光電導和磁穿透深度在鐵基超導的理論研究

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(1)Theoretical Studies of Optical Conductivity and Penetration Depth in Pnictide Superconductors by Hsuan-Hao Fan. A Thesis Submitted in Conformity with the Requirements for the Degree Master of Physics. Graduate Department of Physics National Taiwan Normal University ©Copyright by Hsuan-Hao Fan, 2009..

(2) TITLE: Theoretical Studies of Optical Conductivity and Penetration Depth in Pnictide Superconductors AUTHOR: Hsuan-Hao Fan ADVISOR: Dr. Wen-Chin Wu.

(3) i. Acknowledgements I thank my advisor, Prof. Wen-Chin Wu, very much for his guidance. He inspired me to study the field of superconductivity. Due to these studies, I have learned how to program and how to use the method of the many-body theory. Most importantly, I have learned some knowledge of superconductivity although I still need to work hard on many things in this field. Finally, I thank his encouragement and support. I also thank my senior, Chou-Chun Huang, in our group for his helping to solve the programming problems. Moreover, I greatly appreciate my families for their support. Finally, I am grateful to my girlfriend, Chun-Ping Hsieh, for her accompanying and support..

(4) ii. Abstract We studied the optical conductivity and the penetration depth on pnictide superconductors based on the two-orbital model proposed by S. Raghu et al. Taking into account the D4 symmetry of the crystals, various possible pairing symmetries are considered. We have found that the real part of optical conductivity σ1 (ω ) has a strong peak near zero frequency for the case of nodal gaps. For the case of nodeless gaps, σ 1 (ω ) is essentially zero for ω ≤ 2Δ 0 for the intraband contribution.. Nevertheless, interband contribution seems to dominate in most cases in these materials. In the study of the penetration depth, we have obtained that the penetration depth λ 2 ( 0 ) λ 2 (T ) behaves exponentially down to zero temperature for nodeless gaps, while λ 2 ( 0 ) λ 2 (T ) has linear temperature dependence at very low temperature.. Key Words: iron-based superconductor, pnictides, two-orbital model, optical conductivity, and penetration depth..

(5) iii . Contents 1 Introdction .............................................................................. 1 2 Model ...................................................................................... 7 2.1 Introduction........................................................................................7 2.2 Model Hamiltonian in Normal State .................................................7 2.3 Model Hamiltonian in Superconducting State ................................13. 3 Optical Conductivity ........................................................... 22 3.1 Introduction......................................................................................22 3.2 Theory ..............................................................................................23 3.2.1 Linear Response Theory ............................................................23 3.2.2 Kubo Formula for Optical Conductivity ...................................26 3.3 Results ..............................................................................................34. 4 Penetration Depth ................................................................ 49 4.1 Introduction......................................................................................49 4.2 Theory ..............................................................................................50 4.3 Results ..............................................................................................52. 5 Conclusion ............................................................................ 57 References .....................................................................................................................59.

(6) 1. Chapter 1 Introduction. In the last year, Kamihara et al. showed the existence of superconductivity in a layered iron arsenide material, LaOFeAs, with a transition temperature ( Tc ) of 26 K by doping of fluoride ions at the O. 2-. sites (LaO1-xFxFeAs)1. Note that LaOFeAs has a layered tetragonal crystal structure shown in Fig. 1.1. FeAs layers are interplaced between the layers of La and O. The Fe atoms form a squared lattice with Fe-Fe spacing of 2.853 Α .. Fig. 1.1 By doping F into oxygen sites, electron carriers are injected into FeAs layers. Note that the carrier doping layer is separated from the conduction FeAs layer. Moreover, there are two Fe ions and two As ions in the crystallographic unit cell. (Source: Ref.2).

(7) 2 In the following, we compare the pnictides, which are compounds of the nitrogen group, to the cuprates. Recall that the cuprates and other unconventional superconductors are antiferromagnetic at low doping. Increasing doping will destroy their antiferromagnetism which leads to the superconductivity. Similarly, the parent compound, LaOFeAs, exhibits a spin-density-wave (SDW) instability which is suppressed by doping with electrons to induce superconductivity.3, 4 Fig. 1.2 shows the antiferromagnetic structure of the 122 FeAs materials and the CuO2 plane. Owing to this similarity, many researchers have guessed that the mechanism of superconductivity for pnictides may relate to that of the cuprates. (a). (b). Fig. 1.2 (a) This picture illustrates the antiferromagnetic structure of CaFe2As2. (Source: Ref.5) (b) This picture shows the antiferromagnetic structure of CuO2 plane.. However, there is an important difference between the FeAs materials and cuprates. The parent compound in the cuprates is actually an antiferromagnet, but a special type－a Mott insulator. According to band.

(8) 3 theory, a Mott insulator should conduct, but the charge carriers are localized due to the large Coulomb repulsion between the electrons. In contrast, the iron-pnictides is an antiferromagnetic “SDW” metal where the electrons appear to be more delocalized. The causes of the antiferromagnetic order in a Mott insulator and the SDW are different. The former is that spins can lower their energy if they are antiparallel to their neighbors, and the latter is that a SDW is a collective effect which emerges from an instability of the paramagnetic Fermi surface. Due to the discovery of superconductivity in LaOFeAs, scientists have made amount of analogues mainly by replacing rare earth ions for La. Note that LaOFeAs is known as the “1111” structure because of its formula unit. These materials have been possible to achieve a critical temperature Tc beyond 50K with doping or pressure6-8. We hope that one can find a material with a Tc above the temperature of 77K where nitrogen gas liquefies one day. Then experiments can be performed more easily, and applications can become more economically. In addition, the antiferromagnetism of the undoped FeAs compound is different in structure from that found in the cuprates. These FeAs materials are composed of aligned iron spins which alternate in direction from one row of iron atoms to the next. (See Fig. 1.2) Therefore, this permutation of iron spins lead to a stripe-like structure3, 4. However, the checkerboard pattern of up and down spins were formed in the cuprates. A distortion of the crystal lattice could transfer the magnetic phase from a tetragonal to an orthorhombic structure3, 9. The phase diagrams of.

(9) 4 iron-pnictides and cuprates are shown in Fig. 1.3 and Fig.1.4.. Fig. 1.3 (a) Phase diagram of F doped CeO1-xFxFeAs determined by neutron scattering. (Source: Ref.10) It shows a smooth change from antiferromagnetism (AFM) at low doping to superconductivity (SC) at large dopings. Note that the Néel temperatures TN is the magnetic transition. (b) Phase diagram of F doped LaO1-xFxFeAs shows a discontinuous transformation from spin-density-wave (SDW) antiferromagnetism to superconductivity as a function of F content. Ts is the structural transition. (Source: Ref. 11). Fig. 1.4 The phase diagram of p-type and n-type cuprates in zero magnetic field is plotted. Note that AFM denotes antiferromagnetic phase, SC denotes superconducting phase, TN denotes the ∗ Néel temperature, Tc denotes the superconducting temperature, and T denotes pseudogap transition temperature. (Source: Ref.12).

(10) 5 It is of paramount importance to identify the configuration of the Fermi surface and the superconducting gap. By angle-resolved photoemission (ARPES), it is able to map out the anisotropy in momentum of the superconducting energy gap13. Fig. 1.5 shows the Fermi surface and superconducting gap ( Δ ) of Ba0.6K0.4Fe2As2.. Fig. 1.5 The Fermi surface and superconducting gap. (Δ). of Ba0.6K0.4Fe2As2. determined by ARPES. Superconducting gap size measured at 15K is plotted in the three pieces of Fermi surfaces. The pattern at the bottom is the photoemission intensity around Fermi surface.(Source: Ref.13).

(11) 6 On the experiments of optical measurements14,. 15. and magnetic. penetration depths16, there are not many experiments available so far for pnictide superconductors. We attempt to calculate the optical conductivity and penetration depth for pnictides, based on the two-orbital band model and taking into account possible pairing symmetries arising from the D4 crystal symmetry. We hope that our results will boost more optical and penetration depth experiments done on the pnictides..

(12) 7. Chapter 2 Model. 2.1 Introduction We follow the minimal two-orbital model proposed by S. Raghu et al.17. The model has two orbitals, dxz and dyz, per site on a two-dimensional square lattice. There are several reasons why we follow this model. Most importantly, this model exhibits a Fermi surface similar to that obtained from the band structure calculations. From the first-principle band structure calculations18-21, we have known that superconductivity in the family of Fe-based oxypnictides is associated with the Fe-pnictide layer, and that the density of states (DOS) around the Fermi level gets its maximum contribution from the Fe-3d orbitals. Finally, this model is relatively simple and should be practical to give the phenomenological analysis of experiments.. 2.2 Model Hamiltonian in Normal State The structure of the FeAs layer of all families of Fe-based oxypnictides viewed along the c–axis is illustrated in Fig. 2.2.1(a). The Fe ions form a squared lattice which is interlaced with a second square lattice of As ions. The As ions located in the center of each square of the Fe lattice and are displaced above and below the plane of the Fe ions. Moreover, there are two Fe ions and two As ions in the crystallographic unit cell. As shown by Ref.19, the band structure near the Fermi surface is relatively simple in the unfolded one-Fe first Brillouin zone (BZ) where it.

(13) 8 primarily involves three Fe orbitals, dxz, dyz, dxy. According to the Ref.17, they make an approximation that the role of the dxy orbit can be replaced by a next-near-neighbor (NNN) hybridization between dxz and dyz orbitals. Hence, this model considers only a two-dimensional square lattice with two degenerate orbitals, dxz and dyz, per site.. Fig. 2.2.1 (a) The Fe ions form a square lattice and the crystallographic unit cell contains two Fe and two As ions. The As ions are sit above (solid circles) or below (dashed circles) the plane of the Fe square lattice. (b) A figure showing the hopping parameters of the two-orbital dxz, dyz model on a square lattice. The projections of the dxz (dyz ) orbitals onto the xy plane are plotted in green (white). (Source: Ref.17). The tight-binding part of the Hamiltonian is then written as ⎛ ε x (k ) − μ ε xy ( k ) ⎞ H 0 = ∑ ϕ †kσ ⎜⎜ ⎟ϕ , ε y ( k ) − μ ⎟⎠ kσ kσ ⎝ ε xy ( k ). (2.2.1) where ϕk†σ = ( dx†σ ( k ) , d y†σ ( k ) ) is the creation operator with spin-σ for the orbitals ( d xz , d yz ) , μ is the chemical potential, and.

