arsenic-oxides ReFeAsO1-Ξ΄ (Re = rare-earth metal) without fluorine doping. Ren, Zhi-An, et al. 2008, Euro Physics Letters, Vol. 83. doi: 10.1209/0295-5075/83/17002. 5. Boeri, L., Dolgov, O. V. and Golubov, A. A. Is LaO1-xFxFeAs an electron-phonon superconductor? s.l. : arXiv, March 18, 2008. arXiv: 0803.2703. 6. Cao, Chao, Hirschfeld, P. J. and Cheng, Hai-Ping. Proximity of antiferromagnetism and superconductivity in LaO1βxFxFeAs: effective Hamiltonian from ab initio studies. s.l. : arXiv, May 1, 2008. arXiv: 0803.3236v2. 7. Unconventional pairing originating from disconnected Fermi surfaces in the iron-based superconductor. Kuroki, Kazuhiko, et al. s.l. : IOP Publishing Ltd & Deutsche Physikalische Gesellschaft, February 27, 2009, New Journal of Physics, pp. 1-8. doi: 10.1088/1367-2630/11/2/025017. 8. Minimal two-band model of the superconductivity iron oxypnictides. Raghu, S., et al. s.l. : The American Physical Society, May 8, 2008, Physical Review B, Vol. 77, pp. 1-4. doi: 10.1103/PhysRevB.77.220503. 9. Enhanced Orbital Degeneracy in Momentum Space for LaOFeAs. Zhang, Hai-Jun, et al. Beijing : s.n., 2009, Chinese Physics Letters, Vol. 26. doi: 10.1088/0256-307X/26/1/017401. 10. Unconventional Superconductivity with a Sign Reversal in the Order Parameter of LaFeAsO1-xFx. Mazin, I. I., et al. July 29, 2008, Physical Review Letters, Vol. 101, pp. 1-4. doi: 10.1103/PhysRevLett.101.057003. 11. Ghosh, Haranath. Elementary and collective excitations as probe for order parameter symmetry in Fe-based superconductors. [Unpublished work]. Indore, M.P, India : s.n., April 13, 2010. pp. 1-2. 12. Dirac cone in Iron based superconductors. Hasan, M. Zahid and Bernevig, B. Andrei. 27, Princeton : American Physical Society, March 29, 2010, Physics, Vol. 3, pp. 1-3. doi: 10.1103/Physics.3.27. 13. Pairing Symmetry in a Two-Orbital Exchange Coupling Model of Oxypnictides. Seo, Kangjun, Bernevig, B. Andrei and Hu, Jiangping. s.l. : The American Physical Society, November 14, 2008, Physical Review Letters, Vol. 101, pp. 1-4. doi: 10.1103/PhysRevLett.101.206404. 14. Interplay of staggered flux phase and d-wave superconductivity. Ghosh, Haranath. Indore : The American Physical Society, August 30, 2002, Physical Review B, Vol. 66, pp. 1-9. doi: 10.1103/PhysRevB.66.064530.
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31 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
15. Ghosal, Amit. Old FORTRAN Codes. [Programs] [ed.] Harsh Purwar. Kolkata, W.B, India : s.n., 2009. 16. Observation of Fermi-surface-dependent nodeless superconducting gaps in Ba0.6K0.4Fe2As2. Ding, H, et al. July 14, 2008, Euro Physics Letters, Vol. 83, pp. 1-4. doi: 10.1209/0295-5075/83/47001.
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Appendix
FORTRAN CODES The following FORTRAN-90 code generates data files (βprog6_1.datβ & βprog6_2.datβ) for plotting Fermi surface in the Full Brillouin Zone (FBZ) based on the Raghuβs dispersion (8). The data files contain those values of ππ₯πΏ and ππ¦πΏ (in various rows) for which the Energy νπ
+, νπβ is close to zero.
1 program prog6
2 implicit none
3 real(8) :: eps, kx, ky, pi, EnerN, EnerP, mu
4 eps = 2.d-2
5 pi = 4.d0*datan(1.d0)
6 mu = 1.54d0
7 open(unit=1, file="prog6_1.dat", action="write")
8 open(unit=2, file="prog6_2.dat", action="write")
9 kx = -pi
10 do while(kx<=pi)
11 ky = -pi
12 do while(ky<=pi)
13 if(abs(EnerP(kx,ky,mu)) <= eps) then
14 write(1,*) kx,ky
15 end if
16 if(abs(EnerN(kx,ky,mu)) <= eps) then
17 write(2,*) kx,ky
18 end if
19 ky = ky + pi/500.d0
20 end do
21 kx = kx + pi/500.d0
22 end do
23 close(1)
24 close(2)
25 end program prog6
26
27 real(8) function En(kx,ky)
28 implicit none
29 real(8) :: kx, ky, Ex, Ey
30 En = (Ex(kx,ky) - Ey(kx,ky))/2.d0
31 end function En
32
33 real(8) function Ep(kx,ky)
34 implicit none
35 real(8) :: kx, ky, Ex, Ey
36 Ep = (Ex(kx,ky) + Ey(kx,ky))/2.d0
37 end function Ep
38
39 real(8) function Ex(kx,ky)
40 implicit none
41 real(8) :: t1, t2, t3, kx, ky
42 t1 = -1.0d0
43 t2 = 1.3d0
44 t3 = -0.85d0
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45 Ex = - (2.d0 *t1*dcos(kx) + 2.d0*t2*dcos(ky) + 4.d0*t3*dcos(kx)*dcos(ky))
46 end function Ex
47
48 real(8) function Ey(kx,ky)
49 implicit none
50 real(8) :: t1, t2, t3, kx, ky
51 t1 = -1.0d0
52 t2 = 1.3d0
53 t3 = -0.85d0
54 Ey = - (2.d0*t2*dcos(kx) + 2.d0*t1*dcos(ky) + 4.d0*t3*dcos(kx)*dcos(ky))
55 end function Ey
56
57 real(8) function Exy(kx,ky)
58 implicit none
59 real(8) :: kx, ky, t4
60 t4 = -0.85d0
61 Exy = -4.d0*t4*dsin(kx)*dsin(ky)
62 end function Exy
63
64 real(8) function EnerP(kx, ky, mu)
65 implicit none
66 real(8) :: Ep, kx, ky, En, Exy, mu
67 EnerP = Ep(kx,ky) + dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu
68 end function EnerP
69
70 real(8) function EnerN(kx, ky, mu)
71 implicit none
72 real(8) :: Ep, kx, ky, En, Exy, mu
73 EnerN = Ep(kx,ky) - dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) β mu
74 end function EnerN
The following figure (Figure 10) was plotted using the data files generated by the above code.
