Odd-Frequency Pairing

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DEPARTMENT OF PHYSICS AND ASTRONOMY DIVISION OF MATERIALS THEORY

Black-Schaffer Group – Research project

Superconductivity in M oS

₂

heterostructure :

Odd-Frequency Pairing

Yann Gaucher

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1. Que les r´esultats d´ecrits dans ce rapport sont l’aboutissement de mon travail.

2. Que je suis l’auteur de ce rapport.

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Abstract

We model an two-dimensional system composed by a monolayer of transition metal dichalcogenide, in which superconductivity is proximity-induced. The study of the Green functions of the system enables us to study the presence of odd-frequency pairing : we derive an analytical criterion for nonzero odd-frequency pairing. We then use numerical computations to investigate its magnitude and its dependence on the systems parameters.

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Acknowledgements

I would like to thank Annica Black-Schaffer for her efforts to enable the internship.

I would like to express my gratitude to Christopher Triola for being a committed and benevolent supervisor, and for his guidance along the internship.

Thanks also to Andreas Theiler for his support and the rest of the Black-Schaffer group for welcoming me during four months.

This action benefited from the support of the Chair “innovative Process Materials” led by l’X- Ecole polytechnique and the Fondation de l’Ecole polytechnique and sponsored by SAINT- GOBAIN.

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Introduction

This study was performed during a research internship in the division of Materials Theory of the Uppsala University as a 3rd year student of Ecole Polytechnique. During this stay I was a guest of the Black-Schaffer group, under the direct supervision of Christopher Triola.

The research activity of the internship concerned the study of superconductivity in a given system composed by two two-dimensional materials. In the frame of the BCS theory, superconductors charge carriers are pairs of electrons which form a macroscopic condensate. The function that describes the pairing of electrons is called the anomalous Green function and depends, among other things, on time, hence frequency in the Fourier domain. The aim of my work was to identify the features of the odd-in-time part (referred as ”odd-frequency part”) of this anomalous Green function : as explained later, odd-frequency pairing has been reported to be associated with a lot of uncommon features.

This report presents an introduction to superconductivity and odd-frequency pairing. Then, we present the physical system investigated and the mathematical model used, and we use analytical calculations to obtain a condition to have odd-frequency pairing. Finally, we use numerical computations to get more insight on the behaviour of the anomalous Green function.

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Chapter 1 Introduction to Superconductivity

1.1 Historical perspective

In 1908, the liquefaction of helium enabled Heike Kamerlingh Onnes to reach previously unattainable temperatures. In particular, he was able to measure the conductivity of metals at very low temperatures.In 1911, he discovered a phase transition in mercury, characterized by a drop of direct-current resistivity to zero below a critical temperature T_c of 4, 2K. H.K. Onnes named it ”superconductivity”.

Perfect DC conductivity is not the only feature of superconductivity: instead of capturing magnetic field as a perfect conductor would, it was found (1933, Meissner Ochsenfeld) that superconductors were also characterized by perfect diamagnetism (the Meissner Effect), that is the expulsion of magnetic field. This phase of matter occurs at given conditions of temperature, but also pressure and below a critical magnetic field.

Since its discovery, great efforts have been deployed to develop a theoretical explanation of superconductivity. The London theory (1935) was the first phenomenological description of superconductivity. Their equations replace the Ohm’s law in the superconductor, and explain in particular the Meissner effect. It was followed by the Ginzburg-Landau theory (1950), another phenomenological description of superconductivity based on Landau’s theory of phase transitions. In the Ginzburg-Landau theory, the free energy is expressed in terms of an order parameter that is related to the density of superconducting charge carriers.

But the first succesful microscopic explanation of superconductivity is the Bardeen, Cooper and Schrieffer (BCS, 1957) theory, which allowed the prediction of the critical temperatures of some superconductors.

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1.2 BCS Theory

In the remainder of this work, we will consider ¯h = 1

1.2.1 Overview

The BCS theory [1] explains superconductivity through the pairing of electrons : At low temperature, an attractive interaction can appear between electrons with opposite spins and momentum (~k, ↑) and ( ~−k, ↓)at the vicinity of the Fermi surface, mediated by electron-phonon scattering : the coulomb repulsion is over-screened by the lattice vibrations. The role played by the lattice was suggested by the fact that the critical temperature and the Debye frequency, which characterizes the lattice vibrations, exhibited a similar isotope effect.

From this analysis, Bardeen, Cooper and Schrieffer showed that the pairing amplitude was related to the energy gap and to the critical temperature.

1.2.2 BCS Hamiltonian

We consider the following two body interaction Hamiltonian : HˆI = 1

V X

σ⁰

X

kk’q

Vqc^†_kσc^†_k’σ0ck’-q,σ⁰ck+q,σ

Where c^†_kσ (ckσ)creates (annihilates) an electron with spin σ and momentum k, Vq are the Fourier coefficients of the space-dependant electron-electron interaction, and V is the volume of the system. Bardeen, Cooper and Schrieffer use a simplified Hamiltonian, considering only pairs of electrons with net momentum equal to zero and opposite spins (singlet state). Hence, we only keep pairs such as k’ = −k.

HˆI = _V¹ P

kk’Vk−k⁰c^†_k↑c^†_−k↓ck’↑c−k’↓

Where V_k−k⁰ is attractive for states with energy within a ¯hωD distance from Fermi energy. The original BCS calculation considered a contact interaction with Vk−k⁰ = V0. The 4-fermion term makes an analytic treatment impossible. A mean field approximation is therefore used, neglect- ing the second order fluctuations, : AB ' hAiB + AhBi − hAihBi.

HˆI = _V¹ P

kk’Vk−k⁰(hc^†_k↑c^†_−k↓ick’↑c−k’↓+ c^†_k↑c^†_−k↓hck’↑c−k’↓i)

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The constant term hc^†_k↑c^†_−k↓c_k’↑c−k’↓i is neglected.

