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# Nernst response, viscosity and mobile entropy in vortex liquids

Kamran Behnia

► To cite this version:

Kamran Behnia. Nernst response, viscosity and mobile entropy in vortex liquids. Journal of Physics: Condensed Matter, 2022, 35 (7), pp.074003. 10.1088/1361-648X/aca9b0. hal-04028916

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Submitted on 14 Mar 2023

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# Nernst response, viscosity and mobile entropy in vortex liquids

## Kamran Behnia

Laboratoire de Physique et Etude des Matériaux (CNRS- Sorbonne Université)
ESPCI Paris, PSL University, 75005 Paris, France

E-mail: kamran.behnia@espci.fr

August 2022

**Abstract.** In a liquid of superconducting vortices, a longitudinal thermal gradient generates a transverse electric field. This Nernst signal peaks at an intermediate temperature and magnetic field, presumably where the entropy difference between the vortex core and the superfluid environment is largest. There is a puzzling similarity of the amplitude of this peak across many different superconductors. This peak can be assimilated to a minimum in the viscosity to entropy density ratio of the vortex liquid. Expressed in units of $\hbar/k_B$, this minimum is one order of magnitude larger than what is seen in common liquids. Moreover, the entropy stocked in the vortex core is *not* identical to the entropy bound to a moving magnetic flux line. Due to a steady exchange of normal quasi-particles, entropy can leak from the vortex core. A slowly moving vortex will be peeled off its entropy within a distance of the order of a superconducting coherence length, provided that the $\Delta E_F$ ratio is sufficiently large.

## 1. Introduction

In presence of a magnetic field, the thermoelectric response acquires an off-diagonal component. This is the Nernst coefficient, which is particularly large in dilute metals with highly mobile carriers [1]. An emblematic case is elemental bismuth. The peak Nernst signal in (large and dislocation free crystals of) bismuth approaches 1 V/K at liquid He temperature and a magnetic field of 1 T [2]. Expressed in the natural unit for thermoelectric response ($k_B/e = 8.6 \times 10^{-5} V/K$), this becomes a very large number. Often, the Nernst response is much weaker than this natural unit. However, even when it is small as nV/K, it can be safely measured by many experimentalists.

The focus of the present paper is the Nernst signal generated by the vortex motion in a type II superconductor. This phenomenon was discovered in 1960s [3, 4, 5] and became the subject of several contemporaneous theoretical investigations [6, 7, 8]. A few decades later, it was also observed in superconducting cuprates [9], which host a remarkably broad vortex liquid state [10].

Despite this long history, a recent experimental study by Rischau *et al.* [11] demonstrated that the vortex Nernst signal cannot be quantitatively understood with
available theories. This is in surprising contrast with the case of the Nernst signal generated by the superconducting fluctuations above the critical temperature. In the latter case, the theory originally formulated by Ussishkin, Sondhi and Huse [12] has proven to be a great success. It was confirmed and extended by other theorists [13, 14, 15] and was experimentally confirmed in both conventional [16, 17, 18] and cuprate [19, 20, 21] superconductors (For reviews, see [22, 1, 23]).

Ironically, the exploration of the Nernst effect in this century was initially driven by attributing to vortex-like excitations an unexplained Nernst signal above the critical temperature in undoped cuprates [24]. Thanks to numerous experiments which followed this report, we have nowadays a reasonable understanding of the amplitude of the quasi-particle Nernst signal in metals and the fluctuating Nernst signal of in the normal state of superconductors [1]. The rough amplitude of the anomalous Nernst signal of magnets can also be guessed from their anomalous Hall conductivity [25]. In contrast, we lack an established quantitative account of how mobile vortices give rise to a Nernst response.

The present paper aims to summarize what is known and what is yet to be understood about the measured vortex liquid Nernst signal in various superconductors. It identifies a link between an upper bound to the amplitude of the Nernst signal and a lower bound to the viscosity of the vortex liquid and highlights a distinction between two concepts of entropy of a superconducting vortex: the one statically stocked in its core and the one dynamically carried by the mobile flux line.