Figure 10: Fermi Surface (or Zero Energy Contour) for
Raghu's dispersion (8) in the full Brillouin Zone for π = π.ππ.
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The following FORTRAN-90 code generates data files (βprog7_P.datβ & βprog7_N.datβ) for plotting Fermi surface (or zero energy contour) in the reduced (by folding or translating all points outside the BZ by π,π ) and rotated (by 45π in clockwise direction) Brillouin zone. The data files contain appropriate values of ππ₯πΏ and ππ¦πΏ in various rows.
1 program prog7
2 implicit none
3 real(8) :: eps, kx, ky, pi, EnerN, EnerP, mu, x, y, xp, yp
4 eps = 2.d-2
5 pi = 4.d0*datan(1.d0)
6 mu = 1.54d0
7 open(unit=1, file="prog7_P.dat", action="write")
8 open(unit=2, file="prog7_N.dat", action="write")
9 kx = -pi
10 do while(kx<=pi)
11 ky = -pi
12 do while(ky<=pi)
13 if(abs(EnerP(kx,ky,mu)) <= eps) then
14 x = kx
15 y = ky
16 if(y+x+pi < 0.d0) then
17 x = x + pi
18 y = y + pi
19 else if(y-x-pi > 0.d0) then
20 x = x + pi
21 y = y - pi
22 else if(y+x-pi > 0.d0) then
23 x = x - pi
24 y = y - pi
25 elseif(y-x+pi < 0.d0) then
26 x = x - pi
27 y = y + pi
28 end if
29 xp = x*dcos(pi/4.d0) - y*dsin(pi/4.d0)
30 yp = x*dsin(pi/4.d0) + y*dcos(pi/4.d0)
31 write(1,*) xp, yp
32 end if
33 if (abs(EnerN(kx,ky,mu)) <= eps) then
34 x = kx
35 y = ky
36 if(y+x+pi < 0.d0) then
37 x = x + pi
38 y = y + pi
39 else if(y-x-pi > 0.d0) then
40 x = x + pi
41 y = y - pi
42 else if(y+x-pi > 0.d0) then
43 x = x - pi
44 y = y - pi
45 elseif(y-x+pi < 0.d0) then
46 x = x - pi
47 y = y + pi
48 end if
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49 xp = x*dcos(pi/4.d0) - y*dsin(pi/4.d0)
50 yp = x*dsin(pi/4.d0) + y*dcos(pi/4.d0)
51 write(2,*) xp, yp
52 end if
53 ky = ky + pi/500.d0
54 end do
55 kx = kx + pi/500.d0
56 end do
57 close(1)
58 close(2)
59 end program prog7
60
61 real(8) function En(kx,ky)
62 implicit none
63 real(8) :: kx, ky, Ex, Ey
64 En = (Ex(kx,ky) - Ey(kx,ky))/2.d0
65 end function En
66
67 real(8) function Ep(kx,ky)
68 implicit none
69 real(8) :: kx, ky, Ex, Ey
70 Ep = (Ex(kx,ky) + Ey(kx,ky))/2.d0
71 end function Ep
72
73 real(8) function Ex(kx,ky)
74 implicit none
75 real(8) :: t1, t2, t3, kx, ky
76 t1 = -1.0d0
77 t2 = 1.3d0
78 t3 = -0.85d0
79 Ex = - (2.d0 *t1*dcos(kx) + 2.d0*t2*dcos(ky) + 4.d0*t3*dcos(kx)*dcos(ky))
80 end function Ex
81
82 real(8) function Ey(kx,ky)
83 implicit none
84 real(8) :: t1, t2, t3, kx, ky
85 t1 = -1.0d0
86 t2 = 1.3d0
87 t3 = -0.85d0
88 Ey = - (2.d0*t2*dcos(kx) + 2.d0*t1*dcos(ky) + 4.d0*t3*dcos(kx)*dcos(ky))
89 end function Ey
90
91 real(8) function Exy(kx,ky)
92 implicit none
93 real(8) :: kx, ky, t4
94 t4 = -0.85d0
95 Exy = -4.d0*t4*dsin(kx)*dsin(ky)
96 end function Exy
97
98 real(8) function EnerP(kx, ky, mu)
99 implicit none
100 real(8) :: Ep, kx, ky, En, Exy, mu
101 EnerP = Ep(kx,ky) + dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu
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102 end function EnerP
103
104 real(8) function EnerN(kx, ky, mu)
105 implicit none
106 real(8) :: Ep, kx, ky, En, Exy, mu
107 EnerN = Ep(kx,ky) - dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu
108 end function EnerN
The following figure was plotted using the data generated by the above code.
Figure 11: Fermi Surface (or Zero Energy Contour) for Raghu's dispersion (8)
in the reduced + rotated (by πππ) Brillouin Zone for π = π.ππ.