We set

∆k= _V¹ P

k’Vk−k⁰hck’↑c−k’↓i to obtain :

HˆI = _V¹ P

k∆^∗_kck↑c−k↓+ c^†_k↑c^†_−k↓∆k

The BCS Hamiltonian is the sum of this interaction part and a kinetic part:

Hˆ0 =P

kσ(k− µ)c^†_kσckσ

That takes into account the eigenenergies _k and the chemical potential µ of the electrons. The diagonalisation of ˆH^BCS = ˆH_I + ˆH₀ defines two eigenenergies for each k :

±E_k = ±p(k− µ)²+ |∆_k|². Since the pairing occurs around the Fermi surface, we find a bandgap opening equal to 2∆_k, which can be probed by scanning tunneling microscope.

Further algebra shows that a self-consistent equation, the so-called ”Gap Equation” on ∆_k can be derived from the BCS hamiltonian (under some assumptions, such as isotropic Fermi surface etc.), and the Fermi statistics of its eigenstates:

∆_k= − 1 2V

X

k⁰

V_k-k’∆_k’

Ek’

(1 − 2f (E_k’))

Where f refers to the Fermi distribution. From the gap equation, BCS derive a universal relation between the gap at T = 0K and the critical temperature : 2∆₀ = 3.53k_BT_c. The term hc_k’↑c−k’↓i appears to be proportional to ∆_k, and plays the role of an order parameter. The prediction of the critical temperature and energy gap for some s-wave, low T_csuperconductors, such as Al and N b, are the main successes of BCS theory. However, it should be noted that it is not sure whether this theory and its extensions can provide a satisfactory description of unconventional superconductivity (high T_c superconductors, iron-based superconductors, etc.).

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1.3 Odd-Frequency Pairing

Many examples and definitions of this chapter are developed in the review article on odd- frequency pairing [6].

1.3.1 Pairing Amplitude

As we can see, the quantity hc_k↑c_−k↓i is key to describe superconductivity, since it is directly related to the superconducting gap ∆. Indeed, this correlation between the states k ↑ and −k ↓ can be seen as the ”pairing amplitude” of the Cooper pair. The features of those Cooper pairs determine the properties of the superconductors.

More generally, the correlation function between two electrons at different coordinates is quan- tified by the time-ordered anomalous Green function :

F_σ,σ^α,β0( ~r₁, r₂, t₁− t₂) = h ˆT_tˆc^α_σ( ~r₁, t₁)ˆc^β_σ0( ~r₂, t₂)i

Where T_t is the time ordering operator, (α, σ), (β, σ⁰) refer to the orbital and spin of each electron state, while (r_i, t_i) are their positions and times. It can be seen as the expectation value of forming a Cooper pair from electrons with those coordinates.

1.3.2 SPOT Classification

The study of the symmetries of the anomalous Green function is widely used to understand the superconductors features. Due to the Fermi-Dirac statistics, this pairing amplitude must be odd with regard to the exchange of coordinates at equal times:

F_σ,σ^α,β0( ~r1, ~r2, 0) = −F_σ^β,α₂_,σ₁( ~r2, ~r1, 0)

It has been shown (Berezinskii, 1974,[2]), that this rule can be extended at different times : F_σ,σ^α,β0( ~r1, ~r2, t) = −F_σ^β,α₂_,σ₁( ~r2, ~r1, −t) (1)

Let us define S, P, O and T , the exchange operators for spin, position, orbital and time coordinates :

S ∗ F_σ^α,β₁_,σ₂( ~r₁, ~r₂, t) = F_σ^α,β₂_,σ₁( ~r₁, ~r₂, t) ; P ∗ F_σ^α,β₁_,σ₂( ~r₁, ~r₂, t) = F_σ^α,β₁_,σ₂( ~r₂, ~r₁, t) O ∗ F_σ^α,β₁_,σ₂( ~r₁, ~r₂, t) = F_σ^β,α₁_,σ₂( ~r₁, ~r₂, t) ; T ∗ F_σ^α,β₁_,σ₂( ~r₁, ~r₂, t) = F_σ^α,β₁_,σ₂( ~r₁, ~r₂, −t)

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S - + + - + - - +

P + - + - + - + -

O + + - - + + - -

T + + + + - - - -

Table 1.1: Characterization of the eight symmetry classes

Since these operators are involutions, their eigenvalues are 1 and -1. Equation (1) implies that S ∗ P ∗ O ∗ T = −1. Hence, there are eight symmetry classes of states as showed in the Table 1.1.

For instance, in superconductors with a single band contributing to superconductivity (i.e.

even in orbital exchange), the pairing of s-wave electrons (i.e. even spatial symmetry), at equal times, must be odd in spin (spin-singlet). On the right half of Table 1.1 are shown the symmetry classes that are odd in relative time. Equivalently, their Fourier transforms are odd in frequency. For historic reason, they are called odd-frequency states.

1.3.3 Occurrence of odd-frequency pairing

When Berezinskii [2] described the possibility of odd-frequency superconductivity, he was considering the possibility of having an intrinsic superconducting order parameter ∆(ω) odd in frequency. In this work, we will rather consider odd-frequency pairing appearing in a system where even-ω pairing is the source of superconductivity and the leading instability.

Odd-frequency pairing of this kind has been reported in a variety hetero-structures, as a conse- quence of proximity induced superconductivity[6]. Even-frequency cooper pairs can be trans- formed into odd-frequency pairs. According to the SPOT classification, it requires changing 2 parity components to keep SP OT = −1. It has been found that in superconductor/ferromagnet junction, s-wave, spin triplet pairs tunnel into spin-triplet, odd-ω states (due to rotational spin symmetry breaking) ; in superconductor/normal metal junction, s-wave (even in parity) even-ω spin-singlet transforms into a p-wave (odd in parity) odd-ω spin singlet (due to translational symmetry breaking).

