**1. Introduction**

The recently discovered kagome metal series AV3Sb5 (A = K, Rb, Cs) exhibit topologically nontrivial band structures, chiral charge order, charge density wave (CDW) and superconductivity, presenting a unique platform for realizing exotic electronic states [1–6]. These materials crystallize in the *P*6/*mmm* space group with ideal kagome nets of V atoms which are coordinated by Sb atoms. The kagome layers of CsV3Sb5 are sandwiched by extra antimonene layers and Cs layers, as shown in the inset of Figure 1. With the decrease in temperature, the CDW phase transition takes place at about 94 K for CsV3Sb5 [7–11]. Based on the density functional theory, the first principle calculations show that the CDW observed in this family of compounds is a consequence of the atomic displacement from the high-symmetry positions of the kagome network [12,13]. Meanwhile, the experimental measurements with Raman spectroscopy have also confirmed the dynamical lattice distortions in the CDW phase [14–16]. Then, below about 2.5 K superconductivity is observed and coexists with the CDW order without further structural transitions [17–23].

For a series of hexagonal symmetric layered materials, the electronic band structure and topological properties have already been carried out based on the numerical ab initio calculations [24,25]. Furthermore, it is also well known that the two-dimensional kagome lattice hosts a pair of Dirac bands protected from the lattice symmetry and will trigger the correlated topological states of matter. For the three-dimensional crystal CsV3Sb5,V3*d* and Sb 5*p* orbitals play dominant contributions to the density of states near the Fermi level, and the nontrivial band crossing is extended along a one-dimensional line in the Brillouin zone. Such band structure features are associated with a Z<sup>2</sup> topological index [2,26], and the topological surface states can be easily observed if a direct gap exists for every momentum in this system. Additionally, electronic transport and heat capacity measurements reveal a large Kadowaki–Woods (KW) ratio. It indicates the V-based kagome prototype structure may be of potential interest as a host of correlated electron phenomenon, particularly as a heavy-fermion material.

**Citation:** Han, T.; Che, J.; Ye, C.; Huang, H. Ginzburg–Landau Analysis on the Physical Properties of the Kagome Superconductor CsV3Sb5. *Crystals* **2023**, *13*, 321. https://doi.org/10.3390/ cryst13020321

Academic Editor: Jacek Cwik ´

Received: 21 January 2023 Revised: 10 February 2023 Accepted: 12 February 2023 Published: 15 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Up to now, several theoretical and experimental investigations have already been performed on the pairing symmetry of the kagome compound CsV3Sb5. Multiband structure of this compound was predicted by previous theoretical calculations and then confirmed by angle-resolved photoemission spectroscopy studies [27–31]. Meanwhile, the measurement of nuclear magnetic resonance on this kagome metal showed a Hebel–Slichter coherence peak just below *Tc*, indicating that CsV3Sb5 is an *s*-wave superconductor [11]. Magnetic penetration depth of this system measured by tunneling diode oscillator displayed a clear exponential behavior at low temperatures, which also provides evidence for the nodeless structure in this compound. Furthermore, experimental data on temperature dependence of the superfluid density and electronic specific heat can be well described by two-gap superconductivity scenario [32], which is consistent with the presence of multiple Fermi surfaces in this system. Obviously, to date there is still no general consensus on the form of superconducting order parameter in CsV3Sb5 and further explorations to elucidate this issue are necessary.

The main motivation of the present paper is to identify the form of order parameter in this kagome superconductor. Based on the two-band Ginzburg–Landau (GL) theory, we study the temperature dependence of upper critical field and magnetic penetration depth for this compound. Our results can fit the experimental data well in a broad temperature range, which thus strongly suggests CsV3Sb5 as a two-gap *s*-wave superconductor. We can also obtain the effective mass of the electron in the *c*-axis for the first band as 38*me* and only 0.31*me* for the other band. With this semi-heavy-fermion feature, we can qualitatively understand the experimental value of the KW ratio in CsV3Sb5.

The paper is organized as follows: In the next section, we discuss the two-band GL theory. We derive the formula for the critical temperature and discuss how to properly choose the parameters in the GL theory. In Section 3, we calculate the upper critical field *Hc*<sup>2</sup> for the kagome superconductor CsV3Sb5. Then in Section 4, we work out the magnetic penetration depth for this compound. In Section 5, we discuss the KW ratio and semi-heavy-fermion feature in this material. Finally, Section 6 contains the conclusion of the paper.