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The following FORTRAN β 90 code generates six data files for plotting the Band structures in the full BZ for six different values of the chemical potential. 1 program prog8 2 implicit none 3 real(8)::eps,kx,ky,pi,EnerN,EnerP 4 integer::i,j 5 real(8),dimension(6)::mu 6 character(15)::fil 7 character(1)::str 8 mu(1) = 1.25d0 9 mu(2) = 1.35d0 10 mu(3) = 1.45d0 11 mu(4) = 1.55d0 12 mu(5) = 1.65d0 13 mu(6) = 1.75d0 14 eps=0.02 15 pi=4.d0*datan(1.d0) 16 do j = 1,6 17 write(str,'(i1)') j 18 fil = "prog8_"//str//".dat" 19 open(unit=1,file=fil,status="replace",action="write") 20 i = 0 21 ky = 0.d0 22 kx = 0.d0 23 do while (kx<=pi) 24 i = i+1 25 write(1,*) i,EnerN(kx,ky,mu(j)),EnerP(kx,ky,mu(j)) 26 kx = kx + 0.01d0 27 end do 28 print*,i 29 ky = 0.d0 30 kx = pi 31 do while(ky<=pi) 32 i=i+1 33 write(1,*) i,EnerN(kx,ky,mu(j)),EnerP(kx,ky,mu(j)) 34 ky = ky + 0.01d0 35 end do 36 print*,i 37 kx = pi 38 ky = pi 39 do while(kx>=0) 40 i=i+1 41 write(1,*) i,EnerN(kx,ky,mu(j)),EnerP(kx,ky,mu(j)) 42 kx = kx - 0.01d0 43 ky = ky - 0.01d0 44 end do 45 print*,i 46 close(1) 47 end do 48 end program prog8 49 50 real(8) function En(kx,ky) 51 implicit none 52 real(8)::kx, ky, Ex, Ey 53 En = (Ex(kx,ky) - Ey(kx,ky))/2.d0 54 end function En 55 56 real(8) function Ep(kx,ky) 57 implicit none 58 real(8)::kx, ky, Ex, Ey
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59 Ep = (Ex(kx,ky) + Ey(kx,ky))/2.d0 60 end function Ep 61 62 real(8) function Ex(kx,ky) 63 implicit none 64 real(8) :: t1, t2, t3, kx, ky 65 t1 = -1.0d0 66 t2 = 1.3d0 67 t3 = -0.85d0 68 Ex = - (2.d0 *t1*dcos(kx) + 2.d0*t2*dcos(ky) + 4.d0*t3*dcos(kx)*dcos(ky)) 69 end function Ex 70 71 real(8) function Ey(kx,ky) 72 implicit none 73 real(8) :: t1, t2, t3, kx, ky 74 t1 = -1.0d0 75 t2 = 1.3d0 76 t3 = -0.85d0 77 Ey = - (2.d0*t2*dcos(kx) + 2.d0*t1*dcos(ky) + 4.d0*t3*dcos(kx)*dcos(ky)) 78 end function Ey 79 80 real(8) function Exy(kx,ky) 81 implicit none 82 real(8) :: kx, ky, t4 83 t4 = -0.85d0 84 Exy = -4.d0*t4*dsin(kx)*dsin(ky) 85 end function Exy 86 87 real(8) function EnerP(kx, ky, mu) 88 implicit none 89 real(8) :: Ep, kx, ky, En, Exy, mu 90 EnerP = Ep(kx,ky) + dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu 91 end function EnerP 92 93 real(8) function EnerN(kx, ky, mu) 94 implicit none 95 real(8) :: Ep, kx, ky, En, Exy, mu 96 EnerN = Ep(kx,ky) - dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu 97 end function EnerN
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The following FORTRAN β 90 code generates twelve data files for plotting Band structure in the reduced BZ for six different values of the chemical potential. 1 program prog12 2 implicit none 3 real(8)::kx,ky,pi,EnerN,EnerP 4 integer::i,j 5 real(8),dimension(6)::mu 6 character(15)::fil1,fil2 7 character(1)::str 8 mu(1) = 1.25d0 9 mu(2) = 1.35d0 10 mu(3) = 1.45d0 11 mu(4) = 1.55d0 12 mu(5) = 1.65d0 13 mu(6) = 1.75d0 14 pi=4.d0*datan(1.d0) 15 do j = 1,6 16 write(str,'(i1)') j 17 fil1 = "prog12_"//str//"_a.dat" 18 fil2 = "prog12_"//str//"_b.dat" 19 open(unit=1,file=fil1,status="replace",action="write") 20 open(unit=2,file=fil2,status="replace",action="write") 21 i = 0 22 kx = 0.d0 23 ky = 0.d0 24 do while(ky<=pi/2.d0) 25 i=i+1 26 write(1,*) i,EnerN(kx,ky,mu(j)),EnerP(kx,ky,mu(j)) 27 kx = kx+0.01d0 28 ky = ky+0.01d0 29 end do 30 print*,i 31 kx = pi/2.d0 32 ky = pi/2.d0 33 do while(kx<=pi) 34 i=i+1 35 write(1,*) i,EnerN(kx,ky,mu(j)),EnerP(kx,ky,mu(j)) 36 kx = kx + 0.01d0 37 ky = ky - 0.01d0 38 end do 39 print*,i 40 kx = pi 41 ky = 0.d0 42 do while(kx>=0) 43 i=i+1 44 write(1,*) i,EnerN(kx,ky,mu(j)),EnerP(kx,ky,mu(j)) 45 kx = kx - 0.01d0 46 end do 47 print*,i 48 close(1) 49 i = 0 50 kx = pi 51 ky = pi 52 do while(ky>=pi/2.d0) 53 i=i+1 54 write(2,*) i,EnerN(kx,ky,mu(j)),EnerP(kx,ky,mu(j)) 55 kx=kx-0.01d0 56 ky = ky-0.01d0 57 end do 58 print*,i
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59 kx = pi/2.d0 60 ky = pi/2.d0 61 do while(kx<=pi) 62 i=i+1 63 write(2,*) i,EnerN(kx,ky,mu(j)),EnerP(kx,ky,mu(j)) 64 kx = kx + 0.01d0 65 ky = ky - 0.01d0 66 end do 67 print*,i 68 kx = pi 69 ky = 0.d0 70 do while(ky>=-pi) 71 i=i+1 72 write(2,*) i,EnerN(kx,ky,mu(j)),EnerP(kx,ky,mu(j)) 73 ky = ky - 0.01d0 74 end do 75 print*,i 76 close(2) 77 end do 78 end program prog12 79 80 real(8) function En(kx,ky) 81 implicit none 82 real(8)::kx, ky, Ex, Ey 83 En = (Ex(kx,ky) - Ey(kx,ky))/2.d0 84 end function En 85 86 real(8) function Ep(kx,ky) 87 implicit none 88 real(8)::kx, ky, Ex, Ey 89 Ep = (Ex(kx,ky) + Ey(kx,ky))/2.d0 90 end function Ep 91 92 real(8) function Ex(kx,ky) 93 implicit none 94 real(8) :: t1, t2, t3, kx, ky 95 t1 = -1.0d0 96 t2 = 1.3d0 97 t3 = -0.85d0 98 Ex = - (2.d0 *t1*dcos(kx) + 2.d0*t2*dcos(ky) + 4.d0*t3*dcos(kx)*dcos(ky)) 99 end function Ex 100 101 real(8) function Ey(kx,ky) 102 implicit none 103 real(8) :: t1, t2, t3, kx, ky 104 t1 = -1.0d0 105 t2 = 1.3d0 106 t3 = -0.85d0 107 Ey = - (2.d0*t2*dcos(kx) + 2.d0*t1*dcos(ky) + 4.d0*t3*dcos(kx)*dcos(ky)) 108 end function Ey 109 110 real(8) function Exy(kx,ky) 111 implicit none 112 real(8) :: kx, ky, t4 113 t4 = -0.85d0 114 Exy = -4.d0*t4*dsin(kx)*dsin(ky) 115 end function Exy 116 117 real(8) function EnerP(kx, ky, mu) 118 implicit none 119 real(8) :: Ep, kx, ky, En, Exy, mu
