Theoretical Modeling of Iron-based Superconductors

Young Scientist Research Program – 2010 17th May – 9th July IRON BASED SUPERCONDUCTORS – A THEORETICAL STUDY

1 | H a r s h P u r w a r ( I I S E R – K o l k a t a ) , Y S R P – 2 0 1 0

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



Certificate

It is certified that the work in this project report entitled “Theoretical Modeling of

Iron based Superconductors”, 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



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 – 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)



Contents

CERTIFICATE ...................................................................................................................... 2

ACKNOWLEDGEMENT ........................................................................................................ 3

CONTENTS ......................................................................................................................... 4

INTRODUCTION ................................................................................................................. 5

SUPERCONDUCTIVITY – 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 SC’S ....................................................................................................................... 8

THEORETICAL MODELING OF FE-BASED SC’S .................................................................... 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 – BASED SC’S .................................................................. 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



Introduction

SUPERCONDUCTIVITY – GENERAL DISCUSSION Superconductivity is an electrical resistance of exactly zero which occurs in certain materials below a characteristic temperature 𝑇𝑐 . 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 (−183.15 °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



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 2Δ centered about the Fermi energy, in the set of allowed one-electron levels. Thus an electron of energy 휀 can be accommodated by (or extracted from) a superconductor only if 휀 − 휀𝐹 exceeds Δ. The energy gap Δ increases in size as the temperature drops, leveling off to a maximum value Δ 0 at very low temperatures.

CRITICAL TEMPERATURE The transition to the superconducting state is a sharp one in bulk specimens. Above a critical temperature 𝑇𝑐 the properties of the metal are completely normal; below 𝑇𝑐 , 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 𝑘𝐵𝑇, varies from about 10–7 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, 𝐻𝑐 , and changing abruptly from one state to the other when it is reached.

Type II superconductors: Having two critical fields, 𝐻𝑐1 and 𝐻𝑐2

, being a perfect superconductor

under the lower critical field (𝐻𝑐1) and leaving completely the superconducting state above the

upper critical field (𝐻𝑐2), 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.



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 𝑇𝑐 .

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 𝐹𝑒𝑆𝑒 can have a critical temperature 𝑇𝑐 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 𝐿𝑎𝑂𝐹𝑒𝐴𝑠 are often referred to as the ‘1111’ pnictides. (3)

Crystal structure of LaOFeAs (left) and FeAs layer (right) (4)

The crystalline material 𝐿𝑎𝑂𝐹𝑒𝐴𝑠, stacks iron and arsenic layers, where the electrons flow, between planes of lanthanum 𝐿𝑎 and oxygen 𝑂 . Replacing up to 11 percent of the oxygen with fluorine improved the compound – 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 𝑆𝑟2𝑆𝑐𝐹𝑒𝑃𝑂3 (discovered in 2009) are referred to as the ‘22426’ family. (as 𝐹𝑒𝑃𝑆𝑟2𝑆𝑐𝑂3).

Fe

La

O

As



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 – or variations in the location of electrons around atoms – 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 SC’S Following is a list of some of the recently discovered iron – based superconductors (2006 – 2009) with their reported critical temperatures and the group that discovered it.

S. No.

Compound (Powder/Single

crystals)

Critical Temperature 𝑻𝒄

References

1 𝐿𝑎𝑂𝐹𝑒𝑃 ~5 K Y. Kamihara et al., J. Am. Chem. Soc. 128,

10012 (2006)

2 𝐿𝑎𝑁𝑖𝑂𝑃 ~3 K T. Watanabe et al., Inorg. Chem. 46, 7719

(2007)

3 𝐿𝑎 𝑂1−𝑥𝐹𝑥

− 𝐹𝑒𝐴𝑠 𝑥 = 0.05 − 0.12

26 K Y. Kamihara et al., J. Am. Chem. Soc. 130,

3296 (2008)

4 𝐿𝑎 𝑂1−𝑥𝐶𝑎𝑥2+ 𝐹𝑒𝐴𝑠 0 K

Y. Kamihara et al., J. Am. Chem. Soc. 130, 3296 (2008)

5 𝐿𝑎 𝑂1−𝑥𝐹𝑥 𝑁𝑖𝐴𝑠 2.75 K 𝑥 = 0 3.8 K 𝑥 = 0.1

Z. Li et al., arXiv: 0803.2572

6 𝐿𝑎1−𝑥𝑆𝑟𝑥 𝑂𝑁𝑖𝐴𝑠 2.75 K 𝑥 = 0

3.7 K 𝑥 = 0.1 − 0.2 L. Fang et al., arXiv: 0803.3978

7 𝐿𝑎1−𝑥𝑆𝑟𝑥 𝑂𝐹𝑒𝐴𝑠 25 K 𝑥 = 0.13 H.H. Wen et al., EPL 82, 17009 (2008)

8 𝐶𝑒 𝑂1−𝑥𝐹𝑥 𝐹𝑒𝐴𝑠 41 K 𝑥 = 0.2 G.F. Chen et al., arXiv: 0803.3790

9 𝑃𝑟 𝑂1−𝑥𝐹𝑥 𝐹𝑒𝐴𝑠 52 K 𝑥 = 0.11 Z.-A. Ren et al., 0803.4283

10 𝑁𝑑 𝑂1−𝑥𝐹𝑥 𝐹𝑒𝐴𝑠 52 K 𝑥 = 0.11 Z.-A. Ren et al., 0803.4234

11 𝐺𝑑 𝑂1−𝑥𝐹𝑥 𝐹𝑒𝐴𝑠 36 K 𝑥 = 0.17 P. Cheng et al., arXiv: 0804.0835



12 𝑆𝑚 𝑂1−𝑥𝐹𝑥 𝐹𝑒𝐴𝑠 55 K 𝑥 = 0.1 − 0.2 Z.-A. Ren et al., 0804.2053

R.H. Liu et al., arXiv: 0804.2105

13 𝐸𝑢,𝑇𝑚 𝑂1−𝑥𝐹𝑥 𝐹𝑒𝐴𝑠 No stable 𝑍𝑟𝐶𝑢𝑆𝑖𝐴𝑠

structure G.F. Chen et al., arXiv: 0803.4384

14 𝐵𝑎𝐹𝑒2𝐴𝑠2 ~5 K (SDW at 140 K) M. Rotter, Marcus Tegel et al., PRB

78,020503 (2008)