## 2. The amplitude of $S_{xy}$ and its surprising invariability among superconductors

Experimentally, Nernst effect is measured by applying a thermal gradient along the crystal (let us call it the x-axis) and measuring an electric field in the lateral direction (the y-axis) in presence of a magnetic field perpendicular to both (along the z-axis). The Nernst signal is defined as:

$$S_{xy} = \frac{E_y}{\nabla_x T} \qquad (1)$$

A striking fact about the amplitude of $S_{xy}$ in superconducting crystals or thin films measured in the past half a century is that it peaks to a value in the range of $\mu V/K$. This feature, first highlighted in ref. [11], is illustrated in Fig.1. Despite the fact that the superconducting critical temperature is $\approx 0.3K$ in strontium titanate and $\approx 100K$ in Bi2212, their Nernst peaks, occurring at widely different temperatures and magnetic field, are comparable in magnitude. This is to be contrasted with the magnitude of $S_{xy}$ in metals, where it can vary by many orders of magnitude [26, 1]. The $S_{xy}$ peak in bismuth, a dilute semi-metal hosting high-mobility carriers, is five to eight orders of magnitude larger [27, 28, 29, 2] than in dense metallic perovskytes [30, 31].

Let us examine a possible link between this observation and a general property of liquids.
<figure><img src="image_3.png" /><figcaption>Figure 1. The amplitude of the Nernst signal in various superconductors: Temperature dependence of the largest Nernst signal in several superconductors. Despite three orders of magnitude variation in their critical temperature, these superconductors display a roughly comparable vortex Nernst signal, which ranges between 2 and 11 µV/K. In contrast, the quasi-particle Nernst signal in metals can be many orders of magnitude larger (≈ mV/K) or many orders of magnitude smaller (≈ nV/K), according to the mobility and the Fermi energy of the metal in question.</figcaption></figure>

## 2.1. A lower bound to viscosity of liquids

The shear viscosity of a fluid, $\eta$, quantifies the tangential stress, $\sigma_{xy}$, required to generate a velocity gradient in the fluid [32]:

$$ \sigma_{xy} = \eta \frac{dv}{dz} \qquad (2) $$

$\eta$ is the dynamic viscosity, which is expressed in units of Poise (≡ Pa.s). Divided by the particle mass, $m$ and particle density, $n$ of the fluid, it yields the kinematic viscosity, $\nu = \frac{\eta}{mn}$. The latter is equivalent to momentum diffusivity.

Note that the sign of viscosity can only be positive. It also happens that its amplitude cannot be arbitrarily small [33, 34]. Indeed, the magnitude of kinematic viscosity is bounded by fundamental constants [34]. Trachenko and Brazhkin have demonstrated that, while the amplitude of viscosity of a fluid depends on the microscopic
interactions among its constituent particles, it always exceeds a minimal bound, set by $m$ and the electron mass, $m_e$ [35]:

$$ \nu_{min} = \frac{1}{4\pi} \frac{\hbar}{\sqrt{m_e m}} \qquad (3) $$

The temperature dependence of viscosity in common fluids displays a minimum, which approaches (while exceeding) $\nu_{min}$. The minimum arises because of the competition between a liquid-like regime (where warming diminishes viscosity by enabling particles to hop across energy barriers) and a gas-like regime (where warming strengthens viscosity by shortening their mean-free-path) [35]. In supercritical fluids, this minimum in momentum diffusivity is accompanied by a similar minimum in thermal diffusivity [36].

This minimum has attracted recent attention in the context of the debate on a possible bound to the ratio of viscosity to entropy density in fluids [37]. This gives us an opportunity to compare vortex liquids with common liquids.

