Characterization of Electron Pair Velocity in YBa$_{2}$Cu$_{3}$O$_{7-\textit{$\delta $}}$ Thin Films

TL;DR

This paper models the superconducting phase in YBCO thin films doped with CeO2 nanodots, focusing on electron pair velocity and flux vortex behavior to optimize nanodot deposition parameters.

Contribution

A new model for characterizing the superconducting phase using electron pair work and chemical potential, based on established superconductivity theories.

Findings

Model predicts optimal nanodot density and growth conditions.

Analyzes flux vortex pinning effects on critical current.

Provides theoretical framework for superconducting phase transition.

Abstract

The superconducting phase transition in YBa$_{2}$Cu$_{3}$O$_{7-\textit{$\delta $}}$(YBCO) thin film samples doped with non-superconducting nanodot impurities of CeO$_{2}$ are the focus of recent high-temperature superconductor studies. Non-superconducting holes of the superconducting lattice induce a bound-state of circulating paired electrons. This creates a magnetic flux vortex state. Examining the flow of free-electrons shows that these quantized magnetic flux vortices arrange themselves in a self-assembled lattice. The nanodots serve to present structural properties to constrict the "creep" of these flux vorticies under a field response in the form of a pinning-force enhancing the critical current density after phase transition. In this work, a model for characterizing the superconducting phase by the work done on electron pairs and chemical potential, following the well-known theories of Superconductivity (Bardeen-Cooper-Scheifer \& Ginzburg-Landau), is formulated and tested.A solution to the expression for the magnetic flux, zero net force and pair velocity will generate a setting for the optimal deposition parameters of number density, growth geometry and mass density of these nanodot structures.

Tables

Table 1. Table 1: Substrate Modification: THA (10 pulses), THB (30 pulses) Multilayer: THA1 (10 pulses), THB1 (30 pulses) Haywood  et al. ( 2008 )

Samples	Approx. Diameter	Approx. Height
THA & THA1	4.0-6.0nm	1.77nm
THB & THB1	4.0-6.0nm	4.0nm

Table 2. Table 2: Substrate Modification: THA (10 pulses), THB (30 pulses) Multilayer: THA1 (10 pulses), THB1 (30 pulses) Haywood  et al. ( 2008 )

Samples	Approx. Radius	Approx. Volume
THA & THA1	2.0nm	28.48377${\text{nm}}^3$
	2.5nm	44.50589${\text{nm}}^3$
	3.0nm	64.08849${\text{nm}}^3$
THB & THB1	2.0nm	67.02064${\text{nm}}^3$
	2.5nm	107.71975${\text{nm}}^3$
	3.0nm	150.79644${\text{nm}}^3$

Table 3. Table 3: Approximate Nanodot Mass

Mass (${10}^{-20}$g) at 10 pulses	Mass(${10}^{-20}$g) at 30 pulses
2.055	4.83542
3.211	7.77179
4.624	10.87971
(Average Mass) 3.29667	(Average Mass)7.82897

Equations

$${\mathbf{V}}_{\ast }=\frac{1}{m_{\ast }}\left(\hslash \nabla \varphi -\frac{e_{\ast }}{c}\mathbf{A}\right)$$

$${\mathbf{J}}_{\ast }$$

$$=$$

$${\mathbf{V}}_{\ast }=\frac{{\mathbf{J}}_{\ast }}{n_{\ast }{e}_{\ast }}$$

$$dF=-SdT-\sum _k{\mathbf{Q}}_k{}^{(r)}d{\text{r}}_k{}^{(r)}+\sum _j{\mu }_j\mathbf{d}{N}_j$$

$$F_{G-L}(r)=F_{\text{grad}}(r)+F_{\text{Lattice}}(r)+U_{\text{magnetic}}(r)$$

$$dF=dU-SdT+\sum _j{\mu }_{j}d{N}_j$$

$$dU=TdS-{\mathbf{r}}_{k}d{\text{χ}}_k+d{F}_{G-L}$$

$${\mu }_{\text{tot}}$$

$$V_{\text{spheriod}}$$

$${\mu }_{\text{tot}}$$

$${\mu }_{\ast }={\mu }_e+e_{\ast }{\varphi }_{\ast }=-E$$

$${\mu }_{\text{k}}=\sum _i\frac{\partial }{\partial N_i}\left[\frac{1}{\beta }\left(\log (z)+{\text{βU}}_i\right)-\sum _k{\mathbf{Q}}_k{}^{(r)}d{\text{χ}}_k{}^{(r)}\right]$$