#### **2. Two-Band Anisotropic Ginzburg–Landau Theory**

Taking into account the multi-gap characteristics of V-based superconductors, we can note the two-band GL free energy functional as [33–35].

$$F = \int d^3 \mathbf{r} (f\_1 + f\_2 + f\_{12} + \mathbf{H}^2 / 8\pi),\tag{1}$$

with

$$\begin{split} f\_{i} &= \frac{\hbar^{2}}{2m\_{i}} \left| \left( \partial\_{x} - \frac{2ieA\_{x}}{\hbar c} \right) \Psi\_{i} \right|^{2} + \frac{\hbar^{2}}{2m\_{i}} \left| \left( \partial\_{y} - \frac{2ieA\_{y}}{\hbar c} \right) \Psi\_{i} \right|^{2} + \frac{\hbar^{2}}{2m\_{iz}} \left| \left( \partial\_{z} - \frac{2ieA\_{z}}{\hbar c} \right) \Psi\_{i} \right|^{2} \\ &- a\_{i}(T) |\Psi\_{i}|^{2} + \frac{\beta\_{i}}{2} |\Psi\_{i}|^{4} \end{split} \tag{2}$$

and

$$f\_{12} = \eta\_{12} (\Psi\_1^\* \Psi\_2 + \text{c.c.}).\tag{3}$$

Here, *fi* (*i* = 1, 2) is the free energy density for each band and *f*<sup>12</sup> is the interactionfree energy density. <sup>Ψ</sup>*<sup>i</sup>* <sup>∝</sup> <sup>√</sup>*Ni*Δ*<sup>i</sup>* with *Ni* the density of states at the Fermi level is the superconducting order parameter and *N*1/*N*<sup>2</sup> = 0.79/0.21 from the specific heat data in CsV3Sb5 [32]. *mi* and *miz* denote the effective masses in the *ab*-plane and in the *c*-direction for band *i*. From the measurement of Shubnikov–de Haas oscillations with the magnetic field parallel to the *c*-direction, we have *m*<sup>1</sup> = 0.55*me* and *m*<sup>2</sup> = 0.13*me* [36]. *η*<sup>12</sup> is the Josephson coupling constant. The coefficient *α<sup>i</sup>* is a function of temperature, while *β<sup>i</sup>* is independent of temperature. If the interband interaction is neglected, the functional can be reduced to two independent single-band problems with the corresponding critical temperatures *Tc*<sup>1</sup> and *Tc*2, respectively. Thus, the parameters *α*<sup>1</sup> and *α*<sup>2</sup> can be approximately

expressed as *α<sup>i</sup>* = *αi*0(1 − *T*/*Tci*) with *αi*<sup>0</sup> the proportionality constant [37]. **H** = ∇ × **A** is magnetic field and **A** = (*Ax*, *Ay*, *Az*) is the vector potential.

By minimizing the free energy *F* with Ψ∗ *<sup>i</sup>* , we can obtain the GL equations for the description of the two-band superconductivity

$$
\hat{M}\_{11}\Psi\_1 + \hat{M}\_{12}\Psi\_2 = 0\tag{4}
$$

and

$$
\hat{\mathcal{M}}\_{21}\Psi\_1 + \hat{\mathcal{M}}\_{22}\Psi\_2 = 0,\tag{5}
$$

with

$$\begin{split} \mathcal{M}\_{\vec{n}\vec{i}} &= -\frac{\hbar^2}{2m\_i} \left( \partial\_x - \frac{2ieA\_x}{\hbar c} \right)^2 - \frac{\hbar^2}{2m\_i} \left( \partial\_y - \frac{2ieA\_y}{\hbar c} \right)^2 - \frac{\hbar^2}{2m\_{i\vec{z}}} \left( \partial\_z - \frac{2ieA\_z}{\hbar c} \right)^2 \\ &- a\_{\vec{i}} + \beta\_{\vec{i}} |\Psi\_{\vec{i}}|^2 \end{split} \tag{6}$$

and

$$
\hat{\mathcal{M}}\_{12} = \hat{\mathcal{M}}\_{21} = \eta\_{12}.\tag{7}
$$

By minimizing the free energy *F* with the vector potential **A**, we then obtain the equation for the current **j** = - *jx*, *jy*, *jz* as

$$
\nabla \times \mathbf{H} = 4\pi \mathbf{j} \tag{8}
$$

with

$$j\_x = \frac{e}{\text{i}c} \sum\_i \left(\frac{\hbar}{m\_i} \Psi\_i^\* \partial\_x \Psi\_i - \frac{\hbar}{m\_i} \Psi\_i \partial\_x \Psi\_i^\* - \frac{4\text{i}c}{m\_i c} \Psi\_i^\* \Psi\_i A\_x\right),\tag{9}$$