## 2.2. The viscosity-to-entropy density ratio

Kinematic viscosity, $\eta$, can be expressed in units of Planck constant per volume and entropy density, $s$, in units of Boltzmann constant per volume. Therefore, their ratio, $\eta_s$, can be expressed in units of $\hbar/k_B$ for any fluid It has been noticed that in common fluids this ratio approaches unity, but remains above it (see Fig. 2.a) [38]. This observation was made in the context of the debate following the discovery of a theoretical holographic bound to $\eta_s$ [39, 40].

Eq. 3 defines an explicit minimum for kinematic viscosity. However, one can see that its also implies a minimum in $\eta_s$. Within a numerical factor of the order of unity, the entropy density of a classical liquid is of the order of $k_B$ times its molar density. The dynamic viscosity ($\eta$) is the kinematic viscosity ($\nu$) times the mass density of the liquid and the particle mass ($m$) can be expressed in terms of the proton mass $m_p$. Therefore, Eq. 3 leads to:

$$ \frac{\eta}{s} \geq \frac{1}{4\pi} \sqrt{\frac{m_p}{m_e}} \frac{\hbar}{k_B} \qquad (4) $$

Here, $m_p/m_e = 1836$ is the proton-electron mass ratio. Thus, this inequality accounts for what is seen in Fig. 2.a.

## 2.3. The peak in $S_{xy}$ as a minimum in $\eta_s$

Let us come back to the vortex liquid. The ratio of the quantum of the magnetic flux $\Phi_0 = \hbar/2e$ to the Nernst signal $S_{xy}$ can be expressed in units of $\hbar/k_B$. Let us show that it can be assimilated to the ratio of viscosity to the entropy density of the vortex liquid.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** The viscosity-to-entropy density ratio: a) The ratio of dynamic viscosity, η, to entropy density, s, in two fluids (H₂O and helium) at two different pressures (25 and 100 MPa). The data was extracted from the National Institute of Science and Technology (NIST) reference database [41]. Note the presence of a minimum, which is larger, but of the order of ħ/k<sub>B</sub>. b) The ratio of the quantum of the magnetic flux (Φ<sub>0</sub> = ħ/2e) to the Nernst signal (S<sub>xy</sub> = E<sub>y</sub>/∂V<sub>x</sub>) in several superconductors. This quantity is equivalent to viscosity to entropy density ratio of a vortex liquid (see text). Expressed in units of ħ/k<sub>B</sub>, it has a minimum, which is an order of magnitude larger than what is seen in classical fluids.</figcaption></figure>
The vortex Nernst signal arises due to the drift velocity of magnetic flux lines, $v_x$. Taking the number of vortices per unit area to be $n_V$, the Josephson equation yields the Electric field:

$$E_y = n_V \Phi_0 v_x \qquad (5)$$

This drift velocity along the x-axis is caused by the thermal force exerted by a thermal gradient, which is itself balanced by a viscous force:

$$\nabla_x T s = n_V \eta v_x \qquad (6)$$

Compare this equation with eq. 2, to note that the assumption here is that $\eta$ is a genuine shear velocity and the drift velocity of vortices can be assimilated to a gradient in local fluid velocity generated by thermal stress. Combining these two equations leads to the following expression for the Nernst signal:

$$S_{xy} \equiv \frac{E_y}{\nabla_x T} = \Phi_0 \frac{s}{\eta} \qquad (7)$$

Therefore, the ratio of the Nernst signal to the quantum of the magnetic flux in a given vortex liquid equals the inverse of the ratio of the viscosity to the entropy density.

Plotting $\Phi_0/S_{xy}$ for a few superconductors among those shown in Fig. 1, we can see that the viscosity-to-entropy-density in different vortex liquids presents a minimum of comparable amplitude, which in units of $\hbar/k_B$, exceeds what is seen in common liquids by one order of magnitude (Fig. 2b).

Note the difference in the origin of minimum in the two cases. In common liquids, the non-monotonic temperature dependence is due to viscosity, in the vortex liquids, it is mostly due to the difference in entropy between the two components of the fluid. Nevertheless, the comparison suggests that the observed similarity of the amplitude of $S_{xy}$ across superconductors may be a consequence of a bound, set by fundamental constants, to the ratio of viscosity to entropy density in any liquid of superconducting vortices.

