Results

In previous section, we have discussed how to evaluate the real

conductivity σ ω1

( )

by using the current-current response function. Here we first calculate σ ω1

( )

)

in the normal state. There are two parts of the contribution to σ ω1

(

. One is the intraband contribution, whose results should behave like Drude conductivity at low frequencies. The other is the interband contribution to which σ ω1

( )

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 ω1

( )

. In Fig. 3.3.1, one sees these results clearly. It is worth noting that the width of the Drude-like peak in σ ω1

( )

becomes larger as the scattering rate increases. Moreover, for intraband contribution, σ1^intra

( )

ω increases as Γ decreases. We can see this result clearly from the following asymptotic low ω behavior,

( ) ( ) ( )

small ω. Based on a similar analysis , one sees that the interband contribution, ^σ1^inter

(

^ω

)

, is proportional to Γ for small ω. At small ω, proportional to Γ for small ω. Furthermore, one sees that the curve of

1

( )

σ ω tends to approach some value as ω→0. In fact, σ ω1

(

⁼⁰

)

is the

0.0 0.5 1.0 1.5 2.0 2.5 3.0

direct current (DC) conductivity.

T=0.045T_c; Γ=0.200Δ₀

Fig. 3.3.1 The real part of the optical conductivity, σ ω1

( )

, is plotted for different scattering rate, (a) Γ =0.200Δ₀ and (b) Γ =0.095Δ₀

There is an important sum rule which describes that the area under the 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,

( ) ( )

where the superscripts and n denote the superconducting and normal states respectively.

Fig. 3.3.2 We compare the real part of the optical conductivity, σ ω1

( )

0.3

According to Eq. (3.3.4), the area under the curve of σ ω1

( )

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 ω1

( )

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. indicates the maximum frequency of absorption. Moreover, is the Fermi velocity and is the wave number. In (a), the wavelength is so long that the maximum absorption frequency is smaller than the energy gap threshold . 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)

v q0

v0 q

ωg

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 ω1

( )

later. One is intraband contribution to σ ω1

( )

and the other is interband contribution to

0.0 0.5 1.0 2.0 2.5 3.0

σ ω . Moreover, we don’t show the London current at zero frequency here. Then we plot the case of the 2 2

x y

d ₋ pairing symmetry to illustrate this phenomenon in our case, (see Fig. 3.3.4).

ω/Δ₀ the sum rule holds in both normal and superconducting states. We choose ,

, and the doping 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

2Δ0

. For nodal order parameters which have nodes on the Fermi surface, there are low lying states such that quasiparticles can be excited below

. In Fig. 3.3.6 and Fig. 3.3.7, we plot the curves of σ ω1

( )

for nodal the hole Fermi pockets, while d_xy has nodes on both hole Fermi pockets

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< Δ₀ for nodeless gap. In Fig.

3.3.5, DOS is linear around ω=0. Therefore, we confirm that 2 2

x y

d ₋ and

dxy are nodal gaps.

-1.0 0 1.0

w/Δ

DOS

0

d_x²_-y² d_xy

Fig. 3.3.5 We plot DOS at μ =1.6 for pairing gaps, 2 2

x y

d ₋ and . Here we choose . One can see clearly that DOS is linear near

dxy 0 0.3 0

Δ = ω= for pairing

symmetry 2 2

x y

d ₋ or d_xy. Note that ω= is the Fermi level. 0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Next we consider the nodeless pairing symmetry. Recall that is a fully constant gap and

s0 symmetry, we find that the intraband contribution is zero below

s0

ω= Δ0, while it will dominate at ω= Δ2 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 ω1

( )

even in the superconducting state at low frequency.

Then Fig. 3.3.8 (b) shows that 2 2

sx y pairing gap is a nodeless gap as choosing μ =1.6. That is, for 2 2

sx y pairing symmetry, although it has 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 2 2

sx y pairing symmetry are near the Fermi surface. Hence, it implies that we only need energy less than 2Δ₀ to excite a quasiparticle from the normal state. (See Fig. 3.3.8 (b) and Fig.

3.3.10)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 spectrum is dim) with clear nodeless structure.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

μ =1.6. In the inset, we show the intraband only (though the spectrum is dim) with clear nodeless structure.

total interband intraband

T=0.045T_c Γ=0.095Δ₀

Now we consider the pairing symmetry 2 2

sx y with different dopings.

In Fig. 3.3.11, we can see how the Fermi surface varies as increasing doping. Moreover, the 2 2

sx y pairing symmetry only has nodes on the 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< Δ₀. That is, 2 2

sx y becomes a nodal gap as a doping exceeds μ =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.