It has also been predicted to be ubiquitous in multiband superconductor (that is, several superconducting bands) with interband hybridisation.[3]

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1.3.4 Specific interest of odd-frequency pairing

As far as we know, no general experimental observable related unambiguously to odd-ω pairing have been found yet, but some observables have been proved to be the signatures of odd-ω pairing in specific systems. One can for example measure paramagnetic Meissner effect in superconductor/ferromagnetic heterostructures.[4]

The current research about odd-frequency pairing aims at understanding the underlying mech- anism that originates an odd-, non local in time correlation, identifying its specific properties, particularly compared to conventional superconductivity, and finding the materials where it appears.

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Chapter 2 M oS ₂ - Superconductor Heterostructure

The purpose of this work is to study odd-frequency pairing in a theoretical system composed of two different single-layer materials : one monolayer of a transition metal dichalcogenides (molebdynum disulfide), and one two-dimensional conventional superconductor. A tunnel coupling between those two materials induces superconductivity in the M oS₂ by proximity effect.

In this chapter, we will review the main features of transition metal dichalcogenides (TMD), then present the models of TMD and superconductor

Figure 2.1: schematic view of the system studied

2.1 Transition metal dichalcogenides

2.1.1 Overview

Molybdenum disulfide belongs to a class of materials called transition metal dichalcogenides.

They are composed of transition metal atoms (molybdenum, tungsten, niobium etc.) and chalcogen atoms such as sulfur, selenium or tellurium. They are lamellar solids of layers bound by Van der Waals attraction, and some are of common use in the industry, such as M oS2

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which is used as a mechanical lubricant but also in the fabrication of electronic components.

The micro-mechanical exfoliation process developed in 2004 by A. Geim and K. Novoselov for graphene can be used to single out monolayers of transition metal dichalcogenides, and other ways of fabrication are also developed, namely molecular beam epitaxy or chemical vapor deposition. They have attracted the interest of solid state physicists due to their electronic and optical properties.

2.1.2 Properties of Transition Metal Dichalcogenides

Transition metal dichalcogenides (M X2) monolayers have an hexagonal lattice (”honeycomb”), and a triangular bravais lattice, such as graphene’s, but are composed by 3 atomic plans : one layer of transition metal between two layers of chalcogens. The unit cell is a trigonal prismatic coordination.

Figure 2.2: Left : trigonal prismatic coordination of M X2 ; Right : M X2 lattice [7]

Research on TMD, in particular M oS2 has been active in the past few years, highlighting its multiple electronic properties [9] : Its lack of inversion symmetry creates a band structure characterized by two inequivalent valleys in the first Brillouin Zone K and −K. It has a strong spin-orbit coupling (possible due to lack of inversion symmetry), which leads to valence band splitting, and this splitting has opposite signs in the two ±K valleys (cf. figure 2.3).

This feature limits spin and valley relaxation, since those two degree of freedom are coupled, and enables valley polarization by optical pumping, opening the way to spintronics and

”valleytronics”[9].

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Figure 2.3: Left :First Brillouin zone of M X₂, featuring the reciprocal lattice basis b₁ and b₂, the two inequivalent valleys ±K[5]. Right : spin-splitting of the valence bands at K points [9]

2.2 Theoretical Modeling

2.2.1 Tight binding model for a MoS2 monolayer

Tight-Binding Model

Some definitions and calculations about the tight-binding picture can be found in the appendix A.1. We use the tight-binding model proposed in [5] by G-B. Liu et al. to model the transition metal dichalcogenide. In their work, G-B. Liu et al. propose to use a three band model: the main contributors to valence and conduction bands are the d_z², d_xy and d_x²_−y² orbitals from the transition metal atom (this statement is justified by first-principle calculations). We therefore have a 3 × 3 Hamiltonian :

Hˆ₀^{T M D} = X

α,α⁰,k

E_k^α,α⁰c^†_k,αc_k,α⁰

Where α refers to the d orbitals considered. G-B. Liu et al. also take into account up to the third nearest-neighbour in order to minimize the deviation from the band structure resulting from first-principle calculations (FP) : in E_k^α,α⁰ =P

de^{−i k·d}E_d^α,α⁰ , the sums run over the three first neighbours.

They then fit their result to FP calculations to obtain numerical values for the E_d^α,α⁰ hopping integrals, for different transition metal dichalcogenides. In the following, we will only consider the case of M oS₂, but we shall keep in mind the fact that we could also study W S₂, M oSe₂, W Se₂, M oT e₂ and W T e₂, just by adapting the numerical values of E_d^α,α⁰.

Spin-orbit Coupling

As mentioned in the previous section, spin-orbit coupling is an important feature of M oS₂. The Hamiltonian becomes a 6 × 6 matrix when we take into account the spin degree of freedom.

We consider the first order contribution of spin-orbit coupling, equal to ˆH_SOC = λL · S, where λ is a constant, and Since we are dealing with cubic harmonics, we have :

L_zd_z² = 0, L_zd_xy = −2i d_x²_−y², and L_zd_x²_−y² = 2i d_xy

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It follows that (keeping in mind that ¯h = 1) : ˆH_SOC = λi (c^†_d_xy_,↑c_d

x2−y2,↑− c^†_d_xy_,↓c_d

x2−y2,↓) + h.c.

The diagonalisation of the hamiltonian, with the numerical coefficients given in [5] gives the band structures in the figure 2.4.

Figure 2.4: band structure of M oS2 in the first B.Z

On the left without SOC, on the right with SOC : the valence band spin-splitting at K points is clear.

2.2.2 Tight binding model for the Superconductor

For the superconducting layer in the system (figure 2.1), we consider a simple model of superconductor, with a triangular lattice with the same lattice constant as the M oS₂, essentially for matter of simplicity. It provides the same critical points for the two materials. We again use a tight-binding Hamiltonian, this time with first-nearest neighbors, for simplicity. Its Hamilto- nian can be decomposed between a kinetic part and an ”interaction” part, describing the BCS pairing between electrons.