### 3. Extracting the mobile entropy of vortices from $\alpha_{xy}$

While the Nernst signal, $S_{xy}$ is what is immediately measured in a Nernst experiment, more extensive attention has been given to $\alpha_{xy}$, sometimes called the Nernst conductivity. It is the off-diagonal component of the thermoelectric conductivity tensor defined by $\overline{\rho\alpha} = \overline{S}$, In general one has:

$$S_{xy} = \rho_{xx}\alpha_{xy} + \rho_{xy}\alpha_{xx} \qquad (8)$$

However, in our context, the second term can be safely neglected in most cases. Then $\alpha_{xy}$ can be extracted from the measured $S_{xy}$ and the measured $\rho_{xx}$. Note that in three dimensions, where resistivity is expressed in $\Omega$.m, the units of $\alpha_{xy}$ is A/K.m. But the length disappears in two dimensions, and $\alpha_{xy}$ can be expressed in a natural unit: $\frac{ek_B}{\hbar} \approx 21nA/K$ [16].
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Nernst effect and resistivity in cuprates: Published figures of the Nernst and resistivity data in three cuprate superconductors: Top $Nd_{2-x}Ce_xCuO_4$ (NCCO) [42]; Bottom left: $YBa_2Cu_3O_{7-\delta}$ (YBCO) [43]; $La_{1.92}Sr_{0.08}CuO_4$ (LSCO) [44]. Reprinted with permission; Copyright (1994, 1998, 2002) by the American Physical Society. In the case of NCCO, the negative sign of the Nernst coefficient is due to the choice of a different sign convention.</figcaption></figure>
In the flux flow regime, electrical resistivity is set by the balance between the Lorentz force and a viscous force. Therefore:

$$ \rho_{xx} = \Phi_0 \frac{n_V}{\eta} \qquad (9) $$

By assuming that the viscous force on vortices is the same in a flux flow experiment and in a Nernst experiment, one can link the amplitude of $\alpha_{xy}$ to the mobile entropy of a vortex per unit of length, $S_d$ [45]:

$$ \alpha_{xy} = \frac{S_{xy}}{\rho_{xx}} = \frac{S_d}{\Phi_0} \qquad (10) $$

In three dimensions, $S_d$ is expressed in units of $J.K^{-1}.m^{-1}$. Multiplying this quantity by a length leads to $S_{sheet}^d$, a quantity which can be expressed in units of Boltzmann constant.

### 3.1. Experimentally observed $\alpha_{xy}$ in superconductors

Rischau et al. observed that the peak $\alpha_{xy}$, extracted by dividing the maximum $S_{xy}$ in the vortex state to the resistivity at the same temperature and magnetic field reveals a general, and yet to be understood, trend. Here is a brief review of this point.

**Cuprates** - Numerous studies have been devoted to quantifying either the Ettingshausen [46] and the Nernst [43, 42, 47, 48, 44, 49, 50, 51] effects in the vortex state of the cuprates. Fig. 3 reproduces the data from three different studies, where both the Nernst signal and the resistivity were reported. As seen in Table 1, which summarizes the situation for four different cuprates, if one takes the c-axis lattice parameter to pass from $S^d$ to $S_{sheet}^d$, one finds that, its magnitude varies between 0.7 $k_B$ in LSCO to 4 $k_B$ in Bi2212. Boltzmann constants. In other words, $S_{sheet}^d$ is of the order of magnitude of the Boltzmann constant, despite the fact that at each CuO$_2$ plane, thousands of quasi-particles are associated with the normal core of a vortex.

Note that the relatively large value of $S_{sheet}^d$ in Bi2212 is a consequence of its relatively longer c-axis parameter. Instead of the latter, one may take the distance between copper-oxygen planes (of the order of 0.6 to 0.8 nm) or the superconducting coherence length (between 2 to 7 nm) as the third-dimension distance. Such choices will lead to different magnitudes of $S_{sheet}^d$, which will remain comparable to $k_B$ in order of magnitude. Nevertheless, in all these cases, the difference between compounds appears to be larger than the experimental uncertainty (for a recent discussion of the cuprate data, following ref. [11], see [52]).