(14) 9 ε x ( k ) = −2t1 cos k x − 2t2 cos k y − 4t3 cos k x cos k y , ε y ( k ) = −2t1 cos k y − 2t2 cos k x − 4t3 cos k x cos k y , ε xy ( k ) = −4t4 sin k x sin k y .. (2.2.2) Here t1 is the near-neighbor (NN) hopping parameter between σ -bondings and t2 is the NN hopping between π -bondings. Furthermore, we also introduce next-near-neighbor hopping t4 between different orbitals and t3 between similar ones. In our calculations, we use t1 = −1 , t2 = 1.3 , and t3 = t4 = −0.85 in unit of. t1 throughout this thesis.. The eigenvalues of Eq. (2.2.1) are found to be E± ( k ) = ε + ( k ) − μ ± ε −2 ( k ) + ε xy2 ( k ) ,. (2.2.3). where ε± =. εx ±εy 2. .. One can diagonalize the Hamiltonian by a unitary transformation ⎛u U =⎜ k ⎝ vk. −vk ⎞ ⎟, uk ⎠. (2.2.4). where the elements in U are called “coherence factor”. They satisfy the following property, uk2 + vk2 = 1,. and, in fact, ε− 1⎛ uk2 = ⎜ 1 + 2 2⎜ ε − + ε xy2 ⎝. ⎞ 1 ⎟ = + ε− , ⎟ 2 E+ − E− ⎠. ε− 1 ⎛⎜ 1− 2 2⎜ ε − + ε xy2 ⎝. ⎞ 1 ⎟ = − ε− , ⎟ 2 E+ − E− ⎠. vk2 =. and.

(15) 10 uk vk =. ε xy 2 ε +ε 2 −. 2 xy. =. ε xy E+ − E−. .. (2.2.5). Therefore, we can obtain the diagonalized Hamiltonian and get the Bogliubov transformation in the same time. i.e. 0 ⎞ ⎛ E (k ) H 0 = ∑ γ k†σ ⎜ + ⎟γ E− ( k ) ⎠ kσ kσ ⎝ 0. (2.2.6). where ⎛u. γ kσ = U †ϕ kσ = ⎜ k ⎝ −vk. vk ⎞ ⎛ d xσ ⎟⎜ uk ⎠ ⎝ d yσ. ⎞ ⎟. ⎠. (2.2.7). Moreover, the definition of the normal-state single-particle Green’s function is Gˆ ( k ,τ ) ≡ − Tˆτ ϕkσ (τ ) ϕk†σ ( 0 ) ,. (2.2.8). where Tˆτ is the time-ordering operator. Then we can calculate the one-electron Matsubara Green’s function by the definition, β Gˆ (k ,iωn ) = ∫ dτ eiωnτ Gˆ ( k ,τ ) 0. β Gˆ (k ,iωn ) = − ∫ dτ eiωnτ ϕkσ (τ ) ϕk†σ ( 0 ) 0. ⎡iωn − ε + ( k ) + μ ⎤⎦ 1 + ε − ( k ) σ 3 + ε xy ( k ) σ 1 Gˆ (k ,iωn ) = ⎣ , ⎡⎣iωn − E+ ( k ) ⎤⎦ ⎡⎣iωn − E− ( k ) ⎤⎦. with σ i the Pauli matrices, τ. the imaginary time, and ωn the. Matsubara frequency. The elements of the Green’s function are G11 = G22 =. and. (2.2.9). iωn − ε + + μ + ε − , ⎡⎣iωn − E+ ( k ) ⎤⎦ ⎡⎣iωn − E− ( k ) ⎤⎦. iωn − ε + + μ − ε − , ⎡⎣iωn − E+ ( k ) ⎤⎦ ⎡⎣iωn − E− ( k ) ⎤⎦.

(16) 11 G12 = G21 =. ε xy. ⎡⎣iωn − E+ ( k ) ⎤⎦ ⎡⎣iωn − E− ( k ) ⎤⎦. (2.2.10). .. From Eq. (2.2.8), one can obtain the density of states (DOS), N (ω ) = −. 1. π. ∑ Im ⎡⎣G ( k ,ω + iδ ) + G 11. 22. k. (k ,ω + iδ ) ⎤⎦. N (ω ) = ∑ ⎡⎣δ (ω − E+ ) + δ (ω − E− ) ⎤⎦ .. (2.2.11). k. In Fig. 2.2.2, the DOS is plotted for k x , k y ∈ [ -π , π ] with μ = 1.45 and. N(w). we can clearly see that there are two Van Hove singularities.. -8. -7. -6. -5. -4. -3. -2 -1 w/|t |. 0. 1. 2. 3. 4. 1. Fig. 2.2.2 The density of states of the two-orbital model for k x , k y ∈ [ − π , π ] , with two Van Hove singularities. The dashed line shows the Fermi level corresponding. to μ=1.45..

(17) 12 From E± ( k ) , one can also obtain the Fermi surface. First of all, we plot the band structure of the two-orbital model in Fig. 2.2.3(a). Note that E ( k ) = 0 is the Fermi level and E± ( −k ) = E± ( k ) . Then Fig. 2.2.3(b) is the. Fermi surface on the one Fe cell 1st BZ. We classify the Fermi surface into two parts. One part marked by red solid line is given by E− ( k ) = 0 and the other part represented by blue solid line results from E+ ( k ) = 0 . That is, there are two hole Fermi pockets, α1 and α 2 , and two electron Fermi pockets, β1 and β 2 . In order to compare with ab initio study, we folded the 1st BZ into a smaller one which is due for the crystallographic unit cell with two Fe atoms. In Fig. 2.2.3(b), the dashed square represents this smaller zone. Then we show the folded BZ in Fig. 2.2.3(c) which contains the two Fe atoms and this result is similar to the result obtained from local density approximation (LDA) band structure calculations.. (b). (a). (c) Fig.. 2.2.3. (a). The. band. structure. for. k x , k y ∈ [ 0 , π ] . (b) The Fermi surface on the one Fe 1st BZ. (c) The folded BZ contain two Fe atoms. There are two α surfaces around the point Γ and two elliptical β surfaces around the point M..

(18) 13 Finally,. we. plot. the. band. structure. along. the. path. ( 0, 0 ) → (π , 0 ) → (π , π ) → ( 0, 0 ) in Fig. 2.2.4(a). Fig. 2.2.4(b) shows the band structure in the folded BZ, and it illustrates why the density of states has two Van Hove singularities. Besides, we can see that there is a saddle point in each band in Fig. 2.2.4(b). These results are also qualitatively consistent with the band structure calculations21.. (a). (b). Fig. 2.2.4 (a) The band structure is plotted along the path ( 0, 0 ) → (π , 0 ) → (π , π ) → ( 0, 0 ) (b) The band structure is folded into the folded BZ. The Γ, X, M are defined in the folded BZ as shown in Fig. 2.2.3(c). Note that the dashed line represents the Fermi level.. 2.3 Model Hamiltonian in Superconducting State Now we study the superconducting state for Fe-based superconductors. We consider the following BCS Hamiltonian H BCS = H 0 + ∑ Vkkxx' d x†, k ,↑ d x†,−k ,↓ d x ,−k ',↓ d x , k ',↑ kk'. + ∑Vkkyy' d y†, k ,↑ d y†,−k ,↓ d y ,−k ',↓ d y , k ',↑ , kk '. (2.3.1).

(19) 14 where H 0 is the two-orbital tight-binding model Hamiltonian given in Eq. (2.2.1). For the interaction part, Vkkxx' is the coupling strength between the same orbital dxz and Vkkyy' is the coupling strength between the same orbital dyz. Note that, for simplicity, we do not consider the interorbital pairing to make the problem analytically tractable. Moreover, as pointed out by the Ref.22, the expectation value of the interorbital pairing is negligible even in the strong Hund’s rule coupling. By the mean-field approximation, we obtain the mean-field BCS Hamiltonian, MF H BCS = ∑ ⎡⎣( ε x ( k ) − μ ) d x†, k ,σ d x , k ,σ + (ε y ( k ) − μ )d y†, k ,σ d y , k ,σ kσ. (. +ε xy ( k ) ( d x†, k ,σ d y , k ,σ + d y†, k ,σ d x , k ,σ ) ⎤⎦ − ∑ ⎡ Δ1 ( k ) d x†, k ,↑ d x†,− k ,↓ ⎣ k. ). +Δ 2 ( k ) d y†, k ,↑ d y†,− k ,↓ + h.c.] ,. (2.3.2) where Δ1 ( k ) ≡ −∑Vkkxx' d x ,−k ',↓ d x , k ',↑ k'. and Δ 2 ( k ) ≡ −∑Vkkyy' d y ,−k ',↓ d y , k ',↑ . k'. Letting ξ x ≡ ε x − μ , ξ y ≡ ε y − μ , and making some simplification, we can obtain a brief form of the mean-field Hamiltonean, MF H BCS = ∑ Ψ † ( k )B ( k ) Ψ ( k ) ,. (2.3.3). k. where ⎛ ξ x ( k ) Δ1 ( k ) ε xy ( k ) 0 ⎞ ⎜ ∗ ⎟ 0 Δ1 ( k ) −ξ x ( k ) −ε xy ( k ) ⎟ ⎜ , B (k ) = ⎜ ε xy ( k ) 0 ξ y (k ) Δ2 (k ) ⎟ ⎜⎜ ⎟ −ε xy ( k ) Δ∗2 ( k ) −ξ y ( k ) ⎟⎠ ⎝ 0. (2.3.4). and Ψ ( k ) = ( d x,k ,↑ , d x†,−k ,↓ , d y ,k ,↑ , d y†,−k ,↓ ) is the four component spinor. Note that B ( k ) is a Hermitian matrix..