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
41 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
120 EnerP = Ep(kx,ky) + dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu 121 end function EnerP 122 123 real(8) function EnerN(kx, ky, mu) 124 implicit none 125 real(8) :: Ep, kx, ky, En, Exy, mu 126 EnerN = Ep(kx,ky) - dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu 127 end function EnerN
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
42 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
The following FORTRAN β 90 program calculates the static spin susceptibility in the normal state within two band model using the eigenvectors and eigenvalues calculated for 2 Γ 2 Hamiltonian for various values of the wave vector q. Program generates a data file containing the values of static spin susceptibility in the second column whereas the first column is occupied by arbitrary integers but in order. 1 program prog16 2 implicit none 3 real(8) :: pi,qx,qy,mu,T,x,y,l,m 4 integer :: i 5 complex(8) :: Chi,om 6 T = 100.d0 7 mu = 1.45d0 8 pi = 4.d0*datan(1.d0) 9 om = -(0.d0,1.d-8) 10 open(unit=1,file='prog16.dat',action='write') 11 i=0 12 qx = 0.001d0 13 qy = 0.001d0 14 x=pi 15 y=-pi 16 do while(qx<=pi) 17 i=i+1 18 write(1,*) i,real(Chi(qx,qy,om,mu,T)+Chi(x,y,om,mu,T)) 19 qx = qx + pi/100.d0 20 y = y + pi/100.d0 21 end do 22 print*,i 23 qx=pi 24 qy=0.d0 25 do while(qy<=pi/2.d0) 26 i=i+1 27 write(1,*) i,real(2.d0*Chi(qx,qy,om,mu,T)) 28 qy = qy + pi/100.d0 29 qx = qx - pi/100.d0 30 end do 31 print*,i 32 qx = pi/2.d0 33 qy = pi/2.d0 34 x = pi/2.d0 35 y = pi/2.d0 36 do while(qx>=0.001d0) 37 i=i+1 38 write(1,*) i,real(Chi(qx,qy,om,mu,T)+Chi(x,y,om,mu,T)) 39 qx = qx - pi/100.d0 40 qy = qy - pi/100.d0 41 x = x + pi/100.d0 42 y = y + pi/100.d0 43 end do 44 print*,i 45 46 close(1) 47 end program prog16 48 49 complex(8) function Chi(qx,qy,om,mu,T) 50 implicit none 51 real(8) :: qx, qy, pi, kx, ky, mu, Neum, nF,T,varx,vary 52 complex(8) :: suma, Deno, om, fk, as, Cheb 53 integer :: i,j,v,vp,n 54 pi = 4.d0*datan(1.d0) 55 suma = (0.d0,0.d0)
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
43 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
56 n = 100 57 do v = -1,1,2 58 do vp = -1,1,2 59 as = (0.d0,0.d0) 60 do i =1, n 61 kx=dcos((2.d0*i-1.d0)*pi/(2.d0*n)) 62 do j =1, n 63 ky=dcos((2.d0*j-1.d0)*pi/(2.d0*n)) 64 varx=(kx*dcos(qx) - dsqrt(1.d0-kx**2)*dsin(qx)) 65 vary=(ky*dcos(qy) - dsqrt(1.d0-ky**2)*dsin(qy))
66 fk = (dabs(Neum(kx,ky,varx,vary,v,vp,mu))**2)*(nF(v,varx,vary,mu,T) - nF(vp,kx,ky,mu,T))/Deno(kx,ky,varx,vary,v,vp,om,mu)
67 as=as+fk 68 end do 69 end do 70 Cheb = 2.d0*pi*(1.d0/(n*n))*as 71 suma = suma + Cheb 72 end do 73 end do 74 Chi = -suma/2.d0 75 end function Chi 76 77 real(8) function Norm(v,kx,ky) 78 implicit none 79 real(8)::En, Exy, kx, ky, Norm1,Norm2 80 integer::v 81 Norm1 = Exy(kx,ky)/dsqrt((En(kx,ky)+dsqrt(En(kx,ky)**2+Exy(kx,ky)**2))**2+Exy(kx,ky)**2) 82 Norm2 = Exy(kx,ky)/dsqrt((En(kx,ky)-dsqrt(En(kx,ky)**2+Exy(kx,ky)**2))**2+Exy(kx,ky)**2) 83 if (v .eq. 1) then 84 Norm = Norm1 85 else if(v .eq. -1) then 86 Norm = Norm2 87 end if 88 end function Norm 89 90 real(8) function Neum(kx,ky,qx,qy,v,vp,mu) 91 implicit none 92 real(8) :: Ep,En,Exy,kx,ky,qx,qy,Eak,Ebk,Eakq,Ebkq,Esmart,mu,Norm 93 integer::v,vp 94 Ebk = (En(kx,ky)-dsqrt(En(kx,ky)**2+Exy(kx,ky)**2))/Exy(kx,ky) 95 Eak = (En(kx,ky)+dsqrt(En(kx,ky)**2+Exy(kx,ky)**2))/Exy(kx,ky) 96 Ebkq = (En(qx,qy)-dsqrt(En(qx,qy)**2+Exy(qx,qy)**2))/Exy(qx,qy) 97 Eakq = (En(qx,qy)+dsqrt(En(qx,qy)**2+Exy(qx,qy)**2))/Exy(qx,qy) 98 99 if(v .eq. 1 .and. vp .eq. 1) then 100 Neum = Norm(vp,kx,ky)*Norm(v,qx,qy)*((Eakq*Eak)+1.d0) 101 else if(v .eq. 1 .and. vp .eq. -1) then 102 Neum = Norm(vp,kx,ky)*Norm(v,qx,qy)*((Eakq*Ebk)+1.d0) 103 else if(v .eq. -1 .and. vp .eq. 1) then 104 Neum = Norm(vp,kx,ky)*Norm(v,qx,qy)*((Ebkq*Eak)+1.d0) 105 else if(v .eq. -1 .and. vp .eq. -1) then 106 Neum = Norm(vp,kx,ky)*Norm(v,qx,qy)*((Ebkq*Ebk)+1.d0) 107 end if 108 end function Neum 109 110 complex(8) function Deno(kx,ky,qx,qy,v,vp,om,mu) 111 implicit none 112 real(8) :: kx,ky,qx,qy,mu,Esmart 113 complex(8) :: om 114 integer :: v,vp 115 Deno = om+Esmart(v,qx,qy,mu)-Esmart(vp,kx,ky,mu)