**Other uncommon superconductors** - The Nernst effect in $\kappa$-(BEDT-TTF)$_2$X$_2$ family of organic superconductors was studied by two different groups [53, 55]. Logevenov et al. [53] also reported the resistivity of their samples. In the iron-based superconductor, FeTe$_{0.6}$Se$_{0.4}$, a Nernst peak in the vortex liquid state was detected by Pourret et al. [54] who reported concomitant flux flow resistivity data, too. Finally, the vortex Nernst signal and concomitant resistivity data were reported in the case of superconducting strontium titanate by Rischau et al. [11]. These three data sets are
<table><caption>**Table 1. Extracting vortex entropy per sheet in cuprates**-The maximum Nernst signal, the resistivity corresponding to temperature and magnetic field of the Nernst peak, the c-axis lattice parameter and the extracted entropy per vortex per sheet in the three cuprate superconductors.</caption><thead><tr><th>Compound</th><th>T<sub>c</sub><br>[K]</th><th>N<sup>peak</sup><br>[μ/K]</th><th>ρ<sup>peak</sup><br>[μΩ cm]</th><th>c<br>[nm]</th><th>S<sup>sheet</sup><sub>d</sub><br>[10<sup>-23</sup> J/K]</th></tr></thead><tbody><tr><td>YBa<sub>2</sub>Cu<sub>3</sub>O<sub>7-δ</sub> (12 T) [43]</td><td>92</td><td>2.6</td><td>50</td><td>1.19</td><td>1.3</td></tr><tr><td>Bi<sub>2</sub>Sr<sub>2</sub>Ca Cu<sub>22</sub>O<sub>8+x</sub> (12 T) [43]</td><td>95</td><td>4.1</td><td>50</td><td>3.1</td><td>5.1</td></tr><tr><td>Nd<sub>2-x</sub>Ce<sub>x</sub>CuO<sub>4</sub> (1 T) [42]</td><td>24</td><td>1.6</td><td>20</td><td>1.2</td><td>1.9</td></tr><tr><td>La<sub>1.92</sub>Sr<sub>0.08</sub>CuO<sub>4</sub> (12 T) [44]</td><td>29</td><td>9.1</td><td>260</td><td>1.2</td><td>0.88</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Figure 4. Nernst effect and resistivity in three superconductors:** Temperature dependence of resistivity and the Nernst signal in an organic superconductor [53], in an iron-based superconductor [54] and in strontium titanate [11]. In all cases, the Nernst signal peaks in the vortex liquid state at a fraction of the critical temperature and a fraction of the zero-temperature upper critical field. This data has been used for values expressed in table 2.</figcaption></figure>

reproduced in Fig.4 and summarized in table 2. Here also, one may contest the choice of lattice parameter as the relevant distance to pass from 3D to 2D. Note that the c-axis superconducting coherence is more than one order of magnitude longer in cubic strontium titanate than in cuprates. Therefore the similarity would disappear if one takes it as the relevant perpendicular distance.