(20) 15 Furthermore, we can diagonaliz B ( k ) by unitary transformation and simultaneously obtain the eigenvalues E1 , E2 , E3 , E4 . Note that E1 = − E2 and E3 = − E4 , with E1 , 3 ( k ) =. 1 2. L±. M2+N ,. where 2. 2. L ≡ ξ x2 + ξ y2 + 2ε xy2 + Δ1 + Δ 2 , 2. 2. M ≡ ξ x2 − ξ y2 + Δ1 − Δ 2 ,. and 2 2 N ≡ 4ε xy2 ⎡(ξ x + ξ y ) + Δ1 − Δ 2 ⎤ . ⎢⎣ ⎥⎦. We. now. calculate. the. − Tˆτ Ψ k (τ ) Ψ k† ( 0 ) , where τ. Nambu. (2.3.5) Green’s. function,. G ( k ,τ ) =. is the imaginary time, Tˆτ is the time. ordering operator, and Ψ k (τ ) = ( d x,k ,↑ (τ ) , d x†,−k ,↓ (τ ) , d y ,k ,↑ (τ ) , d y†,−k ,↓ (τ ) ) . We shall take a different approach, that is, equation of motion, from what we did for the normal state. First of all, let us study the derivative of the time of the annihilation operator MF ∂τ Ψ k (τ ) = ⎡⎣ H BCS , Ψ k ⎤⎦ (τ ) , −. where the minus sign means the commutator. One obtains the following results, ∂τ d x , k ,↑ (τ ) = −ξ x ( k ) d x , k ,↑ (τ ) + Δ1 ( k ) d x†,− k ,↓ (τ ) − ε xy ( k ) d y , k ,↑ (τ ) , ∂τ d x†,− k ,↓ (τ ) = Δ1∗ ( k ) d x , k ,↑ (τ ) + ξ x ( k ) d x†,− k ,↓ (τ ) + ε xy ( k ) d y†, −k ,↓ (τ ) , ∂τ d y , k ,↑ (τ ) = −ε xy ( k ) d x , k ,↑ (τ ) − ξ y ( k ) d y , k ,↑ (τ ) + Δ 2 ( k ) d y†,− k ,↓ (τ ) , ∂τ d y†,− k ,↓ (τ ) = ε xy ( k ) d x†, −k ,↓ (τ ) + Δ∗2 ( k ) d y , k ,↑ (τ ) + ξ y ( k ) d y†,− k ,↓ (τ ) .. (2.3.6).

(21) 16 By using Eq. (2.3.6), it is straightforward to get the equation of motion for G ( k ,τ ) : ⎛1 ⎜ 0 ∂τ G ( k ,τ ) = −δ (τ ) ⎜ ⎜0 ⎜ ⎝0. 0 1 0 0. 0 0 1 0. 0 ⎞ 0 ⎞ ⎛ ξ x ( k ) −Δ1 ( k ) ε xy ( k ) ⎟ ⎟ ⎜ −Δ∗ k −ε xy ( k ) ⎟ 0 0 ⎟ ⎜ 1 ( ) −ξ x ( k ) G ( k ,τ ) . − 0 ξ y ( k ) −Δ 2 ( k ) ⎟ 0 ⎟ ⎜ ε xy ( k ) ⎟ ⎟ ⎜ −ε xy ( k ) −Δ∗2 ( k ) −ξ y ( k ) ⎠⎟ 1 ⎠ ⎝⎜ 0. (2.3.7) Furthermore, we make the Fourier transformation of the Eq. (2.3.7) in τ-space to iωn -space by Matsubara frequency summation and collect terms containing G . Then we get DiG ( k , iωn ) = I, where I is the 4 by 4 identity matrix. and Δ1 ( k ) −ε xy ( k ) 0 ⎛ iωn − ξ x ( k ) ⎞ ⎜ ⎟ ∗ Δ1 ( k ) iωn + ξ x ( k ) 0 ε xy ( k ) ⎟ −1 ⎜ D=G = . ⎜ −ε xy ( k ) iωn − ξ y ( k ) 0 Δ2 (k ) ⎟ ⎜⎜ ⎟ ε xy ( k ) iωn + ξ y ( k ) ⎟⎠ 0 Δ ∗2 ( k ) ⎝. (2.3.8) Matrix inversion results in the final form for G ( k , iωn ) , G ( k , iωn ) =. 1 1 (Cij )T = (C ji ), det (D) det (D). (2.3.9). where Cij is the cofactor of the matrix D , CT represents the matrix transpose and det ( D ) = ⎡( iωn ) − E12 ( k ) ⎤ ⎡( iωn ) − E32 ( k ) ⎤ , ⎣ ⎦⎣ ⎦ 2. 2. (2.3.10). with E1 , 3 ( k ) the Eq. (2.3.5). To save the space, we don’t show the final result of G ( k , iωn ) . The DOS of the current system is given by the imaginary part of the.

(22) 17 retarded Green’s function via N (ω ) ≡ ∑ A ( k , ω ) = − k. N (ω ) = −. 1. π. 1. π. ∑ Im ⎡⎣G ( k , iω ) + G ( k , iω )⎤⎦ R 11. R 33. n. n. k. ∑ Im ⎡⎣G ( k , ω + iδ ) + G ( k , ω + iδ )⎤⎦ , 11. (2.3.11). 33. k. where A ( k , ω ) is the spectral function, δ is an infinitesimal positive number, and G11 ( k , ω + iδ ) as well as G33 ( k , ω + iδ ) are the electron components. of. the. superconducting. Green’s. function.. It. is. straightforwardly to obtain. (. ). (. ⎡ ε 2 ( 2ω − ξ − ξ ) − (ω + ξ ) ω 2 − ξ 2 − Δ 2 − (ω + ξ ) ω 2 − ξ 2 − Δ 1 2 xy x y y x x y A (k,ω ) = ⎢ ⎢ E12 − E32 ⎣⎢. 2. ) ⎤⎥ ⎥ ⎦⎥. ⎧ 1 ⎫ 1 ⎡⎣δ ( E3 − ω ) − δ ( E3 + ω ) ⎤⎦ − ⎡δ ( E1 − ω ) − δ ( E1 + ω ) ⎤⎦ ⎬ , A (k,ω ) = × ⎨ ⎣ 2 E1 ⎩ 2 E3 ⎭. (2.3.12) where E1 and E3 defined in Eq. (2.3.5) are eigenvalues of the matrix B ( k ) . This result is the same as that obtained in Ref. . If Δ1 = Δ 2 = Δ , we 23. will get the simple form A (k, ω ) =. ω + E− ( k ). ⎡δ ( E−Δ ( k ) − ω ) − δ ( E−Δ ( k ) + ω ) ⎤ ⎣ ⎦. (k ) ω + E+ ( k ) ⎡δ ( E+Δ ( k ) − ω ) − δ ( E+Δ ( k ) + ω ) ⎤ A (k, ω ) = + Δ ⎦ 2 E+ ( k ) ⎣ 2E. Δ −. , (2.3.13). where E±Δ ( k ) = E±2 ( k ) + Δ . 2. In addition, the function A ( k , ω ) represents the probability that a particle of momentum k occupies a state with energy ω . Thus A ( k , ω = 0 ) corresponds to the probability that a particle of momentum k. resides at the Fermi level. The occupation number of state k is then.

(23) 18 given by nk = ∫ d ω A ( k , ω ) f (ω ),. where f (ω ) = ( e βω + 1). −1. (2.3.14). is the Fermi-Dirac distribution function. Note. that f (ω ) is equal to the step function θ ( −ω ) as T → 0 . From A ( k , ω ) , one can get information about the nodal structure of possible pairing gaps. To study the spectral function near the Fermi level corresponding to each pairing symmetry, we define a function W (k ) ≡ ∫. 0.15|t1 |. −0.15|t1 |. d ω A ( k , ω ) f (ω ) ,. (2.3.15). where f (ω ) is the Fermi-Dirac function. We choose the integration limit “0.15 t1 ” arbitrarily in order to see the nodal structure clearly. W ( k ) indicates how many particles with momentum k occupying the energy levels within [ −0.15, 0.15] . Fig. 2.3.1 shows where the nodes located and how to affect the distribution of particles in the unfolded 1st BZ at T=0. Note that the dark red points indicate the largest value of W ( k ) . On the other hand, the dark red points correspond to larger DOS in k space. Due to the D4 symmetry of the sample, we consider in this thesis two possible d-wave symmetries, d x − y and d xy , as well as three possible 2. 2. s-wave symmetries, sx + y , sx y , and s0 for the pairing gaps. In Fig. 2. 2. 2 2. 2.3.1 (a), (c), (e), and (g), we can see these pairing gaps how to vary in the 1st BZ, but we do not plot the pairing symmetry s0 here. The pairing gap, s0 corresponds to the constant gap Δ ( k ) = Δ 0 . In addition, for simplicity, we assume Δ1 = Δ 2 ≡ Δ . We choose Δ 0 = 0.1 | t1 | , as done in Ref.23. The d x − y = Δ 0 ( cos k x − cos k y ) 2 pairing gap has nodes on the 2. 2. Fermi surface at k x = ± k y , so we can see that quasi-particles mainly.

(24) 19 occupy the hole Fermi pockets in Fig. 2.3.1(b). In Fig. 2.3.1(d), we see that d xy = Δ 0 sin k x sin k y pairing gap has nodes on Fermi surface when k x = 0 or ±π or k y = 0 or ±π . Hence, both hole Fermi pockets and. electron Fermi pockets are occupied by quasi-particles. Fig. 2.3.1(f) shows for sx + y = Δ 0 ( cos k x + cos k y ) 2 pairing gap which has nodes at 2. 2. k x + k y = ±π or k x − k y = ±π . As a result, the hole Fermi pockets are fully. gapped and only electron pockets contribute. Finally, we can see clearly that the sx y = Δ 0 cos k x cos k y pairing gap is 2 2. fully gaped as doping is μ = 1.6 in Fig. 2.3.1(h). However, we also see the evolution of the Fermi-surface topology for different dopings in Fig. 2.3.1(h) and Fig. 2.3.2. As increasing the doping, the electron Fermi pockets become larger and larger, but the hole Fermi pockets become smaller and smaller. Therefore, we can obtain that there are nodes on the electron-like Fermi surface by increasing doping. In Fig. 2.3.2, we can see that sx y merely exhibits nodes on the Fermi surface of the electron 2 2. pockets above a critical doping μ ≈ 2 as a result of the nodes at k x = ± π / 2 or k y = ± π / 2 . The result we obtain is the same as that in. Ref.23. Finally, we don’t show the s0 pairing gap case, because both hole Fermi pockets and electron Fermi pockets are fully gapped as choosing Δ = Δ 0 = 0.1 . In this work, we will focus on these pairing gaps and study. how these pairing gaps affect the optical conductivity (Chapter 3) and the penetration depth (Chapter 4)..