Hˆ^SC = ˆH₀^SC + ˆH_I^SC The kinetic part is given by :

Hˆ₀^SC = t · X

<ij>,σ

a^†_i,σa_j,σ− µX

i,σ

a^†_i,σa_i,σ

Where aj,σ is the annihilation operator for the single-atom state at site j with spin σ, and the sum runs over nearest neighbours. t is the hopping constant between nearest neighbours, and

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µ is the chemical potential. Following the calculations of Appendix A.1, we obtain : Hˆ₀^SC =X

k,σ

ξ_ka^†_k,σa_k,σ

For the considered lattice, ξ_~_k = P

δte^−i~^k·~^δ− µ = 2t [cos(2α) + cos(α + β) + cos(α − β)] − µ, with α = ¹₂k_xa , β =

√3 2 k_ya.

The BCS interaction part of the Hamiltonian is given by : Hˆ_I^SC =X

k

∆^∗_k a−k,↑a_k,↓+ h.c

2.3 Model of proximity-effect

We characterize the proximity effect with finite hopping coefficients between the electronic states in the SC and the M oS₂. The contribution to the Hamiltonian should then take the form :

T =ˆ X

α,k,k⁰,σ,σ⁰

t_α,σ,σ⁰_,k,k⁰c^†_α,σ,ka_k⁰_,σ⁰ + h.c

Several assumptions are made :

• Tunneling conserves momentum, so t_k,k⁰ = 0 if k 6= k⁰ ;

• Tunneling conserves spin (no spin-active interface), so t_σ,σ⁰ = 0 if σ 6= σ⁰ ;

• Tunneling does not depend on spin nor momentum : t = t_α ;

• Tunneling conserves phase, so t_α coefficients are real.

Ultimately, we obtain a tunneling contribution to the Hamiltonian : T =ˆ X

α,k,σ

t_αc^†_α,σ,ka_k,σ+ h.c

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2.3.1 Expression in particle-hole space

Since we want to study the anomalous Green Function, it is useful to work in particle-hole space, that is to say, to express the Hamiltonian in a basis of Nambu spinors. Let us begin with the superconductor Hamiltonian. The Nambu spinors for the superconductor are Ψ_~^†

k =

√1 2

a^†_k,↑ a^†_k,↓ a−k,↑ a−k,↓

. Using fermionic operators commutation relations, we can rewrite Hˆ_I^SC as :

Hˆ_I^SC = 1 2

X

k

∆^∗_k a−k,↑ak,↓− ∆^∗_−k a−k,↓ak,↑+ h.c

We express ˆH₀^SC as : ˆH₀^SC =P

kξ_ka^†_k,↑a_k,↑+ ξ−ka^†_−k,↓a−k,↓

And, using Fermi’s commutation relations again, we can write it : Hˆ₀^SC = 1

2 X

k

ξ_ka^†_k,↑a_k,↑+ ξ_ka^†_k,↓a_k,↓− ξ−ka−k,↑a^†_−k,↑− ξ−ka−k,↓a^†_−k,↓

Where we neglect an additional constant coming from {a^†, a}=1. We express the superconductor Hamiltonian as :

Hˆ^SC =X

k

√1 2

a^†_k,↑ a^†_k,↓ a−k,↑ a−k,↓







ξ_k 0 0 −∆−k

0 ξ_k ∆_k 0

0 ∆^∗_k −ξ−k 0

−∆^∗_−k 0 0 −ξ−k







√1 2





 a_k,↑

a_k,↓

a^†_−k,↑

a^†_−k,↓





 Hˆ^SC =X

k

Ψ_~^†

kH_~^SC

k Ψ_~_k

For the remainder of this work, we assume that ∆_k = ∆ ∈ R and t ≈ 1eV, µ ≈ 1eV and

∆ ≈ 1meV (which corresponds to a critical temperature of ≈ 10K), unless specified otherwise.

The diagonalisation of the superconductor Hamiltonian yields the band structure in figure 2.5.

The M oS₂ Hamiltonian, which is a 6 × 6 matrix ˆH_6×6^{T M D} in the orbitals ⊗ spin basis, becomes a 12 × 12 matrix in Nambu space. We define its Nambu spinors as follow :

Φ^†_k= 1

√2(c^†_1,k,↑c^†_2,k,↑c^†_3,k,↑c^†_1,k,↓c^†_2,k,↓c^†_3,k,↓c_1,−k,↑c_2,−k,↑c_3,−k,↑c_1,−k,↓c_2,−k,↓c_3,−k,↓)

Where 1, 2 and 3 refer to the three d orbitals. We can then rewrite the M oS₂ Hamiltonian,

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Figure 2.5: band structure of the SC in the first B.Z

On the left with an acceptable value of ∆, on the right with a giant ∆, to illustrate the superconducting gap opening.

with the same method as for the superconductor Hamiltonian :

Hˆ^{T M D} =X

k

Φ^†_k

ˆH_6×6^{T M D}(~k) 0 0 − ˆH_6×6^{∗T M D}( ~−k)

!

Φk=X

k

Φ^†_kHˆ_12×12^{T M D}(k)Φk

Similarly, the expression of ˆT in particle-hole space is : T =ˇ X

k

Φ^†_kTˇ12×4Ψk+ h.c.

Where ˇT_12×4 =

ˆT_6×2 0 0 − ˆT_6×2

!

and ˆT_6×2^† = t₁ t₂ t₃ 0 0 0 0 0 0 t₁ t₂ t₃

!

Finally, the total Hamiltonian for the heterostructure is :

H =ˆ X

k

Φ^†_k Ψ^†_k

ˆH_12×12^{T M D}(k) Tˇ Tˇ^† Hˆ_4×4^SC(k)

! Φ_k Ψ_k

!

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Chapter 3 Analytical Work

3.1 Green Functions

The Green function describes the correlation between the fermionic states at different times and positions. work with Matsubara Green functions that are defined in the following way : given a spinor Φ_k in momentum space, the imaginary-time Green function is G(k, τ ) = −hT_τΦ_k(τ )Φ^†_ki where T_τ is the time-ordering operator, and Φ_k(τ ) = e^HτΦ_ke^−Hτ.