15 𝐵𝑎1−𝑥𝐾𝑥𝐹𝑒2𝐴𝑠2

𝑥 = 0.4 ~38 K, 𝑇ℎ𝐶𝑟2𝑆𝑖2

M. Rotter, Marcus Tegel, Dirk Johrendt, PRL 101, 107006 (2008)

16 𝑆𝑟𝐹𝑒2𝐴𝑠2 C.Krellner et al., arxiV 0806.1043

17 𝑆𝑟𝑥𝐾1−𝑥𝐹𝑒2𝐴𝑠2 𝑥 = 0.5 − 0.6

36.5 K Kalyan Sasmal et al., arXiv 0806.1301

18 𝑆𝑟𝑥𝐶𝑠1−𝑥𝐹𝑒2𝐴𝑠2 𝑥 = 0.5 − 0.6

37.2 K Kalyan Sasmal et al., arXiv 0806.1301

19 𝐾𝐹𝑒2𝐴𝑠2 3.8 K

20 𝐶𝑠𝐹𝑒2𝐴𝑠2 2.6 K

21 𝐶𝑎𝐹𝑒2𝐴𝑠2 170 K (Structural,

Magnetic Transition) N. Ni,S. Nandi et al ., PRB 78, 014523 (2008)

22 𝐹𝑒𝑆𝑒 18 K Y. Mizuguchi, F. Tomioka, S. Tsuda, T.

Yamaguchi and Y. Takano, arXiv 0807.4315



Theoretical Modeling of Fe-Based SC’s

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 – 3d orbitals namely - 3𝑑𝑥𝑦 , 3𝑑𝑦𝑧 ,

3𝑑𝑧𝑥 , 3𝑑𝑧2 and 3𝑑𝑥2−𝑦2 .

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 𝑑𝑦𝑧 ,𝑑𝑥𝑧 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,

𝐻0 = 𝜓𝑘 ,𝜍† 𝜖+ 𝑘 − 𝜇 𝟏 + 𝜖− 𝑘 𝜏3 + 𝜖𝑥𝑦 𝑘 𝜏1 𝜓𝑘 ,𝜍

𝑘 ,𝜍

where the wave function 𝜓𝑘 ,𝜍 is given by,

𝜓𝑘 ,𝜍 = 𝐶𝑑𝑦𝑧 ,𝑘 ,𝜍

𝐶𝑑𝑥𝑧 ,𝑘 ,𝜍

𝜓𝑘 ,𝜍 is the annihilation operator for spin 𝜍 electrons in the two orbitals and similarly 𝜓𝑘 ,𝜍† is the creation

operator for spin 𝜍 electrons in the two orbitals. And,

𝐹𝑒 atoms (all in a plane)

𝐴𝑠 atom (slightly below the plane of 𝐹𝑒 atoms)

𝐴𝑠 atom (slightly above the plane of 𝐹𝑒 atoms)



𝜖+ 𝑘 = − 𝑡1 + 𝑡2 cos𝑘𝑥 + cos 𝑘𝑦 − 4𝑡3 cos 𝑘𝑥 cos 𝑘𝑦

𝜖− 𝑘 = 𝑡1 − 𝑡2 cos𝑘𝑥 − cos𝑘𝑦

𝜖𝑥𝑦 𝑘 = −4𝑡4 sin 𝑘𝑥 sin𝑘𝑦

1 is the 2 × 2 identity matrix, 𝜏3 and 𝜏1 and the standard Pauli’s matrices given by,

𝜏1 = 1 00 −1

, 𝜏3 = 0 11 0

𝑡𝑖 ’s are the various tight binding parameters as illustrated in Figure 2.

Figure 2: A schematic showing the hopping parameters of the two-orbital 𝒅𝒙𝒛 − 𝒅𝒚𝒛 model on a square lattice.

Here 𝒕𝟏 is a near neighbor hopping between 𝝈 – orbitals, 𝒕𝟐 is a near neighbor hopping between 𝝅 – orbitals, 𝒕𝟑 is the second neighbor hopping between similar orbitals and 𝒕𝟒 is the second neighbor hopping between different

orbitals.

So the Hamiltonian can be rewritten as,

𝐻0 = 𝐶𝑑𝑦𝑧 ,𝑘 ,𝜍† 𝐶𝑑𝑥𝑧 ,𝑘 ,𝜍

†

𝜖+ 𝑘 − 𝜇 0

0 𝜖+ 𝑘 − 𝜇 +

𝜖− 𝑘 0

0 −𝜖− 𝑘

𝑘 ,𝜍

+ 0 𝜖𝑥𝑦 𝑘

𝜖𝑥𝑦 𝑘 0

𝐶𝑑𝑦𝑧 ,𝑘 ,𝜍


= 𝐶𝑑𝑦𝑧 ,𝑘 ,𝜍† 𝐶𝑑𝑥𝑧 ,𝑘 ,𝜍

†

𝜖+ 𝑘 + 𝜖− 𝑘 − 𝜇 𝜖𝑥𝑦 𝑘

𝜖𝑥𝑦 𝑘 𝜖+ 𝑘 − 𝜖− 𝑘 − 𝜇

𝐶𝑑𝑦𝑧 ,𝑘 ,𝜍


𝑘 ,𝜍

= 𝜓𝑘 ,𝜍† 𝑇 𝑘𝜓𝑘 ,𝜍

𝑘 ,𝜍

Using Bogoliubov transformation we have,

𝜓𝑘 ,𝜍 = 𝐶𝑑𝑦𝑧 ,𝑘 ,𝜍

𝐶𝑑𝑥𝑧 ,𝑘 ,𝜍 =

𝑢𝑘 −𝜍𝑣𝑘𝜍𝑣𝑘 𝑢𝑘

𝛼𝑘 ,𝜍

𝛽𝑘 ,𝜍

⇒ 𝐶𝑑𝑦𝑧 ,𝑘 ,𝜍 = 𝑢𝑘𝛼𝑘 ,𝜍 − 𝜍𝑣𝑘𝛽𝑘 ,𝜍 & 𝐶𝑑𝑥𝑧 ,𝑘 ,𝜍 = 𝜍𝑣𝑘𝛼𝑘 ,𝜍 + 𝑢𝑘𝛽𝑘 ,𝜍

⇒ 𝐶𝑑𝑦𝑧 ,𝑘 ,𝜍† = 𝑢𝑘𝛼𝑘 ,𝜍

† − 𝜍𝑣𝑘𝛽𝑘 ,𝜍† & 𝐶𝑑𝑥𝑧 ,𝑘 ,𝜍

† = 𝜍𝑣𝑘𝛼𝑘 ,𝜍† + 𝑢𝑘𝛽𝑘 ,𝜍

†

Now rewriting the Hamiltonian above in terms of the transformed operators 𝛼𝑘 ,𝜍 and 𝛽𝑘 ,𝜍 we get,