(25) (a). (b). (c). (d). (e). (f). (g). (h). 20. Fig. 2.3.1 We consider these four candidate order parameters here with doping μ = 1.6 at T=0 K. There are two d-wave, (a) d x2 − y 2 and (c) d xy , and two s-wave, (e) sx2 + y 2 and (g) sx2 y 2 . The dashed lines mark the position where nodes located. For each corresponding pairing gap, (b), (d), (f), and (h) show the distribution of quasi-particles in the unfolded 1st BZ..

(26) (a). 21. μ =1.9. -π. 0.9 0.8 -π/2. 0.7. ky. 0.6 0.5. 0. 0.4 0.3. π/2. 0.2 0.1. π -π. -π/2. (b). 0 kx. π/2. π. 0. μ =2.0. -π. 0.9 0.8. -π/2. 0.7. ky. 0.6 0. 0.5 0.4 0.3. π/2. 0.2 0.1. π -π. -π/2. 0 kx. π/2. π. 0. Fig. 2.3.2 We plot the function W ( k ) for pairing gap sx2 y 2 with different dopings at T=0 K. Frame (a) is for μ = 1.9 and frame (b) is for μ = 2.0 . (b) shows that pairing symmetry sx2 y 2 only exhibits nodes on the electron Fermi pockets with μ = 2.0 ..

(27) 22. Chapter 3 Optical Conductivity 3.1 Introduction The prediction of the existence of a energy gap in the quasiparticle excitation spectrum is important in the BCS theory of superconductivity. To excite normal state particles across the gap, particles need to absorb at least 2Δ 0 energy to break a Cooper pair and two quasiparticles are promoted to the excited states which is above the gap. Therefore, one can observe such a gap at low temperature by using optical absorption methods in the far infrared regime. We shall make the long wave length approximation q → 0 in our later calculation. In this chapter, we introduce the theory that we use to calculate the real part of the optical conductivity in section 3.2 and show the results in section 3.3. In section 3.3, we also make the comparison for different pairing symmetries in the real part of the optical conductivity. Moreover, there is an important sum rule which describes that the area under the curve of the real part of the conductivity σ 1 (ω ) must be conserved. That is,. ∫. ∞. 0. σ 1 (ω ) d ω =. ω p2 8. ,. (3.1.1). where ω p is the plasma frequency. This result is proposed by Kubo in 1957.24 This result is correct in both normal state and superconducting state. We will give more details in the following discussions..

(28) 23 3.2 Theory In the following, we show the details about how to derive the optical conductivity by linear response theory. Then we will calculate the current-current response function to obtain the optical conductivity.. 3.2.1 Linear Response Theory Linear response theory is widely used in various branches of physics. It says that the response to a “weak” external perturbation is proportional to the perturbation. Hence, all we have to do is to figure out the proportion corresponding to the physical quantity we are interested in. Consider the Hamiltonian. H0. which is time-independent in. thermodynamic equilibrium. An expectation value of a physical quantity P can be calculated as. P =. 1 1 Tr [ ρ 0 P ] = Z0 Z0. ∑. n P n e − β En ,. (3.2.1.1a). n. ρ0 = e− β H = ∑ e− β H n n = ∑ n n e− β E , 0. 0. n. n. (3.2.1.1b). n. where ρ0 is the density matrix and Z 0 is the partition function. Here n. are the eigenstates of H 0 corresponding to the eigenvalues En . On. the other hand, H 0 n = En n .. (3.2.1.2). Here H 0 describes the system before the perturbation was applied and we assume that an external perturbation is applied to the system at time t = t0 . That is,. H ( t ) = H 0 + H ′ ( t ) θ ( t − t0 ) .. (3.2.1.3).

(29) 24 In order to find the expectation value of the physical quantity P at time t > t0 , we need to find the time evolution of the density operator.. Therefore, we can obtain P ( t ) as P (t ) =. 1 Tr ⎡ ρ ( t ) P ⎤⎦ , Z0 ⎣. (3.2.1.4). where ρ ( t ) = ∑ n ( t ) n ( t ) e− β E ,. (3.2.1.5). n. n. and n ( t ) are the eigenstates of H ( t ) . Moreover, the states n ( t ) are time dependent and governed by the Schrödinger equation i∂ t n ( t ) = H ( t ) n ( t ) .. (3.2.1.6). Note that we regard H ′ as a small perturbation. Since H ′ depends on time, it is convenient to use the interaction picture to represent n ( t ) . i.e. n ( t ) = e − iH 0t nˆ ( t ) = e − iH 0tUˆ ( t , t0 ) nˆ ( t0 ) ,. (3.2.1.7). where n ( t ) is represented in Schrödinger representation and nˆ ( t ) is represented in interaction representation. The relation between n ( t0 ) and nˆ ( t0 ) is nˆ ( t0 ) = eiH 0t0 n ( t0 ) = n .. (3.2.1.8). Besides, Uˆ ( t , t0 ) is a unitary operator and is given by ∞ 1 ⎛1⎞ Uˆ ( t , t0 ) = ∑ ⎜ ⎟ n =0 n ! ⎝ i ⎠. n. ∫. t. t0. dt1. ∫. t. t0. (. dtnTt Hˆ ′ ( t1 ). Hˆ ′ ( tn ). ). ⎛ −i ∫ dt ′Hˆ ′( t ′) ⎞ Uˆ ( t , t0 ) = Tt ⎜ e t0 ⎟⎟ , ⎜ ⎝ ⎠ t. (3.2.1.9). where Tt is the time ordering operator. Here we consider only to linear order in H ′ and omit the higher order terms. That is to say,.

(30) 25 t Uˆ ( t , t0 ) = 1 − i ∫ dt ′Hˆ ′ ( t ′ ).. (3.2.1.10). t0. Note that we use a hat over H ′ or n to mean that it is represented in the interaction picture. Then insert Eq. (3.2.1.10) into Eq. (3.2.1.4) and one can obtain the expectation value of P up to linear order in the perturbation P (t ) =. 1 Z0. ∑e β. − En. n ( t0 ) e− iH 0tUˆ † ( t , t0 ) eiH 0t Pe− iH 0tUˆ ( t , t0 ) eiH 0t n ( t0 ). n. P ( t ) = n ( t0 ) P n ( t0 ) − i ∫ dt ′ t. t0. 1 ∑ e− β En n ( t0 ) Pˆ ( t ) Hˆ ′ ( t ′) − Hˆ ′ ( t ′) Pˆ ( t ) n ( t0 ) Z0 n. t P ( t ) = P 0 − i ∫ dt ′ ⎡⎣ Pˆ ( t ) , Hˆ ′ ( t ′ ) ⎤⎦ , t0 0. (3.2.1.11) where n ( t0 ) are the eigenstates of the time-independent Hamiltonian H 0 . Furthermore, we can define the difference between. P (t ). and P. 0. as δ A ( t ) . That is, δ P ( t ) ≡ P ( t ) − P 0 = −i ∫ dt ′ ⎡⎣ Pˆ ( t ) , Hˆ ′ ( t ′ ) ⎤⎦ . t 0 t. (3.2.1.12). 0. By definition, the retarded correlation function is R ˆ ⎡ˆ ⎤ . CPH ′ ( t , t ′ ) ≡ −iθ ( t − t ′ ) ⎣ P ( t ) , H ′ ( t ′ ) ⎦ 0. (3.2.1.13). Then substitute Eq. (3.2.1.13) into Eq. (3.2.1.12) and one obtains ∞. R δ P ( t ) = ∫ dt ′CPH ′ ( t , t ′ ). t0. (3.2.1.14). This is so called Kubo formula which expresses the linear response to a perturbation, H ′ . If the external perturbation has the form H P′ ′ ( t ) = P′g ( t ) ,. (3.2.1.15). where P ′ is a time-independent operator and g ( t ) is a function of time.

(31) 26 R t , the response function CPH ′ ′ ( t , t ′ ) becomes P R R CPH ′ ′ ( t , t ′ ) = CPP′ ( t − t ′ ) g ( t ′ ) , P. (3.2.1.16). R Since P ′ is independent of time, it is easy to prove that CPP ′ ( t, t ′). depends only on the time difference. When inserting Eq. (3.2.1.16) into Eq. (3.2.1.14) and letting t0 = −∞ , we obtain ∞. R δ P ( t ) = ∫ dt ′CPP ′ ( t − t ′ )g ( t ′ ) .. −∞. (3.2.1.17). By Fourier transformation, Eq. (3.2.1.17) becomes R δ P (ω ) = CPP ′ (ω ) g (ω ) .. (3.2.1.18). In general, the external force can depend on position and direction. That is, an external perturbation is of the form H P′ ( t ) = ∑ ∫ drPν ( r ) gν ( r , t ) .. (3.2.1.19). ν. By Fourier transformation, we obtain R ν δ P (ω ) = ∑ ∫ drCPP ′( r ) ( ω ) g ( r , ω ) .. (3.2.1.20). ν. This result will be used in the following section where we discuss Kubo formula for conductivity.. 3.2.2 Kubo Formula for Optical Conductivity Now we consider charged particles, electrons, in a system and apply an external electromagnetic (EM) field to this system. EM field will induce a current, and the linear response coefficient is related to the optical conductivity. That is to say, once we obtain the linear response coefficient, we can get the optical conductivity immediately. In general, the optical conductivity may be nonlocal in time and space,.

(32) 27 J ie ( r, t ) = ∫ dt ′∫ dr′∑ σ ij ( rt , r′t ′ )E j ( r′, t ′ ) ,. (3.2.2.1). j. where σ ij ( rt , r′t ′ ) is the optical conductivity tensor which represents the current response in the direction eˆ i according to an applied electric field in the direction eˆ j . Since we consider an equilibrium system, the optical conductivity only relates to the difference in time σ ij ( rt , r′t ′ ) = σ ij ( r, r′, t − t ′ ) . Then Fourier transform J ( r,t ) from time domain into the frequency domain. It is written as J ie ( r, ω ) = ∫ dr′∑ σ ij ( r, r′, ω ) Eextj ( r′, ω ).. (3.2.2.2). j. The electric field E is given by E ( r, t ) = −∇rφext ( r, t ) − ∂ t A ext ( r, t ) ,. (3.2.2.3). where φext is the electric potential and Aext is the vector potential. Then choose a gauge φext = 0 to simplify the following expressions. It is always possible by choosing a suitable Aext ( r, t ) in Eq. (3.2.2.3). Moreover, the final result should be independent of the choice of gauge. Therefore, Eq. (3.2.2.3) becomes E ( r, t ) = −∂ t A ext ( r, t ) .. (3.2.2.4). By Fourier transformation in frequency domain, Eq. (3.2.2.4) becomes A ext ( r, ω ) = (1/ iω ) Eext ( r, ω ) .. (3.2.2.5). For this case, we can express electrons in the magnetic field for the kinetic energy by substituting the kinetic momentum p + eA into the canonical momentum p , 2. H =. 1 ⎛ ⎞ drψ σ† ( r ) ⎜ ∇ r + eA ⎟ ψ σ ( r ) ∑ ∫ 2m σ ⎝i ⎠. (3.2.2.6).