Since we are dealing with fermionic operators, we have that G(k, τ + β) = −G(k, τ ) , hence its Fourier transform G(k, ω) = R G(k, τ )e^iωτ dτ will be nonzero only for a discrete set of frequencies, the Matsubara frequencies ω_n = ^(2n+1)π_β , n ∈ N.

We also use the retarded Green function, G^R(k, ω) = lim_η→0R G(k, τ )e^(ω+iη)τdτ . This is usually called ”analytic continuation” from imaginary frequency to the real line (iω_n → ω + 0⁺). For numerical computations, we will take a small η ' 0⁺. It gives us access to the spectral function A(k; ω) = −_π¹ = (T r(G^R(k; ω))), as shown in Fig 3.1, whose integral over the first Brillouin zone is equal to the density of state ρ(k, ω).

3.2 Equation of Motion

The equation of motion of the Green functions can be derived using the Heisenberg equations for each fermionic operator. From the equation of motion we find : G = G_{T M D} G_{T M D−SC}

G_{SC−T M D} G_SC

!

= (z1 − ˆH)⁻¹, where we set z = ω + i η, or z = iω_n, depending on wether we consider retarded or Matsubara Green function.

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Figure 3.1: spectral function (log scale) of M oS2 and the SC in the first B.Z.

Vertical axis : energy in eV.

We use the equation of motion to determine F_{T M D}which is such that G_{T M D} = G_{T M D} F_{T M D} F¯_{T M D} G¯_{T M D}

!

Detailed calculations can be found in the appendix A.2.

We obtain an expression for G_{T M D} :

G_{T M D} = [z 1_12×12− ˆH_{T M D}− ˆΣ]⁻¹

Where ˆΣ ≡ ˇT [z 1_6×6− ˆH_SC]⁻¹Tˇ^†.

In the literature, ˆΣ is usually referred to as the ”self-energy”.

3.3 Sufficient Condition for odd-frequency pairing

With the expression for G_{T M D}derived from the equation of motion, we can study the anomalous retarded Green function F_{T M D}, and particularly its even- and odd-frequency components, F_{T M D}^even and F_{T M D}^odd defined by ^F^{T M D}^(z)+F₂ ^{T M D}^(−z), and ^F^{T M D}^(z)−F₂^{T M D}^(−z). In this section, we derive an analytical criterion for odd-frequency pairing from the equation of motion, detailed calculations can be found in A.3.4.

From the equation of motion, we obtain that F_{T M D} is bloc off-diagonal, and we rewrite it the following way :

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F_{T M D} = 0 F^↑↓

F^↓↑ 0

!

, where F_i,j^σσ⁰ = −hT_τc_i,σ(τ )c_j,σ⁰(0)i.

We set : W =ˆ _z₂_−ξ¹2

k−∆²∆







t₁t₁ t₁t₂ t₁t₃ t₁t₂ t₂t₂ t₂t₃ t₃t₁ t₃t₂ t₃t₃







We find that, in the weak-coupling approximation ( ˇT → 0), F_odd^↑↓ 6= 0 if : W ˆˆH_↑,3×3^{T M D}(~k) − ˆH_↑,3×3^{T M D}(~k) ˆW 6= 0

Symmetrically :F_odd^↓↑ 6= 0 if : W ˆˆH_↓,3×3^{T M D}(~k) − ˆH_↓,3×3^{T M D}(~k) ˆW 6= 0

This criterion will be generally valid : since is diagonalizable, with eigenvalues 0,0 and t²₁+t²₂+t²₃, the commutant will have a dimension of 5 and ˆH_3×3^{T M D}, whose upper-diagonal coefficients are linearly independent [5], evolves in a 6-dimensional space.

We have a general criterion for nonzero odd-frequency pairing. We note that this criterion cannot be satisfied by a single-band model.

Previous work on this kind of heterostructure (Triola al, [8]), considering a single-band model for the TMD, had put to light a set of general necessary conditions to have a nonzero odd- frequency pairing, such as spin-orbit coupling in the TMD for a spin-triplet superconductor, or ferromagnetism in the TMD for a spin singlet superconductor etc. This sufficient condition shows that the multiband nature is sufficient to odd-frequency pairing, and that the conditions determined in [8] are not necessary anymore as long as you have a multiband model. It is also consistent with previous results on multiband superconductivity where odd-frequency pairing was found to be ubiquitous as soon as there is interband hybridisation [3]. In our multiband model, we should expect to find odd-ω pairing in the system. We will now use numerical analysis to gain some insight about those odd-ω pairing amplitude.

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Chapter 4 Numerical results

4.1 Methodology

Our analytical work provides us with a condition to find odd-frequency pairing in the system, but it gives us no indication about its magnitude and the influence of the parameters of the system. The aim of the numerical investigation is therefore to identify the relationships between some parameters of the model, such as spin-orbit coupling in the TMD, the chemical potentials of M oS₂ and the superconductor, the coupling coefficients ˆT , and the absolute and relative importance of odd-frequency pairing. For a given set of these parameters, two approach are used :

• We integrate the norm of the 8 symmetry part of the anomalous Green function defined in table 1.1 over the first brillouin zone and sum over Matsubara frequencies. We can have access to the relative contributions of each class.

• We compute the anomalous Green function, and its odd-ω and even-ω parts, and we plot their norm : ||F_{T M D}|| =P

i,j|(F_{T M D})_i,j|. as in fig 4.3.

We keep in mind that, with the spectral function A, we can compute the local density of state and the spectral density, two quantities that could be probed experimentally by scanning tunneling microscopy (STM) and Angle-resolved photo-emission spectroscopy. It gives us a way to simulate experiments, and could have enabled us to find an observable related to odd-ω.