-1.5 -1.0 -0.5 0 0.5 1.0 1.5 w/Δ₀

DOS

μ^=1.6 μ^=1.8 μ^=2.0 μ^=2.2 μ^=2.4

Fig. 3.3.12 Density of states (DOS) with different doping is plotted. We can see clearly that sx y2 2 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

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

2Δ0

1

( )

σ ω near ω=0, we plot σ ω1

( )

in the interval ω Δ ∈0

[ ]

0,1 . (See Fig. 3.3.14(a)) Then we find that σ ω1

( )

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.

0.25

Then let’s consider the 2 2

x y

s ₊ pairing symmetry. Recall Fig. 2.3.1(f) which indicates that 2 2

x y

s ₊ pairing symmetry has nodes on the electron Fermi pockets and fully gap on the hole Fermi pockets. In Fig. 3.3.15, we also see that the 2 2

x y

s ₊

)

pairing symmetry has nodes on the Fermi surface.

Therefore, σ ω1

(

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, 2 2

x+y

s pairing symmetry still has similar behavior at 2 0

ω = Δ . (See Fig. 3.3.16)

-1.25 -1.00 -0.75 -0.50 -0.25 0 0.25 0.50 0.75 1.00 1.25 w/Δ₀

DOS

Fig. 3.3.15 We plot DOS for the pairing symmetrysx2₊y2. We can see that the pairing symmetry sx2₊y2 has nodes on the Fermi surface.

Finally, it is worth noting that the interband contribution to σ ω1

( )

is proportional to . The reason is similar to the normal state. We can obtain that from the following relation,

Γ

σ ω is proportional to . Therefore, this result is true for both normal and superconducting state.

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 n_s in the Drude model, as the relaxation time τ becomes large. where n_s 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 n_s. i.e.,

This is the well-known London Equation. Note that the last term in Eq.

(4.1.4) is given by ^E

( )

^{ω =iω ωA}

( )

. In the following section, we will use another method to derive the penetration depth.

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,

(

^,

)

²

(

^,

) (

^,

)

^, where the paramagnetic term is expressed as

( )

_{( ) ( )}

( )

and the diamagnetic term is represented as

( )

¹

( )

^.

d

ij n ij

m δ

∏ q = q (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

∏ⁿij

(

q^,ω =⁰

)

= ∏^npij

(

q^,ω=⁰

)

+ ∏ij^d

( )

q =0. (4.2.5)

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 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,

2 ²

(

1 4πe _ij 0,

λ ^{= ∏ q→ 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 . In the superconducting state, the penetration depth is given by

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.

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,

(

^,

)

²

(

^,

) ( )

where ∏ qxx^p

(

^,ω is the paramagnetic current-current correlation function.

We first consider in the imaginary time domain and then Fourier transform it into the Matsubara frequency domain,

∏_xxp approximation. Therefore, one can obtain

→0 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

( ) ( ) ( )

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 is set to zero.

1 2

Δ = Δ = Δ

Δ

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,

temperature for different pairing gaps (see Fig. 4.3.2). We also plot the frequency dependence of the density of states (DOS) at (see Fig. 4.3.1). One sees clearly that and

s ₊ have nodes on the Fermi surface. Furthermore, as mentioned for the nodal structure of these pairing gaps in Chapter 2, we have known that 2 2

x y

s ₊ exhibits nodes on the electron Fermi pockets,

while 2 2

x y

d ₋ exhibits nodes on the hole Fermi pockets, and d_xy exhibits nodes on both the hole and electron Fermi pockets. One can refer to Fig.

2.3.1 for details.

Note that the temperature dependence of the penetration depth is predicted to be different for different pairing gaps. For instance, for a local clean-limit BCS superconductor whose pairing gap is fully gapped over the Fermi surface, ^⎡_⎣^λ

( ) ( )

^{T λ 0 −1⎤}_⎦ should decrease to zero exponentially as T^{−1 2}e^{−Δ( )0 kT} as . This result was pointed out by Ref.²⁸. On the other hand, when pairing gaps exhibit nodes on the Fermi surface,

T →0

( ) ( )

^{T 0 1}

λ λ −

⎡⎣ _⎤⎦ should go to zero as a power low

(

T T_c

)

ⁿ, where n depends on if the nodes are points or lines, etc.²⁷ In particular,

( ) ( )

⁰

λ λ −

⎡⎣ T 1⎤⎦ is proportional to T T_c at low temperature for the line nodes. Thus a power law temperature dependence of ^λ

( )

^T at the lowest temperatures results from vanishing of the order parameter on the Fermi surface or strong impurity effects for conventional s-wave gapless superconductivity. It is worth noting that ⎡⎣λ