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
44 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
116 end function Deno 117 118 real(8) function nF(v,kx,ky,mu,T) 119 implicit none 120 real(8) :: Esmart,kx,ky,T,mu,kb 121 integer :: v 122 kb = 0.8617d-4 123 ! nF = (1.d0-dtanh(Esmart(v,kx,ky,mu)/(2.d0*kb*T)))/2.d0 124 nF=1.d0/(dexp(Esmart(v,kx,ky,mu)/(kb*T)) + 1.d0) 125 end function nF 126 127 real(8) function Esmart(v,kx,ky,mu) 128 implicit none 129 real(8) :: kx,ky,mu,EnerP,EnerN 130 integer :: v 131 if(v .eq. 1) then 132 Esmart = EnerP(kx,ky,mu) 133 else if (v .eq. -1) then 134 Esmart = EnerN(kx,ky,mu) 135 end if 136 end function Esmart 137 138 real(8) function En(kx,ky) 139 implicit none 140 real(8)::kx, ky, Ex, Ey 141 En = (Ex(kx,ky) - Ey(kx,ky))/2.d0 142 end function En 143 144 real(8) function Ep(kx,ky) 145 implicit none 146 real(8)::kx, ky, Ex, Ey 147 Ep = (Ex(kx,ky) + Ey(kx,ky))/2.d0 148 end function Ep 149 150 real(8) function Ex(kx,ky) 151 implicit none 152 real(8) :: t1, t2, t3, kx, ky 153 t1 = -1.0d0 154 t2 = 1.3d0 155 t3 = -0.85d0 156 Ex = -(2.d0 *t1*kx + 2.d0*t2*ky + 4.d0*t3*kx*ky) 157 end function Ex 158 159 real(8) function Ey(kx,ky) 160 implicit none 161 real(8) :: t1, t2, t3, kx, ky 162 t1 = -1.0d0 163 t2 = 1.3d0 164 t3 = -0.85d0 165 Ey = -(2.d0*t2*kx + 2.d0*t1*ky + 4.d0*t3*kx*ky) 166 end function Ey 167 168 real(8) function Exy(kx,ky) 169 implicit none 170 real(8) :: kx, ky, t4 171 t4 = -0.85d0 172 Exy = -4.d0*t4*dsqrt(1-kx**2)*dsqrt(1-ky**2) 173 end function Exy 174 175 real(8) function EnerP(kx, ky, mu) 176 implicit none
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
45 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
177 real(8) :: Ep, kx, ky, En, Exy, mu 178 EnerP = Ep(kx,ky) + dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu 179 end function EnerP 180 181 real(8) function EnerN(kx, ky, mu) 182 implicit none 183 real(8) :: Ep, kx, ky, En, Exy, mu 184 EnerN = Ep(kx,ky) - dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu 185 end function EnerN
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
46 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
The following FORTRAN β 90 program also calculates the static spin susceptibility as a function of the chemical potential for three wave vectors q and correspondingly generates three data files containing values of π and ππ π, 0 in various rows. 1 program prog19 2 implicit none 3 real(8) :: pi,mu,T,dmu 4 integer :: i 5 complex(8) :: Chi,om 6 T = 100.d0 7 pi = 4.d0*datan(1.d0) 8 om = -(0.d0,1.d-8) 9 open(unit=1,file='prog19_1.dat',action='write') 10 open(unit=2,file='prog19_2.dat',action='write') 11 open(unit=3,file='prog19_3.dat',action='write') 12 13 mu = 0.5d0 14 dmu = 0.02d0 15 do while(mu<=3.1d0) 16 write(1,*) mu,real(Chi(1.d-4,1.d-4,om,mu,T)) 17 write(2,*) mu,real(Chi(1.d-4,pi,om,mu,T)) 18 write(3,*) mu,real(Chi(pi,pi,om,mu,T)) 19 mu=mu+dmu 20 print*,mu 21 end do 22 23 close(1) 24 close(2) 25 close(3) 26 end program prog19 27 28 complex(8) function Chi(qx,qy,om,mu,T) 29 implicit none 30 real(8) :: qx, qy, pi, kx, ky, mu, Neum, nF,T,varx,vary 31 complex(8) :: suma, Deno, om, fk, as, Cheb 32 integer :: i,j,v,vp,n 33 pi = 4.d0*datan(1.d0) 34 suma = (0.d0,0.d0) 35 n = 150 36 do v = -1,1,2 37 do vp = -1,1,2 38 as = (0.d0,0.d0) 39 do i = 1,n 40 kx=dcos((2.d0*i-1.d0)*pi/(2.d0*n)) 41 do j = 1,n 42 ky=dcos((2.d0*j-1.d0)*pi/(2.d0*n)) 43 varx=(kx*dcos(qx) - dsqrt(1.d0-kx**2)*dsin(qx)) 44 vary=(ky*dcos(qy) - dsqrt(1.d0-ky**2)*dsin(qy))
45 fk = (dabs(Neum(kx,ky,varx,vary,v,vp,mu))**2)*(nF(v,varx,vary,mu,T) - nF(vp,kx,ky,mu,T))/Deno(kx,ky,varx,vary,v,vp,om,mu)
46 as=as+fk 47 end do 48 end do 49 Cheb = 2.d0*pi*(1.d0/(n*n))*as 50 suma = suma + Cheb 51 end do 52 end do 53 Chi = -suma/2.d0 54 end function Chi 55
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
47 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