**Historical Nernst data on superconducting niobium and its alloys** - We saw that in many superconductors, explored during the last few decades, the amplitude of $\alpha_{xy}$ leads to extracting an entropy per vortex per sheet of the order of $k_B$. However, as
<table><caption>**Table 2.** Extracting vortex entropy per sheet in three other superconductors-The maximum Nernst signal, the resistivity corresponding to temperature and magnetic field of the Nernst peak, the c-axis lattice parameter and the extracted entropy per vortex per sheet in the three superconductors.</caption><thead><tr><th>Compound</th><th>T<sub>c</sub><br>[K]</th><th>N<sup>peak</sup><br>[μ/K]</th><th>ρ<sup>peak</sup><br>[μΩ cm]</th><th>c<br>[nm]</th><th>S<sup>sheet</sup><sub>d</sub><br>[10<sup>-23</sup>J/K]</th></tr></thead><tbody><tr><td>κ-(ET)<sub>2</sub>Cu[N(CN)<sub>2</sub>]Br (2 T)[53]</td><td>11</td><td>6.1</td><td>380</td><td>2.9</td><td>0.96</td></tr><tr><td>FeTe<sub>0.6</sub>Se<sub>0.4</sub> (24 T) [54]</td><td>14</td><td>4</td><td>48</td><td>0.58</td><td>0.96</td></tr><tr><td>SrTi<sub>1-x</sub>Nb<sub>x</sub>O<sub>3</sub> (0.06 T) [11]</td><td>0.35</td><td>11</td><td>100</td><td>0.39</td><td>0.89</td></tr></tbody></table>

one can see in tables 1, and 2, the resistivity of the normal state in these superconductors is relatively large. One may wonder if the observation holds for superconductors with a higher conductivity.

Most historical explorations of the Nernst effect did not explicitly report the measured amplitude of $S_{xy}$. They focused on the measured transverse voltage and often extracted $S^d$.

In the case of niobium, Huebener and Seher [56] studied high-purity foils and found a Nernst signal as large as $S_{xy} \sim \mu V/K$ in samples with a residual resistivity as low as 20 nΩcm. This would imply an entropy per vortex per sheet exceeding $k_B$ by several orders of magnitude.

In another study, de Lange and Otter [57] quantified the magnitude of the vortex transport entropy, $S_d$ in a niobium alloy (Nb<sub>80</sub>Mo<sub>20</sub>) and found that it peaks to $S_d \approx 10^{-11} J/K.m$. Combined with a lattice parameter of 0.3 nm, this yields a sheet entropy, $S_d^{sheet}$ exceeding 200 $k_B$.

Thus, niobium differs from the superconductors listed in the two tables. This suggests that $S_d^{sheet} \approx k_B$ is a lower bound. On the other hand, in order of magnitude, the amplitude of $S_{xy}$ in this low-resistivity superconductor is comparable to those listed above.

## 3.2. Theory of the vortex transport entropy

As mentioned in the introduction, as early as its discovery, the vortex Nernst effect was a subject of theoretical investigations [6, 7, 8]. In 2010, Sergeev and co-workers [58], reviewing these earlier theories argued that magnetization currents do not transfer the thermal energy and therefore, the vortex transport entropy is much smaller than what was believed to be.

Sergeev et al. [58] contested the validity of the following expression for the vortex transport entropy derived by Stephen [6]:

$$S_d^{EM} = -\frac{\phi_0 \partial H_{c1}}{4\pi \partial T} \qquad (11)$$

This expression assumes that the entropy of the vortex is set by the temperature derivative of the energy cost to introduce a vortex at the lower critical field, $H_{c1}$. But,
according to Sergeev et al. [58], such an assumption implies that supercurrents transport entropy. They derived the following expression for the vortex transport entropy :

$$S_d^{\text{core}} \approx -\pi \xi^2 \frac{\partial H_c^2}{\partial T} \frac{H_c^2}{8\pi} \qquad (12)$$

This theoretical expression for $S^d$ failed to explain the case of a BCS superconductor with a vortex liquid regime and well-established material-dependent parameters. Indeed, SrTi$_{1-x}$Nb$_x$O$_3$ at optimal doping (x=0.01) is such a superconductor. This is a s-wave superconductor [59, 60] where both the lower and the upper critical fields have been experimentally measured [61] ($H_{c1}(0)= 4.8$ Oe; $H_{c2}(0)= 240$. Inserting these numbers in Eq. 12, one finds $S_d^{EM} \approx 5.2 \times 10^{-12}$ J/K. m.

However, the experimental $S_d$ is $\approx 2.3 \times 10^{-14}$ J/K. m (equal to the ratio of $S_{sheet}^d$ and c in Table 2). Thus, there is a fifty-fold discrepancy between theory and experiment.