$${\mu }_k=-\frac{\partial }{\partial N_i}\left[\sum _k{\mathbf{Q}}_k{}^{(r)}{\text{dχ}}_k{}^{(r)}\right]$$

$${\mathbf{F}}_{\ast ,\text{net}}=-\nabla \left({\mu }_k+{\mu }_{\ast }\right)$$

$${\mathbf{F}}_{\ast ,\text{net}}$$

$$=$$

$${\Phi }_k=\frac{\left({\mathbf{Q}}_k\cdot {\mathbf{\xi }}_{\ast }\right)}{{\mathbf{J}}_{\ast }\cdot \left(\pi \left(\frac{{\mathbf{\chi }}_{\beta }}{2}\right){}^2\right)}+{\Phi }_0$$

$$\Phi$$

$$J_{\ast }$$

$$\simeq$$

$${\omega }_{\ast }\approx \frac{\left(-\nabla \left(e_{\ast }{v}_{\ast }{B}_a\text{sinγ}\right)\cdot {\xi }_{\ast }\right)}{{\pi }^2\left(\frac{{\chi }_{\beta }}{2}\right){}^4\cdot B_a}\frac{1}{n_{\ast }{e}_{\ast }}$$

Full text

I Introduction

High-Temperature Superconductivity (HTS) is the coherent ordering of macroscopic quantum states for valence band electrons. HTS occurs at temperatures higher than that of liquid Nitrogen (  77K). These coherent or paired valence band electrons, referred to as Cooper PairsBardeen, Cooper, and Schrieffer (1957) account for the persistent electrical currents within the (a,b)-surfaces of the lattice structure for the conducting material. Modifications with non-superconducting properties can serve to restrict the motion of magnetic flux vorticies via a pinning-force as well as enhance the critical current density. High-Temperature Superconductivity is a very promising field of study given it’s marriage between macroscopic quantum phenomenology and emerging technologies on the mesoscale and nanoscale. A number of complications arise experimentally; including the cost of creating each sample superconductor (e.g. laser ablation, doping agents, vacuum environments, etc.) and the current limitations (i.e. physical properties measurements, SQUID, and again laser ablation, etc.). These complications generate difficulties when attempting to characterize the superconducting compound. Multiple techniques are employed to overcome some of these difficulties, using a variety of sample growth methods (i.e. single $\&$ multi-layer growth modes), introducing structual impurities into the superconductor that strengthens the overall electrodynamics of the system and other experimental techniques to help characterize the sample.

The work described here extends the understanding of the characterization of superconducting compounds in terms of the lattice modifications, nanodot impurities, and applying the modifactions as a basis for further characterizing the superconducting sample in terms of the interaction between the nanodot and the superconducting electrons. Using the description Bardeen, Cooper, and Schrieffer (1957) and the derived thermodynamic formulation of the superconducting system by Ginzburg and Landau Ginzburg (1955), a new theoretical approximation of the electron pair velocity is presented. A formulation of the new model for the system is given by a variation in the electron pair velocity from a ficticious force generated by the presence of a nanodot. The model is tested using the results from (T.Haywood et al.)Haywood et al. (2008). A comparison is made between experimental and theoretical velocity calculations using growth geometry and total chemical potential. This referenced work contained a very good basis in regards to having two different $\text{CeO}_{2}$ deposition methods, substrate modifications and multilayer growth, while using the stable Volmer-Weber growth mode introducing 3-dimensional surface modifactions.

This model gives insight into how the current density for a doped high-temperature superconductor will be modified and tuned based on the dynamics and density of the nanodots themselves. Electron pair velocities can be calculated using the current density, collective charge of the superconducting pair and the number density of the superfluid from the referenced work above. Haywood et al. (2008); Beasley (2009); Kondo et al. (2009).