56 real(8) function Norm(v,kx,ky) 57 implicit none 58 real(8)::En, Exy, kx, ky, Norm1,Norm2 59 integer::v 60 Norm1 = Exy(kx,ky)/dsqrt((En(kx,ky)+dsqrt(En(kx,ky)**2+Exy(kx,ky)**2))**2+Exy(kx,ky)**2) 61 Norm2 = Exy(kx,ky)/dsqrt((En(kx,ky)-dsqrt(En(kx,ky)**2+Exy(kx,ky)**2))**2+Exy(kx,ky)**2) 62 if (v .eq. 1) then 63 Norm = Norm1 64 else if(v .eq. -1) then 65 Norm = Norm2 66 end if 67 end function Norm 68 69 real(8) function Neum(kx,ky,qx,qy,v,vp,mu) 70 implicit none 71 real(8) :: Ep,En,Exy,kx,ky,qx,qy,Eak,Ebk,Eakq,Ebkq,Esmart,mu,Norm 72 integer::v,vp 73 Ebk = (En(kx,ky)-dsqrt(En(kx,ky)**2+Exy(kx,ky)**2))/Exy(kx,ky) 74 Eak = (En(kx,ky)+dsqrt(En(kx,ky)**2+Exy(kx,ky)**2))/Exy(kx,ky) 75 Ebkq = (En(qx,qy)-dsqrt(En(qx,qy)**2+Exy(qx,qy)**2))/Exy(qx,qy) 76 Eakq = (En(qx,qy)+dsqrt(En(qx,qy)**2+Exy(qx,qy)**2))/Exy(qx,qy) 77 78 if(v .eq. 1 .and. vp .eq. 1) then 79 Neum = Norm(vp,kx,ky)*Norm(v,qx,qy)*((Eakq*Eak)+1.d0) 80 else if(v .eq. 1 .and. vp .eq. -1) then 81 Neum = Norm(vp,kx,ky)*Norm(v,qx,qy)*((Eakq*Ebk)+1.d0) 82 else if(v .eq. -1 .and. vp .eq. 1) then 83 Neum = Norm(vp,kx,ky)*Norm(v,qx,qy)*((Ebkq*Eak)+1.d0) 84 else if(v .eq. -1 .and. vp .eq. -1) then 85 Neum = Norm(vp,kx,ky)*Norm(v,qx,qy)*((Ebkq*Ebk)+1.d0) 86 end if 87 end function Neum 88 89 complex(8) function Deno(kx,ky,qx,qy,v,vp,om,mu) 90 implicit none 91 real(8) :: kx,ky,qx,qy,mu,Esmart 92 complex(8) :: om 93 integer :: v,vp 94 Deno = om+Esmart(v,qx,qy,mu)-Esmart(vp,kx,ky,mu) 95 end function Deno 96 97 real(8) function nF(v,kx,ky,mu,T) 98 implicit none 99 real(8) :: Esmart,kx,ky,T,mu,kb 100 integer :: v 101 kb = 0.8617d-4 102 ! nF = (1.d0-dtanh(Esmart(v,kx,ky,mu)/(2.d0*kb*T)))/2.d0 103 nF=1.d0/(dexp(Esmart(v,kx,ky,mu)/(kb*T)) + 1.d0) 104 end function nF 105 106 real(8) function Esmart(v,kx,ky,mu) 107 implicit none 108 real(8) :: kx,ky,mu,EnerP,EnerN 109 integer :: v 110 if(v .eq. 1) then 111 Esmart = EnerP(kx,ky,mu) 112 else if (v .eq. -1) then 113 Esmart = EnerN(kx,ky,mu) 114 end if 115 end function Esmart 116
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
48 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
117 real(8) function En(kx,ky) 118 implicit none 119 real(8)::kx, ky, Ex, Ey 120 En = (Ex(kx,ky) - Ey(kx,ky))/2.d0 121 end function En 122 123 real(8) function Ep(kx,ky) 124 implicit none 125 real(8)::kx, ky, Ex, Ey 126 Ep = (Ex(kx,ky) + Ey(kx,ky))/2.d0 127 end function Ep 128 129 real(8) function Ex(kx,ky) 130 implicit none 131 real(8) :: t1, t2, t3, kx, ky 132 t1 = -1.0d0 133 t2 = 1.3d0 134 t3 = -0.85d0 135 Ex = -(2.d0 *t1*kx + 2.d0*t2*ky + 4.d0*t3*kx*ky) 136 end function Ex 137 138 real(8) function Ey(kx,ky) 139 implicit none 140 real(8) :: t1, t2, t3, kx, ky 141 t1 = -1.0d0 142 t2 = 1.3d0 143 t3 = -0.85d0 144 Ey = -(2.d0*t2*kx + 2.d0*t1*ky + 4.d0*t3*kx*ky) 145 end function Ey 146 147 real(8) function Exy(kx,ky) 148 implicit none 149 real(8) :: kx, ky, t4 150 t4 = -0.85d0 151 Exy = -4.d0*t4*dsqrt(1-kx**2)*dsqrt(1-ky**2) 152 end function Exy 153 154 real(8) function EnerP(kx, ky, mu) 155 implicit none 156 real(8) :: Ep, kx, ky, En, Exy, mu 157 EnerP = Ep(kx,ky) + dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu 158 end function EnerP 159 160 real(8) function EnerN(kx, ky, mu) 161 implicit none 162 real(8) :: Ep, kx, ky, En, Exy, mu 163 EnerN = Ep(kx,ky) - dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu 164 end function EnerN
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
49 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
The following FORTRAN β 90 code calculates band gap for one orbital per site using the tight binding model at various temperatures and generates a data file (βprog4.datβ) containing band gap Ξ and corresponding Temperature π in various rows. 1 program prog4
2 implicit none
3 real(8) :: D, T, dT, eps, D0, mu, ChebD, Dpr
4 integer :: i, co
5 open(unit = 1, file = 'prog4.dat', action = 'write')
6 T = 0.d0
7 dT = 0.5d0
8 eps = 1.0d-7
9 D = 1.0d-2
10 Dpr = 2.d0
11 mu = 0.2d0
12 co = 0
13 do while(abs(D-Dpr) > eps .or. co < 90)
14 co = co + 1
15 Dpr=D
16 D0 = 2.0d0
17 D =1.0d-2
18 i=0
19 do while(abs(D-D0) > eps)
20 D0 = D
21 D = ChebD(D0, T, mu)
22 i=i+1
23 end do
24 print*,T,D,i
25 write(1,*) T,D
26 T = T + dT
27 end do
28 close(1)
29 end program prog4
30
31 real(8) function ChebD(D,Temp,mu)
32 implicit none
33 real(8) :: D, Temp, mu, as, pi, fk, t, xi, yi, ep, kb, V
34 integer :: n, i, j
35 t = 0.8d0
36 V = 0.26d0
37 kb = 0.8617d-4
38 pi = 4.d0 * datan(1.d0)
39 as = 0.d0
40 n = 100
41 do i = 1,n
42 xi=dcos((2.d0*i-1.d0)*pi/(2.d0*n))
43 do j = 1,n
44 yi=dcos((2.d0*j-1.d0)*pi/(2.d0*n))
45 ep = (-2.d0*t*(xi+yi)) β mu
46 if (abs(ep)<=0.02) then
47 fk=dtanh(dsqrt(ep**2+D**2)/(2.d0*kb*Temp))/dsqrt(ep**2+D**2)
48 else
49 fk = 0.d0
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
50 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
50 end if
51 as = as + fk
52 end do
53 end do
54 ChebD = (2.d0*pi*(1.d0/(n*n))*as)*V*D
55 end function ChebD
Figure 12: Band gap versus Temperature for π = π.π, tight binding parameter π = π.π eV
and potential π½ = π.ππ eV.
