*3.2. The Upper Critical Field Parallel to the ab-Plane*

In this subsection, we will study the nucleation of superconductivity with the magnetic field **H** applied in the *ab*-plane. We set **H** = (0, *H*, 0) and take **A** = (0, 0, −*Hx*). Similarly, we look for a solution with the form (15). Close to the upper critical field we can also obtain the linearized GL Equations (16) and (17), but the diagonal element of the *M*ˆ -matrix changes into

$$\hat{M}\_{\text{ii}} = -\frac{\hbar^2}{2m\_i} \frac{d^2}{dx^2} + \frac{1}{2} m\_i \omega\_i'^2 \left(\mathbf{x} + \mathbf{x}\_0'\right)^2 - \mathbf{a}\_{i\prime} \tag{28}$$

where *ω <sup>i</sup>* = 2*eH*/*c* <sup>√</sup>*mimiz* and *<sup>x</sup>* <sup>0</sup> = *hck* ¯ *<sup>z</sup>*/2*eH*.

If *η*<sup>12</sup> = 0, analogous to the analysis of the last subsection, the harmonic oscillator eigenvalues are

$$E'\_{i,n} = (n+1/2)\hbar\omega'\_i - a\_i. \qquad (n = 0, 1, 2, \dots) \tag{29}$$

If *η*<sup>12</sup> = 0, we cannot get an exact result due to the mixing between the minimum and higher-level eigenfunctions. We thus follow a variational approach. We look for a solution in the form

$$f\_1(\mathbf{x}) = c\_1 g\_1(\mathbf{x}) = c\_1 \left(\frac{b\_1}{\pi}\right)^{1/4} e^{-b\_1 \mathbf{x}^2/2} \tag{30}$$

and

$$f\_2(\mathbf{x}) = c\_2 \mathbf{g}\_2(\mathbf{x}) = c\_2 \left(\frac{b\_2}{\pi}\right)^{1/4} e^{-b\_2 \mathbf{x}^2/2},\tag{31}$$

with *b*<sup>1</sup> and *b*<sup>2</sup> the variational parameters. Introducing *Dij*<sup>=</sup> *gi*|*M*<sup>ˆ</sup> *ij*|*gj*, detailed calculations give

$$D\_{11} = \frac{\hbar^2 b\_1}{4m\_1} + \frac{e^2 H^2}{m\_{1z} c^2 b\_1} - a\_{1\prime} \tag{32}$$

$$D\_{22} = \frac{\hbar^2 b\_2}{4m\_2} + \frac{e^2 H^2}{m\_{2z}c^2 b\_2} - a\_2 \tag{33}$$

and

$$D\_{12} = D\_{21} = \eta\_{12} (b\_1 b\_2)^{1/4} \left(\frac{2}{b\_1 + b\_2}\right)^{1/2}.\tag{34}$$

Then we can transform Equations (16), (17) and (28) into

$$D\_{11}c\_1 + D\_{12}c\_2 = \varepsilon' c\_1 = 0\tag{35}$$

and

$$D\_{21}c\_1 + D\_{22}c\_2 = \mathfrak{e}'c\_2 = 0.\tag{36}$$

Let *ε* denote the eigenvalue of the *D*-matrix. The upper critical field corresponds to the minimum eigenvalue *ε min* = 0, and it is available from Equations (35) and (36) as

$$
\varepsilon'\_{\min} = \frac{1}{2} \left[ \left( D\_{11} + D\_{22} \right) - \sqrt{\left( D\_{11} - D\_{22} \right)^2 + 4D\_{12}D\_{21}} \right]. \tag{37}
$$

Minimizing *ε min* with respect to *b*<sup>1</sup> and *b*<sup>2</sup>

$$
\frac{\partial \epsilon'\_{\min}}{\partial b\_1} = 0 \qquad \text{and} \qquad \frac{\partial \epsilon'\_{\min}}{\partial b\_2} = 0,\tag{38}
$$

and combining with

$$
\varepsilon'\_{\min} \left( H^{\parallel ab}\_{c2}, b\_1, b\_2 \right) = 0,\tag{39}
$$

we can obtain the upper critical field *Hab <sup>c</sup>*<sup>2</sup> at an arbitrary temperature.