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4.2 Anomalous Green Function

The following figures show the spectral function of M oS₂ when coupled to the SC, compared to each spectral function alone 4.1. We can see that the Spectral function features several gaps openings near spectral functions of M oS₂ and the SC of figure 4.1, such as the one circled in red and black. On the next figure4.2, the norm of the pairing function is significantly high only around the Fermi level, which shows that only the gaps circled in black are superconducting gaps, when the one circled in red are hybridisation gaps.

Figure 4.1: Left : superposition of spectral functions of TMD and SC without tunnel coupling; Right :Spectral function of the TMD with tunnel coupling

Horizontal axis : ~k along the axes ΓK, KM and MΓ of the first B.Z.

Vertical axis : energy ω.

Figure 4.2: Log of the anomalous Green function ||F_{T M D}||

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We plot the odd-ω and even-ω parts of the anomalous Green function in figure4.3. Consistently with the analytical criterion, we see that the odd-ω component is finite. We also see that it is several orders of magnitude smaller than the even part.

Figure 4.3: Left : Log of ||F_{T M D}^even || ; Right : Log of ||F_{T M D}^odd ||

4.3 Dependence on the model parameters

4.3.1 Spin-orbit coupling

In this section, we study the effect of spin-orbit coupling on the anomalous Green function. We can tune the spin-orbit coupling intensity by changing the value of λ.

The previous analysis of the shape of FT M D shows that the only non-zero spin pairs are ^{|↑↓i+|↓↑i}^√₂ (which is spin triplet) and ^{|↑↓i−|↓↑i}^√₂ (which is spin singlet). Figure 4.4 shows the relative contribution of the spin-singlet components to the anomalous Green function F_{T M D}^singlet/FT M D : it is equal to one for λ = 0, which means that there is no spin-triplet pairing, but it decreases when λ increases. The explanation is straightforward : if no spin-orbit coupling is present, the spin degeneracy and the Pauli principle imply that : F^↑↓ = −F^↓↑.

The influence of spin-orbit coupling on the relative amplitude of the odd-ω pairs is limited (only changes its contribution to less than 20% , and does not exhibit any unambiguous tendency, as we can see on figure 4.5. We see that spin-orbit coupling is not necessary to odd-ω pairing in multiband M oS2.

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Figure 4.4: Relative contribution of the spin-singlet pairs to the anomalous Green function.

Horizontal axis : spin-orbit coupling magnitude λ. The red dot stands for 0.073, the real value in [5]

Figure 4.5: Relative contribution of the spin-singlet pairs to the anomalous Green function.

Horizontal axis : spin-orbit coupling magnitude λ.

4.3.2 Chemical Potential

We can tune the chemical potential in the M oS₂ monolayer or in the superconductor, which will shift the band structure of one of the materials, and therefore modifiy the electronic density around the Fermi level, changing the values of the anomalous Green function 4.7. The total

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pairing amplitude increases when the electronic density around the Fermi level (E = 0eV ) increases. No explicit pattern seems to emerge from the comparison of the spectral function and the magnitude of odd-ω pairing. 4.6.

Figure 4.6: Red : normalized magnitude of the anomalous Green function, Blue : relative magnitude of odd-ω pairing. Horizontal axis : M oS2 chemical potential in eV.

Figure 4.7: Spectral function of the TMD for several values of its chemical potential. From left to right, top to bottom : 1.40 eV; 1.85 eV; 2.00 eV; 2.19 eV

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4.3.3 Dependence on the tunneling coefficients

To investigate the influence of tunneling coefficients, we keep t²₁ + t²₂ + t²₃ = 1, and we change their relative magnitudes. We compute the anomalous Green function for various values of : ~t =





 t₁ t₂ t₃





=







sin(θ)cos(φ) sin(θ)sin(φ)

cos(θ)





around Bloch’s sphere4.8. We can see that the relative contribution of odd-ω pairing changes a lot, from 0.085 to 0.215. The behaviour is not symmetric, the extremes are reached for finite values of each t-coefficient. Again, no obvious pattern emerges in the spectral function. It shows that odd-ω pairing is very sensitive to the tunneling coefficients.

Figure 4.8: Relative contribution of odd-ω pairs to the anomalous Green function.

Horizontal axis : θ. Vertical axis : Φ.

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Conclusion

The analytical work proposed a general criterion for a non-zero odd-frequency part of the anomalous Green function in multiband transition metal dichalcogenides hetero-structures, and the numerical investigation confirmed that odd-ω pairing should occur in M oS₂-superconductor 2D system. The numerical investigation nonetheless shows that odd-ω pairing is generally quite weak compared to even-ω pairing, but it is possible to change this ratio by tuning some parameters of the system. It has an influence on experimental observables such as spectral function and local density of states, but no unambiguous correspondence between their behaviour and the magnitude of odd-ω pairing was found yet.

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Appendix A Appendix

A.1 Tight-Binding Model

To determine the electronic states in a periodic crystal, one can use a tight binding model.

The idea underlying any tight-binding model is that the electrons are strongly bound to their atoms, and that the contribution of the interactions with other atomic potentials can be treated as a perturbation. We express our Hamiltonian in a basis of electronic states which are highly localized (“single-atom orbitals”) : electrons are considered occupying a linear combination of single-atom orbitals, and the overlap between neighbouring atomic orbitals which originates the perturbation is described by hopping terms quantifying the energy contribution of delocalizing between several single-atom states.

Let us consider a general Tight-Binding model. We assume that the crystal is periodic, we call r_i the lattice vectors, c_i,αand c^†_i,α are the creation and annihilation operators for the single-atom state |φ^α_ii = φ^α(r − r_i) of the orbital α at position r_i.