Compared to Eq. 11, Eq. 12 leads to a downward revision to the amount of entropy carried by a vortex [58]. Nevertheless, what it yields is still more than one order of magnitude larger than what the experiment finds. This discrepancy, together with the empirical observation that $S_{sheet}^d \approx k_B$, motivates us to explore other reasons for a theoretical overestimation of vortex transport entropy. This is the subject of the next section ‡.

## 4. Distinguishing between vortex entropy at rest and in motion

The experimental Nernst data and the theoretical expectation (Eq. 12) are confronted following a chain of reasoning, which contains a contestable assumption: The entropy stocked in the core remains intact when transported by a mobile vortex.

In general, it is true that the magnitude of $\alpha_{xy}$ quantifies the ratio of entropy to the magnetic flux of a mobile carrier [1] (See also [64]). When the two opposing forces exerted on the carrier (the thermal force on its entropy and the Lorentz force on its magnetic flux) cancel each other, there is a finite $\alpha_{xy}$. It represents the ratio of a charge density flow to a perpendicular thermal gradient. This picture gives an account of the expressions for $\alpha_{xy}$ generated by electronic quasi-particles or fluctuating Cooper pairs, given the amount of entropy or magnetic flux they carry with themselves [1].

Now, a superconducting vortex is bound to a magnetic flux and its core has an excess of entropy. Therefore, one would naively expect to see that its generate a finite $\alpha_{xy}$ corresponding to the ratio of the entropy inside the core to the quantum of magnetic flux. However, this overlooks the fact that in this mesoscopic entity, the entropy and the magnetic flux are not necessarily bound to each other.

Fig.5a is a sketch of a superconducting vortex in presence of a thermal gradient. Around the vortex core, Cooper pairs whirl around the magnetic field with a velocity which increases as the core is approached. The core begins when this velocity becomes unsustainable and the pair breaks up. In presence of a thermal gradient, because of the

<footnote>‡ For alternative proposals following the experimental observation reported in ref. [11], see [62] and [63].</footnote>
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Vortex core and normal fluid in presence of thermal gradient: a) A cooper pair in the immediate vicinity of a vortex core breaks up on the hot side of the vortex. b) As the vortex drifts under the influence of a thermal gradient, quasi-particles inside the core mix with those outside. c) When the vortex moves a distance of the order of its core radius, it can be totally stripped off its stocked entropy as a consequence of the exchange of inside and outside quasi-particles.</figcaption></figure>

temperature dependence of the superconducting gap, the depairing is favored on the hot side. A vortex drifting from the hot side to the cold side of the sample has thus a leaking tail. There is a steady transformation of peripheral Cooper pairs to core quasi-particles. Moreover, quasi-particles are not restricted to the normal core. At fine temperature, a fraction of the total electronic density outside the vortex core is the normal fluid, which also diffuses heat in response to the thermal gradient.

The failure of Eq.12 can be traced to the fact that because of the slow drift of the vortices, the mobile magnetic flux will be stripped of its entropy during a distance of the order of the core diameter (i.e. the superconducting coherence length, ξ) (See Fig.5b, c).

Let us make a rough estimate of how slow a vortex should move in order to be stripped of its stocked entropy. The first relevant quantity is the tunneling rate between inside and outside of the vortex core. This rate is ≈ $\frac{\Delta^2}{\hbar E_F}$. The second is the number of quasi-particles stocked inside the core. In the most naive approach, the vortex core, can be assimilated to a normal-state cylinder of radius ξ and the number of quasi-particles will be ≈ $k_F^2 \xi^2$. In a more sophisticated approach, what the core contains are ≈ $k_F \xi$ Andreev bound states [65, 66]. Therefore, one can estimate the time it takes to entirely reconfigure the vortex to be of the order of $k_F \xi \times \frac{\hbar E_F}{\Delta^2}$.