II Magnetic Flux and Critical Current Density Distributions

It is known that magnetic flux through a ring of supercurrent will become quantized thus, creating a magnetic flux vortex from the torsion effects of the supercurrentsAbrikosov (1957). The normal zones of Cerium Oxide $\left(\text{CeO}_{2}\right.$) deposited onto the thin film samples through laser ablation serve as field penetration sites permittes magnetic flux lines to pass through the sample, in a “swiss cheese”-like structure. The creation of these magnetic flux vortices introduces a vortex state in the sample, existing between the lower and upper critical field limits where $H_{\text{lower}}<H_{\text{vortex}}<H_{\text{upper}}$. The Vortex state of YBCO follows the Abrikosov Vortex lattice theory for the anisotropic surfaces of type-II superconductors Abrikosov (1957). Focusing on the magnetic flux penetrating the sample we look into how this combination of flux and lattice hole effects the flow of paired electrons, inertially. A solution to the expression for the magnetic flux, zero net force and pair velocity will generate a setting for the optimal deposition parameters of number density, growth geometry and mass density of these nanodot structures. From the dimensional analysis describing magnetic flux, one can derive a relationship between work and current density. The standard unit of measure for magnetic flux is normally a Weber (Wb) or a Tesla square meter $\left(T\cdot m^{2}\right)$. These units can be simplified into fundamental terms with respect to the M$\cdot$K$\cdot$S system of measure $\text{Wb}=T\cdot m^{2}=\text{Kg}\cdot m^{2}/{s^{2}\cdot A}$. Now that the magnetic flux is recast in to standard units of length, mass and time, an expression describing the same physical action will be constructed that corresponds to the magnetic flux units of measure. Including the magnetic field effects with the current density, the velocity of each pair is now expressed in terms of the magnetic vector potential (A) and the quantum mechanical representation of the potential energy of the state $(\hbar\nabla\phi)$Bardeen, Cooper, and Schrieffer (1957):

[TABLE]

With the inclusion of the electron pair mass and quantization via ($\hbar\nabla\phi$) in equation (1) for the canonical velocity of the electron pair suggests that the quantum mechanical operations for this coherent state of electrons is of a macroscopic nature, corresponding to an inertial response with respect to the mass term.

The expectation values, probability amplitude, and average densities are associated to observable values and not probabilistic in nature. This macroscopic quantum mechanical expression for the canonical velocity of paired electrons gives rise to the inertial dynamics of the pair themselves. This states that the critical current density of the pairs, and fundamentally the pair velocity, is reactive to some external inertial force acting on the center of mass of the pair. The critical current density is stated as Bardeen, Cooper, and Schrieffer (1957):

[TABLE]

with $\left(e_{*}\right)$ the charge, $\left(m_{*}\right.$) the mass, and $\left|\psi|^{2}\equiv n_{*}\right.$ is the number density of the electron pairs; and (c) is the speed of light. A modification to this distribution of the supercurrent density due to the presence of nanodots suggests that the nanodots, normal zones of the lattice. Introducing a pseudo-potnetial well that the paired electrons fall into will thus change the supercurrent densityKondo et al. (2009). From the analysis of magnetic flux the “electronic hole” that is made from the presence of nanodots serves as an enclosed area that can be determined. The superconducting electrons can circumvent the enclosed surface, strengthening the supercurrent density.

[TABLE]

enclosing a single fluxonTozan (2010).