(33) 28 By expanding the square in Eq. (3.2.2.6), the integrand becomes ψ σ† ( r ). 2. 1 ⎛ e e2 † ⎞ † ∇ + ∇ + ∇ + ψ r ψ r i A A i ψ r ψ σψ σ ( r ) A 2 [ r r ⎟ σ ( ) σ ( ) r] σ ( ) ⎜ 2m ⎝ i 2mi 2m ⎠. (3.2.2.7) Note that the first term is the kinetic energy when the applied magnetic field is turned off. The second and third terms come from the applied magnetic field. Then integrate by part and one can get ⎧e ⎫ e2 † H = T − ∑ ∫ dr ⎨ A i ⎡⎣( ∇ψ σ† ( r ) )ψ σ ( r ) −ψ σ† ( r ) ( ∇ψ σ ( r ) ) ⎤⎦ + ψ σ ( r )ψ σ ( r ) A 2 ⎬ . 2m σ ⎩ 2mi ⎭. (3.2.2.8) Hence, the variations of H is ⎧e ⎫ e2 ⎡( ∇ψ σ† ( r ) )ψ σ ( r ) −ψ σ† ( r ) ( ∇ψ σ ( r ) ) ⎤ − Aψ σ† ( r )ψ σ ( r ) ⎬ i δ A. ⎣ ⎦ m ⎩ 2mi ⎭. δ H = −∑ ∫ dr ⎨ σ. (3.2.2.9) One can compare the results to those found in analytical mechanics. The variations δ H due to variations δ A is given by δ H = − ∫ dr J e iδ A = e ∫ dr J iδ A .. (3.2.2.10). For electrons, we have J e = −eJ . There are two terms in the current density operator denoted respectively by the paramagnetic and diamagnetic contribution, J p and J d . That is, Jσ ( r ) = Jσp ( r ) + Jσd ( r ) ,. (3.2.2.11). where J σp ( r ) =. and. ⎡ψ σ† ( r ) ( ∇ψ σ ( r ) ) − ( ∇ψ σ† ( r ) )ψ σ ( r ) ⎤ ⎣ ⎦ 2mi. (3.2.2.12).

(34) 29 J σd ( r ) =. e † e ψ σ ( r )ψ σ ( r ) A ( r ) = ρ ( r ) A ( r ) . m m. (3.2.2.13). It is noted that ρ ( r ) = ψ σ† ( r )ψ σ ( r ) is the density operator. Next, go back to Eq. (3.2.2.10), δ A = A − A 0 where A denotes the total vector potential and A 0 denotes the vector potential in equilibrium before opening the “small” perturbation δ A . In other words, A ( r,t ) = A 0 ( r ) + δ A ( r, t ) .. (3.2.2.14). Then substitute the total vector potential A into Eq. (3.2.2.11), one can obtain Jσ ( r , t ) = Jσp ( r , t ) +. e A ( r, t ) ρ ( r ) m. Jσ ( r, t ) = J σp ( r , t ) +. e ⎡ A 0 ( r ) + δ A ( r , t ) ⎤⎦ ρ ( r ) . m⎣. (3.2.2.15). Using Eq. (3.2.2.15) and Eq. (3.2.2.14), we can rewrite the Eq. (3.2.2.10) of the form ⎧ ⎩. e ⎫ ⎡⎣ A 0 ( r ) + δ A ( r, t ) ⎤⎦ ρ ( r ) ⎬ i δ A ( r, t ) m ⎭. ⎡ ⎣. e ⎤ ρ ( r ) A 0 ( r ) ⎥ i δ A ( r, t ) m ⎦. δ H ( t ) = e ∫ dr ⎨Jσp ( r, t ) + δ H ( t ) = e ∫ dr ⎢ Jσp ( r, t ) +. δ H ( t ) ≡ e ∫ dr J 0 ( r ) i δ A ( r, t ) .. (3.2.2.16) Here we consider only to the first order of δ A , because we are merely interested in the linear response. It is worth noting that Eq. (3.2.2.16) is similar to what we obtained in Eq. (3.2.1.19). Then we can obtain the formulation of the linear response in frequency space by Fourier transform, namely, δ H ( ω ) = e ∫ dr J 0 ( r ) i δ A ( r , ω ) .. (3.2.2.17).

(35) 30 In fact, δ A ( r, ω ) is Aext ( r, ω ) in Eq. (3.2.2.5). It is true that δ A ( r, ω ) actually represents the small perturbation. Therefore, substitute δ A ( r, ω ) by Eq. (3.2.2.5) and Eq. (3.2.2.17) becomes δ H (ω ) =. e dr J 0 ( r ) i Eext ( r , ω ) . iω ∫. (3.2.2.18). Finally, we want to find the expectation value of the current operator, i.e., e A ext ( r, ω ) ρ ( r ) . m. J ( r , ω ) = J 0 ( r, ω ) +. In Eq. (3.2.2.19), to linear order in. A ext ,. (3.2.2.19) we can replace. ρ ( r ) ≈ ρ 0 ( r ) = n ( r ) in the last term. For the first term in Eq. (3.2.2.19),. make use of the linear response theory. By Eq. (3.2.1.20) mentioned in section 3.2.1, one obtains the general Kubo formula in frequency domain, i.e., δ J i0 ( r, ω ) = ∫ dr′∑ CJR (r )J i 0. j. j 0. ( r ′). (ω ). e j Eext ( r′, ω ) . iω. (3.2.2.20). As a result, there is no current before the external potential applied. ( i.e. J0. 0. = 0 or δ J 0 = J 0 − J 0. 0. = J 0 ) Hence, Eq. (3.2.2.19) is rewritten. of the form J i0 ( r, ω ) = ∫ dr′∑ CJRi ( r )J j ( r′) (ω ) j. 0. 0. e j Eext ( r′, ω ) . iω. (3.2.2.21). As indicated above, the total current density is e ⎡e ⎤ j ′ n ( r ) δ ( r − r′ ) δ ij ⎥ Eext J i ( r, ω ) = ∑ ∫ dr′ ⎢ CJRi ( r )J j ( r′) (ω ) + (r ,ω ) . iω m ⎣ iω 0 0 ⎦ j. (3.2.2.22) Introduce ∏ijR tensor which is equal to CJR (r )J (r′) for notation simplicity i 0. j 0. and use J e = −e J . Then compared with the definition in Eq. (3.2.2.2), we obtain the linear response formula for the optical conductivity.

(36) 31 σ ij ( r, r′, ω ) =. ie 2. ω. ∏ ijR ( r , r′, ω ) −. e2 n ( r ) δ ( r − r′ ) δ ij , iω m. (3.2.2.23). Notice that the retarded current-current correlation function in the time domain is given by ∏ijR ( r , r ′, t − t ′ ) = CJRi ( r ) J j (r′) ( t − t ′ ) = −iθ ( t − t ′ ) ⎡⎣ Jˆ i0 ( r, t ) , Jˆ 0j ( r′, t ′ ) ⎤⎦ . 0 0 0. (3.2.2.24) It must further be noted that J 0 = J p when no magnetic field was applied before turning on the perturbation. That is, A 0 = 0 and then Eq. (3.2.2.24) becomes ∏ijR ( r , r ′, t − t ′ ) = −iθ ( t − t ′ ) ⎡⎣ Jˆ ip ( r, t ) , Jˆ pj ( r′, t ′ ) ⎤⎦ . 0. (3.2.2.25). As was mentioned above, one can study the retarded current-current correlation function instead of finding the optical conductivity of a given system. In order to obtain a more brief form, we can rewrite Eq. (3.2.2.22) in a way similar to that in Ref.25. That is, e ⎡ ⎤ j ′ J i ( r, ω ) = ∑ ∫ dr′ ⎢eCJRi ( r )J j ( r′) (ω ) + n ( r ) δ ( r − r′ ) δ ij ⎥ A ext (r ,ω ). m ⎣ 0 0 ⎦ j. (3.2.2.26) Similarly, we use J e = −e J and Eq. (3.2.2.26) becomes ⎡ ⎤ j e2 J ie ( r, ω ) = −∑ ∫ dr′ ⎢e 2CJRi (r )J j (r′) (ω ) + n ( r ) δ ( r − r′ ) δ ij ⎥ A ext ( r′, ω ) . 0 0 m j ⎣ ⎦. (3.2.2.27) Then Fourier transform Eq. (3.2.2.27) from real space into momentum space, 1 ⎡ ⎤ j J ie ( q, ω ) = −e2 ∑ ⎢CJRi ( q )J j (q ) (ω ) + n ( q ) δ ij ⎥ A ext ( q, ω ) 0 0 m ⎦ j ⎣ J ie ( q, ω ) ≡ −e2 ∏ij ( q, ω ) A ejxt ( q, ω ) ,. (3.2.2.28).