𝐻0 = 𝜖+ 𝑘 + 𝜖− 𝑘 − 𝜇 𝑢𝑘𝛼𝑘 ,𝜍 † − 𝜍𝑣𝑘𝛽𝑘 ,𝜍

† 𝑢𝑘𝛼𝑘 ,𝜍 − 𝜍𝑣𝑘𝛽𝑘 ,𝜍

𝑘 ,𝜍

+ 𝜖+ 𝑘 − 𝜖− 𝑘 − 𝜇 𝜍𝑣𝑘𝛼𝑘 ,𝜍† + 𝑢𝑘𝛽𝑘 ,𝜍

† 𝜍𝑣𝑘𝛼𝑘 ,𝜍 + 𝑢𝑘𝛽𝑘 ,𝜍

𝑘 ,𝜍

+ 𝜖𝑥𝑦 𝑘 𝜍𝑣𝑘𝛼𝑘 ,𝜍† + 𝑢𝑘𝛽𝑘 ,𝜍

† 𝑢𝑘𝛼𝑘 ,𝜍 − 𝜍𝑣𝑘𝛽𝑘 ,𝜍

𝑘 ,𝜍

+ 𝜖𝑥𝑦 𝑘 𝑢𝑘𝛼𝑘 ,𝜍 † − 𝜍𝑣𝑘𝛽𝑘 ,𝜍

† 𝜍𝑣𝑘𝛼𝑘 ,𝜍 + 𝑢𝑘𝛽𝑘 ,𝜍

𝑘 ,𝜍

Simplifying and collecting terms we have,

𝐻0 = 𝜖+ 𝑘 + 𝜖− 𝑘 − 𝜇 𝑢𝑘2 + 𝜖+ 𝑘 − 𝜖− 𝑘 − 𝜇 𝑣𝑘

2 + 2𝜍𝜖𝑥𝑦 𝑘 𝑢𝑘𝑣𝑘 𝛼𝑘 ,𝜍† 𝛼𝑘 ,𝜍

𝑘 .𝜍

+ 𝜖+ 𝑘 + 𝜖− 𝑘 − 𝜇 𝑣𝑘2 + 𝜖+ 𝑘 − 𝜖− 𝑘 − 𝜇 𝑢𝑘

2 − 2𝜍𝜖𝑥𝑦 𝑘 𝑢𝑘𝑣𝑘 𝛽𝑘 ,𝜍† 𝛽𝑘 ,𝜍

𝑘 ,𝜍

+ −2𝜍𝜖− 𝑘 𝑢𝑘𝑣𝑘 + 𝜖𝑥𝑦 𝑘 𝑢𝑘2 − 𝑣𝑘

2 𝛼𝑘 ,𝜍† 𝛽𝑘 ,𝜍 + 𝛽𝑘 ,𝜍

† 𝛼𝑘 ,𝜍

𝑘 ,𝜍

Now we demand the coefficient of the cross terms (last term) equal to zero to get, 2𝜍𝜖− 𝑘 𝑢𝑘𝑣𝑘 = 𝜖𝑥𝑦 𝑘 𝑢𝑘

2 − 𝑣𝑘2

And from the normalization condition we already have, 𝑢𝑘

2 + 𝑣𝑘2 = 1

Solving these two equations for 𝑢𝑘 and 𝑣𝑘 yields,

𝑢𝑘 2 =

1

2 1 ±

𝜖− 𝑘

𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘

𝑣𝑘 2 =

1

2 1 ∓

𝜖− 𝑘


Also,

2𝑢𝑘𝑣𝑘 = 𝜖𝑥𝑦 𝑘


Substituting these values back into the Hamiltonian we get a diagonalized Hamiltonian,

𝐻0 = 𝜖+ 𝑘 + 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘 − 𝜇 𝛼𝑘 ,𝜍† 𝛼𝑘 ,𝜍

𝑘 ,𝜍

+ 𝜖+ 𝑘 − 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘 − 𝜇 𝛽𝑘 ,𝜍† 𝛽𝑘 ,𝜍

𝑘 ,𝜍

Representing these eigenvalues as,

𝜺𝒌± = 𝝐+ 𝒌 ± 𝝐−𝟐 𝒌 + 𝝐𝒙𝒚𝟐 𝒌 − 𝝁

⇒ 𝐻0 = 휀𝑘+𝛼𝑘 ,𝜍

† 𝛼𝑘 ,𝜍 + 휀𝑘−𝛽𝑘 ,𝜍

† 𝛽𝑘 ,𝜍

𝑘 ,𝜍

= 𝛼𝑘 ,𝜍† 𝛽𝑘 ,𝜍

† 휀𝑘

+ 00 휀𝑘

− 𝛼𝑘 ,𝜍

𝛽𝑘 .𝜍

𝑘 ,𝜍

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 휀𝑘

+ and label it as 휀𝑘𝑒 and similarly 휀𝑘

− as

휀𝑘ℎ . Following is a three dimensional surface plot showing the electron and hole like dispersions in the two

directions 𝑘𝑥 and 𝑘𝑦 in momentum space.



Figure 3: Electron and hole like dispersions as a function of 𝒌 in momentum space for our choice of 𝝁 = 𝟏.𝟒𝟓.

Upper curve represents 𝜺𝒌𝒆 and lower 𝜺𝒌

𝒉.

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,

휀𝑘𝑒 = 휀𝑘

+ = 𝜖+ 𝑘 + 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘 ≈ 0 {Red dots}

휀𝑘ℎ = 휀𝑘

− = 𝜖+ 𝑘 − 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘 ≈ 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 𝜇 = 1.45 and other tight binding parameters 𝑡1 = −1.0 eV, 𝑡2 = 1.3 eV and 𝑡3 = 𝑡4 = −0.85 eV. Blue colored lines (dots) corresponds to the hole pockets and red corresponds to electron pockets.



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 𝛼, 𝛽 and 𝛾 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,



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 휀𝑘𝑒 and 휀𝑘

ℎ that the topology of the FS is very much dependent on the chemical potential 𝜇 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 𝜇 .

Fermi surfaces in full Brillouin zone for 𝝁 = (top left to right) 𝟏.𝟐𝟓, 𝟏.𝟑𝟓, 𝟏.𝟒𝟓; (bottom left to right) 𝟏.𝟓𝟓, 𝟏.𝟔𝟓

and 𝟏.𝟕𝟓. Blue colored lines (dots) correspond to hole pockets 𝜺𝒌𝒉 ≈ 𝟎 and red to electron pockets 𝜺𝒌

𝒆 ≈ 𝟎 .

And these fold to give,



Fermi surfaces in folded or reduced Brillouin zone for 𝝁 = (top left to right) 𝟏.𝟐𝟓, 𝟏.𝟑𝟓, 𝟏.𝟒𝟓; (bottom left to

right) 𝟏.𝟓𝟓, 𝟏.𝟔𝟓 and 𝟏.𝟕𝟓. 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 𝜇 is increased (decreased).