A drifting vortex with a velocity of $v_L$ will see its core content replaced if the drift
time along a distance of $\xi$ is longer than this time. This will happen if :

$$ \frac{\xi}{v_L} > k_F \xi \times \frac{\hbar E_F}{\Delta^2} \qquad (13) $$

Neglecting numerical factors of the order of unity, one finds that the inequality (13) eventually leads to:

$$ \frac{v_L}{v_F} < \left( \frac{\Delta}{E_F} \right)^2 \qquad (14) $$

Numerical prefactors of the order of unity have been neglected. Therefore the vortex core will be stripped off its quasi-particles and entropy if the drift velocity is much smaller than the Fermi velocity, $v_F$, multiplied by the square of the ratio of the superconducting gap to the Fermi energy $\frac{\Delta}{E_F}$.

Now, let us consider the numbers. The measured electric field in a Nernst experiment is of the order of $\approx 10^{-3}$Vm$^{-1}$ in a magnetic field of $\approx 1$ T Therefore, the flux lines drift with a velocity of $10^{-3}$ m/s. The Fermi velocity, on the other hand, is of the order of $10^5$ m/s. This is a discrepancy of 8 orders of magnitude. It allows inequality (14) to hold, provided that $\frac{\Delta}{E_F}$ does not fall below a very small number. Interestingly, all superconductors listed in tables 1 and 2 have relatively large $\frac{\Delta}{E_F}$ ratios (in the range of $10^{-1}$ to $10^{-2}$). On the other hand, in an elemental superconductor such as Nb, $\frac{\Delta}{E_F}$ can be as small as $10^{-4}$. In the latter case, the vortex may not be totally stripped from the entropy stocked in their core.

### 4.1. An irreducible entropy per vortex ?

Is there a lower boundary to the entropy of a moving vortex? If the whole content of the vortex core get replaced during a movement as short as $\xi$, why should there be any mobile entropy at all?

In search of a possible answer to this question, let us recall what Volovik [67] pointed out about vortices in fermionic superfluids. He stated that analogs of the black hole horizon can occur in 'liquids moving with velocities exceeding the local maximum attainable speed of quasiparticles. Then an inner observer, who uses only quasiparticles as a means of transferring the information, finds that some regions of space are not accessible for observation. For this observer, who lives in the quantum liquid, these regions are black holes.'[67]

It is tempting to speculate that a lower boundary of $k_B \ln 2$ arises is a consequence of the event horizon surrounding a vortex. An information barrier between observers inside and outside would ensure the survival of this last bit of information when the vortex constituents are totally replaced.

## 5. Concluding remarks

The transport properties of the vortex liquids are often more complex than what it seems at the first sight. This is true not only for the Vortex Nernst response, but also for flux flow resistivity [68] and the Hall response in the vortex liquid regime [69].
A satisfactory theory of the Nernst response in the vortex liquid is yet to be
formulated. This is in contrast with the case of the Nernst signal in the normal state
of a superconductor, where the theory has been tested by multiple experimental studies
[1, 70]. Here, I argued that the rough amplitude of the Nernst signal can be understood
provided that the vortex liquid respects a bound to $\frac{\eta}{s}$ (the viscosity-to entropy density
ratio), which is observed by other liquids. Moreover, the entropy inside a vortex core
does not remain bound to the magnetic flux line during the drift if the ratio of the
superconducting gap to the Fermi energy is above reasonable threshold. This may
explain why the entropy per vortex is much smaller in superconductors with a larger
$\frac{\Delta}{E_F}$ ratio, but not in superconducting niobium.

The subject deserves further theoretical and experimental investigations.

6. Acknowlegements

I am grateful to Benoît Fauqué, M. V. Feigel’man, S. A. Hartnoll, A. Kapitulnik, S. A. Kivelson, K. Trachenko, C.W. Rischau, and G. E. Volovik for discussions. This work was supported by the Agence Nationale de la Recherche (ANR-18-CE92-0020-01; ANR-19-CE30-0014-04).

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