iit masters thesis powerpoint presentation

condensed matter physics

1. Study of superconductivity in MoTe2 monolayer using BdG method Susmit Walve Integrated MSc Physics Supervisor: Prof. Tulika Maitra

2. Goal of the project To model the band structure of Td-MoTe 2 using the tight-binding model. We also use crystal symmetry to minimize the number of independent hopping parameters Apply the constructed Hamiltonian to model s-wave superconductivity in the material using the Bogoliubov -de Gennes method.

3. Hamiltonian for triangular MoTe 2 The crystal structure of MoTe 2 is shown in the image The crystal follows mirror symmetry and breaks inversion symmetry Elements of point symmetry group  : Identity operation : Rotation by 120 o : Reflection through bisector of A 1 – A 2 : Reflection through bisector of A 2 – A 3 : Reflection through bisector of A 3 – A 4  

4. Meaning of crystal symmetry The Hamiltonian being symmetric under some coordinate transformation means that the functional form of the Hamiltonian doesn’t change. Symmetric under x → -x

5. Tight binding basis states The first principle band structure exhibits majority occupation by d orbitals near the Fermi level. z-symmetric orbitals {d z 2 , d x 2 , d xy } and z-antisymmetric orbitals { d xz , d yz } don’t mix in any band. Hence we will use the three z symmetric orbitals Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides, Gui-Bin Liu et al =0 From Bloch theorem

6. Hamiltonian matrix Now that we know the basis states of the system, we can write the matrix elements of the 3 x 3 Hamiltonian. For example, suppose we wish to write the element H 12 (k) = ⟨ψ 1 |H|ψ 2 ⟩ where

7. Hamiltonian matrix Now suppose we wish to calculate the term

8. Hamiltonian matrix Similarly we can write such transformations for all nearest neighbors in order to obtain the following Each state transformed using symmetry operations into state  

9. Hamiltonian matrix Since Hamiltonian is symmetric under the operations, they will commute and cancel

10. Td-MoTe 2 Reflection symmetry across x direction Another translation symmetry apart from lattice translations  

11. First principle plot Unlike previous crystal, all d- orbitals as well as p x orbitals contribute to the band formation We need to consider a 14 band model to consider all the orbitals.

12. Reducing hopping parameters

13. Band fitting To fit the tight binding bands to the first principle data, I have written a function in MATLAB to find the sum of the squares of the difference between the first principle data and the 14 eigenvalues of the Hamiltonian.

14. Including SOC Transition metals have a strong spin-orbit coupling and to get accurate results it is necessary to extend the basis to include spin . This is in angular momentum basis. We can transform this into the atomic basis using a unitary matrix.

15. Including SOC S is a vector of pauli matrices. After direct multiplication we finally get The final Hamiltonian including the SOC term would be

16. SOC We calculate the Hamiltonian matrix using the parameters obtained earlier and add the SOC terms to it and then diagonalize it.

17. Hamiltonian in second quantization The procedure we have developed till now is sufficient for band calculation but if we want to include interactions , we need to use second quantization. The exact second quantized Hamiltonian would be

18. Hamiltonian in second quantization We use the fact that the V is significant only when k’ = -k , k’’’ = -k’’ and   Now we use the mean field approximation in order to obtain the final Hamiltonian where

19. Superconductivity using BdG equations Since we wish to use the properties of grand canonical ensemble eventually, we will use a modified Hamiltonian   The Hamiltonian is not diagonal in the electron creation annihilation operators. We will choose a modified basis in which we will work with a quasi-particle which is a superposition of holes and electrons . In the new basis

20. Superconductivity using BdG equations Finally we obtain the following equations We can can write these equations in matrix form

21. Superconductivity using BdG equations

22. Calculating μ when ∆ is known Suppose we know the ∆ for the system, we can calculate its partition function Using the partition function, we can also calculate the Landau potential

23. Calculating μ when ∆ is known Landau potential has the following property We get the following intrinsic equation of μ . We need to find the chemical potential that satisfies the condition on the number of electrons per unit cell

24. Density of states for the superconductor We will use a trick to calculate the density of states The overall number of electrons can be calculated by We will set E at which we want to calculate the DOS to be the Fermi level. Suppose there are n electrons filled in the band structure at E F =E. Now suppose I modify E F = E+dE there would be n+dn electrons filled in the band structure. Thus dn would be the number of states present between E and E+dE

25. Density of states for the superconductor The DOS equation would be Another way to interpret this density of states is that the system is filled with quasi-particles which are in a superposition of electron and hole states hence we need to consider the probability of the quasi-particle being hole or electron.

26. Density of states for the superconductor The following plot is obtained for two different ∆ values. The ∆ =0 plot overlaps with the DOS plot for the normal state.

27. Calculating ∆ when μ is known When μ is known, we use the definition of ∆ to calculate its matrix iteratively

28. Calculating ∆ and μ when both are unknown Set a random value of ∆ and μ

29. Calculating ∆ and μ when both are unknown Trace of ∆ matrix Determinant of ∆ matrix

30. Thanks