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,



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 𝜒 for the tight binding model (8) can be written as,

𝜒𝑠𝑡 𝑞, 𝑖Ω = 𝑑𝜏𝑒𝑖Ωτ TτSs −q, τ . St q, 0 𝛽

0

Due to the existence of two degenerate orbitals in the model, the spin susceptibility also has two orbital indices 𝑠 and 𝑡. Here,

𝑆𝑠 𝑞 =1

2 𝜓𝑠𝛼

† 𝑘 + 𝑞 𝜍 𝛼𝛽𝜓𝑠𝛽 𝑘

𝑘

is the spin operator for the orbital labeled by 𝑠. The physical spin susceptibility is given by,

𝜒𝑆 𝑞, 𝑖Ω = 𝜒𝑠𝑡 𝑞, 𝑖Ω

𝑠,𝑡

The one loop contribution to the spin susceptibility can be obtained as,

𝜒𝑆 𝑞, 𝑖Ω = −𝑇

2𝑁 Tr 𝐺 𝑘 + 𝑞, 𝑖𝜔𝑛 + 𝑖Ω 𝐺 𝑘, 𝑖𝜔𝑛

𝑘 ,𝜔𝑛

𝜒𝑆 𝑞, 𝑖Ω = −1

2𝑁

𝑘 + 𝑞, 𝜈 𝑘, 𝜈′ 2

𝑖Ω + 𝐸𝜈 ,𝑘+𝑞 − 𝐸𝜈 ′ ,𝑘 𝑛𝐹 𝐸𝜈 ,𝑘+𝑞 − 𝑛𝐹 𝐸𝜈 ′ ,𝑘

𝑘 ,𝜈 ,𝜈 ′

Here 𝐸𝜈 ,𝑘 , 𝜈 = ±1 is the eigenvalue of the upper and lower band of the dispersions derived above

and 𝑘, 𝜈 is the corresponding normalized eigenvector. 𝑁 is the total no. of integration or summation

points and 𝑛𝐹 𝐸𝜈 ,𝑘 is the Fermi distribution function given by,

𝑛𝐹 𝐸 =1

exp 𝛽𝐸 + 1

with 𝛽 = 1 𝑘𝐵𝑇 where 𝑘𝐵 is the Boltzmann constant and 𝑇 is the temperature. The eigenvalues have already been calculated in the above sections,

𝐸+1,𝑘 = 휀𝑘+ = 𝜖+ 𝑘 + 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘 − 𝜇

𝐸−1,𝑘 = 휀𝑘− = 𝜖+ 𝑘 − 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘 − 𝜇



The eigenvectors were calculated using Wolfram Mathematica and then were normalized to give,

𝑘, +1 = 𝑁+1 𝜖− 𝑘 + 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘

𝜖𝑥𝑦 𝑘 , 1

𝑘,−1 = 𝑁−1 𝜖− 𝑘 − 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘

𝜖𝑥𝑦 𝑘 , 1

with 𝑁+1 and 𝑁−1 being the normalization constants given by,

𝑁+1 = 𝜖𝑥𝑦2 𝑘

𝜖− 𝑘 + 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘 2

+ 𝜖𝑥𝑦2 𝑘

𝑁−1 = 𝜖𝑥𝑦2 𝑘

𝜖− 𝑘 − 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘 2

+ 𝜖𝑥𝑦2 𝑘

Following plot shows the variation of real part of static 𝜔 = 0 spin susceptibility with the wave vector 𝑞 for 𝜇 = 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 𝜒𝑆 𝑞, 0 occurs around 𝑞 = 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 𝝌𝑺 𝒒,𝟎 as a function of wave vector (q) for

𝝁 = 𝟏.𝟒𝟓 at 𝟏𝟎𝟎 deg. K.



Figure 5: Variation of Real part of SSS 𝝌𝑺 with 𝝁 for three different wave vectors 𝒒 as indicated in the figure.

As could be seen in the figure above there is a sudden change in the static spin susceptibility for 𝑞 = 0,0 at 𝜇 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.



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 𝑀𝑛𝐹2 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 𝑇𝑆𝐷𝑊 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 ↑ and with spin ↓ is the same everywhere; the spatial variation of the total charge density is given by,

𝜌 𝒓 = 𝜌↑ 𝒓 + 𝜌↓ 𝒓 and only reflects the periodicity of the crystal lattice. The development of a SDW violates the translational invariance; now the charge density 𝜌↑ ↓ has a sinusoidal modulation,

𝜌↑ ↓ 𝒓 =1

2𝜌0 𝒓 1 ± 𝜍0 cos𝑸. 𝒓

with 𝜍0 the amplitude and 𝑸 the wave vector of the SDW. The wavelength 𝜆 = 2𝜋 𝑄 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 𝐿. In fact, the ratio 𝑛𝜆 𝐿 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 𝑸 = 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 𝑄. The case is most obvious for one dimension, where the Fermi surface consists of two parallel planes at ±𝑘𝐹 . In two or three dimensions a complete



nesting by just a single 𝑄-vector is not possible any more, but different parts of the Fermi surface can be mapped by different 𝑄 vectors in a more or less perfect way. The spatial modulation of the electron spin density leads to a superstructure, and an energy gap 2Δ 𝑇 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 𝑇𝑆𝐷𝑊 . A typical example of a one-dimensional metal which undergoes a SDW transition is the Bechgaard salt 𝑇𝑀𝑇𝑆𝐹 2𝑃𝐹6.

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 𝑄 as nesting vector 𝜋, 0 for the 2 band model,

휀𝑘+𝑄𝑒 = 휀𝑘

ℎ nested hole pockets

휀𝑘+𝑄ℎ = 휀𝑘

𝑒 nested electron pockets

Also these points satisfy, 휀𝑘ℎ ≈ 0 or 휀𝑘

𝑒 ≈ 0 so as to lie on the FS (or zero energy surface).

Showing nesting of Fermi surface for 𝝁 = 𝟏.𝟓𝟑 & 𝚫𝑺𝑫𝑾 = 𝟎.𝟎𝟐 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,𝟎 .

Actual nested points of the Fermi surface in FBZ for 𝝁 = 𝟏 .𝟓𝟑 & 𝚫𝑺𝑫𝑾 = 𝟎.𝟎𝟐 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.