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
51 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
The following FORTRAN β 90 code generates 12 output files containing values of ππ₯πΏ and ππ¦πΏ in rows for
plotting Fermi surface in the spin density wave state for 6 different values of chemical potential π in the reduced Brillouin zone. 1 program prog34 2 implicit none 3 real(8)::kx,ky,Dsdw,T,pi,eps,x,y 4 logical::Enested,Pnested 5 real(8),dimension(6)::mu 6 character(15)::fil1,fil2 7 character(1)::str 8 integer :: i 9 mu(1) = 1.25d0 10 mu(2) = 1.35d0 11 mu(3) = 1.45d0 12 mu(4) = 1.53d0 13 mu(5) = 1.65d0 14 mu(6) = 1.75d0 15 pi=4.d0*datan(1.d0) 16 do i=4,4 17 write(str,'(i1)') i !Converting a number to a string. 18 fil1 = 'prog34_'//str//'_a.dat' 19 fil2 = 'prog34_'//str//'_b.dat' 20 open(unit=1,file=fil1,status="replace",action="write") 21 open(unit=2,file=fil2,status="replace",action="write") 22 kx=-pi 23 do while(kx<=pi) 24 ky=-pi 25 do while(ky<=pi) 26 if(Pnested(kx,ky,mu(i)).eqv..true.) then 27 x = kx 28 y = ky 29 if(y+x+pi<0.d0) then 30 x = x + pi 31 y = y + pi 32 else if(y-x-pi>0.d0) then 33 x = x + pi 34 y = y - pi 35 else if(y+x-pi>0.d0) then 36 x = x - pi 37 y = y - pi 38 else if(y-x+pi<0.d0) then 39 x = x - pi 40 y = y + pi 41 end if 42 write(1,*) x,y 43 end if 44 if(Enested(kx,ky,mu(i)).eqv..true.) then 45 x = kx 46 y = ky 47 if(y+x+pi<0.d0) then 48 x = x + pi 49 y = y + pi 50 else if(y-x-pi>0.d0) then 51 x = x + pi 52 y = y - pi 53 else if(y+x-pi>0.d0) then 54 x = x - pi 55 y = y - pi 56 else if(y-x+pi<0.d0) then
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
52 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
57 x = x - pi 58 y = y + pi 59 end if 60 write(2,*) x,y 61 end if 62 ky=ky+pi/500.d0 63 end do 64 kx=kx+pi/500.d0 65 end do 66 close(1) 67 close(2) 68 end do 69 end program prog34 70 71 logical function Pnested(kx,ky,mu) 72 implicit none 73 real(8):: kx,ky,mu,eps,pi 74 real(8)::EnerP,EnerN 75 eps = 0.05d0 76 pi = 4.d0*datan(1.d0) 77 if(abs(EnerP(kx,ky,mu)*EnerN(kx+pi,ky,mu))<=eps**2) then 78 Pnested=.true. 79 Else 80 Pnested=.false. 81 end if 82 end function Pnested 83 84 logical function Enested(kx,ky,mu) 85 implicit none 86 real(8):: kx,ky,mu,eps,pi 87 real(8)::EnerP,EnerN 88 eps = 0.05d0 89 pi = 4.d0*datan(1.d0) 90 if(abs(EnerN(kx,ky,mu)*EnerP(kx+pi,ky,mu))<=eps**2) then 91 Enested=.true. 92 Else 93 Enested=.false. 94 end if 95 end function Enested 96 97 real(8) function EnerP(kx, ky, mu) 98 implicit none 99 real(8) :: Ep, kx, ky, En, Exy, mu 100 EnerP=Ep(kx,ky)+dsqrt((En(kx,ky))**2+(Exy(kx,ky))**2)-mu 101 end function EnerP 102 103 real(8) function EnerN(kx, ky, mu) 104 implicit none 105 real(8) :: Ep, kx, ky, En, Exy, mu 106 EnerN=Ep(kx,ky)-dsqrt((En(kx,ky))**2+(Exy(kx,ky))**2)-mu 107 end function EnerN 108 109 real(8) function En(kx,ky) 110 implicit none 111 real(8)::kx, ky, Ex, Ey 112 En = (Ex(kx,ky) - Ey(kx,ky))/2.d0 113 end function En 114 115 real(8) function Ep(kx,ky) 116 implicit none 117 real(8)::kx, ky, Ex, Ey
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
53 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
118 Ep = (Ex(kx,ky) + Ey(kx,ky))/2.d0 119 end function Ep 120 121 real(8) function Ex(kx,ky) 122 implicit none 123 real(8) :: t1, t2, t3, kx, ky 124 t1 = -1.d0 125 t2 = 1.3d0 126 t3 = -0.85d0 127 Ex = -(2.d0*t1*dcos(kx)+2.d0*t2*dcos(ky)+4.d0*t3*dcos(kx)*dcos(ky)) 128 end function Ex 129 130 real(8) function Ey(kx,ky) 131 implicit none 132 real(8) :: t1, t2, t3, kx, ky 133 t1 = -1.d0 134 t2 = 1.3d0 135 t3 = -0.85d0 136 Ey = -(2.d0*t2*dcos(kx)+2.d0*t1*dcos(ky)+4.d0*t3*dcos(kx)*dcos(ky)) 137 end function Ey 138 139 real(8) function Exy(kx,ky) 140 implicit none 141 real(8) :: kx, ky, t4 142 t4 = -0.85d0 143 Exy = -4.d0*t4*dsin(kx)*dsin(ky) 144 end function Exy
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
54 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
This FORTRAN β 90 code calculates band gap for the spin density wave state using the model described in above sections at various temperatures and generates six data files corresponding to six different values of chemical potential π . The data files contain SDW gap ΞSDW and corresponding Temperature π in various rows. 1 program prog20 2 implicit none 3 real(8) :: T,dT,eps,Dsdw,Dpre,ChebDsdw,D0 4 integer :: i,co 5 real(8),dimension(6)::mu 6 character(15)::fil1,fil2 7 character(1)::str 8 integer :: z 9 mu(1) = 1.45d0 10 mu(2) = 1.48d0 11 mu(3) = 1.51d0 12 mu(4) = 1.54d0 13 mu(5) = 1.57d0 14 mu(6) = 1.60d0 15 dT = 2.d0 16 eps = 1.0d-7 17 do z = 1,6 18 write(str,'(i1)') z !Converting a number to a string. 19 fil1 = 'prog20_'//str//'.dat' 20 open(unit=1,file=fil1,status="replace",action="write") 21 T = 0.d0 22 D0=2.d0 23 Dsdw = 1.0d-2 24 do while(abs(D0-Dsdw) > eps .or. T < 100) 25 D0=Dsdw 26 Dpre = 2.0d0 27 Dsdw = 1.d-2 28 i=0 29 do while (abs(Dsdw-Dpre)>eps) 30 Dpre = Dsdw 31 Dsdw = ChebDsdw(Dpre, T, mu(z)) 32 i=i+1 33 end do 34 print*, T, Dsdw, i 35 write(1,*) T, Dsdw 36 T = T + dT 37 end do 38 close(1) 39 end do 40 end program prog20 41 42 real(8) function ChebDsdw(Dsdw,Temp,mu) 43 implicit none 44 real(8) :: U,kb,Dsdw,Temp,mu,pi,as,kx,ky,t1,t2,t3,t4 45 real(8) :: EnerP,EnerN,fk,eps,x,y,EsdwNP,EsdwNN,EsdwPP,EsdwPN 46 integer :: n,i,j 47 logical::Pnested,Nnested 48 eps = 2.d-2 49 U = 0.34d0 50 kb = 0.8617d-4 51 pi = 4.d0*datan(1.d0) 52 as = 0.d0 53 n = 150 54 do i =1,n 55 kx=dcos((2.d0*i-1.d0)*pi/(2.d0*n))