We choose the GL parameters *m*1*<sup>z</sup>* = 38*me* and *m*2*<sup>z</sup>* = 0.31*me* to fit the experimental data. By numerically solving three nonlinear Equations (38) and (39), we plot the theoretical result of *Hab <sup>c</sup>*<sup>2</sup> as the dotted line in Figure 1. Note that our calculation is in agreement with the experimental measurement of *Hab <sup>c</sup>*<sup>2</sup> in temperature down to 0.2*Tc*.

## **4. Calculation on the Magnetic Penetration Depth of CsV**3**Sb**<sup>5</sup>

Now we begin to calculate the magnetic penetration depth for this two-band superconductor. In the presence of the weak fields, the solution takes the form [39]

$$\Psi\_i(\mathbf{r}) = |\Psi\_i|e^{i\varphi\_i(\mathbf{r})},\tag{40}$$

where |Ψ*i*| is constant. If the external field is applied parallel to the *c*-axis, we can set **A** = - *Ax*, *Ay*, 0 , and without loss of generality we consider the phase factor *ϕ<sup>i</sup>* as a function of *x* and *y*. From Equation (8), we get

$$\frac{\nabla \times \mathbf{H}}{4\pi} = \frac{2e\hbar}{m\_1c} |\Psi\_1|^2 \left(\nabla\varphi\_1 - \frac{2e}{\hbar c}\mathbf{A}\right) + \frac{2e\hbar}{m\_2c} |\Psi\_2|^2 \left(\nabla\varphi\_2 - \frac{2e}{\hbar c}\mathbf{A}\right). \tag{41}$$

Then following the standard procedure in Ref. [39], we can rewrite Equation (41) as

$$
\nabla^2 \mathbf{H} - \left(\frac{16\pi c^2}{m\_1 c^2} |\Psi\_1|^2 + \frac{16\pi c^2}{m\_2 c^2} |\Psi\_2|^2\right) \mathbf{H} = 0. \tag{42}
$$

Therefore, we can obtain the magnetic penetration depth in the *ab*-plane as

$$
\lambda\_{ab} = \left(\frac{m\_1 m\_2 c^2}{16\pi m\_2 e^2 |\Psi\_1|^2 + 16\pi m\_1 e^2 |\Psi\_2|^2}\right)^{1/2}.\tag{43}
$$

Similarly, the magnetic penetration depth along the *c*-axis is given by

$$
\lambda\_c = \left(\frac{m\_{1z}m\_{2z}c^2}{16\pi m\_{2z}c^2|\Psi\_1|^2 + 16\pi m\_{1z}c^2|\Psi\_2|^2}\right)^{1/2}.\tag{44}
$$

We take *<sup>β</sup>*<sup>1</sup> = 1.3 × <sup>10</sup>−<sup>2</sup> meV·μm3 to fit the experimental data on the magnetic penetration depth. First of all, |Ψ1| and |Ψ2| as function of temperature can be numerically obtained from Equations (12) and (13). Then from Equations (43) and (44), we plot *λab* and *λ<sup>c</sup>* as function of temperature in Figure 2. From Figure 2, we can see that our theoretical calculation can fit the experimental data well almost in the whole temperature range.

**Figure 2.** Magnetic penetration depth along the *c*-axis (solid line) and in the *ab*-plane (dotted line) as function of temperature. The experimental data are from Ref. [43].

At this stage, we would also like to point out that Gupta et al. also tried to fit the experimental data of the magnetic penetration depth with the *d*-wave model [43]. However, compared with the two-gap *s*-wave model, the *d*-wave model does not describe the data well, which provides further evidence for the nodeless structure in this compound.