Figure 7: Nested points of the FS in the RBZ for 𝝁 = 𝟏.𝟓𝟑 and 𝚫𝑺𝑫𝑾 = 𝟎.𝟎𝟐 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 – BASED SC’S The Hamiltonian for the two bands per site model in spin density wave (SDW) state can be written as,

𝐻 𝑆𝐷𝑊 = 휀𝑘𝑒𝐶𝑘 ,𝜍

† 𝐶𝑘 ,𝜍 + 휀𝑘ℎ𝑓𝑘 ,𝜍

† 𝑓𝑘 ,𝜍 + Δ𝑆𝐷𝑊 𝐶𝑘 ,𝜍† 𝑓𝑘+𝑄,𝜍 + 𝑓𝑘+𝑄,𝜍

† 𝐶𝑘 ,𝜍

𝑘 ,𝜍𝑘 ,𝜍

+ 휀 𝑘𝑒𝐶𝑘+𝑄,𝜍

† 𝐶𝑘+𝑄,𝜍 + 휀 𝑘ℎ𝑓𝑘+𝑄,𝜍

† 𝑓𝑘+𝑄,𝜍

𝑘 ,𝜍

+ ΔSDW 𝐶𝑘+𝑄,𝜍† 𝑓𝑘 ,𝜍 + 𝑓𝑘 ,𝜍

† 𝐶𝑘+𝑄,𝜍

𝑘 ,𝜍

where 𝐶𝑘 ,𝜍 , 𝐶𝑘+𝑄,𝜍 , 𝑓𝑘 ,𝜍 and 𝑓𝑘+𝑄,𝜍 are the annihilation operators. 𝐶𝑘 ,𝜍 destroys an electron with

the wave vector 𝑘 and spin 𝜍 while 𝑓𝑘 ,𝜍 destroys a hole with wave vector 𝑘 and spin 𝜍. Similarly we have

four creation operators which are essentially the complex conjugate of the annihilation operators. 휀𝑘𝑒 and

휀𝑘ℎ are as derived earlier in the previous section,

휀𝑘𝑒 = 휀𝑘

+ = 𝜖+ 𝑘 + 𝜖−2 𝑘 + 𝜖𝑥𝑦

2 𝑘 − 𝜇………… 1

휀𝑘ℎ = 휀𝑘

− = 𝜖+ 𝑘 − 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘 − 𝜇………… 2

where,

𝜖+ 𝑘 = − 𝑡1 + 𝑡2 cos𝑘𝑥 + cos 𝑘𝑦 − 4𝑡3 cos 𝑘𝑥 cos 𝑘𝑦

𝜖− 𝑘 = − 𝑡1 − 𝑡2 cos𝑘𝑥 − cos 𝑘𝑦

𝜖𝑥𝑦 𝑘 = −4𝑡4 sin 𝑘𝑥 sin𝑘𝑦

𝑡𝑖 ’s 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 휀 𝑘𝑒 and 휀 𝑘

ℎ , which are also

functions of wave vector, 𝑘. However, it is important to note that 휀 𝑘𝑒 and 휀 𝑘

ℎ are such that,

휀 𝑘𝑒 = 휀𝑘+𝑄

𝑒 𝑎𝑛𝑑 휀 𝑘ℎ = 휀𝑘+𝑄

ℎ

with 𝑄 being the nesting vector equal to 𝜋, 0 or 0,𝜋 . The above Hamiltonian can be expressed in a matrix form as,

𝐻 𝑆𝐷𝑊 = 𝜓𝑘 ,𝜍† 𝐻 𝑘𝜓𝑘 ,𝜍

𝑘 ,𝜍



with

𝜓𝑘 ,𝜍 =

𝐶𝑘 ,𝜍

𝑓𝑘 ,𝜍

𝐶𝑘+𝑄,𝜍

𝑓𝑘+𝑄,𝜍

𝐻 𝑘 =

휀𝑘𝑒 0 0 Δ𝑆𝐷𝑊

0 휀𝑘ℎ Δ𝑆𝐷𝑊 0

0 Δ𝑆𝐷𝑊 휀 𝑘𝑒 0

Δ𝑆𝐷𝑊 0 0 휀 𝑘ℎ

Now, the electron and hole like dispersions can be rewritten as,

휀𝑘𝑒 = 휀𝑛𝑛 𝑘 + 휀𝑒

𝑛 𝑘 휀𝑘ℎ = 휀𝑛𝑛 𝑘 + 휀ℎ

𝑛 𝑘

where 휀𝑛𝑛 𝑘 is the contribution to the energies, 휀𝑘𝑒 and 휀𝑘

ℎ , coming from the non-nested

portions/regions of the Fermi surface. Similarly 휀𝑒𝑛 𝑘 and 휀ℎ

𝑛 𝑘 are the contributions to 휀𝑘𝑒 and 휀𝑘

ℎ from the nested regions of the FS respectively. These are functions of wave vector, 𝑘 and also depend on the tight binding parameters and may be expressed as follows,

휀𝑛𝑛 𝑘 = − 𝑡1 + 𝑡2 cos𝑘𝑦 − 𝜇

휀+𝑛 𝑘 = − 𝑡1 + 𝑡2 cos 𝑘𝑥 − 4𝑡3 cos 𝑘𝑥 cos 𝑘𝑦 + 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘

휀−𝑛 𝑘 = − 𝑡1 + 𝑡2 cos 𝑘𝑥 − 4𝑡3 cos 𝑘𝑥 cos 𝑘𝑦 − 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘

And similarly, 휀 𝑘𝑒 = 휀𝑛𝑛 𝑘 − 휀ℎ

𝑛 𝑘 휀 𝑘ℎ = 휀𝑛𝑛 𝑘 − 휀𝑒

𝑛 𝑘 Now the above Hamiltonian 𝐻 𝑘 can be rewritten as,

𝐻 𝑘 =

휀𝑛𝑛 𝑘 + 휀𝑒𝑛 𝑘 0 0 Δ𝑆𝐷𝑊

0 휀𝑛𝑛 𝑘 + 휀ℎ𝑛 𝑘 Δ𝑆𝐷𝑊 0

0 Δ𝑆𝐷𝑊 휀𝑛𝑛 𝑘 − 휀ℎ𝑛 𝑘 0

Δ𝑆𝐷𝑊 0 0 휀𝑛𝑛 𝑘 − 휀𝑒𝑛 𝑘

Diagonalizing the Hamiltonian in the mean field approximation using Bogoliubov transformation,

𝜓𝑘 ,𝜍 =

𝐶𝑘 ,𝜍

𝑓𝑘 ,𝜍

𝐶𝑘+𝑄,𝜍

𝑓𝑘+𝑄,𝜍

=

−𝑢𝑘 0 0 𝑣𝑘0 𝛼𝑘 𝛽𝑘 00 −𝛽𝑘 𝛼𝑘 0𝑣𝑘 0 0 𝑢𝑘

𝛾𝑘 ,𝜍𝑐

𝛿𝑘 ,𝜍𝑐

𝛿𝑘 ,𝜍𝑣

𝛾𝑘 ,𝜍𝑣

Functions 𝑢𝑘 and 𝑣𝑘 are real and symmetric in the momentum space 𝑢−𝑘 = 𝑢𝑘 , 𝑣−𝑘 = 𝑣𝑘 and they fulfill the normalization condition,