(37) 32 where ∏ij ( q, ω ) is the current-current correlation function and i and j denote the Cartesian coordinates. The correlation function is composed of two components: a paramagnetic term ∏ijp ( q, ω ) and a diamagnetic term ∏ijd ( q ) . Notice that ∏ijp ( q, ω ) depends on frequency and momentum q ,. while ∏ijd ( q ) depends only on frequency. Thus, ∏ij ( q, ω ) can be represent as ∏ij ( q, ω ) = ∏ijp ( q, ω ) + ∏ijd ( q ) ,. (3.2.2.29). where the paramagnetic term is given by ∏ijp ( q, ω ) = CJRi ( q) J j (q ) (ω ) 0. 0. (3.2.2.30). and the diamagnetic term is ∏ ijd ( q ) =. 1 n ( q ) δ ij . m. (3.2.2.31). It is worth noting that ∏ijd ( q ) is a real function. This result is important for later calculation. Next, we derive an expression for the optical conductivity represented by Nambu Green’s function. From Eq. (3.2.2.28) and Eq. (3.2.2.5), one can obtain J ie ( q, ω ) = −. e2 j ∏ij ( q, ω ) Eext ( q, ω ) = σ ij ( q, ω ) Eextj ( q, ω ) . (3.2.2.32) iω. As mentioned above, one can find the optical conductivity through finding the correlation function. Moreover, σ ij ( q, ω ) is a complex function σ ij ( q, ω ) = σ 1 ( q, ω ) + iσ 2 ( q, ω ) ,. (3.2.2.33). where σ 1 ( q, ω ) is the real part and σ 2 ( q, ω ) is the imaginary part. Here we focus on the study of the real optical conductivity σ 1 ( q, ω ) . That does.

(38) 33 not lose the generality, because we can always obtain the imaginary one, σ 2 ( q, ω ) , by Kramers-Kroning relations. From Eq. (3.2.2.32), (3.2.2.29),. and (3.2.2.33), we get the relation between σ 1 ( q, ω ) and ∏ij ( q, ω ) , σ 1 ( q, ω ) = −. e 2 Im ∏ ij ( q, ω ). ω. =−. e 2 Im ∏ijp ( q, ω ). ω. .. (3.2.2.34). Note that the diamagnetic correlation function, ∏ijd ( q ) , is purely real function, so it does not matter in the calculation of σ 1 ( q, ω ) . Therefore, we need to evaluate the paramagnetic correlation function to obtain the real part of the optical conductivity. Now, we can Fourier transform Eq. (3.2.2.12) into the momentum space, so it is given by J ip ( q ) = ∑ψ k†,σ γî ( k + q 2 )ψ k +q ,σ ,. (3.2.2.35). k ,σ. where ψ k†σ = ( d x†σ ( k ) , d y†σ ( k ) ) in the two-orbital model we adopt and the EM vertex function, γî ( k ) is represented as γî ( k ) =. ∂H . ∂ki. (3.2.2.36). Then the paramagnetic term, ∏ijp can be written in terms of the paramagnetic current operators. That is, ∏ijp ( q,τ ) = − Tˆτ J ip ( q,τ ) J pj ( −q, 0 ) ,. (3.2.2.37). where Tˆτ is the imaginary time ordering operator, τ is the imaginary time, and J p is the paramagnetic current operator in Heisenberg picture. Furthermore, we can Fourier transform Eq. (3.2.2.37) from imaginary time domain into Matsubara frequency domain and then we obtain 1 N. ∫. ∏ ijp ( q, iΩ ) = −. 1 N. ∏ ipj ( q, iΩ ) =. β. 0. dτ eiΩτ ∏ijp ( q,τ ). ∫. β. 0. dτ eiΩτ Tˆτ J ip ( q,τ ) J pj ( −q, 0 ) ,. (3.2.2.38).

(39) 34 where Ω = 2nπ β is a bosonic Matsubara frequency and β = 1 kBT . Substituting the finite temperature Green’s function, Eq. (3.2.2.38) can be given by ∏ijp ( q, iΩ ) =. 2 Nβ. ⎡. ⎛. q. ∑ Tr ⎢⎣γˆ ( k ) Gˆ ⎜⎝ k + 2 , iω i. k ,m. m. q ⎞ ⎛ ⎞⎤ + iΩ ⎟ γˆ j ( k ) Gˆ ⎜ k − , iωm ⎟ ⎥ , 2 ⎠ ⎝ ⎠⎦. (3.2.2.39) where ωm = ( 2m + 1) π β is a fermionic Matsubara frequency and the factor of 2 comes from sum over all spin. For the normal state, γî ( k ) is expressed as γî ( k ) = ∂H 0 ∂ki. where H 0 is given by Eq. (2.2.1).. However, for the superconducting state, γî ( k ) is expressed as ⎫ ∂H ( Δ1 = 0, Δ 2 = 0 ) 1 ⎧ ∂H , γˆ0 ⎬ or γî ( k ) = iγˆ0 , (3.2.2.40) ∂ k 2 ⎩ ∂ki i ⎭. γî ( k ) = ⎨. where H is given by Eq. (2.3.4), { } is the anticommutator, and γˆ0 is given by ⎛1 0 ⎜ 0 −1 γˆ0 = ⎜ ⎜0 0 ⎜ ⎝0 0. 0 0⎞ ⎟ 0 0⎟ . 1 0⎟ ⎟ 0 −1 ⎠. (3.2.2.41). Eq. (3.2.2.39) is the same as the expression in Ref.23. Finally, we take the limit iωm → ω + iΓ and q → 0 for the response to an electric field, so we obtain the real part of the optical conductivity, σ 1 (ω ) . Notice that Γ is the scattering rate. In the following section, we will show σ 1 (ω ) with respect to each pairing gap for superconducting state. Moreover, we also show σ 1 (ω ) for the normal state.. 3.3 Results In previous section, we have discussed how to evaluate the real.

(40) 35 conductivity σ 1 (ω ) by using the current-current response function. Here we first calculate σ 1 (ω ) in the normal state. There are two parts of the contribution to σ 1 (ω ) . One is the intraband contribution, whose results should behave like Drude conductivity at low frequencies. The other is the interband contribution to which σ 1 (ω ) should increase at higher frequency to certain ω values. The scattering rate may be a small quantity in the absence of impurities and at low temperatures. Therefore, intraband contribution should be dominant at low ω for σ 1 (ω ) . In Fig. 3.3.1, one sees these results clearly. It is worth noting that the width of the Drude-like peak in σ 1 (ω ) becomes larger as the scattering rate increases. Moreover, for intraband contribution, σ 1intra (ω ) increases as Γ decreases. We can see this result clearly from the following asymptotic low ω behavior, σ 1intra (ω ) ∝. Γ ⎡⎣ Ei ( k + q 2 ) − Ei ( k − q 2 ) + ω ⎤⎦ + Γ 2 2. (3.3.1). ,. where i = +, − . As taking the limit q → 0 , ⎡⎣ Ei ( k + q 2 ) − Ei ( k − q 2 ) ⎤⎦ is much smaller than Γ . Therefore, σ 1intra (ω ) is proportional to 1 Γ for small ω . Based on a similar analysis , one sees that the interband contribution, σ 1inter (ω ) , is proportional to Γ for small ω . At small ω , σ 1inter (ω ) ∝. Γ ⎡⎣ Ei ( k + q 2 ) − E j ( k − q 2 ) + ω ⎤⎦ + Γ 2 2. (3.3.2). ,. where i, j = +, − and i ≠ j . Since ⎣⎡ Ei ( k + q 2 ) − E j ( k − q 2 ) ⎦⎤ >> Γ and ⎡⎣ Ei ( k + q 2 ) − E j ( k − q 2 ) ⎤⎦ >> ω. for. small. ω ,. σ 1inter (ω ). is. thus. proportional to Γ for small ω . Furthermore, one sees that the curve of σ 1 (ω ) tends to approach some value as ω → 0 . In fact, σ 1 (ω = 0 ) is the.

(41) 36 direct current (DC) conductivity. T=0.045Tc; Γ=0.200Δ0 0.060 0.055. (a) σ dc. 0.050 0.045 0.040. σ1. 0.035 0.030 0.025 0.020 0.015. total interband intraband. 0.010 0.005 0.000 0.0. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. ω/Δ0. T=0.045Tc; Γ=0.095Δ0. (b) 0.30. total interband intraband. σdc. 0.25. σ1. 0.20. 0.15. 0.10. 0.05. 0.00 0.0. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. ω/Δ0. Fig. 3.3.1 The real part of the optical conductivity, σ 1 (ω ) , is plotted for different scattering rate, (a) Γ = 0.200Δ 0 and (b) Γ = 0.095Δ 0 . Here we choose We choose Δ 0 = 0.3 , 2Δ 0 = 3.52k BTc , and the doping μ = 1.6 ..

(42) 37 There is an important sum rule which describes that the area under the curve of the conductivity σ 1 (ω ) must be conserved24,. ∫. ∞. 0. σ 1 (ω ) d ω =. π ne 2 2m. =. ω p2 8. (3.3.3). ,. where n is the electron density and ω p is the plasma frequency. We also confirm this result in Fig. 3.3.2. Although we only plot Fig. 3.3.2 in the low frequency, we still observe this fact phenomenologically. Ref.24 proved that the sum rule must be hold whether in the normal state or in the superconducting state. Therefore, we obtain the result,. ∫ (σ (ω ) − σ (ω ) ) dω = 0, ∞. 0. s 1. (3.3.4). n 1. where the superscripts s and n denote the superconducting and normal states respectively. T=0.045Tc. Γ=0.095Δ0. 0.30. Γ=0.200Δ0. 0.25. σ1. 0.20. 0.15. 0.10. 0.05. 0.00 0.0. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. ω/Δ0. Fig. 3.3.2 We compare the real part of the optical conductivity, σ 1 (ω ) with different scattering rate. We can see the variation clearly. We choose Δ 0 = 0.3 , 2Δ 0 = 3.52k BTc , and the doping μ = 1.6 ..

(43) 38 According to Eq. (3.3.4), the area under the curve of σ 1 (ω ) in the superconducting state should have the same value as in the normal state. Ref.26 pointed out that the missing area at the finite frequencies as a delta function at ω = 0 . In Fig. 3.3.3, we can see how σ 1 (ω ) varies from the normal state to the superconducting state as gap exists for long- and short-wavelength electromagnetic wave respectively.. Fig. 3.3.3 The frequency-dependent conductivity for (a) long- and (b) short-wavelength electromagnetic wave transforms from normal state to superconducting state. The dashed line denotes the normal state. v0 q indicates the maximum frequency of absorption. Moreover, v0 is the Fermi velocity and q is the wave number. In (a), the wavelength is so long that the maximum absorption frequency is smaller than the energy gap threshold ω g. Therefore, all of the strength is absorbed by delta function at zero frequency. It results in a full London current. In (b), the short wavelength leads to that the maximum absorption frequency is larger than the energy gap. Thus, the strength of the delta function is less. That is to say, the London current is weakened. (Source: Ref.26). Fig. 3.3.3 shows the one-band case, but we consider the two-band case here. Therefore, we will see two parts of the σ 1 (ω ) later. One is intraband contribution to σ 1 (ω ) and the other is interband contribution to.