$$j\_y = \frac{e}{\text{i}c} \sum\_i \left( \frac{\hbar}{m\_i} \Psi\_i^\* \partial\_y \Psi\_i - \frac{\hbar}{m\_i} \Psi\_i \partial\_y \Psi\_i^\* - \frac{4\text{i}c}{m\_i c} \Psi\_i^\* \Psi\_i A\_y \right) \tag{10}$$

and

$$\dot{y}\_z = \frac{c}{\text{i}\varepsilon} \sum\_i \left( \frac{\hbar}{m\_{iz}} \Psi\_i^\* \partial\_z \Psi\_i - \frac{\hbar}{m\_{iz}} \Psi\_i \partial\_z \Psi\_i^\* - \frac{4\text{i}\varepsilon}{m\_{iz}c} \Psi\_i^\* \Psi\_i A\_z \right). \tag{11}$$

Equations (4), (5) and (8) are the fundamental GL equations for the two-gap superconductors. In the absence of fields and gradients, Equations (4) and (5) give

$$\left(-\varkappa\_1 + \beta\_1 |\Psi\_1|^2\right)\Psi\_1 + \eta\_{12}\Psi\_2 = 0\tag{12}$$

and

$$
\eta\_{12} \Psi\_1 + \left( -\alpha\_2 + \beta\_2 |\Psi\_2|^2 \right) \Psi\_2 = 0. \tag{13}
$$

$$\text{At } T \to T\_{c\prime} \text{ we get}$$

$$
\alpha\_1(T\_\mathfrak{c})\alpha\_2(T\_\mathfrak{c}) = \eta\_{12}^2. \tag{14}
$$

With *Tc* = 2.5 K for CsV3Sb5, we can easily get *η*<sup>12</sup> from the equation above.

In principle, we can derive the parameters in our GL theory from microscopic twoband BCS theory. Following Ref. [35], if we compare the microscopic forms of *α<sup>i</sup>* and *βi*, we can obtain two useful relations *α*10/*α*<sup>20</sup> = *Tc*1/*Tc*<sup>2</sup> and *β*<sup>1</sup> = *β*2. Since two superconducting gaps appear at 1.6*kBTc* and 0.63*kBTc* [32], we can approximate *Tc*1/*Tc*<sup>2</sup> as 1.6/0.63 ≈ 2.5. In addition, it has been proven that the ratio of energy gaps at zero temperature is equal to that at critical temperature [38], and according to Equations (12)–(14) the ratio of energy gaps at *Tc* can be written as | - Ψ1/ <sup>√</sup>*N*<sup>1</sup> / - Ψ2/ <sup>√</sup>*N*<sup>2</sup> | = (*N*2/*N*1)[*α*2(*Tc*)/*α*1(*Tc*)]. Then with simple algebra, we can obtain *Tc*<sup>1</sup> = 2.4 K from this condition.

## **3. Calculation on the Upper Critical Field of CsV**3**Sb**<sup>5</sup>

#### *3.1. The Upper Critical Field Parallel to the c-Axis*

Now let us solve the problem of the nucleation of superconductivity in the presence of a field **H**. With the magnetic field along the *c*-axis, the vector potential **A** can be chosen as **A** = (0, *Hx*, 0). Since the vector potential depends only on *x*, similar to the single-band case, we can look for solution with the form

$$
\Psi\_i = e^{ik\_y y} e^{ik\_z z} f\_i(x). \tag{15}
$$

Near the upper critical field, the quartic terms in Equation (2) can be ignored, so the linearized two-band GL equations take the form

$$
\hat{M}\_{11}f\_1(x) + \hat{M}\_{12}f\_2(x) = 0\tag{16}
$$

and

$$
\hat{M}\_{21} f\_1(\mathbf{x}) + \hat{M}\_{22} f\_2(\mathbf{x}) = 0 \tag{17}
$$

with

$$\hat{M}\_{\text{ii}} = -\frac{\hbar^2}{2m\_i}\frac{d^2}{dx^2} + \frac{1}{2}m\_i\omega\_i^2(\mathbf{x} - \mathbf{x}\_0)^2 - \mathbf{a}\_i + \frac{\hbar^2}{2m\_{iz}}k\_z^2. \tag{18}$$

Here *ω<sup>i</sup>* = 2*eH*/*mic* and *x*<sup>0</sup> = *hck* ¯ *<sup>y</sup>*/2*eH*. Thus, inclusion of the factor *e*i*kyy* only shifts the location of the minimum of the effective potential. This is unimportant for the present, but it will become important when we deal with superconductivity near surfaces of finite samples [39,40]. We can also set *kz* = 0 if we only consider the upper critical field [39].