𝑢𝑘2 + 𝑣𝑘

2 = 1 Same goes for 𝛼𝑘 and 𝛽𝑘 . Rewriting we have,

𝐶𝑘 ,𝜍 = −𝑢𝑘𝛾𝑘 ,𝜍𝑐 + 𝑣𝑘𝛾𝑘 ,𝜍

𝑣 ⟹ 𝐶𝑘 ,𝜍† = −𝑢𝑘 𝛾𝑘 ,𝜍

𝑐 †

+ 𝑣𝑘 𝛾𝑘 ,𝜍𝑣

†

𝑓𝑘 ,𝜍 = 𝛼𝑘𝛿𝑘 ,𝜍𝑐 + 𝛽𝑘𝛿𝑘 ,𝜍

𝑣 ⟹ 𝑓𝑘 ,𝜍† = 𝛼𝑘 𝛿𝑘 ,𝜍

𝑐 †

+ 𝛽𝑘 𝛿𝑘 ,𝜍𝑣

†

𝐶𝑘+𝑄,𝜍 = −𝛽𝑘𝛿𝑘 ,𝜍𝑐 + 𝛼𝑘𝛿𝑘 ,𝜍

𝑣 ⟹ 𝐶𝑘+𝑄,𝜍† = −𝛽𝑘 𝛿𝑘 ,𝜍

𝑐 †

+ 𝛼𝑘 𝛿𝑘 ,𝜍𝑣

†

𝑓𝑘+𝑄,𝜍 = 𝑣𝑘𝛾𝑘 ,𝜍𝑐 + 𝑢𝑘𝛾𝑘 ,𝜍

𝑣 ⟹ 𝑓𝑘+𝑄,𝜍† = 𝑣𝑘 𝛾𝑘 ,𝜍

𝑐 †

+ 𝑢𝑘 𝛾𝑘 ,𝜍𝑣

†



To calculate 𝑢𝑘 , 𝑣𝑘 , 𝛼𝑘 and 𝛽𝑘 we rewrite the Hamiltonian in terms of the transformed operators to get,