#### **5. KW Ratio and the Semi-Heavy–Fermion System**

In this section, we would like to discuss the KW ratio and semi-heavy-fermion feature in the compound CsV3Sb5. Since the discovery by Steglich et al. of superconductivity in the high-effective-mass (∼100*me*) electrons in CeCu2Si2, the search for and characterization of such heavy-fermion systems has been a rapidly growing field of study [44]. In a Fermi liquid, the electronic contribution to the heat capacity has a linear temperature dependence *Cel*(*T*) = *γT*, and at low temperatures the resistivity varies as *ρ*(*T*) = *ρ*<sup>0</sup> + *AT*2. This is observed experimentally when electron–electron scattering, which gives rise to the quadratic term, dominates over electron–phonon scattering in the process. In a number of typical transition metals, we have *<sup>A</sup>*/*γ*<sup>2</sup> ≈ 0.09 × <sup>10</sup>−<sup>5</sup> <sup>μ</sup><sup>Ω</sup> cm mol<sup>2</sup> K2 mJ−<sup>2</sup> even though *γ*<sup>2</sup> varies by an order of magnitude across the materials studied. Meanwhile, it was found in many heavy-fermion compounds *<sup>A</sup>*/*γ*<sup>2</sup> reaches 1.0 × <sup>10</sup>−<sup>5</sup> <sup>μ</sup><sup>Ω</sup> cm mol<sup>2</sup> K2 mJ−<sup>2</sup> despite the large mass renormalization. Because of this remarkable behavior *A*/*γ*<sup>2</sup> has become known as the KW ratio, and large value of this ratio is treated as a robust signature of heavy-fermion systems [45,46].

With the effective masses in the *ab*-plane (*m*<sup>1</sup> = 0.55*me*, *m*<sup>2</sup> = 0.13*me*) [36] and those along the *c*-direction (*m*1*<sup>z</sup>* = 38*me*, *m*2*<sup>z</sup>* = 0.31*me*) from the two-band GL theory, we can expect that the first band in CsV3Sb5 will show the heavy-fermion properties, while the other band can be treated as the normal metal. Meanwhile, for this kagome crystal we have resistivity coefficient *<sup>A</sup>* = 2.3 × <sup>10</sup>−<sup>3</sup> <sup>μ</sup><sup>Ω</sup> cm K−<sup>2</sup> from the electronic transport measurement and Sommerfeld factor *γ* = 20 mJ mol−<sup>1</sup> K−<sup>2</sup> from the specific heat data [32]. It is thus reasonable that CsV3Sb5, as a semi-heavy-fermion compound, possesses a medium KW ratio *<sup>A</sup>*/*γ*<sup>2</sup> ≈ 0.58 × <sup>10</sup>−<sup>5</sup> <sup>μ</sup><sup>Ω</sup> cm mol<sup>2</sup> K2 mJ−<sup>2</sup> between 0.09 × <sup>10</sup>−<sup>5</sup> and 1.0 × <sup>10</sup>−<sup>5</sup> <sup>μ</sup><sup>Ω</sup> cm mol2 K2 mJ<sup>−</sup>2.

#### **6. Conclusions**

In summary, based on the two-band anisotropic GL theory, we studied the temperature dependence of upper critical field and magnetic penetration depth for the kagome superconductor CsV3Sb5. Our theoretical results fit the experimental data in a broad temperature range, pointing to the existence of two-gap *s*-wave superconductivity in this system. From the large anisotropy of effective masses in the first band, we also suggest that CsV3Sb5 is a semi-heavy-fermion compound. The possible mechanism of the semi-heavy-fermion state and other problems for these kinds of materials are reserved for further investigations.

**Author Contributions:** Conceptualization, T.H., J.C., C.Y. and H.H.; methodology, T.H., J.C., C.Y. and H.H.; software, T.H.; validation, T.H., J.C., C.Y. and H.H.; formal analysis, T.H., J.C., C.Y. and H.H.; investigation, T.H., J.C., C.Y. and H.H.; resources, H.H.; data curation, T.H., J.C., C.Y. and H.H.; writing—original draft preparation, T.H.; writing—review and editing, T.H., J.C., C.Y. and H.H.; visualization, T.H.; supervision, J.C., C.Y. and H.H.; project administration, H.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