At *η*<sup>12</sup> = 0, we can obtain the solutions to Equations (16) and (17) immediately by noting that, for each band, it is the Schro¨dinger equation for a particle bound in a harmonic oscillator potential. The resulting harmonic oscillator eigenvalues are

$$E\_{i,n} = (n+1/2)\hbar\omega\_i - \mathfrak{a}\_i. \qquad (n = 0, 1, 2, \dots) \tag{19}$$

If *η*<sup>12</sup> = 0, Equations (16) and (17) describe a system of two coupled oscillators. We can set the form of the solutions as

$$f\_1(\mathbf{x}) = c\_1 \left(\frac{b}{\pi}\right)^{1/4} e^{-bx^2/2} \tag{20}$$

and

$$f\_2(x) = c\_2 \left(\frac{b}{\pi}\right)^{1/4} e^{-bx^2/2}.\tag{21}$$

Here *c*1, *c*<sup>2</sup> are constants, and *b* = 2*πH*/Φ<sup>0</sup> with the magnetic flux quantum Φ<sup>0</sup> = *πhc*¯ /*e*. Thus, for *n* = 0, we can transform Equations (16) and (17) into

$$E\_{1,0}c\_1 + \eta\_{12}c\_2 = \varepsilon c\_1 = 0\tag{22}$$

and

$$
\pi \eta\_{12} c\_1 + E\_{2,0} c\_2 = \varepsilon c\_2 = 0. \tag{23}
$$

with *ε* the eigenvalue of the matrix.

Then we can obtain the upper critical field parallel to the *c*-axis from the minimum energy eigenvalue

$$\varepsilon\_{\min} = \frac{1}{2} \left[ \left( E\_{1,0} + E\_{2,0} \right) - \sqrt{\left( E\_{1,0} - E\_{2,0} \right)^2 + 4\eta\_{12}^2} \right] = 0,\tag{24}$$

which can be simplified as

$$E\_{1,0}E\_{2,0} = \eta\_{12}^2.\tag{25}$$

With *Ei*,0 <sup>=</sup> *<sup>h</sup>*¯ *<sup>ω</sup>i*/2 <sup>−</sup> *<sup>α</sup><sup>i</sup>* and *<sup>ω</sup><sup>i</sup>* <sup>=</sup> <sup>2</sup>*eH<sup>c</sup> <sup>c</sup>*2/*mic*, we get from Equation (25)

$$
\left(\frac{\hbar e}{m\_1 c} H\_{c2}^{\parallel c} - \alpha\_1\right) \left(\frac{\hbar e}{m\_2 c} H\_{c2}^{\parallel c} - \alpha\_2\right) = \eta\_{12}^2. \tag{26}
$$

Simple algebra shows that the upper critical field can be expressed as

$$H\_{c2}^{\parallel c} = \frac{\Phi\_0}{2\pi\hbar^2} \left[ \left( m\_1\alpha\_1 + m\_2\alpha\_2 \right) + \sqrt{\left( m\_1\alpha\_1 - m\_2\alpha\_2 \right)^2 + 4m\_1m\_2\eta\_{12}^2} \right].\tag{27}$$

Single crystals of CsV3Sb5 can be synthesized via a self-flux growth method [41–43]. In order to prevent the reaction of Cs with air and water, all the preparation processes are performed in an argon glovebox. After high temperature reaction in the furnace, the excess flux is removed by water and a millimeter-sized single crystal can be obtained. The asgrown CsV3Sb5 single crystals are stable in the air. Then electrical transport measurements can be carried out in a Quantum Design physical property measurement system (PPMS-14T), and magnetization measurements can be performed in a SQUID magnetometer (MPMS-5T). For CsV3Sb5, the experimental data of the upper critical field can be measured following these steps and then shown in Figure 1.

To fit the experimental measurement, we choose the GL parameter *α*<sup>10</sup> = 0.11 meV. According to Equation (27), we plot the theoretical result of *H<sup>c</sup> <sup>c</sup>*<sup>2</sup> as the solid line in Figure 1. Note that the experimental data are almost linear and our calculation fits the experimental measurement well.

**Figure 1.** Upper critical field *H<sup>c</sup> <sup>c</sup>*<sup>2</sup> (solid line) and *<sup>H</sup>ab <sup>c</sup>*<sup>2</sup> (dotted line) as function of temperature. The experimental data are from Ref. [43]. The inset shows the schematic crystal structure of CsV3Sb5.