We can construct the fermionic operators for the Bloch states with the following Fourier transforms : c_k,α = P

j e^{−i k·r}^jc_j,α and c^†_k,α = P

j e^{i k·r}^jc^†_j,α. The periodicity enables us to define E_d^α,α⁰, the hopping integral between orbitals α and α⁰ of two sites distant from d : E_d^α,α⁰ = hφ^α(r)| ˆH |φ^α⁰(r − d)i

The Hamiltonian is therefore given by : H =ˆ P

j,d,α,α⁰E_d^α,α⁰c^†_j,αc_j+d,α⁰ H =ˆ P

j,d,α,α⁰E_d^α,α⁰P

k,k⁰e^{i (k−k’)·r}^je^{−i k’·d}c^†_k,αc_k’,α⁰ We can rearrange the terms : H =ˆ P

α,α⁰,k(P

de^{−i k·d}E_d^α,α⁰)P

k⁰c^†_k,αc_k’,α⁰(P

je^{i (k−k’)·r}^j)

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Since P

je^{i (k−k’)·r}^j 6= 0 only for k-k’ in the reciprocal lattice, that is for identical Bloch states with vectors k and k⁰, we identify E_k^α,α⁰ =P

de^{−i k·d}E_d^α,α⁰ to get : H =ˆ P

α,α⁰,kE_k^α,α⁰c^†_k,αc_k,α⁰

A.2 Dyson Equation

We can express the equation of motion with the contributions from the TMD and the superconductor :

z1_12×12− ˆH_{T M D} − ˇT

− ˇT^† z 1_4×4− ˆH_SC

! G_{T M D} G_{T M D−SC} G_{SC−T M D} G_SC

!

= 1 (1)

Equation (1) leads to :

( [z 1_12×12− ˆH_{T M D}] G_{T M D}− ˇT G_{SC−T M D} = 1 [z 1_6×6− ˆH_SC] G_{SC−T M D} − ˇT^†G_{T M D} = 0

Hence :

G_{SC−T M D} = [z 1_6×6− ˆH_SC]⁻¹Tˇ^†G_{T M D}

=⇒ 1 = [z 1_12×12− ˆH_{T M D}− ˇT [z 1_6×6− ˆH_SC]⁻¹Tˇ^†] G_{T M D} We set

Σ ≡ ˇˆ T [z 1_6×6− ˆH_SC]⁻¹Tˇ^† So that

G_{T M D} = [z 1_12×12− ˆH_{T M D}]⁻¹+ [z 1_12×12− ˆH_{T M D}]⁻¹ΣGˆ _{T M D} (2) G_{T M D} = [z 1_12×12− ˆH_{T M D}− ˆΣ]⁻¹

Equation (2) is usually referred to as the Dyson equation, and ˆΣ as the ”self-energy”.

A.3 Sufficient criterion for odd-frequency pairing

A.3.1 Calculation of ˆ Σ

We introduce the Pauli matrices σ_0,1,2,3 to express ˆΣ : σ₀ = 1 0 0 1

!

, σ₂ = 0 −i i 0

!

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[z 1_4×4− ˆH_SC] = (z − ξ_k) ˆσ₀ i∆ˆσ₂

−i∆ˆσ₂ (z + ξ_k)ˆσ₀

!

[z 1_4×4− ˆH_SC]⁻¹ = _z₂_−ξ¹2 k−∆²

(z + ξ_k) ˆσ₀ −i∆ˆσ₂ i∆ˆσ₂ (z − ξ_k)ˆσ₀

!

We can express the ”self-energy” matrix : Σ = ˇˆ T [z 1_6×6− ˆH_SC]⁻¹Tˇ^†= _z2−ξ¹²_k−∆²

(z + ξ_k) ˆT ˆT^† i∆ ˆT σ₂Tˆ^†

−i∆ ˆT ˆσ₂Tˆ^† (z − ξ_k) ˆT ˆT^†

!

Given the expressions of ˆT and σ_i , we find that:

i ˆT σ₂Tˆ^†= 0 ρˆ

−ˆρ 0

!

and : ˆT ˆT^†= ρ 0ˆ 0 ρˆ

!

Where we define ˆρ with the tunneling coefficients ti : ˆρ =













: ˆρ =













Finally : ˆΣ = _z2−ξ¹²_k−∆²







(z + ξ_k) ˆρ 0 0 ∆ ˆρ

0 (z + ξ_k) ˆρ −∆ˆρ 0 0 −∆ˆρ (z − ξ_k) ˆρ 0

∆ ˆρ 0 0 (z − ξ_k) ˆρ





 (3)

A.3.2 Block decomposition of G

_{T M D}

We are interested in the anomalous Green function F_{T M D} which is such that

G_{T M D} = G_{T M D} F_{T M D} F¯_{T M D} G¯_{T M D}

!

Let us decompose ˆH_{T M D} in particle-hole space according to [5], we let the different spins

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components appear :

Hˆ_12×12^{T M D} =

ˆH_6×6^{T M D}(~k) 0 0 − ˆH_6×6^{∗T M D}(−~k)

!

=







Hˆ_↑,3×3^{T M D}(~k) 0 0 0

0 Hˆ_↓,3×3^{T M D}(~k) 0 0

0 0 − ˆH_↑,3×3^{∗T M D}(−~k) 0

0 0 0 − ˆH_↓,3×3^{∗T M D}(−~k)







Where, according to the expression of ˆH given in [5], ˆH_↑,3×3^{T M D}(~k) = ˆH_↓,3×3^{∗T M D}(−~k) (4)

Using equation (2), we obtain : G_{T M D} =







Aˆ↑ 0 0 − ˆW 0 Aˆ↓ Wˆ 0 0 Wˆ Bˆ↑ 0

− ˆW 0 0 Bˆ↓







−1

Where :











Aˆ↑ = z 1 − ˆH_↑,3×3^{T M D}(~k) − _z2−ξ¹_k²−∆²(z + ξk) ˆρ Wˆ = _z2−ξ¹_k²−∆²∆ ˆρ

Aˆ_↓ = z 1 − ˆH_↓,3×3^{T M D}(~k) − _z₂_−ξ¹2

k−∆²(z + ξ_k) ˆρ Bˆ_↑ = z 1 + ˆH_↑,3×3^{∗T M D}(−~k) − _z₂_−ξ¹2

k−∆²(z − ξ_k) ˆρ Bˆ_↓ = z 1 + ˆH_↓,3×3^{∗T M D}(−~k) − _z₂_−ξ¹2

k−∆²(z − ξ_k) ˆρ

We deduce the following equations on F_{T M D}and G_{T M D}:









 G

ˆA↑ 0 0 Aˆ↓

!