(44) 39 σ 1 (ω ) . Moreover, we don’t show the London current at zero frequency. here. Then we plot the case of the d x − y pairing symmetry to illustrate 2. 2. this phenomenon in our case, (see Fig. 3.3.4). dx -y 2. 2. 0.15 s. σ1 n. σ1. σ1. 0.10. 0.05. 0.00 0.0. 0.5. 1.0. Fig. 3.3.4 We plot σ (ω ) and σ s 1. 1.5. n 1. (ω ). 2.0. 2.5. 3.0. ω/Δ0. for d x2 − y 2 pairing symmetry to illustrate that. the sum rule holds in both normal and superconducting states. We choose Δ 0 = 0.3 , 2Δ 0 = 3.52k BTc , and the doping μ = 1.6 .. In the rest of this section, we show the results of σ 1 (ω ) for the superconducting state with different pairing symmetries. For the nodal order parameters, one is able to see that there is a continuously increasing spectrum at low ω as ω decreases and a strong peak develops near ω = 0 . For nodal order parameters which have nodes on the Fermi surface,. there are low lying states such that quasiparticles can be excited below 2Δ 0 . In Fig. 3.3.6 and Fig. 3.3.7, we plot the curves of σ 1 (ω ) for nodal d x2 − y 2 and d xy . The results are reasonable because d x2 − y 2 has nodes on. the hole Fermi pockets, while d xy has nodes on both hole Fermi pockets.

(45) 40 and electron Fermi pockes. Furthermore, we can recognize whether the pairing gap is nodal gap or nodeless gap from the density of states (DOS). For nodal gap, the curve of the DOS is linear near ω = 0 . However, the curve of the DOS is essentially zero as ω < Δ 0 for nodeless gap. In Fig. 3.3.5, DOS is linear around ω = 0 . Therefore, we confirm that d x − y and 2. 2. d xy are nodal gaps.. d. 2. 2. x -y. d. DOS. xy. -1.0. 0 w/Δ. 1.0 0. Fig. 3.3.5 We plot DOS at μ = 1.6 for pairing gaps, d x2 − y 2 and d xy . Here we choose Δ 0 = 0.3 . One can see clearly that DOS is linear near ω = 0 for pairing symmetry d x2 − y 2 or d xy . Note that ω = 0 is the Fermi level..

(46) 41. dx -y 2. 2. 0.026 0.024 0.022 0.020 0.018. σ1. 0.016 0.014 0.012. Γ=0.095Δ0. 0.010. T=0.045Tc. 0.008 0.006. total intraband interband. 0.004 0.002 0.000 0.0. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. ω/Δ0. Fig. 3.3.6 σ 1 (ω ) with d x2 − y 2 pairing symmetry is plotted with Γ = 0.095Δ 0 , T = 0.045Tc , and the doping μ = 1.6 . The curve of σ 1 (ω ) varies to low frequency continuously. dxy 0.026 0.024 0.022 0.020 0.018. σ1. 0.016 0.014 0.012. Γ=0.095Δ0. 0.010. T=0.045Tc. 0.008 0.006. total interband intraband. 0.004 0.002 0.000 0.0. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. ω/Δ0. Fig. 3.3.7 σ 1 (ω ) with d xy pairing symmetry is plotted with Γ = 0.095Δ 0 , T = 0.045Tc , and the doping μ = 1.6 . The curve of σ 1 (ω ) varies to low frequency continuously..

(47) 42 Next we consider the nodeless pairing symmetry. Recall that s0 is a fully constant gap and s x y has nodes at k x = ± π 2 and k y = ± π 2 . In 2 2. Fig. 3.3.8 (a), we can see that s0 is a fully gap. Therefore, for s0 pairing symmetry, we find that the intraband contribution is zero below ω = 2Δ 0 , while it will dominate at ω = 2Δ 0 . (See Fig. 3.3.9) Because the interband contribution in the normal state does not vanish at low frequency, it still contributes to σ 1 (ω ) even in the superconducting state at low frequency. Then Fig. 3.3.8 (b) shows that s x y pairing gap is a nodeless gap as 2 2. choosing μ = 1.6 . That is, for s x y pairing symmetry, although it has 2 2. nodes at k x = ± π 2 and k y = ± π 2 , the nodal lines do not cross the Fermi surface at this particular doping. (See Fig. 2.3.1 (h)) It is worth noting that, however, the nodal lines of s x y pairing symmetry are near the 2 2. Fermi surface. Hence, it implies that we only need energy less than 2Δ 0 to excite a quasiparticle from the normal state. (See Fig. 3.3.8 (b) and Fig. 3.3.10) (a). (b). DOS -1.5. sx2y2. DOS. s0. -1.0. -0.5. 0 w/Δ 0. 0.5. 1.0. 1.5. -1.5. -1. -0.5. 0 w/Δ 0. 0.5. 1.0. 1.5. Fig. 3.3.8 DOS is plotted for different pairing symmetries, (a) s0 and (b) s x2 y 2 . Here we choose the doping μ = 1.6 ..

(48) 43. s0 0.015 0.014. total interband intraband. 0.013 0.012 0.010. 0.000014. 0.009. 0.000012. 0.008. 0.000010. 0.007 0.006. Γ=0.095Δ0. 0.005. T=0.045Tc. 0.004. intraband. 0.000008. σ1. σ1. 0.011. 0.000006. 0.000004. 0.000002. 0.003 0.000000. 0.0. 0.002. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. ω/Δ0. 0.001 0.000 0.0. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. ω/Δ0. Fig. 3.3.9 σ 1 (ω ) with s0 pairing symmetry is plotted with Γ = 0.095Δ 0 , T = 0.045Tc , and the doping μ = 1.6 . In the inset, we show the intraband only (though the spectrum is dim) with clear nodeless structure.. sx y. 0.018 0.017 0.016 0.015 0.014 0.013 0.012 0.011 0.010 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000 0.0. total interband intraband. 0.000030. intraband. 0.000025. 0.000020. Γ=0.095Δ0. σ1. σ1. 2 2. 0.000010. T=0.045Tc. Fig. 3.3.10 σ 1 (ω ). 0.000015. 0.000005. 0.000000 0.0. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. ω/Δ0. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. ω/Δ0. with. s x2 y 2. pairing symmetry is plotted with. Γ = 0.095Δ 0 ,. T = 0.045Tc , and the doping μ = 1.6 . In the inset, we show the intraband only (though the spectrum is dim) with clear nodeless structure..

(49) 44 Now we consider the pairing symmetry s x y with different dopings. 2 2. In Fig. 3.3.11, we can see how the Fermi surface varies as increasing doping. Moreover, the s x y pairing symmetry only has nodes on the 2 2. Fermi surface above a critical doping μ = 2 . Then we can also obtain the same results from looking at the Fig. 3.3.12. As a doping exceeds μ = 2 , DOS is linear near ω = 0 , otherwise, DOS is essentially zero as ω < Δ 0 . That is, s x y becomes a nodal gap as a doping exceeds μ = 2 . 2 2. Fig. 3.3.11 We plot the evolution of the Fermi surface topology for different doping. Note that the dashed green line denotes the position of nodes..

(50) 45. DOS. μ=1.6 μ=1.8 μ=2.0 μ=2.2 μ=2.4. -1.5. -1.0. -0.5. 0 w/ Δ 0. 0.5. 1.0. 1.5. Fig. 3.3.12 Density of states (DOS) with different doping is plotted. We can see clearly that s x2 y2 has nodes on the Fermi surface above critical doping μ = 2.0 . Here we choose Δ 0 = 0.3 .. Furthermore, Fig. 3.3.11 shows that α band and β band are closer and closer as increasing doping. Therefore, we only need energy less than 2Δ 0 to excite a quasiparticle from the normal state. In Fig. 3.3.13, one. can see this result. In order to see the behavior of σ 1 (ω ) near ω = 0 , we plot σ 1 (ω ) in the interval ω Δ 0 ∈ [ 0,1] . (See Fig. 3.3.14(a)) Then we find that σ 1 (ω ) has a strong peak near ω = 0 as increasing doping above μ = 2.0 . In Fig. 3.3.14(b), we only show the intraband contribution with. clear structure although the spectrum is dim..

(51) 46 0.25. μ=1.6 μ=1.8 μ=2.0 μ=2.2 μ=2.4. 0.2. σ 1(w). 0.15. 0.1. 0.05. 0. 0. 1. 2. 3. w/ Δ 0. Fig. 3.3.13 σ 1 (ω ) for pairing symmetry s x2 y2 is plotted with different doping. Here we choose Δ 0 = 0.3 . (a). (b) -4. 2.5. 0.018. μ=1.6 μ=1.8 μ=2.0 μ=2.2 μ=2.4. 0.017. x 10. μ=1.6 μ=1.8 μ=2.0 μ=2.2 μ=2.4. 2. 0.016. 1.5. σ1. σ 1(w). intra. 0.015. 0.014. 1. 0.013 0.5. 0.012. 0.011. 0. 0. 1 w/ Δ0. 0. 1. 2. 3. w/Δ 0. Fig. 3.3.14 (a) We plot σ 1 (ω ) for the pairing symmetry s x2 y 2 with different doping in the interval. ω ∈ [ 0,1] . (b) Intraband contribution to σ 1 (ω ) is plotted with different doping..

(52) 47 Then let’s consider the sx + y pairing symmetry. Recall Fig. 2.3.1(f) 2. 2. which indicates that sx + y pairing symmetry has nodes on the electron 2. 2. Fermi pockets and fully gap on the hole Fermi pockets. In Fig. 3.3.15, we also see that the sx + y pairing symmetry has nodes on the Fermi surface. 2. 2. Therefore, σ 1 (ω ) near the zero frequency is like d-wave cases due to nodal line across the Fermi surface. Because the hole Fermi pockets are fully gapped, sx + y 2. 2. pairing symmetry still has similar behavior at. DOS. ω = 2Δ 0 . (See Fig. 3.3.16). -1.25. -1.00. -0.75. -0.50. -0.25. 0 w/ Δ 0. 0.25. 0.50. 0.75. 1.00. 1.25. Fig. 3.3.15 We plot DOS for the pairing symmetry sx2 + y 2 . We can see that the pairing symmetry sx2 + y 2 has nodes on the Fermi surface..