𝐻 𝑆𝐷𝑊 = 휀𝑛𝑛 𝑘 + 휀𝑒𝑛 𝑘 −𝑢𝑘 𝛾𝑘 ,𝜍

𝑐 †

+ 𝑣𝑘 𝛾𝑘 ,𝜍𝑣

† −𝑢𝑘𝛾𝑘 ,𝜍

𝑐 + 𝑣𝑘𝛾𝑘 ,𝜍𝑣

𝑘 ,𝜍

+ 휀𝑛𝑛 𝑘 + 휀ℎ𝑛 𝑘 𝛼𝑘 𝛿𝑘 ,𝜍

𝑐 †

+ 𝛽𝑘 𝛿𝑘 ,𝜍𝑣

† 𝛼𝑘𝛿𝑘 ,𝜍

𝑐 + 𝛽𝑘𝛿𝑘 ,𝜍𝑣

𝑘 ,𝜍

+ 휀𝑛𝑛 𝑘 − 휀ℎ𝑛 𝑘 −𝛽𝑘 𝛿𝑘 ,𝜍

𝑐 †

+ 𝛼𝑘 𝛿𝑘 ,𝜍𝑣

† −𝛽𝑘𝛿𝑘 ,𝜍

𝑐 + 𝛼𝑘𝛿𝑘 ,𝜍𝑣

𝑘 ,𝜍

+ 휀𝑛𝑛 𝑘 − 휀𝑒𝑛 𝑘 𝑣𝑘 𝛾𝑘 ,𝜍

𝑐 †

+ 𝑢𝑘 𝛾𝑘 ,𝜍𝑣

† 𝑣𝑘𝛾𝑘 ,𝜍

𝑐 + 𝑢𝑘𝛾𝑘 ,𝜍𝑣

𝑘 ,𝜍

+ Δ𝑆𝐷𝑊 𝑣𝑘 𝛾𝑘 ,𝜍𝑐

†+ 𝑢𝑘 𝛾𝑘 ,𝜍

𝑣 † −𝑢𝑘𝛾𝑘 ,𝜍

𝑐 + 𝑣𝑘𝛾𝑘 ,𝜍𝑣

𝑘 ,𝜍

+ Δ𝑆𝐷𝑊 −𝛽𝑘 𝛿𝑘 ,𝜍𝑐

†+ 𝛼𝑘 𝛿𝑘 ,𝜍

𝑣 † 𝛼𝑘𝛿𝑘 ,𝜍

𝑐 + 𝛽𝑘𝛿𝑘 ,𝜍𝑣

𝑘 ,𝜍

+ Δ𝑆𝐷𝑊 𝛼𝑘 𝛿𝑘 ,𝜍𝑐

†+ 𝛽𝑘 𝛿𝑘 ,𝜍

𝑣 † −𝛽𝑘𝛿𝑘 ,𝜍

𝑐 + 𝛼𝑘𝛿𝑘 ,𝜍𝑣

𝑘 ,𝜍

+ Δ𝑆𝐷𝑊 −𝑢𝑘 𝛾𝑘 ,𝜍𝑐

†+ 𝑣𝑘 𝛾𝑘 ,𝜍

𝑣 † 𝑣𝑘𝛾𝑘 ,𝜍

𝑐 + 𝑢𝑘𝛾𝑘 ,𝜍𝑣

𝑘 ,𝜍

Simplifying we get,

𝐻 𝑆𝐷𝑊 = 휀𝑛𝑛 𝑘 + 휀𝑒𝑛 𝑘 𝑢𝑘

2 + 휀𝑛𝑛 𝑘 − 휀𝑒𝑛 𝑘 𝑣𝑘

2 − 2ΔSDW 𝑢𝑘𝑣𝑘 𝛾𝑘 ,𝜍𝑐

†𝛾𝑘 ,𝜍𝑐

𝑘 ,𝜍

+ 휀𝑛𝑛 𝑘 + 휀𝑒𝑛 𝑘 𝑣𝑘

2 + 휀𝑛𝑛 𝑘 − 휀𝑒𝑛 𝑘 𝑢𝑘

2 + 2ΔSDW 𝑢𝑘𝑣𝑘 𝛾𝑘 ,𝜍𝑣

†𝛾𝑘 ,𝜍𝑣

𝑘 ,𝜍

+ 휀𝑛𝑛 𝑘 + 휀ℎ𝑛 𝑘 𝛼𝑘

2 + 휀𝑛𝑛 𝑘 − 휀ℎ𝑛 𝑘 𝛽𝑘

2 − 2ΔSDW 𝛼𝑘𝛽𝑘 𝛿𝑘 ,𝜍𝑐

†𝛿𝑘 ,𝜍𝑐

𝑘 ,𝜍

+ 휀𝑛𝑛 𝑘 + 휀ℎ𝑛 𝑘 𝛽𝑘

2 + 휀𝑛𝑛 𝑘 − 휀ℎ𝑛 𝑘 𝛼𝑘

2 + 2ΔSDW 𝛼𝑘𝛽𝑘 𝛿𝑘 ,𝜍𝑣

†𝛿𝑘 ,𝜍𝑣

𝑘 ,𝜍

+ −2휀𝑒𝑛 𝑘 𝑢𝑘𝑣𝑘 + Δ𝑆𝐷𝑊 𝑣𝑘

2 − 𝑢𝑘2 𝛾𝑘 ,𝜍

𝑐 †𝛾𝑘 ,𝜍𝑣 + 𝛾𝑘 ,𝜍

𝑣 †𝛾𝑘 ,𝜍𝑐

𝑘 ,𝜍

+ 2휀ℎ𝑛 𝑘 𝛼𝑘𝛽𝑘 + Δ𝑆𝐷𝑊 𝛽𝑘

2 − 𝛼𝑘2 𝛿𝑘 ,𝜍

𝑐 †𝛿𝑘 ,𝜍𝑣 + 𝛿𝑘 ,𝜍

𝑣 †𝛿𝑘 ,𝜍𝑐

𝑘 ,𝜍

Putting the coefficient of non-diagonal term (last two terms) to zero we have, Δ𝑆𝐷𝑊 𝑢𝑘

2 − 𝑣𝑘2 = −2휀𝑒

𝑛 𝑘 𝑢𝑘𝑣𝑘 Δ𝑆𝐷𝑊 𝛼𝑘

2 − 𝛽𝑘2 = 2휀ℎ

𝑛 𝑘 𝛼𝑘𝛽𝑘 And from the normalization condition we have,

𝑢𝑘2 + 𝑣𝑘

2 = 1 𝛼𝑘

2 + 𝛽𝑘2 = 1

Solving for 𝑢𝑘 and 𝑣𝑘 from the above equations we get,

𝑢𝑘 2 =

1

2

1 ∓휀𝑒𝑛 𝑘

휀𝑒𝑛 𝑘

2+ Δ𝑆𝐷𝑊

2

𝑣𝑘 2 =

1

2

1 ±휀𝑒𝑛 𝑘

휀𝑒𝑛 𝑘

2+ Δ𝑆𝐷𝑊

2



Also,

2𝑢𝑘𝑣𝑘 =Δ𝑆𝐷𝑊

휀𝑒𝑛 𝑘

2+ Δ𝑆𝐷𝑊

2

And solving for 𝛼𝑘 and 𝛽𝑘 we get,

|𝛼𝑘 2 =

1

2

1 ±휀ℎ𝑛 𝑘

휀ℎ𝑛 𝑘

2

+ Δ𝑆𝐷𝑊2

𝛽𝑘 2 =

1

2

1 ∓휀ℎ𝑛 𝑘

휀ℎ𝑛 𝑘

2

+ Δ𝑆𝐷𝑊2

Also,

2𝛼𝑘𝛽𝑘 =Δ𝑆𝐷𝑊

휀ℎ𝑛 𝑘

2

+ Δ𝑆𝐷𝑊2

Using these values of 𝑢𝑘 , 𝑣𝑘 , 𝛼𝑘 , 𝛽𝑘 and rewriting the Hamiltonian we get,

𝐻 𝑆𝐷𝑊 = 휀𝑘𝑛𝑛 − 휀𝑒

𝑛 𝑘 2

+ Δ𝑆𝐷𝑊2 𝛾𝑘 ,𝜍

𝑐 †𝛾𝑘 ,𝜍𝑐

𝑘 ,𝜍

+ 휀𝑘𝑛𝑛 + 휀𝑒

𝑛 𝑘 2

+ Δ𝑆𝐷𝑊2 𝛾𝑘 ,𝜍

𝑣 †𝛾𝑘 ,𝜍𝑣

𝑘 ,𝜍

+ 휀𝑘𝑛𝑛 − 휀ℎ

𝑛 𝑘 2

+ Δ𝑆𝐷𝑊2 𝛿𝑘 ,𝜍

𝑐 †𝛿𝑘 ,𝜍𝑐

𝑘 ,𝜍

+ 휀𝑘𝑛𝑛 + 휀ℎ

𝑛 𝑘 2

+ Δ𝑆𝐷𝑊2 𝛿𝑘 ,𝜍

𝑣 †𝛿𝑘 ,𝜍𝑣

𝑘 ,𝜍

where, 휀𝑘𝑛𝑛 = 휀𝑛𝑛 𝑘 = − 𝑡1 + 𝑡2 cos 𝑘𝑦 − 𝜇

휀𝑒𝑛 𝑘 = − 𝑡1 + 𝑡2 cos 𝑘𝑥 − 4𝑡3 cos 𝑘𝑥 cos 𝑘𝑦 + 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘

휀ℎ𝑛 𝑘 = − 𝑡1 + 𝑡2 cos 𝑘𝑥 − 4𝑡3 cos 𝑘𝑥 cos 𝑘𝑦 − 𝜖−2 𝑘 + 𝜖𝑥𝑦2 𝑘

The four obtained eigenvalues or quasi particle energies in spin density wave state can be summarized as,

𝑬𝒆±𝑺𝑫𝑾 𝒌 = 𝜺𝒌

𝒏𝒏 ± 𝜺𝒆𝒏 𝒌

𝟐+ 𝚫𝑺𝑫𝑾

𝟐

𝑬𝒉±𝑺𝑫𝑾 𝒌 = 𝜺𝒌

𝒏𝒏 ± 𝜺𝒉𝒏 𝒌

𝟐

+ 𝚫𝑺𝑫𝑾𝟐

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 couldn’t be completed due to shortage of time.