+ F 0 Wˆ

− ˆW 0

!

= 1

G 0 − ˆW Wˆ 0

! + F

ˆB_↑ 0 0 Bˆ_↓

!

= 0 Which lead to :

F =

ˆA↑ 0 0 Aˆ↓

!−1

0 Wˆ

− ˆW 0

!



0 Wˆ

− ˆW 0

! ˆA↑ 0 0 Aˆ↓

!−1

0 Wˆ

− ˆW 0

! +

ˆB↑ 0 0 Bˆ↓

!



−1

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A.3.3 Shape of F

_{T M D}

The equation (5) shows that F_{T M D} is bloc anti-diagonal, we rewrite it the following way : F_{T M D} = 0 F^↑↓

F^↓↑ 0

! .

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A.3.4 Weak coupling

For simplicity, we consider a weak-coupling system, so that we can find an analytical criterion for nonzero odd-frequency pairing when ~t = (t₁, t₂, t₃) → 0 :

At the lowest order in ||~t|| : F =

ˆA↑ 0 0 Aˆ↓

!⁻¹

0 Wˆ

− ˆW 0

! ˆB↑ 0 0 Bˆ↓

!⁻¹ (6)

And :











Aˆ↑ = z 1 − ˆH_↑,3×3^{T M D}(~k) Wˆ = _z2−ξ¹_k²−∆²∆ ˆρ Aˆ↓ = z 1 − ˆH_↓,3×3^{T M D}(~k)

Bˆ↑ = z 1 + ˆH_↑,3×3^{∗T M D}(−~k) = z 1 + ˆH_↓,3×3^{T M D}(~k) Bˆ↓ = z 1 + ˆH_↓,3×3^{∗T M D}(−~k) = z 1 + ˆH_↑,3×3^{T M D}(~k) Which directly leads to : F = 0 F^↑↓

F^↓↑ 0

!

= 0 Aˆ⁻¹_↑ W ˆˆB_↓⁻¹

− ˆA⁻¹_↓ W ˆˆB_↑⁻¹ 0

!

Where F_i,j^σσ⁰ = −hT_τc_i,σ(τ )c_j,σ⁰(0)i. We want to identify an odd-z term : F^↑↓= ˆA⁻¹_↑ W ˆˆB_↓⁻¹

We single out ˆW : ˆA↑F^↑↓Bˆ↓ = ˆW

We multiply on the left by ˆB↓ , and on the right by ˆA↑. Since ˆB↓Aˆ↑ = z²1 − ˆH_↑,3×3^{T M D}(~k)², we obtain :

[z²1 − ˆH_↑,3×3^{T M D}(~k)²]F^↑↓[z²1 − ˆH_↑,3×3^{T M D}(~k)²] =[z 1 + ˆH_↑,3×3^{T M D}(~k)] ˆW [z 1 − ˆH_↑,3×3^{T M D}(~k)]

We note that the matrices on each side of F_odd^↑↓ are even, and we can divide the right term of the equation in odd and even parts :

[z²1 − ˆH_↑,3×3^{T M D}(~k)²]F^↑↓[z²1 − ˆH_↑,3×3^{T M D}(~k)²] = [z²1 − ˆH_↑,3×3^{T M D}(~k) ˆW ˆH_↑,3×3^{T M D}(~k)]+

z[ ˆH_↑,3×3^{T M D}(~k) ˆW − ˆW ˆH_↑,3×3^{T M D}(~k)]

According to the previous equation, we know that F_odd^↑↓ 6= 0 if : W ˆˆH_↑,3×3^{T M D}(~k) − ˆH_↑,3×3^{T M D}(~k) ˆW 6= 0 Symmetrically : F_odd^↓↑ 6= 0 if :

W ˆˆH_↓,3×3^{T M D}(~k) − ˆH_↓,3×3^{T M D}(~k) ˆW 6= 0 Thus, we obtain an analytical condition for odd-frequency pairing.

Odd-Frequency Pairing

Superconductivity in M oS

heterostructure :

Odd-Frequency Pairing

Yann Gaucher

D´ eclaration d’int´ egrit´ e relative au plagiat

Contents

Acknowledgements

Introduction

Chapter 1

Introduction to Superconductivity

1.1 Historical perspective

1.2 BCS Theory

1.2.1 Overview

1.2.2 BCS Hamiltonian

1.3 Odd-Frequency Pairing

1.3.1 Pairing Amplitude

1.3.2 SPOT Classification

1.3.3 Occurrence of odd-frequency pairing

1.3.4 Specific interest of odd-frequency pairing

Chapter 2

M oS 2 - Superconductor Heterostructure

2.1 Transition metal dichalcogenides

2.1.1 Overview

2.1.2 Properties of Transition Metal Dichalcogenides

2.2 Theoretical Modeling

2.2.1 Tight binding model for a MoS2 monolayer

2.2.2 Tight binding model for the Superconductor

2.3 Model of proximity-effect

2.3.1 Expression in particle-hole space

Chapter 3

Analytical Work

3.1 Green Functions

3.2 Equation of Motion

3.3 Sufficient Condition for odd-frequency pairing

Chapter 4

Numerical results

4.1 Methodology

4.2 Anomalous Green Function

4.3 Dependence on the model parameters

4.3.1 Spin-orbit coupling

4.3.2 Chemical Potential

4.3.3 Dependence on the tunneling coefficients

Conclusion

Appendix A Appendix

A.1 Tight-Binding Model

A.2 Dyson Equation

A.3 Sufficient criterion for odd-frequency pairing

A.3.1 Calculation of ˆ Σ

A.3.2 Block decomposition of G

A.3.3 Shape of F

A.3.4 Weak coupling

M oS ₂ - Superconductor Heterostructure