III Chemical Potential of the Normal-Superconducting State Interaction

To modify the description of the thermodynamic dependence of the critical current density and the magnetic flux threading the superconductor in the presence of nanodots, a reformulation of the fundamental free-energy expression is neededGinzburg (1955); Beasley (2009):

[TABLE]

The free-energy expression from the Ginzburg-Landau theoryGinzburg (1955); Beasley (2009) says:

[TABLE]

Here the standard entropy (S) and temperature (T) terms are held constant for this thermodynamic state. With a force and coordinate in thermodynamic state (r), $\sum_{k}\boldsymbol{Q}_{k}{}^{(r)}\boldsymbol{\text{dr}}_{k}{}^{(r)}$ is the amount of work from the nanodot interacting with the system of electron pairs and $\sum_{j}\mu_{j}\boldsymbol{d}N_{j}$ is the chemical potential with respect to the number density (N). Utilizing the work-done on a system of particles combined with the quantized magnetic flux quasiparticles called FluxonAbrikosov (1957), one can formulate a description of the free-energy interaction of these Fluxon with the supercurrent density surrounding them in terms of the chemical potential and nanodot number density. For the system acted upon by an interacting potential, the free-energy is:

[TABLE]

where,

[TABLE]

With respect to the thermodynamics of the superconducting sample the chemical potential of all interacting particles and quasiparticles must be included. The number density of interacting particles and each of their chemical, or electro-chemical, potentials can alter the dynamics of the thermodynamic system. The energy of the paired electrons is just simply their electro-chemical potential in this quantum limit. Considering interactions that occur the total chemical potential is:

[TABLE]

Where $\beta,z,U_{i}$ are the lattice parameters, with the magnetic influence ($m\cdot B$), paired electron potential ($\phi_{*}$), and fluxon/nanodot extent ($\boldsymbol{\chi}_{k}$). The total chemical potential of the entire system suggests that there are other quasiparticles at play interacting with the paired electrons comprising up the supercurrent. Simplifying this total chemical potential in equation (9) we have $\mu_{\text{tot}}=\mu_{k}+\mu_{*}$. Where $\mu_{k}$ is the chemical potential of the nanodot and $\mu_{*}=\mu_{e}+e_{*}\phi_{*}$ is the electro-chemical potential for the electron pair in terms of the thermodynamic chemical potential of the pair and the electrostatic potential for charged particles.

Considering the dimensions of the nanodots as $\left(\boldsymbol{\chi}_{\beta},\boldsymbol{\chi}_{\gamma}\right)$, where these are the respective diameter (with plane-polar symmetry) and height of the nanodots, we can assume that the geometry of the nanodots follow that of a spheroid.

The average volume of each Cerium Oxide nanodots can be calculated using the following equation for a spheroid with plane-polar symmetry,

[TABLE]

Cerium Oxide with a mass density of $\sim$ 7.2148$\times 10^{-22}\text{g/nm}^{3}$ gives an average mass of the nanodots based on the density of Cerium Oxide and the average volume of the nanodots. With this property we can calculate approximate masses for the 10 pulse and 30 pulse Cerium Oxide volumes, respectively.

IV Inertial Response of the Electron Pair

An expression describing the magnetic flux through the superconducting-normal lattice zones can be given as, using the equation above:

[TABLE]

This total chemical potential simplifies to $\mu_{\text{tot}}=\mu_{k}+\mu_{*}$.

[TABLE]

and

[TABLE]

Since the lattice structure of YBCO is periodic with respect to the electron pairs with temperature equal to zero, an approximation for the chemical potential governing the nanodots can be made in the form of the work:

[TABLE]

Equation (14) neglects the magnetic dipole moment and field because of the hole like behavior of the nanodots. For the nanodots acting as electron holes one can approximate these as a neutral mass. Utilizing the work-done from the perspective of the nanodots is an unconventional choice. From the BCS theory the paired electrons have a velocity, refering to eqaution (1) above. With the inclusion of the mass, the paired electrons respond to a force giving an acceleration, $\frac{\boldsymbol{\nabla}\mu_{*}}{m_{*}}=\frac{d}{dt}\left[\boldsymbol{v}_{*}\right]$. The electron pairs respond to an inertial force. We see that it is obvious in these units that magnetic flux is merely the amount of work per current. A net force can be expressed from the interaction of the electron pair and the nanodots:

[TABLE]

Here the work-done is in a thermodynamic energy state (r) operating within the canonical momentum space of the system, and as usual the electrochemical potential arises for the electron pairsParks and Groff (1967); Henley (2010). Like all systems in equilibrium, this net force must equal to zero satisfying the conservation of energy and momentum of the interaction per the 2nd Law of Thermodynamics. Using this, the magnetic flux can be expressed as