(53) 48 sx +y 2. 0.016 0.015 0.014 0.013. 2. total interband intraband. 0.012 0.011 intraband. 0.00020. 0.010. 0.00015. 0.008 0.007 0.006 0.005. σ1. σ1. 0.009. 0.00010. Γ=0.095Δ0 0.00005. 0.004. T=0.045Tc. 0.003. 0.00000 0.0. 0.002. 0.5. 1.0. 2.0. 2.5. 3.0. ω/Δ0. 0.001 0.000 0.0. 1.5. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. ω/Δ0. Fig. 3.3.16 σ 1 (ω ) with sx2 + y 2 pairing symmetry is plotted with Γ = 0.095Δ 0 , T = 0.045Tc , and doping μ = 1.6 .. Finally, it is worth noting that the interband contribution to σ 1 (ω ) is proportional to Γ . The reason is similar to the normal state. We can obtain that from the following relation, σ 1inter (ω ) ∝. Γ ⎡⎣ E. Δ i. ( k + q 2 ) − E ( k − q 2 ) + ω ⎤⎦ + Γ 2 Δ j. 2. ,. (3.3.5). where i, j = +, − . Since ⎣⎡ EiΔ ( k + q 2 ) − E Δj ( k − q 2 ) ⎦⎤ >> ω , Γ for small ω , σ 1 (ω ) is proportional to Γ . Therefore, this result is true for both normal. and superconducting state..

(54) 49. Chapter 4 Penetration Depth. 4.1 Introduction In 1933, Meissner and Ochsenfeld observed that metals in the superconducting state are perfect diamagnets. That is, these metals expel magnetic fields completely when the temperature and the external magnetic field are not too large. After two years, F. London and H. London proposed a theoretical explanation of the Meissner effect. In their analysis, they gave the ordinary AC conductivity for an electron gas of density ns in the Drude model, as the relaxation time τ becomes large. That is, J (ω ) = σ (ω ) E (ω ) ,. σ (ω ) = i. (4.1.1). ns e2 , mω. (4.1.2). where ns is denoted as the density of superconducting electrons. Moreover, they also derived the relation between the penetration depth λ and the density of superconducting electrons ns . i.e., 1/ 2. ⎛ m ⎞ λ =⎜ . 2 ⎟ ⎝ 4π ns e ⎠. (4.1.3). According to Eq. (4.1.1), (4.1.2), and (4.1.3), one can obtain J (ω ) =. i. 1. ω 4πλ. 2. E (ω ) = −. 1 4πλ 2. A (ω ) .. (4.1.4). This is the well-known London Equation. Note that the last term in Eq. (4.1.4) is given by E (ω ) = iω A (ω ) . In the following section, we will use another method to derive the penetration depth..

(55) 50 4.2 Theory In this section, we will derive a general expression for the penetration depth in terms of the EM response function that we have obtained in Chapter 3. To begin, we recall the current density in the momentum space derived in Chapter 3. That is, j J ie ( q, ω ) = −e2 ∏ij ( q, ω ) A ext ( q, ω ) ,. (4.2.1). where ∏ij ( q, ω ) is the current-current response function which we have been obtained in Eq. (3.2.2.28). Moreover, ∏ij ( q, ω ) is composed of two parts: one is the paramagnetic term ∏ijp ( q, ω ) and the other is the diamagnetic term ∏ijd ( q ) . In other words, ∏ij ( q, ω ) = ∏ijp ( q, ω ) + ∏ijd ( q ) ,. (4.2.2). where the paramagnetic term is expressed as ∏ijp ( q, ω ) = CJRi (q )J j ( q) (ω ) 0. 0. (4.2.3). and the diamagnetic term is represented as ∏ ijd ( q ) =. 1 n ( q ) δ ij . m. (4.2.4). According to Ref.25, 27, the diamagnetic term in superconducting state is not different from the one in the normal state, because the diamagnetic is just proportional to the density of electrons. It is worth noting that the total current induced by a static field can be ignored. This result is reasonable, because the paramagnetic and diamagnetic terms almost cancel out for normal metals. In fact, Ref.23 has an explicit proof on this. Therefore, we can write the response function in the normal state as ∏ijn ( q, ω = 0 ) = ∏ijnp ( q, ω = 0 ) + ∏ijd ( q ) = 0.. (4.2.5).

(56) 51 Here we consider a system to which a static magnetic field is applied. As a result, ω = 0 in the above equation. Furthermore, the London theory points out that the current is of the form Je (q ) = −. 1 4πλ 2. A ext ( q ) ,. (4.2.6). where λ is the penetration depth. This equation is sometimes called London’s equation. Comparing Eq. (4.2.1) to Eq. (4.2.6), we obtain the relation between the penetration depth and the response function, 1. λ2. = 4π e 2 ∏ij ( q → 0, ω = 0 ) .. (4.2.7). Note that we take the limit q → 0 and ω = 0 in the above equation, since we are interested in a static external EM wave with a very long wavelength compared to the sample size. Next we consider the normal state and the superconducting state respectively. In the normal state, the penetration depth is infinite because of ∏ijn ( q → 0, ω = 0 ) = 0 . In the superconducting state, the penetration depth is given by 1. λ2. = 4π e 2 ∏ ijs ( q → 0, ω = 0 ) = 4π e 2 ⎡⎣ ∏ ijsp ( 0, 0 ) − ∏ ijnp ( 0, 0 ) ⎤⎦ . (4.2.8). We have used Eq. (4.2.5) in the second equality above. Therefore, we can evaluate the penetration depth in the superconducting state through calculating the paramagnetic response function in the normal state and the superconducting state. In the subsequent section, we will study further the penetration depth in superconducting state and show the result of the response function in detail..

(57) 52 4.3 Results In order to get penetration depths, we need to evaluate the paramagnetic term of the response function in normal state and superconducting state. In our case, we consider a system to which an external static magnetic field along the x-axis is applied. Thus, one can obtain the relation between the paramagnetic current and the correlation function by linear response theory. That is, J xp ( q, ω ) = −e2 ∏ xxp ( q, ω ) A x ( q, ω ) ,. (4.3.1). where ∏ xxp ( q, ω ) is the paramagnetic current-current correlation function. We first consider ∏ xxp in the imaginary time domain and then Fourier transform it into the Matsubara frequency domain, ∏ xxp ( q, iΩ ) = −. 1 N. ∫. β. 0. dτ eiΩτ J xp ( q,τ ) J xp ( −q, 0 ) .. (4.3.2). Then we can calculate ∏ xxp ( q, iΩ ) using Eq. (3.2.2.39) and take the limit ω = 0 due to the static field and q → 0 because of the long wavelength. approximation. Therefore, one can obtain ∏ npxx ( q → 0, ω = 0 ) in normal state, 2 ∏ ( q → 0, ω = 0 ) = N p xx. 2. ⎧⎪⎛ ∂E+ ⎞ ∂f ( E+ ) ⎛ ∂E− ⎞ ∂f ( E− ) +⎜ ⎟ ⎟ ∑k ⎨⎜ ⎝ ∂k x ⎠ ∂E− ⎩⎪⎝ ∂k x ⎠ ∂E+ 2. 2 8 ⎣⎡ f ( E+ ) − f ( E− ) ⎦⎤ ⎛ ∂ε − ∂ε xy ⎞ ⎫⎪ − ε− ∏ ( q → 0, ω = 0 ) + ⎜ ε xy ⎟ ⎬. 3 ∂k x ∂k x ⎠ ⎪ ( E+ − E− ) ⎝ ⎭ p xx. (4.3.3) Notice that the first two terms are intraband contribution and the last term is the interband contribution. Similarly, we can evaluate ∏ spxx ( q → 0, ω = 0 ) in the superconducting state. The result is.

(58) 53 ⎡⎛ ∂E ⎞ 2 ∂f ( E+Δ ) ⎛ ∂E ⎞ 2 ∂f ( E−Δ ) ⎤ ∑k 2 ⎢⎢⎜ ∂k + ⎟ ∂E Δ + ⎜ ∂k − ⎟ ∂E Δ ⎥⎥ + − ⎝ x⎠ ⎣⎝ x ⎠ ⎦ 2 2 ∂ε xy ⎞ ⎧⎪ ξ + E+ + Δ ⎛ ∂ε − 8 p Δ ∏ xx ( q → 0, ω = 0 ) + − ε− ε ⎟ × ⎨ ⎡ 2 f ( E+ ) − 1⎦⎤ 3 ⎜ xy ∂k x ⎠ ⎪⎩ ⎣ E+Δ ξ + ( E+ − E− ) ⎝ ∂k x 2 ξ + E− + Δ ⎫⎪ p Δ ⎡ ⎤ ∏ xx ( q → 0, ω = 0 ) − ⎣ 2 f ( E− ) − 1⎦ (4.3.4) ⎬, E−Δ ⎪⎭ 1 ∏ ( q → 0, ω = 0 ) = N p xx. where ξ+ ≡. ξx + ξ y 2. 2. = ε + − μ and E±Δ = E±2 + Δ .. (4.3.5). Here we consider Δ1 = Δ 2 = Δ . It is clear to see that the first two terms is similar to the first two terms in Eq. (4.3.3) and there are extra terms for the interband contribution. One can easily verify that Eq. (4.3.4) reduce to Eq. (4.3.3) when Δ is set to zero. After obtaining ∏ npxx ( q → 0, ω = 0 ) and ∏ spxx ( q → 0, ω = 0 ) , we can evaluate the penetration depth by Eq. (4.2.8). That is, 1. λ (T ). 2. = 4π e 2 ⎡⎣∏ ijsp ( 0, 0 ) − ∏ ijnp ( 0, 0 ) ⎤⎦ .. Now we plot the penetration depth. λ 2 ( 0 ) λ 2 (T ). (4.3.6) versus the. temperature for different pairing gaps (see Fig. 4.3.2). We also plot the frequency dependence of the density of states (DOS) at T = 0.005Tc (see Fig. 4.3.1). One sees clearly that s0 and sx y pairings are fully gapped 2 2. on the Fermi surface at low temperature as choosing μ = 1.6 , while d x − y , 2. d xy , and sx2 + y 2. 2. have nodes on the Fermi surface. Furthermore, as. mentioned for the nodal structure of these pairing gaps in Chapter 2, we have known that sx + y exhibits nodes on the electron Fermi pockets, 2. 2.