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 𝜇 = 1.45 & 1.53 in the reduced Brillouin zone taking the spin density wave gap Δ𝑆𝐷𝑊 equal to 0.02 eV. It can be seen from the results derived above that the quasi particle energies in SDW state,

𝐸𝑒±𝑆𝐷𝑊 𝑘 = 휀𝑘

𝑛𝑛 ± 휀𝑒𝑛 𝑘

2+ Δ𝑆𝐷𝑊

2

𝐸ℎ±𝑆𝐷𝑊 𝑘 = 휀𝑘

𝑛𝑛 ± 휀ℎ𝑛 𝑘

2

+ Δ𝑆𝐷𝑊2

depend on the temperature through Δ𝑆𝐷𝑊 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 𝜇 = 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 𝚫𝑺𝑫𝑾 = 𝟎.𝟎𝟐 eV and 𝝁 = 𝟏.𝟒𝟓 (left) & 𝟏.𝟓𝟑 (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 𝝁 = 𝟏.𝟓𝟑 & 𝚫𝑺𝑫𝑾 = 𝟎.𝟎𝟐 eV.

SOLUTION OF GAP EQUATION IN SDW STATE The band gap Δ𝑆𝐷𝑊 equation for spin density wave state as derived in (11) can be written as,

ΔSDW = −2𝑈𝑖𝑛𝑡𝑒𝑟 Δ𝑆𝐷𝑊 1

𝐸𝑘𝑒− tanh

𝛽𝐸𝑘𝑒−

2 +

1

𝐸𝑘𝑒+ tanh

𝛽𝐸𝑘𝑒+

2

𝑘

+ Δ𝑆𝐷𝑊 1

𝐸𝑘ℎ− tanh

𝛽𝐸𝑘ℎ−

2 +

1

𝐸𝑘ℎ+ tanh

𝛽𝐸𝑘ℎ+

2

𝑘



Here, in the nested region we have,

𝐸𝑘𝑒 ℎ

±

= ± 휀𝑘𝑒 ℎ

2+ Δ𝑆𝐷𝑊

2

whereas in the non nested region we have,

𝐸𝑘𝑒 ℎ

±

= 휀𝑘𝑒 ℎ

These simplified forms of 𝐸𝑘𝑒 ℎ

±

are obtained, assuming that there is perfect/complete nesting of the Fermi

surface in which case 휀𝑘𝑛𝑛 = 0 and so 휀𝑘

𝑒 ℎ = 휀𝑒 ℎ 𝑛 𝑘 .

The above spin density wave gap equation gives the temperature dependence of the band gap or SDW gap through 𝛽 which is equal to 1 𝑘𝐵𝑇 , 𝑘𝐵 being the Boltzmann constant in eV/ 𝑜K and 𝑇 is the temperature

in 𝑜𝐾. In this equation the summation over 𝑘 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 Δ𝑆𝐷𝑊 is taken to be zero. The constant outside the summation 𝑈𝑖𝑛𝑡𝑒𝑟 is inter – orbital interaction potential. Following is the temperature dependence of the spin density wave gap for chemical potential 𝜇 = 1.53 and 𝑈𝑖𝑛𝑡𝑒𝑟 = −0.34 eV.

Figure 8: Spin density wave gap versus temperature for 𝝁 = 𝟏.𝟓𝟑 and 𝑼𝒊𝒏𝒕𝒆𝒓 = −𝟎.𝟑𝟒 eV. It can be seen that the

critical temp. for SDW state for such a system is close to 160 deg. K.



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 𝑇𝑐

𝑆𝐷𝑊 .

Figure 9: SDW gap versus Temperature (T) for various values of 𝝁 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,

𝜒𝑆 𝑞, 𝑖Ω = −1

2𝑁

𝑘 + 𝑞, 𝜈 𝑘, 𝜈′ 2

𝑖Ω + 𝐸𝜈 ,𝑘+𝑞 − 𝐸𝜈 ′ ,𝑘 𝑛𝐹 𝐸𝜈 ,𝑘+𝑞 − 𝑛𝐹 𝐸𝜈 ′ ,𝑘

𝑘 ,𝜈 ,𝜈 ′

Here 𝑘, 𝜈 is the eigenvector of the 4 × 4 Hamiltonian defined above for the SDW state. As we have four eigen vectors here 𝜈 can take values -2, -1, 1, and 2 (just convention). 𝐸𝜈 ,𝑘 is the corresponding eigenvalue. 𝑁 and 𝑛𝐹 remain same – number of summation points and Fermi function. The summation here in SDW

state is carried out only for those values of 𝑘 which nest. This again makes 𝜒𝑆 dependent on the extent of nesting of the FS. Due to the shortage of time this calculation couldn’t 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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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.



Appendix

FORTRAN CODES The following FORTRAN-90 code generates data files (‘prog6_1.dat’ & ‘prog6_2.dat’) for plotting Fermi surface in the Full Brillouin Zone (FBZ) based on the Raghu’s dispersion (8). The data files contain those values of 𝑘𝑥𝐿 and 𝑘𝑦𝐿 (in various rows) for which the Energy 휀𝑘

+, 휀𝑘− 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


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



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


73 EnerN = Ep(kx,ky) - dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) – 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 𝝁 = 𝟏.𝟓𝟒.



The following FORTRAN-90 code generates data files (‘prog7_P.dat’ & ‘prog7_N.dat’) 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 45𝑜 in clockwise direction) Brillouin zone. The data files contain appropriate values of 𝑘𝑥𝐿 and 𝑘𝑦𝐿 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



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)


60

61 real(8) function En(kx,ky)

62 implicit none


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


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


101 EnerP = Ep(kx,ky) + dsqrt((En(kx,ky))**2 + (Exy(kx,ky))**2) - mu



102 end function EnerP

103

104 real(8) function EnerN(kx, ky, mu)

105 implicit none


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 𝟒𝟓𝒐) Brillouin Zone for 𝝁 = 𝟏.𝟓𝟒.



The following FORTRAN – 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



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



The following FORTRAN – 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



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



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



The following FORTRAN – 90 program calculates the static spin susceptibility in the normal state within two band model using the eigenvectors and eigenvalues calculated for 2 × 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)



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)



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



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



The following FORTRAN – 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 𝜇 and 𝜒𝑆 𝑞, 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



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



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



The following FORTRAN – 90 code calculates band gap for one orbital per site using the tight binding model at various temperatures and generates a data file (‘prog4.dat’) containing band gap Δ and corresponding Temperature 𝑇 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)


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)) – 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



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 𝝁 = 𝟎.𝟐, tight binding parameter 𝒕 = 𝟎.𝟖 eV

and potential 𝑽 = 𝟎.𝟐𝟔 eV.



The following FORTRAN – 90 code generates 12 output files containing values of 𝑘𝑥𝐿 and 𝑘𝑦𝐿 in rows for

plotting Fermi surface in the spin density wave state for 6 different values of chemical potential 𝜇 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



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



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



This FORTRAN – 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 𝜇 . The data files contain SDW gap ΔSDW and corresponding Temperature 𝑇 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))



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



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



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



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Theoretical Modeling of Iron-based Superconductors

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Transcript of Theoretical Modeling of Iron-based Superconductors