( ) ( )

T λ ^{0 −1}⎤⎦αT Tc implies

Here we make the first order approximation. Then, in Fig. 4.3.2, we can see clearly that ^λ2

( )

^{0 λ2}

( )

^T decay exponentially for and s₀ 2 2

sx y cases due to the nodeless order parameter. Moreover, the nodal order parameters exhibit linear dependence at low temperature. T

-2.0 -1.0 0.0 1.0 2.0 w/Δ₀

DOS

d_x2 -y² d_xy s_x2+y² s_x2

y² s₀

Fig. 4.3.1 We plot the density of states (DOS) at μ =1.6 for different order parameters. ω= represents the Fermi level. Note that all of these cases are 0 symmetric about ω= . 0

Fig. 4.3.2 Penetration depth λ²

( )

⁰ λ²

( )

^T is plotted for different pairing gaps at doping μ=1.6 and gap size Δ =₀ 0.1. The temperature dependence of is chosen to be

( )

^T

Δ

( )

^{T Δ0tanh}

(

^{π 2 Tc T −1}

)

Δ = as in Ref.²⁹. Note that the

penetration depths of the dx2₋y2, d_xy, and sx2₊y2 pairing symmetries have linear temperature dependence at low temperature, as for nodal order parameters. While

and

s0 sx y2 2 pairing symmetries have exponentially decay behaviors. Moreover, all cases should decrease to zero as T = . T_c

0 0.06 0.12 0.18 0.24 0.30 0.36 0.42 0.48 0.54 0.60

0 0.2 0.4 0.6 0.8 1

T/T_c λ2 (0)/λ2 (T)

d_x2-y²

d_xy s_x2

y²

s_x2+y²

s₀

Chapter 5 Conclusion

In summary, we used the two-orbital model to study the optical conductivity and penetration depth in this thesis. By using the linear response theory, we have studied the spectral function, the DOS, the real part of the optical conductivity σ ω1

( )

, and the penetration depth λ in both the normal and the superconducting states with different pairing symmetries. We have obtained that the curves of σ ω1

( )

have a strong peak around ω=0 for nodal gaps for the intraband contribution. For nodeless gaps, the threshold for the intraband absorption is at ω= Δ2 ₀; below ω = Δ2 ₀, ^{σ ω}₁

( )

is negligibly small. The interband contribution, however, dominates in many cases! Moreover, σ ω1

( )

for the interband contribution is proportional to Γ for both normal and superconducting state at small ω . Then we have found that the penetration depth

( ) ( )

2 2

λ 0 λ T behaves exponentially down to zero temperature for nodeless gaps, while ^λ2

( )

^{0 λ2}

( )

^T has linear temperature dependence at very low temperature.

In this thesis, we have considered only the case, Δ = Δ = Δ_{1 2} . However, some experiments^29-31 claimed that some superconductors have two different gaps, . Therefore, we can extend our study for the two-gap superconductors in the future. We do not considered the impurity scattering in

Δ ≠ Δ1

1

( )

2

σ ω in a self-consistent theory, but using only a phenomenological damping rate Γ. A detailed self-consistent impurity study is important to figure out how the σ ω1

( )

behaves quantitatively in

the real materials. Finally, from Eq. (3.2.2.33) and Eq. (4.1.4), one can obtain the result,

( )

₂ ¹ ₂,

σ ω 4 (5.1)

=ω πλ

where σ ω2

( )

is the imaginary part of the optical conductivity and λ is the penetration depth. We shall study the relation between σ ω2

( )

and λ in the near future.

References:

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2 H. Takahashi, K. Igawa, K. Arii, Y. Kamihara, M. Hirano, and H. Hosono, Nature 453, 376 (2008).

3 J. Dong, et al., EPL 83, 27006 (2008).

L. Wang, Phys. Rev. Lett. 100, 247002 (2008).

8 Z.-A. Ren, et al., EPL 82, 57002 (2008).

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Takata, A. L. Shluger, and H. Hosono, Supercond. Sci. Technol. 21, 125028 (2008). X. H. Chen, and C. Bernhard, Phys. Rev. Lett. 101, 097011 (2008).

15 J. Yang, D. Huvonen, U. Nagel, T. Room, N. Ni, P. C. Canfield, S. L. Bud'ko, J.

P. Carbotte, and T. Timusk, Phys. Rev. Lett. 102, 187003 (2009).

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28 J. Halbritter, Z. Phys. 243, 201 (1971).

29 D. V. Evtushinsky, et al., New J. Phys. 11, 055069 (2009).

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