[TABLE]

Magnetic flux is in terms of the current density, an equivalent inertial force and the coherence length describing the size of the electron pairs related to the displacement the pairs should experience from the work. As can be seen, $\left(\boldsymbol{J}_{*}\right)$ is the respective supercurrent density of the sample at a specific temperature, ($\pi\left(\frac{\boldsymbol{\chi}_{\beta}}{2}\right){}^{2}$) is the cross-sectional area of the nanodots keeping the radial symmetry of the geometry. While $\left(\boldsymbol{\xi}_{*}\right.$) is the characteristic superconducting coherence length, $\left(\boldsymbol{Q}_{k}\right)$ the force induced by the magnetic flux on a charged particle, and $\left(\boldsymbol{\Phi}_{0}\right)$ the quantum of magnetic flux (Fluxon, 2.0678 x$10^{-15}\text{Wb}$). This force arises from the potential energy that the nanodot creates on the surface of the superconducting state in momentum space. Without exploring the entire effective field theory for superconductivity only an approximation of the characterized average velocity of paired electrons can be made. Using the fundemental laws that governs this electromagnetic interaction we can approximate or generalize the expression for current density in equation (3) to be $\boldsymbol{J_{*}}=n_{*}e_{*}\boldsymbol{v}_{*}\simeq\frac{\boldsymbol{I}}{\pi\boldsymbol{s}^{2}}$. For an approximation of a simple, homogeneous applied magnetic field $B_{a}$ (assuming no applied magnetic field excitaions) with magnetic flux ($\Phi$) through an enclosed current carrying loop of radius S, we can use the solution of

[TABLE]

From here we can solve for the current density and then the velocity of the electron pairs from equation (16).

[TABLE]

Equations (18) and (19) makes this approximation in terms of the induced force portrayed by the Lorentz force. The electric field contribution is negligible due to the macroscopic electrodynamics explained through the London theory.

[TABLE]

This approximation in equation (20) unfortunately gives a fairly wide range of percent error, percent error $\approx(0.253-7.895)\times 100\%$, due to the lack of a temperature dependance on the supercurrent density approximation from the quantum mechanical field theory. Figures (3) and (4) below show that these theoretical values closely equal in orders of magnitude to the experimental values with some percent error in calculation.

The error in the multilayer growth method is higher due to the approximation methods used for the $\text{CeO}_{2}$ nanodots. We can see that there is a stronger relationship between the nanodots and the paired electrons in terms of their velocity. This suggests that theses magnetic flux vortices penetrating at the normal zones of the sample offer more than expected of them. These normal zones may offer an optimization to the superconducting sample instead of a defect in structure.

V Conclusion

This description of the variation of the superconducting electron pair velocity is incomplete, however it demonstrates a mechanism in terms of further characterizing high-temperature superconductors. Characterization in terms of the respective nanodot densities and geometries deem critical to the enhancement of supercurrent density. The chemical potential and work-done from a constant thermodynamic energy state offer a method of describing an induced force that arises from lattice modifications via single and multi-layer Volmer-Weber growth modes. Equation (20) provides an expression of the modified average electron pair velocity which can be viewed as a predicted quantity. The term ($V_{*}$) in the expression serves as a the velocity of a control sample with unmodified lattice structure (absence of nanodots). While ($\omega_{*}$) is the predicted modified velocity of the superconducting electrons under an applied magnetic field ($B_{a}$) with nanodot diameter ($\chi_{\beta}$). Using this expression one can calculate a predicted average velocity and thus supercurrent density at (T = 5K) before deposition of any lattice modifications (within a 25 percent error).

Next steps will inlcude a richer description of the temperature dependence of the superconducting state to allow for a scalable description of the velocity with respect to the state’s effective temperature. Further correlating this description with the Abrikosov Vortex lattice theory [abrikosov, 1957] and continued research on the subject matter will generate interesting results to the study of theses magnetic singularities (Fluxon) in high-temperature superconductors. The overall effective field theory governing this interaction is to be explored in greater detail.