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
55 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
56 do j =1,n 57 ky=dcos((2.d0*j-1.d0)*pi/(2.d0*n)) 58 if((kx+ky>=0.d0).and.(Pnested(kx,ky,mu).eqv..true..or.Nnested(kx,ky,mu).eqv..true.)) then 59 t1 = dtanh(EsdwPN(kx,ky,mu,Dsdw)/(2.d0*kb*Temp))/EsdwPN(kx,ky,mu,Dsdw) 60 t2 = dtanh(EsdwPP(kx,ky,mu,Dsdw)/(2.d0*kb*Temp))/EsdwPP(kx,ky,mu,Dsdw) 61 t3 = dtanh(EsdwNN(kx,ky,mu,Dsdw)/(2.d0*kb*Temp))/EsdwNN(kx,ky,mu,Dsdw) 62 t4 = dtanh(EsdwNP(kx,ky,mu,Dsdw)/(2.d0*kb*Temp))/EsdwNP(kx,ky,mu,Dsdw) 63 fk = t1+t2+t3+t4 64 Else 65 fk = 0.d0 66 end if 67 as=as+fk 68 end do 69 end do 70 ChebDsdw = (2.d0*pi*(1.d0/(n*n))*as)*2.d0*U*Dsdw 71 end function ChebDsdw 72 73 real(8) function EsdwPP(kx,ky,mu,Dsdw) 74 implicit none 75 real(8)::kx,ky,mu,Dsdw,EnerP 76 logical::Pnested 77 if(Pnested(kx,ky,mu).eqv..true.) then 78 EsdwPP=dsqrt(EnerP(kx,ky,mu)**2+Dsdw**2) 79 else 80 EsdwPP=EnerP(kx,ky,mu) 81 end if 82 end function EsdwPP 83 84 real(8) function EsdwNP(kx,ky,mu,Dsdw) 85 implicit none 86 real(8)::kx,ky,mu,Dsdw,EnerN 87 logical::Nnested 88 if(Nnested(kx,ky,mu).eqv..true.) then 89 EsdwNP=dsqrt(EnerN(kx,ky,mu)**2+Dsdw**2) 90 else 91 EsdwNP=EnerN(kx,ky,mu) 92 end if 93 end function EsdwNP 94 95 real(8) function EsdwPN(kx,ky,mu,Dsdw) 96 implicit none 97 real(8)::kx,ky,mu,Dsdw,EnerP 98 logical::Pnested 99 if(Pnested(kx,ky,mu).eqv..true.) then 100 EsdwPN=-dsqrt(EnerP(kx,ky,mu)**2+Dsdw**2) 101 else 102 EsdwPN=EnerP(kx,ky,mu) 103 end if 104 end function EsdwPN 105 106 real(8) function EsdwNN(kx,ky,mu,Dsdw) 107 implicit none 108 real(8)::kx,ky,mu,Dsdw,EnerN 109 logical::Nnested 110 if(Nnested(kx,ky,mu).eqv..true.) then 111 EsdwNN=-dsqrt(EnerN(kx,ky,mu)**2+Dsdw**2) 112 else 113 EsdwNN=EnerN(kx,ky,mu) 114 end if 115 end function EsdwNN 116
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
56 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
117 logical function Pnested(kx,ky,mu) 118 implicit none 119 real(8):: kx,ky,mu,eps,pi 120 real(8)::EnerP,EnerN 121 eps = 0.05d0 122 pi = 4.d0*datan(1.d0) 123 if(abs(EnerP(kx,ky,mu)*EnerN(-kx,ky,mu))<=eps**2) then 124 Pnested=.true. 125 else 126 Pnested=.false. 127 end if 128 end function Pnested 129 130 logical function Nnested(kx,ky,mu) 131 implicit none 132 real(8):: kx,ky,mu,eps,pi 133 real(8)::EnerP,EnerN 134 eps = 0.05d0 135 pi = 4.d0*datan(1.d0) 136 if(abs(EnerN(kx,ky,mu)*EnerP(-kx,ky,mu))<=eps**2) then 137 Nnested=.true. 138 else 139 Nnested=.false. 140 end if 141 end function Nnested 142 143 real(8) function EnerP(kx, ky, mu) 144 implicit none 145 real(8) :: Ep, kx, ky, En, Exy, mu 146 EnerP = Ep(kx,ky) + dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu 147 end function EnerP 148 149 real(8) function EnerN(kx, ky, mu) 150 implicit none 151 real(8) :: Ep, kx, ky, En, Exy, mu 152 EnerN = Ep(kx,ky) - dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu 153 end function EnerN 154 155 real(8) function En(kx,ky) 156 implicit none 157 real(8)::kx, ky, Ex, Ey 158 En = (Ex(kx,ky) - Ey(kx,ky))/2.d0 159 end function En 160 161 real(8) function Ep(kx,ky) 162 implicit none 163 real(8)::kx, ky, Ex, Ey 164 Ep = (Ex(kx,ky) + Ey(kx,ky))/2.d0 165 end function Ep 166 167 real(8) function Ex(kx,ky) 168 implicit none 169 real(8) :: t1, t2, t3, kx, ky 170 t1 = -1.0d0 171 t2 = 1.3d0 172 t3 = -0.85d0 173 Ex = - (2.d0 *t1*kx + 2.d0*t2*ky + 4.d0*t3*kx*ky) 174 end function Ex 175 176 real(8) function Ey(kx,ky) 177 implicit none
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
57 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
178 real(8) :: t1, t2, t3, kx, ky 179 t1 = -1.0d0 180 t2 = 1.3d0 181 t3 = -0.85d0 182 Ey = - (2.d0*t2*kx + 2.d0*t1*ky + 4.d0*t3*kx*ky) 183 end function Ey 184 185 real(8) function Exy(kx,ky) 186 implicit none 187 real(8) :: kx, ky, t4 188 t4 = -0.85d0 189 Exy = -4.d0*t4*dsqrt(1.d0-kx**2)*dsqrt(1.d0-ky**2) 190 end function Exy
Young Scientist Research Program β 2010 17th May β 9th July IRON BASED SUPERCONDUCTORS β A THEORETICAL STUDY
58 | H a r s h P u r w a r ( I I S E R β K o l k a t a ) , Y S R P β 2 0 1 0
NOTES / CORRECTIONS / COMMENTS